/
Text
Inlerp I Ii n f
Lin r r I rs
'"
by S. G. KREIN
Ju. I. PETUNIN
E. M. SEMENOV
Volume 54
TRANSLATIONS OF
MATHEMATICAL MONOGRAPHS
American Mathematical Society
Interpolation
of
linear Operators
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
VOLUME 54
Interpolation of
Linear Operators
\J
by S. G. KREIN
Ju. I. PETUNIN
E. M. SEMENOV
American Mathematical Society · Providence · Rhode Island
HHTEPnOJIflQHfl JIHHEflHbIX OnEP ATOPOB
C. r. KPEHH, 10. H. nETYHHH H E. M. CEMEHOB
H3M TEJIbCTBO «HAYKA»
r JIABHAB PEMKQHB
WH3HKO-MATEMATHQECKOR JIllTEP ATYPbI
MOCKBA 1978
Translated from the Russian by J. Szucs
Translation edited by Lev J. Leifman
1980 Mathematics Subject Classification.
Primary 46E30, 46E35; Secondary 44AI5, 42CI0.
ABSTRACf. The book is devoted to an important direction in functional analysis: interpolation
theory for linear operators. The main methods for constructing interpolation spaces are ex-
pounded and their properties are studied. These methods allow one to look at a number of
theorems and inequalities of classical analysis from a new standpoint. Interpolation theory for
operators has numerous applications in Fourier series, approximation theory, partial differential
equations, etc. Some of them are developed in the book.
Library of Congress Cataloging in Publication Data
Krein, S. G. (Selim Grigor'evich), 1917-
Interpolation of linear operators.
(Translations of mathematical monographs; 54)
._--
Translation of: Interpoliatsiia lineinykh operatorov.
Bibliography: p.
Includes indexes.
......-
1. Linear operators. I. Petunin, IUrii Ivanovich. II. Semenov, E. M. III.
Title. IV. Series.
QA329.2.K7313
ISBN 0-82184505-7
ISSN 0065-9282
515.7'246
81-20637
AACR2
Copyright @ 1982 by the American Mathematical Society
TABLE OF CONTENTS
FOREWORD TO THE AMERICAN EDITION
FOREWORD
CHAPTER I. IMBEDDED, INTERMEDIATE, AND INTERPOLA-
TION BANACH SPACES
I. Imbedding of Banach spaces
2. Dual spaces of imbedded Banach spaces
3. Intermediate Banach spaces
4. Interpolation spaces and interpolation triples
CHAPTER II. INTERPOLATION IN SPACES OF MEASURABLE
FUNCTIONS
Introduction
1. Positive functions on a semiaxis and their dilation functions
2. Rearrangements of measurable functions
3. Operators in the Banach couple L 1(0, (0), Loo(O, (0)
4. Symmetric spaces. Interpolation between L 1 and Loo
5. Lorentz and Marcinkiewicz spaces
6. Operators of weakened and weak type
7. The singular Hilbert operator
8. Interpolation theorems for spaces with different measures
9. Applications to the theory of orthogonal series
CHAPTER III. SCALES OF BANACH SPACES
I. Scales of Banach spaces. Related spaces
2. Maximal and minimal normal scales
3. The scale of Holder spaces
v
. .
Vll
IX
1
I
6
9
18
39
39
46
58
77
90
107
124
150
156
173
187
187
192
200
VI
CONTENTS
CHAPTER IV. INTERPOLATION METHODS 209
Introduction 209
1. The complex method of interpolation 214
2. The methods of constants and means (% and }methods) 246
CHAPTER V (BY S. G. KREIN). INTERPOLATION IN SPACES
OF SMOOTH FUNCTIONS 283
1. Interpolation spaces constructed from an unbounded operator
and a smoothing approximation process 283
2. Trace theory 309
3. Spaces of smooth functions of n variables 322
NOTES ON THE LITERATURE 349
BIBLIOGRAPHY 355
SUBJECT INDEX 373
NOTATION INDEX 375
FOREWORD TO lHE AMERICAN EDrnON
This edition includes Chapter V, "Interpolation in spaces of smooth func-
tions", written at the same time as the preceding chapters but not included in
the Soviet edition for technical reasons. This chapter expounds the abstract
scheme for constructing interpolation spaces by means of an unbounded
operator in a Banach space, and the corresponding approximation process.
Starting points of this theory were papers by J. L. Lions, Lions and J. Peetre,
and P. Grisvard, which then were extended by other authors. As an applica-
tion we consider only Sobolev and Besov spaces.
The reader can get acquainted with other families of spaces in the books
referred to in the foreword to the Soviet edition.
The authors express their sincere gratitude to the editor of the translation,
Dr. L. J. Leifman, for his penetrating remarks that helped in eliminating a
number of shortcomings.
vii
FOREWORD
The present book is devoted to the systematic exposition of a chapter in
functional analysis that has appeared and developed in the past two decades
and has found applications in various fields.
The basic objects of classical functional analysis were operators acting
from one Banach space (or later from a topological linear space) into another.
The spaces themselves were considered as given in advance. The change of
this ideology was facilitated to a significant extent by the imbedding theorems
of S. L. Sobolev, in which a number of fundamental theorems and inequali-
ties of analysis were interpreted as assertions concerning the imbedding of
one Banach space into another. Imbedding theorems arose in connection with
problems of the theory of partial differential equations, in which for the study
of smoothness of solutions a series of spaces is introduced; for the study of
the behavior near the boundary of the domain or near some singular points
other types of spaces are introduced, the study of values of solutions on
manifolds of smaller dimension is performed in still other spaces, etc. The
abundance of various spaces required a detailed study of the interrelations
between these spaces. Thus a new level of abstraction appeared, on which the
Banach spaces themselves are considered as elements of some category. The
interpolation theory for linear operators expounded in the book is to a great
extent connected with such an approach.
The first interpolation theorem in operator theory was obtained by M.
Riesz in 1926 in the form of an inequality for bilinear forms. A sharpening
and operator formulation of it were given by G. O. ThoOO. An essential
further step was the interpolation theorem of J. Marcinkiewicz (1939), whose
proof was published by A. Zygmund in 1956. In the fifties important
generalizations of the Riesz- Thorin and Marcinkiewicz theorems were ob-
tained by E. M. Stein and G. Weiss. However, all these and other communi-
cations were concerned with 4 spaces or spaces similar to them. The
IX
x
FOREWORD
development of general interpolation theorems for families of abstract Hilbert
and Banach spaces began in 1958 independently in several countries. The first
publications are due to J. L. Lions (1958-1960), E. Gagliardo (1959-1960), A.
P. Calderon (1960), and S. G. Krein (1960). The work of J. A Peetre played
an essential role in the sequel. Several methods have been created for
obtaining interpolation theorems, which have deep interrelations. Moreover,
it became clear fairly soon that the interpolation properties of spaces inter-
mediate between two Banach spaces are consequences of the functoriality of
the methods of construction. Therefore, the main emphasis has been shifted
to the study of properties of intermediate interpolation spaces obtained by
various methods, and to their realization. Along with this, in the work of W.
Orlicz, A. P. Calderon, G. G. Lorentz, E. M. Semenov, and others deep
results have been obtained concerning the interpolation of linear operators in
spaces of measurable functions.
It is impossible to expound all results of interpolation theory for linear
operators in one book. We have tried to illuminate only some of the main
directions in its development: the real and complex methods of constructing
interpolation spaces, the method of scales of Banach spaces, and interpola-
tion in spaces of measurable functions. Supplementary information is con-
tained in remarks and references.
In the development of interpolation theory for operators many new general
notions of functional analysis have emerged. These notions and their interre-
lations are studied in the first chapter of the book. The exposition is based
essentially on the work of N. Aronszajn and E. Gagliardo. To read this
chapter one needs to know only the basic principles of functional analysis.
The second chapter, devoted to interpolation in spaces of measurable
functions, makes up a significant portion of the book. It can be read
independently of the first chapter, from which only the simplest definitions
are needed. The chapter contains a theorem describing all interpolation
spaces between L} and Loo, and a theorem which is a further extension of the
Marcinkiewicz theorem. The exposition is pursued as far as concrete applica-
tions, for example, the theory of orthogonal series: convergence properties of
Fourier series and the basis property of a function system are studied.
Moreover, the chapter contains much auxiliary material from the theory of
functions which is discussed little in the literature. Decreasing rearrangements
of measurable functions are studied in detail, function spaces symmetric in
the sense of E. M. Semenov, and in particular, Lorentz and Marcinkiewicz
spaces, are discussed (in the foreign literature similar spaces are called
invariant with respect to permutations). Sharpenings of classical inequalities
of analysis (the Hardy-Littlewood, Hilbert, and other inequaities) are given.
FOREWORD
Xl
In the third chapter the theory of scales of Banach spaces, developed
mainly in the publications of S. G. Krein and Ju. I. Petunin, is expounded.
The prerequisite material for this is contained in the first chapter. Important
properties of the scales, in particular their "almost" interpolation properties
are also expounded in the fourth chapter. In the last section of the third
chapter properties of the classical scale of Holder spaces important in
applications are studied in detail.
In the fourth chapter two methods of constructing inteqx>lation spaces
enjoying the largest number of applications are described in detail: the
method of complex interpolation proposed independently by A. P. Calderon
and J. L. Lions and extensively developed by Calderon, and the method of
constants and averages due to J. L. Lions and J. Peetre. The latter method is
expounded in the more general form which it acquired in the work of V. I.
Dmitriev (who took the most active part in writing the corresponding
section). The fourth chapter can be read independently from the second and
third chapters.
The book does not include interpolation theory in spaces of smooth
functions and its applications. * This theory developed under the influence of
the work on imbedding theorems by S. L. Sobolev, S. M. Nikol'skll and their
students and followers. The abstract theory did not rise immediately and
easily to the level of concrete imbedding theorems obtained by special means.
However, now such a theory has been created. Its exposition apparently
needs another book. One can get acquainted with it partly in the book [7] by
P. L. Butzer and H. Berens. It is expounded more completely in Hans
Triebel's very recent book Interpolation theory, function spaces, differential
operators (published by VEB Deutscher Verlag Wiss., Berlin, 1977, and by
North-Holland, 1978).(1) One can get acquainted with the applications of this
theory to the study of boundary value problems for partial differential
equations in the book of J. L. Lions and E. Magenes [27] and in Triebel's
book mentioned above. At the end of the book there is a bibliography
covering, in addition, the indicated part of interpolation theory.
As we noted above, some parts of the book were written by V. I. Dmitriev.
I. Ja. Sneiberg provided us with invaluable help. He participated in writing 1
of Chapter IV and read a significant portion of the book. His critical remarks
· Editor's note. For this translation a new Chapter V was added by the authors to cover this
subject.
e)The authors are grateful to Professor Triebel for making the manuscript of this book
available.
XlI
FOREWORD
enabled us to remove a number of inaccuracies and improve some proofs.
The authors express their gratitude to both of them.
Finally, we thank all participants of the Voronezh seminar on interpolation
theory for linear operators, and, in particular, M. S. Braverman, A. A.
Dmitriev, E. A. Pavlov, P. A. Kucment, and A. A. Sedaev, for their constant
help in the preparation of the book.
The authors
CHAPTER I
IMBEDDED, IN1ERMEDIA TE,
AND INTERPOLATION BANACH SPACES
1. Imbedding of Banach spaces
1. Imbedded Banach spaces.
DEFINITION 1.1. We shall say that a Banach space EI is imbedded in a
Banach space Eo if the following conditions are satisfied:
1 0 . x E EI implies that x E Eo.
2 0 . The space Eo induces a vector space structure on E I coinciding with the
structure of E I.
3 0 . There exists a constant C OI such that
IIxil EO COlli xl lEI
(1.1)
for all x EEl.
The smallest possible value of the constant C OI in (1.1) is called the
imbedding constant of EI in Eo.
Sometimes the term "imbedding" is used in a wider sense. Instead of
conditions 1 0 and 2 0 , it is required that there exist an injective linear mapping
j (the imbedding operator) mapping EI into Eo, and then condition (1.1) is
written in the form IIjxll Eo Colllxll EI.
In such a situation we shall always identify EI with its imagejEI.
Condition 3 0 can be formulated in the following equivalent form: if X n x
in E I , then X n x in Eo. In this form the definition of imbedding can be
carried over to topological linear spaces, and then we say that EI is algebrai-
cally and topologically imbedded in Eo.
DEFINITION 1.2. The space EI is densely imbedded in Eo if conditions 1 0 -3 0
hold, and also
4 0 . The set E I is dense in Eo.
1
2
I. INTERPOLATION SPACES
The space E 1 is compactly imbedded in Eo if conditions 1 0 _3 0 hold, and
also
50. Every set bounded in the norm of E 1 is relatively compact in Eo.
In what follows we shall denote the imbedding of E 1 in Eo by the symbol
E 1 c Eo, assuming that the symbol c means not only set-theoretic inclusion,
but imbedding having the properties 2 0 and 3 0 .
If the space E 1 is imbedded in Eo, then on E 1 we can introduce a new
norm:
IIxllk l = C o1 11 x llE I .
Then the space Er equipped with this norm is isomorphic with E 1 , and
II x "Eo II x II k I.
In connection with this we introduce the following definition.
DEFINITION 1.3. We shall say that E 1 is normally imbedded in Eo if E 1 is
dense in Eo and the imbedding constant C 01 does not exceed one, i.e.
II x II Eo II x II E I.
We consider some examples of normalized imbedding of Banach spaces.
Let Eo = C(O, 1) be the space of continuous functions and E 1 = C(1)(O, 1)
the space of continuously differentiable functions. Then C(I)(O, 1) is normally
imbedded in C(O, 1), so that C (1)(0, 1) c C(O, 1) and
II xII C(O 1) = max Ix(t)1 II xII C<I)(O 1)
, t E [0, I] ,
= max Ix(t)1 + max Ix'(t)l.
t E [0, I] t E [0, I ]
Besides, by the Weierstrass theorem the set M of algebraic polynomials is
dense in C(O, 1), and consequently C(1)(O, 1) :J M is also dense in C(O, 1).
Finally, by Arzela's theorem, C(1)(O, 1) is compactly imbedded in C(O, 1).
We may show analogously that the space c(n)(o, 1) of n times continuously
differentiable functions is normally and compactly imbedded in c(m)(o, 1) if
n > m.
Another example of normally imbedded spaces is furnished by the spaces
Lp.
Let G be a bounded domain in n-space. Consider the space 4 consisting of
the real-valued or complex-valued functions that are pth power summable in
G (1 p < 00). We denote by La the space 4 withp = 2/(1 - a) in which
a norm is introduced by the formula
II x II La = (mes G)( a - 1)/2 11 x II -4 .
1. IMBEDDED BANACH SPACES
3
The space L/3 is normally imbedded in L a if /3 > a. Indeed, La :J L/3 for
a < /3, and by the Holder inequality we have
IlxllL" = (mes G)(a-I)/2(f G 1x (t)1 2 /(I-a) dtr-a)/2
..;; (mes G)(a-I)/2(mes G)(p-a)/2(f G 1x (t)1 2 /(I-P) dtr-p)/2
= Ilxil LP (x E L/3).
The set of simple measurable functions (linear combinations of characteris-
tic functions) is dense in any space La, -1 a < 1, and therefore L /3 is
dense in La.
It is not difficult to verify that the imbedding of L/3 in L a does not have the
property of compactness.
It can be shown analogously that the sequence space is normally
imbedded in lq if p < q.
The following circumstance should be kept in mind: the space EI can be
densely imbedded in the space Eo, but the closure of the ball of EI in Eo may
contain no interior points in the sense of the norm of Eo. Moreover, the
following assertion is well known:
LEMMA 1.1. If a Banach space EI is imbedded in a Banach space Eo and does
not coincide with it, then the closure in Eo of any ball of EI is nowhere dense in
Eo.
PROOF. We assume the contrary. Let the closure SI O of the unit ball SI of EI
contain the ball 02r(x O ) with center at the point Xo and radius 2r (in the sense
of the norm of Eo). Then the point (y - xo)/2 E S? , where y E 02r(xo), runs
over the ball Or with center at zero. Consequently, the closure of any ball
S contains the ball 0kr. Let z be any element of Or. There exists an XI E SI
such that IIz - XI II Eo r/2. Hence there is an x 2 E SI/2 such that
IIz - XI - x211Eo r /4. Continuing this process, we construct a sequence of
elements x n E S21-n such that
II z - X I - . . . - X n II Eo r2 - n.
Then z = Lr X k , and this series converges in the norm of EI. Therefore
z EEl. We have arrived at a contradiction to the assumption that EI =F Eo.
2. Relative completion. Let Eo and EI be a couple of imbedded Banach
spaces (E I c Eo). Denote by EOI the collection of all elements of Eo which
are limits in Eo of sequences of elements from E} bounded in the norm of E I :
X E E ol : X = lim X n (in Eo) and Ilxnll E I R. (1.2)
n --+ 00
4
I. INTERPOLATION SPACES
Obviously E01 is a linear manifold in Eo. We can introduce a norm in it by
putting
IIxllo1 = inf R,
where the infimum is taken over those R for which there exist sequences X n
with property (1.2). We verify that IIxll o 1 has the properties of a norm. For
this we note that (1.1) implies that IIxnll Eo Co111xnllEI C o1 R, and since
X n x in Eo, we also have II xII Eo C o1 R. This implies that "xii Eo
Co111xllo1' and, in particular, that IIxllo1 = 0 only if x = O. It is obvious that
IIAxll01 = IAI II x ll o1 . Moreover, if the sequences {x n } and {Yn} have properties
(1.2) relative to the elements x andy, respectively, with constants R and R 1 ,
then X n + Y n x + Y in Eo, and IIxn + YnllE I R + R 1 .
We obtain from this that II x + Y 1101 R + R 1 , and then, taking the in-
fimum on the right side, that II x + Y 1101 II X 1101 + II Y 1101.
We note that for x E E 1 we may choose X n = x, and from the definition of
the norm in E01 it will follow that II x ll o1 IIxil EI.
The construction of E01 may be ascribed the following geometrical mean-
ing. Let x E E01 and IIxllo1 = r. By definition, the element x belongs to the
closure in Eo of any ball of E 1 with radius R > r. Taking a sequence Rk r
and for every R k , choosing an appropriate sequence of elements in E 1 with
property (1.2), we can construct a sequence {x } eEl such that xk x in Eo
and IIx IIEI r as k 00. Then x k = rx (lIx IIE)-l x in Eep and IIxkll EI =
r. Thus, x belongs to the closure in Eo of the ball (and even of the sphere) of
radius r of E 1 and does not belong to the closure of balls of smaller radius.
Hence, the ball of E01 is the closure in Eo of the ball of E 1 with the same
radius.
We prove that the normed space E 01 is complete.
Let {x(k)} be a Cauchy sequence in E 01 . By the inequality II x II Eo <:
Co111xllo1' it is Cauchy in Eo. Let x(k) x in Eo. For any £ > 0 and suffi-
ciently large m and I we have IIx(m) - X(l)II01 E. This means that X(m) - x(l)
belongs to the closure in Eo of the ball of radius £ of the space E 1 . However,
x(m) - x(l) x(m) - x in Eo as I 00. Therefore, the element x(m) - x
belongs to the same closure, i.e. Ilx(m) - xllo1 E. Thus, x(k) x in E 01 as
k 00, and E01 is complete.
Summing up, we may say that we have constructed a Banach space E01
such that E 1 C E01 c Eo; moreover, IIxllo1 IIxll EI (x EEl) and
II xII Eo C 01 II Xllo1 (x E E 01 ). (1.3)
From (1.3) it follows that if E 1 is normally imbedded in Eo, so is E 01 .
The Banach space E01 is called the relative completion of E 1 with respect to
Eo.
1. IMBEDDED BANACH SPACES
5
The following facts are important.
LEMMA 1.2. If EI does not coincide with Eo, then its relative completion EOl
does not coincide with Eo either.
PROOF. By Lemma 1.1 the closure in Eo of any ball Sr of EI is nowhere
dense in Eo. Therefore EOl = U r>O Sr is a set of the first category in Eep and
consequently does not coincide with Eo.
A
LEMMA 1.3. The completion EOl of EOl with respect to Eo coincides with EOl
itself.
A
PROOF. If Y E E ol , then there is a sequence of elements Yk E EOl such that
II Yk 1101 = II Y II EOI and Yk Y in Eo. For every Yk there exists a sequence
x k) Yk in Eo as n 00 and such that Ilx k)IIEI = IIYkllE ol = IIYlli ol . But
then we can construct a sequence x :) such that x ) Y in Eo and IIx )II EI =
II Y II E ol ' and consequently Y E E ol . Moreover, from the above it follows that
Ilyllol IIYIIE ol ' and since the reverse inequality is always true (see (1.3)), we
have II Y 1101 = II Y II E . The lemma is proved.
01
A simple example of relative completion can be obtained by setting
Eo = LI(O, 1) and EI = C(O, 1). Then it is easy to see that EOl = Loo(O, 1).
Similarly, if Eo = C(O, 1) and E I = C 1(0, 1), then EOl = HI (0, 1) is the space
of functions satisfying a Lipschitz condition.
The following assertions concerning the relation between the spaces EI and
Eo are consequences of the definition of relative completion.
LEMMA 1.4. In order that EI be isometrically imbedded in EOl it is necessary
and sufficient that the ball of E I be closed (in E I) in the topology induced by the
norm of Eo.
PROOF. If IIxliol IIxll EI for some x EEl' then there exists a sequence
x n x in Eo such that Ilxnll EI = IIxliol = a. This means that the ball of
radius a in EI is not closed in the norm of Eo. Its limit point x has norm
greater than a.
Conversely, if the ball of radius a is not closed, then there exists a point x
with II x II E I > a and a sequence X n x in Eo such that II X n II EI a. But then
IIxliol a < IIxII EI .
LEMMA 1.5. In order that EI be a closed subset of EOl it is necessary and
sufficient that the closure of a ball of EI in the topology induced in EI by the
norm of Eo be a bounded set in E I.
6
I. INTERPOLATION SPACES
By the Banach theorem, EI is closed in EOl if and only if the norms II xII E
1
and Ilxli ol are equivalent on EI. After this remark the proof of Lemma 1.5 can
be carried out similarly to that of Lemma 1.4.
DEFINITION 1.4. The space EI is said to be complete with respect to Eo if EOl
coincides with EI isometrically.
By Lemma 1.3, EOl is complete with respect to Eo. In particular, LcYJ(O, 1) is
complete with respect to LI(O, 1).
LEMMA 1.6. If EI C F C Eo, then the completion EF,I of EI with respect to F
is imbedded in EOl with imbedding constant not exceeding one. The completion
of EF,I with respect to Eo coincides with EOl isometrically.
PROOF. If x E EF,I' then there exists a sequence X n E EI with IIxnll EI =
IIxilE such that X n x in F. In view of the imbedding F C Eo we then have
F.I
X n x in Eo, and consequently x E EOl and II x II EOI II x II E F . 1 .
Moreover, the unit ball of the completion of EF,I with respect to Eo is the
closure in Eo of the unit ball of EF,I; the unit ball of EI is dense in the latter
ball in the norm of F, and consequently in the norm of Eo as well. Thus, the
unit ball of the completion of EF,I with respect to Eo coincides with the
closure in Eo of the unit ball of E I , i.e. with the unit ball of EOl.
COROLLARY 1. If EI is complete with respect to EfP then it is complete with
respect to F.
For example, Loo(O, 1) is complete with respect to all spaces Lp(O, 1),
1 p < 00.
COROLLARY 2. If EF,I is complete with respect to EfP then the completion of
E F,I coincides with that of EOl.
2. Dual spaces of imbedded Banach spaces
1. Dual spaces and relative completion. If the space EI is imbedded in the
space Eo, then the restriction to EI of every continuous linear functional f(x)
defined on Eo induces a functional on EI in a natural manner. This functional
is continuous in the norm of EI. Indeed,
If(x)1 IlfllE o IIxil Eo ColllfllE o Ilxll EI .
Thus, a linear mapping of the space Eo into E; is obtained. If EI is not
dense in Eo, then there exists a nonzero functional in E which identically
vanishes on E I , and consequently is mapped into zero. In this case the
mapping is not injective. If, on the other hand, EI is dense in EfP then the
mapping is injective, and Eo can be considered imbedded in E;. Then, the
imbedding constant does not exceed COl.
2. DUALS OF IMBEDDED SPACES
7
The quantity
sup If(x)1 = IlfilE'
xEE I IIxll EI I
will be a seminorm on E in the first case and a norm on E; in the second
case.
Now we give another characterization of the relative completion EOI.
(j E E )
LEMMA 2.1. The completion EOI of E I with respect to Eo consists of all
elements of Eo inducing functionals on Eo, bounded in the seminorm II fll E I '
according to the formula x(f) = f(x).
PROOF. If x E E ol , then there exists a sequence X n such that X n x in Eo
and IIxnll EI = IIxllol. Let f E E . We have f(xn) f(x) and If(xn)1
IIfllE I IIxnll EI = IIfllE I Ilxli ol . This implies that If(x)1 IlfllE I IIxliol also
holds. Thus, the functional x(f) is bounded on E in the seminorm II fll E 1 '
and its norm does not exceed II x 1101.
Now let x E Eo and Ix(f)1 = If(x)1 CHfllE I (f E E ). We show that x
belongs to the closure in Eo of the ball Sc of radius C of EI. Otherwise there
would exist a functional fo E E such that sUPYEs)fo(y)1 < fo(x) or C Ilfoll EI
<fo(x), which contradicts the initial assumption. Thus, x E EOI and IIxll ol
C.
From the proof of the lemma we obtain the following corollary.
COROLLARY 1. The norm of the functional x(f) with respect to the seminorm
II fll E. is equal to II x 1101:
IIxll ol = sup If(x)l.
1 E EO, 11111 Ei <: I
(2.1 )
DEFINITION 2.1. A linear manifold M' of continuous linear functionals on a
Banach space E is said to be normative if
IIxllE = sup If(x)1 (x E E).
1EM',1I1I1E' <: I
Formula (2.1) and Lemma 1.4 imply the following theorem.
THEOREM 2.1. In order that the restrictions to EI of all functionals from E
form a normative set for E I it is necessary and sufficient that E I be imbedded
isometrically in E ol , or, what is the same, a ball of EI be closed in EI in the
topology induced by the norm of Eo.
2. Dual spaces of densely imbedded spaces. If E I is densely imbedded in Eep
then, as we have seen above, E is imbedded in E;; however, this imbedding
may not be dense. For example, II = EI is normally imbedded in Co = Eo.
8
I. INTERPOLATION SPACES
The dual spaces Eo = II and E; = 100 are not densely imbedded. Indeed, the
elemen t (1, 1, . . . ) E 100 is at distance one from the linear manifold II.
We recall that a linear manifold M' c E; is said to be total if the condition
f(xo) = 0 (xo EEl) for allf E M' implies that Xo = fJ.(I) If EI is dense in EfJ'
then Eo is a total linear manifold in E;. Indeed, if Xo E EI and Xo =1= 0, then
Xo E Eo, and so there is a functionalf o E Eo such thatfo(x o ) = IlxollEo =1= O.
If EI is reflexive, then every total manifold is dense in E;, and so in this
case Eo is densely imbedded in E;.
THEOREM 2.2. If EI is densely imbedded in Eo, then Eo is complete with
respect to E;.
PROOF. If f belongs to the completion (Eo)1 of Eo with respect to E;, then
there exists a sequence h E Eo such that h finE; and IIkll Eo = IIfll(EO>..
Then for any x E EI we have fn(x) f(x). The sequence of the linear
functionals In E Eo is uniformly bounded and converges to f on the set EI
dense in Eo. By the Banach-Steinhaus theorem this implies that f E Eo and
the sequence fn weakly converges to f on Eo. Moreover, II fll E' limll h II E' =
o _ 0
IIfll(EO).. Since in the case of a completion the reverse inequality always holds,
we have IIfilE o = IIfll(E o )..
THEOREM 2.3. A reflexive space EI is complete with respect to any other space
Eo in which it is imbedded.
PROOF. Without loss of generality we may assume that EI is dense in E
since otherwise Eo could be replaced by the closure of EI in Eo without
changing the completion EOl. Under this assumption, the reflexivity of EI
implies that Eo is densely imbedded in E;. Then by Theorem 2.2 the space
E{' = EI is complete with respect to Eo', and by Corollary 1 of Lemma 1.6
the space EI is complete with respect to Eo. The theorem is proved.
If Eo is densely imbedded in E;, then it is a normative set on E I , and
therefore Lemma 2.1 and formula (2.1) imply the following assertion.
LEMMA 2.2. If EI is densely imbedded in Eo and Eo is densely imbedded in E;,
then E 1 is isometrically imbedded in EOl. The space EOl can be isometrically
imbedded in E{' in a natural manner, and in this natural identification we may
assume that EOl = Eo n E{'.
Now if Eo is not densely imbedded in E;, we can consider its closure i; in
E;. This is a subspace of E;. In this case we may reformulate Lemma 2.1 and
Corollary 1 as follows.
( 1 )8 is the origin of the space.
3. INTERMEDIATE BANACH SPACES
9
LEMMA 2.3. If EI is dense in Eo, then EOl consists of all elements x in Eo
inducing continuous linear functionals on E{ by the formula x(f) = f(x). The
mapping x = x(f) is an isometric imbedding of EOl in the dual space (E;)'.
DEFINITION 2.2. If EI is dense in Eo, and in the natural imbedding the space
EI coincides with the whole space (E;)', then the couple of spaces E I , Eo is
said to be reflexive.
It is clear from Lemma 2.3 that for a reflexive couple Eo, EI the space EI is
complete with respect to Eo.
3. Intermediate Banach spaces
1. Sum and intersection of spaces in a Banach couple.
DEFINITION 3.1. A Banach couple* is two Banach spaces A and B algebrai-
cally and topologically imbedded in a separated topological linear space ct.
This nleans that A and B are linear manifolds in ct and the topology
induced on A and B by the topology of (f is weaker than the original
topologies of the normed spaces A and B.
It is obvious that a couple of imbedded Banach spaces Eo and E I , where
EI c E oI , can be considered as a special case of a Banach couple, where the
role of the topological space (f is played by the Banach space Eo.
With any Banach pair we may canonically associate a couple of imbedded
Banach spaces in the following way:
1. The space A n B consists of the elements common to A and B; the
norm is introduced by
" x" A n B = max ( " x " A' "x " B) (x E A n B). (3.1 )
2. The space A + B consists of elements of the form x = u + v, where
u E A, v E B and is equipped with the norm
" x II A + B = inf { " u II A + "v" B } , (3.2)
where the infimum is taken over all elements u E A and v E B whose sum is
equal to x.
The first of these spaces is called the intersection of the spaces of the
Banach couple, and the second is called the SUln of the spaces of the Banach
couple.
We show that these are Banach spaces. Let XI' x 2 , . .. be a Cauchy
sequence in A n B. By virtue of the definition (3.1) of the norm, the sequence
· Editor's note. In the Western literature it is usually called a compatible couple of Banach
spaces.
10
I. INTERPOLATION SPACES
{x n } is Cauchy in both A and B. The completeness of A and B implies that
X n X A (in A) and X n X B (in B). Since A is topologically imbedded in (t, X n
converges to X A in the topology of (t, and similarly X n X B in the topology
of (t. This implies that X A = x B = x, because the topology of (t is separated.
Consequently, we obtain that X n x E A n B in the norms II . IIA and
II . II B' and therefore in the norm of A n B.
To prove the completeness of A + B we construct a Banach space isomet-
ric to it. To this end, in the product A X B of A and B with norm
II ( u, v) II A x B = II u II A + II v II B we consider the subspace L consisting of the
elements of the form w = (z, -z), z E A n B, and the quotient space
(A X B)j L. If two elements (u, v) and (UI,V I ) belong to the same coset
modulo L, then u + v = u l + VI' and consequently to this whole coset there
corresponds a single element x = U + V E A + B. Conversely, for any ele-
ment x E A + B the pairs (u, v) corresponding to all possible representations
x = u + v form a coset of the space A X B modulo L. Moreover, it is clear
from the definition of the norm (3.2) that the norm of an element x E A + B
is equal to the norm of the corresponding coset in the quotient space
(A X B)j L. Thus, A + B is isometric to (A X B)j L. As is well known (see
[6], Fascicule de resultats, 5.5), the quotient space of a Banach space modulo
a (closed) subspace of it is a Banach space, and so A + B is also a Banach
space. From what has been said it is clear that if A n B = fJ, then A + B is
isometric with the direct product of A and B.
We note that each of the spaces A and B is contained in A + B, and
moreover, by virtue of the inequalities
II x II A + B II x II A + II fJ II B = II x 1\ A f or x E A,
(3.3)
and
II x II A + B II fJ II A + II x II B = II x II B for x E B,
(3.4)
the spaces A and B are imbedded in A + B, and the imbedding constants do
not exceed one. It follows from the preceding inequalities that IlxIIA+B
IIxllAnB holds for x E A n B, which shows that A n B is imbedded in
A + B.
In the case of a couple of imbedded Banach spaces E I C Eo the space
Eo n EI coincides with EI as a set, and the space Eo + EI with Eo. Moreover,
these spaces are respectively isomorphic (Eo n EI with EI and Eo + EI with
Eo). Indeed, if Ilxil Eo Colllxll E.' then for x E EI
Ilxll EI max(llxII Eo ' IIxll E ) = IIxilEonEI max(l, Col)IIxIl E .,
3. INTERMEDIATE BANACH SPACES
11
and for x E EO
min{ 1, 1/ C OI } IIxli Eo min{ 1, 1/ C OI } xl +v (II ull Eo + II vii EJ
x l + v (II u II Eo + II v II E.) = II X II Eo + E I II X II Eo
(the latter by virtue of (3.3) and (3.4)). Thus, Eo n EI and Eo + EI form a
couple of imbedded spaces isomorphic to EI and E respectively.
We consider the geometric structure of the unit balls SI(A n B) and
SI(A + B) of A n B and A + B. It is easy to see that SI(A n B) is the
intersection of the unit balls SI(A) and SI(B). In eed, if X E SI(A n B), then
IIxliA 1 and IlxliB 1, and therefore x E SI(A) and x E SI(B). Con-
versely, if x E SI(A) n SI(B), then IIxli A 1 and IIxll B 1, and so IIxliA nB
= max { II x II A' II x II B} 1.
The open ball SP(A + B) coincides with the convex hull of the open balls
SP(A) and S (B) of A and B:
SP(A + B) = conv(Sp(A), SP(B)). (3.5)
Indeed, let IlxliA +B < 1. Then there exist u and v such that x = u + v and
lIuli A + IIvll B < 1. The elements u l = (liuli A + IIvIIB)u/lluIiA and VI =
(liuliA + IIvIlB)v/llvil B belong to the balls SP(A) and SP(B), respectively.
Moreover,
x = u + v = p,U I + (1 - p,)v l , where JL = lIuli A / (liuli A + IIvIlB)
and consequently, x E conv(Sp(A), S (B)). Conversely, from the relation
x E conv(Slo(A), SP(B)) it follows that x = JLU + (1 - JL)v (u E S?(A), v E
° .
SI (B), 0 JL 1), and hence
IlxliA +B II JLUIIA + 11(1 - JL)vll B < 1,
i.e. x E SP(A + B).
We note some more properties of the sum and intersection of spaces in a
Banach couple.
LEMMA 3.1. For any x E A
dA(x, A n B) = d A +B(X, B),
(3.6)
where dE(x, L) denotes the distance in E from the element x to the linear
manifold L.
PROOF. From (3.3) and the inclusion B :) A n B it follows immediately
that dA(.x, A n B) dA+B(X, B). We establish the reverse inequality. If y E
B, then x - yEA + Band
II x - y II A + B = inf ( II u II A + II v II B)
x-y=u+v
inf II u II A = inf II x - y - v II A . (3.7)
x-y=u+v x-y=u+v
12
I. INTERPOLATION SPACES
Since x E A and x - y - v E A, we havey + v EA. On the other hand,
y E B and v E B, and so y + v E B. Then (3.7) implies that
IIx - yllA+B inf IIx - ZIIA = dA(x, A n B).
zEAnB
Taking the infimum of the left side over y E B, we obtain (3.6).
COROLLARY 1. A n B is dense in A if and only if B is dense in A + B.
Indeed, if A n B is dense in A, then dA(x, A n B) = 0, and hence also
dA+B(X, B) = 0 for any x E A. But then dA+B(x, B) = 0 for any x E A + B
as well, i.e. B is dense in A + B. The converse is obvious.
COROLLARY 2. If A n B is dense in both A and B, then it is cknse in A + B.
Indeed, A n B is dense in B, and the latter is dense in A + B by virtue of
what has been said above. Since II . IIA +B II . II B' the space A n B is dense
in A + B.
Now we assume that A n B is not dense in A. Then for any e > 0 there
exists a nonzero element x E A such that dA(x, A n B) (1 - e)lIxIIA. Then
we obtain from (3.6) that
(1 - e)lIxll A dA+B(x, B) IlxIIA+B.
Comparing this inequality with (3.3), we conclude that the imbedding con-
stant of A in A + B is equal to one in this case. The following deeper
assertion is true.
LEMMA 3.2. If the imbedding constant of the space A in the space A + B is
less than one, then A is imbedded in B.
PROOF. Let the imbedding constant be equal to q < 1. For x E A with
Ilxli A < 1 we have IIxIIA+B < q. By the definition of the norm in A + B
there exists a representation x = u l + VI' where II uIIIA + IIvIII B < q, and
consequently lIudlA < q and IIvlIIB < q. We apply the same arguments to the
element U I and obtain the representation U I = U 2 + V 2 ' where IIu 2 11A < q2 and
IIV211B < q2. Continuing this process, we arrive at the equality x = un +
L Vk' where Ilunli A < qn and IIvkli B < qk. Then
n
X - Vk = IIunll A + B IIunii A < qn o.
k=l A+B
On the other hand, Lf' Vk converges in B. Since B is imbedded in A + B, it
follows that x = Lf' Vk E B. Hence A is contained in B.
From the above it follows that
q
II xii B 1 IIxII A ,
-q
i.e., A is imbedded in B.
3. INTERMEDIATE BANACH SPACES
13
The last part of the proof of the lemma could be omitted if we took
account of the following lemma.
LEMMA 3.3. If the space A in a Banach couple A, B is a linear manifold in B,
then A is imbedded in B.
PROOF. If A is contained in B, then the linear spaces B and A + B
coincide. By (3.4) we have IlxIIA+B IIxilB. Since B and A + B are Banach
spaces, by the Banach theorem on the inverse operator the inequality IIxil B
ClixIIA+B holds. If now x E A, then IlxilB ClixIIA+B CllxliA.
COROLLARY. If the spaces A and B in the Banach couple A, B coincide as
linear spaces, then A and B are isomorphic.
In the sequel we need the following result.
LEMMA 3.4. Given a Banach couple A, B, if there exists a subset MeA n B
dense in A on which the inequality II x II B C II x II A is satisfied, then A is
imbedded in B with imbedding constant not exceeding C.
PROOF. Let Xo E A. We consider the open balls IIx - xoil A < IIxoilA and
IlxliA < (I + e)lIxoil A . They intersect, and so there exists at least one ele-
ment Xl E M in their intersection. Then IIxo - xIII A < llxoilA and
Ilxlil A < (I + e)IIxoIi A . By repeating this reasoning for Xo - Xl' we find an
element X 2 such that
IIx o - Xl - x 2 11A < 2- 2 11 x oilA and IIx211A < 2- 2 (1 + €)llxoIl A .
Continuing this process, we obtain a sequence of elements Xl' X2, . .. from M
such that IIx o - LT xkll A < 2- m llx o ll A and IIxmll A < 2- m (1 + e)IIxoil A . Then
Xo = Lr X k and Lf'llxkll A < (I + e)lIxoII A . By the hypothesis of the theorem
we have
00 00
Ilxkll B C IIxkli A < C(I + e)IIxoII A ,
1 1
consequently Lf' x k converges in B, and by virtue of the imbedding of A and
B in (f its sum is also equal to xo. Consequenty Xo E Band
00
IIxoil B Ilxkll B < C(I + e)llxoil A .
1
Since € was arbitrary, the lemma is proved.
DEFINITION 3.2. Two Banach couples A, Band C, D are said to be
isomorphic if there exists an isomorphism of A + Band C + D whose
restrictions to A and B are isomorphisms of A and C and Band D,
respectively.
14
I. INTERPOLATION SPACES
2. The dutd spaces of a sum and an intersection. In order to illuminate the
structure of the dual spaces of a sum and an intersection of spaces of a
Banach couple we recall that the dual of the product A X B of Banach spaces
with norm
lI(u, v)IIAxB = lIull A + IIvIlB'
is the space A I X B ' with norm
II ( f, g) II A ' x B' = max ( II f II A" II gliB').
Similarly, if we introduce the norm
(3.8)
(3.9)
II(u, v)IIAXB = max(llull A , IIvIl B ),
in A X B, then the norm
II (f, g) II A' x B' = II f II A' + II g II B'
(3.10)
(3.11 )
arises in the dual space A' X B ' .
We note that in both cases the duality between A X B and A' X B' can be
given by the relation
< (u, v), (1, g) > = f( u) - g( v ). (3.12)
In subsection I we have already spoken of the fact that for a Banach
couple A, B the space A + B is isometric to the quotient space of A X B with
norm (3.8) modulo the subspace L consisting of the elements of the form
(z, -z), where z E A n B. According to general theorems, the dual of a
quotient space is isometric to the orthogonal complement of the subspace. We
explain of which functionals from A' X B' the orthogonal complement L.L of
L consists. By (3.12), if (f, g) E L.L, thenf(z) = -g(z) for z E A n B.
Consequently, the space (A + B)' dual to the sum A + B is isometric to
the subspace L.L off A' X B' with norm (3.9), consisting of all functionals
(f, g) withf(z) = -g(z) for z E A n B.
The value of the functional (f, g) E L.L at x E A + B can be calculated
by (3.12), where x = u + v (u E A, v E B).
Similarly, if we introduce the norm (3.10) in A X B, then L becomes
isometric to A n B. The dual space of a subspace of a Banach space is
isometric to the quotient space of the dual modulo the orthogonal comple-
ment of the subspace. Hence, the space (A n B)' dual to A n B is isometric
to the quotient space of A I X B' with norm (3.11) modulo L .L .
The picture becomes significantly simpler if we assume that A n B is dense
in both A and B. As shown in 2.1, in this case the spaces A' and B' are
naturally imbedded in (A n B)', and consequently form a Banach couple. If
now (f, g) E L.L, then g coincides with -f as an element of (A n B)', and
since f E A' and f E B ' , we have f E A' n B'. Conversely, every functional
3. INTERMEDIATE BANACH SPACES
15
f E A' n B' induces a pair (f, -f) E L.L. The norm of (f, g) in A' X B' with
norm (3.9) coincides with the norm of fin A' n B'.
Similarly, the quotient space A' x B' / L.L of A' X B' with norm (3.11)
modulo L.L, which, as we have shown, consists of the pairs (1, -f), f E A' n
B', is isometric to the sum A' + B'. Thus, we have arrived at the following
assertion.
THEOREM 3.1. If A n B is dense in the spaces of the Banach couple A, B,
then A' and B' form a Banach couple; the dual space (A + B)' of A + B is
isometric to A' n B', and the dual (A n B)' of A n B is isometric to A' + B'.
We also elucidate how the duality between A + B and A' n B' and
between A n B and A' + B' can be given. If x E A + Band f E A' n B',
then x = u + v (u E A, v E B), and according to (3.12) we have
<x,f) = «u, v), (f, -f) = f(u) + f(v),
i.e. the functional <x, f> coincides with the linear extension of f to A + B. If
now z E A n Band f E A' + B', then f = g + h (g E A', h E B'), and
according to (3.12) we have
<z,f> = «z, -z), (g, h) = g(z) - h(-z) = f(z),
i.e., <z, f> coincides with the restriction of f to A n B.
3. Intermediate spaces.
DEFINITION 3.3. The Banach space E is said to be intermediate for the
spaces of the Banach couple A, B if the imbeddings
A n BeE c A + B (3.13)
are obtained. We recall that c means algebraic and continuous imbedding
here; however, by Lemma 3.3 in order to verify that the Banach space E is
intermediate it is sufficient to know that it is algebraically and continuously
imbedded in (f, contains A n B, and is contained in A + B.
To illustrate the above definitions, we consider a few simple examples of
intermediate Banach spaces.
It is easy to see that any space Lp(O, 1) is an intermediate space between
40(0, 1) and Lpl(O, 1) (Po < PI) if Po P PI.
Indeed, in 1.3 we showed that Lp.(O, 1) c Lp(O, 1) c Lpo(O, 1) and, more-
over,
Il x ll 40 (O,I) IIxll4(o,l) (x E Lp(O, 1)),
II x II 4(0.1) II x II !PI(O, I) (x E 4(0, 1)).
For the Holder spaces Hao and Hal (ao < a l ), for example, the Holder
spaces Ha are intermediate if ao a a l (see Chapter III, 3).
16
I. INTERPOLATION SPACES
Let Lpo(O, (0) and Lp.(O, (0) (Po <PI) be the spaces of measurable func-
tions whose modulus is pith power summable (i = 0, 1) on the semiaxis.
Neither space Lp;(O, (0) is imbedded in the other: it is obvious that Lpo(O, (0)
n Lp.(O, (0) =1= Lp;(O, (0). Nevertheless, 4(0, (0) is intermediate between
40(0, (0) and Lp.(O, (0) if Po P PI. Indeed, consider the separated topo-
logical vector space S(O, (0) consisting of all measurable functions defined on
the semiaxis with the topology of convergence in measure (see [11], Chapter
III, 2). Then each of the spaces Lp,(O, (0) (i = 0, 1) is algebraically and
topologically imbedded in the topological vector space S(O, (0), and so these
spaces form a Banach couple.
We show that
Lpo(O, (0) n Lp.(O, (0) c Lp(O, (0) c Lpo(O, (0) + 4.(0, (0). (3.14)
Indeed, let x E Lpo(O, (0) n Lp.(O, (0). We write
Xl ( t) = { ( t)
if Ix(t)1 1,
if Ix(t)1 < 1,
and x 2 (t) = x(t) - xl(t). Then XI' X 2 E Lpo(O, (0) n Lp.(O, (0). Since P PI'
we have XI E Lp(O, (0). From the condition Po P it also follows that
X 2 E Lp(O, (0). Thus X E 4(0, (0), i.e. Lpo(O, (0) n 4.(0, (0) is contained in
Lp(O, (0), and therefore, as we pointed out above, is imbedded in it. The
second imbedding in (3.14) can be verified similarly.
In the next chapter we shall study the intermediate spaces for the Banach
couple LI (0, (0), Loo(O, (0) in detail.
4. The sum and intersection of a family of Banach spaces. In this subsection
I
we shall denote by E c F the imbedding of the Banach space E in the
Banach space F satisfying the condition IIxllF IIxllE (x E E).
Let It be a separated topological linear space, and let E; (i E I) be an
arbitrary family of Banach spaces algebraically and topologically imbedded
in It.
The intersection of the family E; is, by definition, the Banach space R E;
satisfying the following conditions:
I
1) R E; c E'i for any) E I.
I I
2) If F is a Banach space such that F C E'i for all) E I, then FeR E;.
The sum of the family E; is, by definition, the Banach space E;
algebraically and topologically imbedded in It and such that the following
conditions are satisfied:
I
1) E'i c E; for any) E I.
3. INTERMEDIATE BA..NACH SPACES
17
2) If the Banach space G algebraically and topologically imbedded in (t is
1 1
such that c G for all) E I, then E; c G.
We show that R E; always exists. We consider the linear subset R E;
consisting of all elements x belonging to all spaces E; for which
II x II (R) E. = sup II x II E. < 00. (3.15)
I iEI I
The number on the left has the properties of a norm. We show the
completeness of the space R E; in this norm. Let X n be a Cauchy sequence of
elements of R E; in the norm (3.15). This sequence is Cauchy in every space
E;, and so it has a limit in E;. Since all spaces E; are imbedded in (f, these
limits coincide with some element x En; E;. Moreover,
IIx - xnll(R)E. = sup II X - xnll E . = sup lim Ilxm - xnll E .
'. '. m--+oo '
I I
sup supllx m - xnll E ; = sup Ilxm - xnllrmE; O asn oo.
m>n i m>n
1
Thus R E; is a Banach space. Now if F c for all) E I and x E F, then
IIxllrm = sup IIxil Ei IIxilF < 00,
;
1
and therefore, x ERE;, i.e. FeR E;.
For the construction of the sum E; we make the additional assumption
1
that there exists a Banach space W imbedded in (f and such that E; C W for
all i E I. As the linear system E; we take the set of all elements x in (f
representable in the form
X= U;
;EI
(U E E;), where II u;II E. < 00.
,
iEI
(3.16)
It follows from the last condition that there are only countably many
summands in L;ef U; different from zero. Moreover, since
lIu;11 w Ilu;llE. < 00,
,
iEI ;EI
(3.17)
the series L;ef U; converges absolutely in W, and hence converges also in W.
We now introduce the norm
II xlI 'UI E: = inf II uill E;'
'C/ ;EI
(3.18)
where the infimum is taken over all possible representations of x in the form
I
(3.16). We leave the verification of the properties of a norm to the reader.
Inequality (3.17) implies that
II xII w IIxll Ei'
18
I. INTERPOLATION SPACES
and therefore the space Ej is imbedded in W, and consequently also in .
We prove the completeness of this space. Let xo, Xl' . .. be a Cauchy
sequence in the norm (3.18). Without loss of generality we assume that this
sequence is such that
00
Xo = 0, Ilx k - xk-llllgJ Ei < 00.
k=1
By the definition of the norm (3.18) there exist representations X k - Xk-l =
j u jk such that
1
II U ik II '" II x k - x k - Iii \gI n + 2 k ·
TIlen we have
00
Ilujkll E . < 00.
,
iEI k=1
(3.19)
This inequality implies in particular that the series v j = k ujk converge
absolutely in the spaces Ej, and V j E Ej. From (3.19) we obtain that
L II Vjll Ei < 00,
iEI
and so the element X = jEI V j E Ej is defined. Next,
x - x k = X - f (XI - XI_I) = { Vi - f U iI } '
/=1 iEI /=1
and therefore
k
IIx - xkll Ei V j - Ui/
iEI /=1 E;
00
= Ui/
iEI /=k+ 1
E;
00
II Ui/ II Ej 0 as k 00
/=k+1 iEI
by virtue of the convergence of (3.19). Thus, the space Ej is complete.
1
If now the Banach space G is such that Ej C G, then as W we can take G,
1
and, as above, show that Ej c G.
4. Interpolation spaces and interpolation triples
1. Interpolation triples of Banach spaces. Let A, Band C, D be two Banach
couples.
DEFINITION 4.1. A linear mapping T acting from the space A + B to
C + D is called a bounded operator from the couple A, B to the couple C, D if
4. INTERPOLATION SPACES AND TRIPLES
19
the restriction of T to the space A (B) is a bounded operator from A (B) to C
(D).
We denote by L(AB, CD) the linear space of all bounded operators from
the couple A, B to the couple C, D. This is a Banach space in the norm
II TIIL(AB,cD) = max(11 TIIA-+c' II TIIB-+D). (4.1)
Indeed, if the operators Tn form a Cauchy sequence in L(AB, CD), then
their restrictions to A and C converge in L(A, C) and L(B, D) to operators
T' and T", respectively, which obviously coincide on A n B. Then the
sequence Tn converges in L(AB, CD) to a uniquely defined operator T acting
from A + B to C + D according to the formula Tx = T' u + T" v (x = u +
v, u E A, v E B).
LEMMA 4.1. A bounded operator T from the couple A, B to the couple C, D
generates a bounded operator from A + B to C + D and from A n B to
C n D, and
II TII A + B-+C+ D II TII L(AB,CD),
II TIIAnB-+cnD II TIIL(AB,cD).
( 4.2)
( 4.3)
PROOF. For x E A + B, by the definition of the norm in the sum of spaces
of a Banach couple we have
II Txll C+D inf (II Tull C + II Tv II D)
x=u+v
xl +v (II TIIA-+c lIull A + II TIIB-+D IIvIl B ) II TIIL(AB,CD) IIxIl A + B ,
which implies (4.2). If x E A n B, then
II Txll C nD = max(1I Txll c, II Txll D)
max(1I TIIA-+C IIxHA' II TIIB-+D IIxII B ) II TIIL(AB,cD) IIxIi AnB .
We establish an auxiliary assertion.
LEMMA 4.2. Let G and H be Banach spaces, and E and F Banach spaces
imbedded in G and H, respectively. If a bounded linear operator T from G to H
maps E into F, then the restriction of T to E is a bounded operator from E to F.
PROOF. It is sufficient to show that the restriction of T to E is a closed
operator. Let X n x in E and TX n y in F. Then X n x in G and TX n Y
in H. Consequently y = Tx; the operator T is closed, and thus bounded.
COROLLARY. If A, Band C, D are two Banach couples, and a bounded linear
operator from A + B to C + D maps A and B to C and D respectively, then
this operator is bounded from the couple A, B to the couple C, D.
20
I. INTERPOLATION SPACES
DEFINITION 4.2. Let A, Band C, D be two Banach couples, and E and F
spaces intermediate between A and Band C and D, respectively. The triple
(A, B, E) is called an interpolation triple relative to (C, D, F) if every
bounded operator from A, B to C, D maps E to F.
If A coincides with C, B with D, and E with F, then E is called an
interpolation space between the spaces of the Banach couple A, B.
REMARK. From Lemmas 4.1 and 4.2 it follows immediately that in the
preceding conditions every operator T E L(AB, CD) is a bounded- operator
from E to F.
LEMMA 4.3. If a triple (A, B, E) is an interpolation triple relative to
(C, D, F), then there exists a constant c > 0 (interpolation constant) such that
II TII E--+F cll TII L(AB,CD) (4.4)
for any T E L(AB, CD).
PROOF. To every operator T E L(AB, CD) we assign its restriction to the
space E: <P(T) = TIE. As we proved above, <P(T) is a bounded operator from
E to F. Consequently a linear mapping <P of the Banach space L(AB, CD) to
the space L( E, F) is defined. This mapping is closed. Indeed, let Tn T in
L(AB, CD) and <P(Tn) S in L(E, F). From the first relation and Lemma
4.1 it follows that Tnx Tx in C + D for any x E A + B, and in particular
for x E E. From the second we obtain that Tnx Sx in F for any x E E.
Consequently, in view of the imbedding FcC + D we have Tx = Sx for
x E E, i.e. <P(T) = TIE = S. Since <P is closed, it is bounded, which is
equivalent to (4.4). The lemma is proved.
If E is an interpolation space between A and B, then (4.4) becomes the
following inequality: II Txll E cll TIIL(AB B) IIxIl E . In this case E can be
equipped with a new, equivalent, norm
1 II Txll E
Ilxll = - c sup .
TEL(AB,AB) II TII L(AB,AB)
For this norm we have the inequality II TII --+E II TIIL(AB B)' i.e. the interpo-
lation constant is equal to 1.
We give an example of an intermediate space which is not an interpolation
space. Let A = 1 2 be the Hilbert space of sequences x = ( 1' 2' . . . ) with the
norm IIxli/2 = ( 1 ;12)1/2, and B = If the Hilbert space consisting of all
sequences x for which
{ 00 } 1/2
IIxli/ = i(I 2i_112 + 1 2J) < 00.
4. INTERPOLATION SPACES AND TRIPLES
21
We consider the intermediate Hilbert space E whose elements are all se-
quences satisfying the condition
IIxllE = { (i1 2i-112 + 1 2J) } 1/2 < 00.
1 1
It is obvious that BeE c A.
Now we define the operators Tn by the equality
Tn x = ( 1' . . . , 2n-2' 2n' 2n-l' 2n+2' . . . ).
It is obvious that these operators act in the spaces A and B, and have norm
1 there, i.e. II Tn II L(AB B) = 1. Calculating the effect of Tn on X n for which
; = 0 for i =1= 2n and 2n = 1, we see that II TnllE-+E vn . If E were an
interpolation space, this would contradict Lemma 4.3.
It is useful to have in mind the following circumstance.
LEMMA 4.4. Let a triple (A, B, E) be an interpolation triple relative to
- -
(C, D, F). If E and F denote the completiom of E and F relative to A + B and
-
C + D, respectively, then the triple (A, B, E) is an interpolation triple relative
-
to (C, D, F).
PROOF. If x E E, then there exists a sequence X n x in A + B such that
Ilxnll E = II xIIi. Then for any operator T E L(AB, CD) we have TX n Tx in
C + D, and II Txnli F ' II TIIE-+Fllxnll E = II TIIE-+Fllxlli. From this it follows
-
that Tx E Fand IITxlli ' IITIIE-+Fllxlli.
We have proved the lemma and the inequality II Tlli-+i ' II TIIE-+F.
2. Good interpolation triples. * The type of an interpolation triple.
DEFINITION 4.3. Let (A, B, E) be an interpolation triple relative to
(C, D, F). We shall say that the triple (A, B, E) is a good interpolation triple
relative to (C, D, F) as applied to A and C if for all operators T E
L(AB, CD)
II T II E-+F ' cp( II TIIA-+c, II TII B-+D)
and cp(A, /l) 0 as A 0 uniformly in /l on every finite interval.
THEOREM 4.1. Let (A, B, E) be a good interpolation triple relative to the
triple (C, D, F) as applied to A and C, and let there exist a sequence of
finite-dimemional operators Pn acting in the spaces C and D, strongly converg-
ing to I in C and uniformly bounded in D. If the operator T E L(AB, CD) is
bounded as an operator from B to D and compact as an operator from A to C,
then it is compact as an operator from E to F.
· Editor's note. The original Russian term is BnOJIHe mrrepnoJIJlIJBoHHhle TpOBxu.
22
I. INTERPOLATION SPACES
PROOF. Let SI be the unit ball of A. The closure of the set TS I is compact
in C. The operators Pn uniformly converge to the identity" operator on
compact sets. Therefore
II(Pn - I) TIIA-+c 0 as n 00.
The norms II(Pn - I)TIIB-+D are uniformly bounded, by the hypotheses of
the theorem. Then
II(Pn - I)TIIE-+F < cp(II(P n - I)TIIA-+c, II(Pn - I)TIIB-+D) O
as n 00. Thus, T acting from E to F is the uniform limit of finite-dimen-
sional operators, and hence is compact.
DEFINITION 4.4. The triple (A, B, E) is said to be of interpolation type a
(0 < a < 1) relative to the triple (C, D, F) if it is an interpolation triple and
the ineq uali ty
II TIIE-+F < ell TII -=:c II TII -+D
(4.5)
holds.
In the case where A coincides with C, B with D, and E with F, E is said to
be an interpolation space of type a between A and B.
If the constant c in (4.5) is equal to one, then (A, B, E) is said to be a
normalized interpolation triple of type a relative to (C, D, F).
It can happen that an interpolation space between A and B is not of type a
for any a E [0, 1].
If (A, B, E) is an interpolation triple of type a, with a < 1, relative to
(C, D, F), then it is a good interpolation triple as applied to A and C, and
Theorem 4.1 holds.
Theorems which establish that one triple of Banach spaces is an interpola-
tion triple relative to another are called interpolation theorems. Historically,
the first interpolation theorem was obtained by M. Riesz and G. O. Thorin,
and the whole theory of interpolation for linear operators began to develop in
the direction of generalizations of this theorem. Here we give a general
formulation of this first interpolation theorem.
THE RIEsz-THORIN THEOREM. Let (QI' L I , Ill)' and (Q2' 2' p,0 be two mea-
sure spaces, and let 4(Q;) (i = 1, 2; p 1) be Banach spaces of complex-val-
ued functions, pth power summable with respect to Il;. Then
(Lpo(QI)' L p1 (QI)' Lp(QI» is a normalized interpolation triple of Banach spaces of
type a relative to the triple (Lqo(Q;), L q1 (Q;), Lq(Q;), if
l/p = (1 - a)/po + a/PI' l/q = (1 - a)/qo + a/ql (0 < a < 1).
( 4.6)
4. INTERPOLATION SPACES AND TRIPLES
23
The proof of this theorem is based on the well-known three lines theorem
in the theory of analytic functions and can be found in [50], Vol. II, Chapter
XII, 1.
We recall that, for the triple, to be a normalized interpolation triple of type
a means that for every linear operator A acting boundedly from 4;(QI) to
Lq;(Q:J and having norms M;, i = 0, 1, the inequality
II TxllLin2) Md-aMfllxllL,,(n t ) (4.7)
holds under conditions (4.6).
3. Optimal interpolation spaces. If E is an intermediate space for the Banach
couple A, B, and C, D is another Banach couple, then there arises the
question of describing the spaces F intermediate between C and D such that
(A, B, E) is an interpolation triple relative to the triple (C, D, F). It is clear
that if F has this property, then every space FI intermediate between C and D
in which F is imbedded has the same property. In this connection, we may
wish to find a space F which is minimal in a certain sense. It is clear that
every space F must contain all elements of C + D of the form Tx, where x
runs over E and T over the set L(AB, CD). Besides, F must contain the linear
span of elements of the form Tx. If we could introduce a Banach space
structure in this span so that it became intermediate between C and D, then
this Banach space would solve our problem. Below we shall accomplish, in a
more complicated way, the program outlined here.
The unit ball in L(AB, CD) will be denoted by 'lTI(AB, CD). Let T belong
to 'lTI(AB, CD). Consider the kernel N(T) of T and the range T(E). Since Tis
continuous as an operator from E to C + D, the kernel N(T) is closed in E.
The operator T induces a canonical one-to-one correspondence between the
quotient space EI N(T) and the range T(E). This allows us to carry the
Banach space structure of EI N(T) over to the linear space T(E). We denote
by Fr(E) the resulting Banach space. According to the above, the formula for
the norm of Fr<E) has the form
II y II Fr<E) = infll xii E' (4.8)
where the infimum is taken over all x E E for whichy = Tx.
The Banach space Fr(E) is imbedded in C + D. Indeed, according to (4.2)
we have
lIy II c+ D = II Txll c+ D II TII L(AB,CD) II xII A +B IIxli A +B (1 I m( E» IIxll E'
where II m(E) denotes the imbedding constant of E in A + B.
By (4.8) this implies that
m(E)lIyllc+D IIYIIFr<E).
24
I. INTERPOLATION SPACES
In what follows, we denote by kG, where G is a Banach space, the Banach
space with norm IIxllkG = kllxlIG (k > 0, x E G).
With this notation, the preceding inequality can be written as
1
Fr(E) c m(E)( C + D).
( 4.9)
We have obtained a family of Banach spaces FrC..E) imbedded in the
Banach space m(E)(C + D). According to 3.4, we may construct the sum of
spaces
F(E) = lW Fr(E).
T E '1Tl(AB, CD)
1
It follows from (4.9) that F(E) c m(E)(C + D).
We now show that C n D is imbedded in F(E). For this, from any element
y E C n D we construct a one-dimensional operator TaX = <x,I>y, where
I is a functional from the space (A + B)'. According to what was said in 3.2,
the restrictions of I to A and B are linear functionals on these spaces, and
II III (A +B)' = max{ IIIIIA" 11111 B'}. Then we obtain
_ II TaXllc _ 1 II(x)1 _ 1
II TolI A --+ c - sup IlxliA - Ilyll csup IIxll A - II iliA' Ily II c,
and, similarly, II TollB--+D = IIIIIB'IIYIID. Therefore
II Toll L(AB,CD) = max {IIIII A' II y II c, Ilfll B' II y II D}.
We choose a = max{IIIIIA,IIYllc, IIIIIB,IIYIID}. Then II TolIL(AB,cD) = 1 and
To E '1T 1 (AB, CD). Hence <x,I>y E Fro(E) for any x E A + B, and there-
fore y E Fr (E) c F(E). Thus, C n D is contained in F(E). To obtain the
o
imbedding inequality for any e we choose an element xe so that
II xe II A + B (1/ (m( E) + e» II xe II E
(such a choice is possible since l/m(E) is the imbedding constant of E in
A + B) and a functionalk(x) so that Ilkll(A+B)' = 1 and <xe,k> = IlxeIIA+B.
Then To(axe/lixeIIA+B) = y, and therefore
lIyIlFTo(E) IIaxe/lixeIlA+BIIE a(m(E) + e). (4.10)
Next,
a max(llyllc, IIYIID)max(IIIIIA" IIIIIB').
According to what was said in 3.2, the last factor coincides with II III (A +B)"
and is equal to one. Therefore a IIYllcnD' and (4.10) gives
IIYIIF(E) lIyIlFTo(E) (m(E) + e)IIYllcnD.
4. INTERPOLATION SPACES AND TRIPLES
25
By the arbitrariness of € this implies the following imbedding:
1
m(E)(C n D) c F(E).
We have arrived at the following assertion.
THEOREM 4.2. Let E be an intermediate space for the Banach couple A, B,
and let C, D be another Banach couple. There exists a space F(E) intermediate
for the couple C, D and having the property that (A, B, E) is an interpolation
triple relative to (C, D, F) if and only if F( E) c F.
PROOF. The space F(E) contains all spaces Fr<E), and consequently all
elements of the form Tx (x E E, T E 'lT1(AB, CD)). Hence (A, B, E) is an
interpolation triple relative to (C, D, F(E)). From this it follows immediately
that (A, B, E) is an interpolation triple relative to (C, D, F) if F :J F(E).
By construction, F(E) consists of elements of the form
z = y;, Llly;IIFT.(E) < 00. (4.11)
,
By the definition of the norm in Fr(E) there exist elements x; such that
,
y; = T;x; and Ilx;IIE Ily;IIFTo(E) + 1/2;. Then llx;IIE < 00. If now
,
(A, B, E) is an interpolation triple relative to (C, D, F), then by (4.11) we
have
L Ily;IIF = L lIT;x;IIF L 11T;IIE--+Fllx;IIE cL Ilx;IIE < 00,
and so z E R. Thus, F(E) c F.
REMARK. F(E) is an interpolation space relative to the couple C, D.
Indeed, let S E 'lT1(CD, CD) and T E 'lTI(AB, CD). We have IISTIIL(AB,CD)
1. If now z E F(E), then (4.11) holds, and since y; converges in C + D,
we have Sz = Sy;.
We calculate
IISy;IIFSTo(E) = inf Ilxll E inf IIxllE = Ily;IIFT.(E).
, Sy; = ST;x Yi = T;x '
It then follows from (4.11) that IISY;IIFsT.(E) converges, and consequently
,
Sz E F(E).
Thus, every operator from L(CD, CD) maps F(E) into F(E), and our
assertion is proved.
We note that if the space EI intermediate between A and B is contained in
E, then (A, B, £1) is an interpolation triple relative to (C, D, F(E»), and
therefore by Theorem 4.2 we have F(E 1 ) C F(E).
Now let A, B be a Banach couple, and F an intermediate space in the
Banach couple C, D. We can attempt to construct a largest intermediate
space between A and B that together with A and B forms an interpolation
( 4.12)
26
I. INTERPOLATION SPACES
triple relative to (C, D, F). Let T E 'lT 1 (AB, CD). Consider the pre image
T - 1( F) of F under the mapping of A + B into C + D by means of the
operator T. On this preimage we introduce the norm
IIxIl ET - 1 = max{m'(F)lIxIl A + B , II TxII F },
where m'(F) is the imbedding constant of C n D in F. With this norm Er-I
is a Banach space. Indeed, if {x n } is a Cauchy sequence in Er-I, then it is also
a Cauchy sequence in A + B, and consequently it converges in A + B to an
element x, and {Tx n } is Cauchy in F and hence converges in F and in C + D
to an element y. Since IITIIA+B C+D ' 1, we have y = Tx, and {x n }
converges to x in Er-I.
The space Er-I is intermediate between A and B. It is obvious that
1
Er-I c m'(F)(A + B). Let x E A n B. If IIxIIET-1 = m'(F)llxIlA+BII, then
II xII Er- I ,m'(F)lIxIi AnB . If, on the other hand, Ilx1l ET - 1 = IITxIIF, then by
(4.3) we have
IIxli Er - 1 ,m'(F)IITxllcnD ,m'(F)lIxIIAnB. (4.13)
1 1
Thus, m'(F)A n BeEr-I c m'(F)(A + B). Now we denote by E(F) the
intersection of the family of the spaces Er-I for all T E 'lTl(AB, CD). We
obviously have the imbeddings
1 1
m'(F)A n B c E(F) c m'(F)(A + B).
We recall that the norm in E(F) is defined by
IIxll E(F) = sup max{ m'(F)llxIi A +B' II Txll F}. (4.14)
T E'1Tl(AB,CD)
This formula can be simplified. Let Xo E E(F). We construct a linear
functional f E (A + B)' such that Ilfll(A+B)' = 1 and <xo,f> = IIxoII A + B .
Using the fact that m'(F) is the imbedding constant of C n D in F, we
choose y E C n D so that lIyllF> (m'(F) - e)lIyllcnD. We consider the
operator Tox = <xo,f>Y. If we choose a = max{lIfIlA' lIyllc' IlfIIB,IIYIID}'
then To E '1T 1 (AB, CD) (see p. 24). Here we have a , lIyllcnD' and
II T oXollF = lIxoIIA+BIlYIIF
llxoIIA+B(m'(F) - e)IIYllcnD ;;> (m'(F) - e)lIxoII A + B .
From this it follows that
sup II TXoll F m'(F)lIxoIi A +B'
TE'1Tl(AB,CD)
and therefore (4.14) can be written in the form
II x II E(F) = sup II Tx II F.
T E '1Tl(AB,CD)
( 4.15)
4. INTERPOLATION SPACES AND TRIPLES
27
In passing we have proved that E(F) consists of all elements x E A + B
for which
sup II Txll F < 00.
TE'1Tl(AB,CD)
THEOREM 4.3. Let A, B be a Banach couple, and F an intermediate space f?r
the Banach couple C, D. There exists a space E( F) intermediate between A and
B having the property that (A, B, E) is an interpolation triple relative to
(C, D, F) if and only if E c E(F).
PROOF. It is obvious that (A, B, E(F)) is itself an interpolation triple
relative to (C, D, F), since for x E E(F) we have Tx E F for any T E
L(AB, CD) by the definition of E(F). This implies that for E c E(F) the
triple (A, B, E) is an interpolation triple relative to (C, D, F). Conversely, if
(A, B, E) is an interpolation triple relative to (C, D, F), then by (4.15) for
x E E we have
II x II £( F) = sup II T x II F
T E'1Tl(AB,CD)
sup {II TII £--+FII xii £} cll xII £,
T E'1Tl(AB,CD)
where c is the interpolation constant for E and F, i.e., E c E(F).
REMARK. E(F) is an interpolation space between A and B.
Indeed, if S E L(AB, AB), then for T E 'lTl(AB, CD) we have TS E
L(AB, CD), and II TSIIL(AB,cD) IISIIL(AB B). Thus for x E E(F) we have
TSx E F and
II Sxll £(F) = sup II TSxl1 F
T E'1Tl(AB,CD)
= II S II L(AB,AB)
sup 1 TSx
T E'1Tl(AB,CD) II S II L(AB,AB) F
sup II Txll F = II S II L(AB,AB) II x II £(F)'
T E'1Tl(AB,CD)
II S II L(AB,AB)
which proves our assertion.
We note that if the space F 1 intermediate between C and D contains F,
then (A, B, E(F» is an interpolation triple relative to (C, D, F 1 ), and so by
Theorem 4.3 we have
E(F) C E(F 1 ).
( 4.16)
DEFINITION 4.5. Let (A, B, E) be an interpolation triple of spaces relative
to (C, D, F). It is said to be an optimal interpolation triple if the conditions
that (A, B, E) is an interpolation triple relative to (C, D, F) and E :) E and
- --
F c F imply that E = E and F = F.
28
I. INTERPOLATION SPACES
In other words, (A, B, E) is an optimal interpolation triple relative to
(C, D, F) if E cannot be enlarged and F cannot be decreased with preserva-
tion of the interpolation property.
THEOREM 4.4. In order that (A, B, E) be an optimal interpolation triple
relative to (C, D, F) it is necessary and sufficient that F = F(E) and E =
E(F).
PROOF. If (A, B, E) is an optimal interpolation triple relative to (C, D, F),
then by Theorem 4.2 we have F(E) c F. Since (A, B, E) is an interpolation
triple relative to (C, D, F(E», from the definition of the optimal interpola-
tion property it follows that F(E) = F. Similarly, Theorem 4.3 implies that
E = E(F).
Now let F = F(E) and E = E(F), and let the triples (A, B, E) and
-
(C, D, F) be as in Definition 4.5. Then (A, B, E) is an interpolation triple
relative to (C, D, F). It follows then from Theorem 4.2 that F :J F(E) = F,
and so F = F. We may show similarly that E = E.
COROLLARY. If (A, B, E) is an optimal interpolation triple relative to
(C, D, F), then E and F are interpolation spaces between A, B and C, D,
respectively .
This follows immediately from Theorem 4.4 and the remarks after Theo-
rems 4.2 and 4.3.
THEOREM 4.5. Let E and F be intermediate between the spaces of the Banach
couples A, B and C, D, respectively. Then the following assertions are true:
1. (A, B, E(F(E») is the optimal interpolation triple relative to (C, D, F(E».
2. (A, B, E(F» is the optimal interpolation triple relative to (C, D, F(E(F)).
- - -
3. If (A, B, E) is the optimal interpolation triple relative to (C, D, F), E :J E
and F :J F, then
F(E) c F c F(E(F»,
E(F(E» c E c E(F).
PROOF. If (A, B, E) is an interpolation triple relative to (C, D, F), and if
E :J E(F(E» and F c F(E), then (A, B, E) is an interpolation triple relative
t (C, D, F(E», and by Theorem 4.3 we have E c E(F(E», and hence
E = E(F(E». Since (A, B, E) is an interpolation triple relative to
(C, D, F(E», by the same theorem we have E c E(F(E» = E, and there-
fore (A, B, E) is an interpolation triple relative to (C, D, F), from which by
Theorem 4.2 we obtain that F:J F(E), and hence F = F(E). We ha e
proved the optimal interpolation property of (A, B, E(F(E») relative to
4. INTERPOLATION SPACES AND TRIPLES
29
(C, D, F(E». The analogous verification works for the triples (A, B, E(F)
and (C, D, F(E(F»).
Now let the triples (A, B, E) and (C, D, F) have the properties indicated in
assertion of of the theorem. Then (A, B, E) and (A, B, E) are interpolation
triples relative to (C, D, F) and (C, D, F), respectively, and therefore by
- -
Theorems 4.2 and 4.3 the imbeddings F(E) c F and E c E(F) are true. By
(4.16) from the first of these imbeddings we obtain that E(F(E» c E(F), but
by Theorem 4.4 from the condition of optimality we have E( F) = E, and
- - - -
therefore E(F(E» c E. Similarly, E c E(F) implies that F = F(E) c
F(E(F».
THEOREM 4.6. If (A, B, E) is the optimal interpolation triple relative to
(C, D, F), then (C, D, F) is an interpolation triple for (A, B, E).
PROOF. We know from the remark to Theorem 4.2 that F is an interpola-
tion space relative to C and D with some constant c F . By Theorem 4.4 we
have E = E(F). If x E F and Tl E L(CD, AB), then by (4.15) we have
II T1xll E = II T1xll E(F) = sup 111T 1 X II F
TE7Tl(AB,CD)
sup cFII1TIIIL(CD,CD)lIxIIF
TE7Tl(AB,CD)
cF11 T1IIL(CD,AB)llxIi F .
4. Complemented subcouple of a BllIUlCh couple. Let A and B be a Banach
couple of spaces imbedded in a separated topological linear space , and M
and N subspaces of A and B, respectively. Then the spaces M and N are
imbedded in , and consequently form a Banach couple, which is called a
subcouple of A, B.
DEFINITION 4.6. The subcouple M, N is said to be complemented if there
exists a bounded projection operator P in A + B projecting A onto M and B
onto N.
The operator P is of course bounded in A and B. The space A n B is
projected by it onto M n N.
Let E be an intermediate space between A and B. The image PE of E
contains M n N and is contained in M + N. Reasoning in the same way as
in subsection 3, in this image we can introduce a Banach space structure by
making it isometric to the quotient space of E modulo the kernel of P. Using
the notation introduced in subsection 3, we denote the resulting Banach space
by Fp(E). Then IIYIIFp(E) = infy=PxllxIIE. The space Fp(E) is intermediate
between M and N (see Lemma 3.3).
30
I. INTERPOLATION SPACES
If now F is an intermediate space between M and N, then we may consider
the inverse image P -IF of F and introduce the norm II xII E p -l(F) =
max{lIxIlA+B, IIPxIIF} in it. As we saw in subsection 3, the Banach space
Ep-I(F) thus obtained is intermediate between A and B.
Now let E be an interpolation space between A and B. Denote by C E the
interpolation constant in the inequality
II TII E E CEil TII L(AB,AB).
Since P E L(AB, AB), the operator P is bounded in E and
II P II E-+E CEil P II,
where for brevity IIPII denotes IIPIIL(AB B). In this case the image PE is a
subspace in E and the norm in PE as a subspace of E is equivalent to the
norm in the space Fp(E). Indeed, if y E PE c E, theny = Py, and therefore
IIYIIFp(E) lIyllE. Next, for Px = y (x E E) we have IIYIIE IIPIIE-+EllxIl E ,
and consequently
Ilyll (E) IIYIIE inf (IIPIIE-+EllxIl E ) CEIIPIiIlYIiF (E). (4.17)
p Px=y p
Thus, up to isomorphism we may assume that Fp(E) is a subspace of E.
LEMMA 4.5. If E is an interpolation space between A and B, then Fp(E) is an
interpolation space between M and N, and (A, B, E) is an interpolation triple
relative to (M, N, Fp(E».
PROOF. If T E L(MN, MN), then TP E L(AB, AB). In view of the inter-
polation property of E for y E Fp(E) c E we have TPy = Ty E E. The
range of T lies in M + N, and therefore Ty = PTy c PE = Fp(E) and
II TYIIFp(E) II TyllE = II TPyIlE CEil TPIIL(AB B)IIYIIE
CEIITIIL(MN,MN)IIPIIIlYIlE. (4.18)
Taking account of (4.17), we finally obtain that
II Ty II Fp(E) (CEIIP 11)2 11 TII L(MN,MN)IIYII Fp(E).
Now let T E L(AB, MN); then T E L(AB, AB), and so, if x E E, then
Tx E E. Next, PTx = Tx E PE = Fp(E) and
II TxIlFp(E) II Txll E CEil TIIL(AB,MN)lIxII E . (4.19)
COROLLARY. The space Fp(E) is isomorphic to the space F(E) constructed
from the triples (A, B, E) according to Theorem 4.2.
Indeed, by the construction of F(E) the membership relation P E
L(AB, MN) implies that Fp(E) is imbedded in F(E). On the other hand,
. INTERPOLATION SPACES AND TRIPLES
31
since (A, B, E) is an interpolation triple relative to (M, N, Fp(E), by Theo-
rem 4.2 we have F(E) c Fp(E). Thus, up to isomorphism, F(E) = Fp(E).
If F is an interpolation space between M and N, then in general Ep-I(F) is
not an interpolation space between A and B. We explain this via an example.
We set A = LI(O, 1) and B = Loo(O, 1). Then A + B = LI(O, 1). As M and N
we take the subspaces of LI(O, 1) and Loo(O, 1) consisting of all functions
vanishing for t E [1/2, 1]. The couple M, N is a complemented subcouple of
A, B. Let F be an interpolation space between M and N, and let F not
coincide with M + N. The inverse image P -IF consists of all functions on
[0, 1] whose restriction to [0, 1/2] belongs to F and whose restriction to
[1/2, 1] belongs to LI(I/2, 1). Let the functionyo(t) from M + N not belong
to F. We define the following function: xo(t) = 0 for 0 <; t 1/2, and
xo(t) = yo(1 - t) for 1/2 < t 1. It is obvious that Xo E P -IF. Denote by T
the operator defined by the equality Tz(t) = z(1 - t) (z E LI(O, 1». This
operator has norm 1 in LI(O, 1) and in Loo(O, 1). Next, PTxo = Yo E F, and
consequently Tx o fl:. P -IF. Thus, P -IF is not an interpolation space.
We may consider the part of the inverse image P - IF studied in subsection
3, namely, the intersection E(F) of the inverse images T-1F of all operators
T in '7T I (AB, MN). As is shown in the remark to Theorem 4.3, this is an
interpolation space between A and B. We recall that the norm in this space
can be introduced by (4.15); namely,
Ilxll E(F) = supll Txll F. (4.20)
We note that the interpolation constant for the space E(F) is equal to 1
(see p. 27). If F is an interpolation space between M and N, then it is
imbedded in E(F). Indeed, if T E 7T I (AB, MN), then T E 'TTI(MN, MN), and
so for x E Fwe have
IlxIIE(F) CFllxII F ,
where C F is the interpolation constant in the inequality
IITIIF-+F CFIITIIL(MN,MN). (4.21)
We also note that E(F) c P -I(F), and therefore PE(F) = F.
It will sometimes be convenient to introduce an equivalent norm in E(F)
by the formula
Il x ll1-(F) = sup IIPTxIIF.
T E 7Tl(AB,AB)
( 4.22)
If T E 7T I (AB, AB), then IIPTIIL(AB,MN) < liP II II TIIL(AB B) < liP II, and
therefore IIPII-lpT E 7T I (AB, MN). From this we obtain that
Ilxll E(F) sup II II P 11-IPTxll F = II P 1I-lll x ll1-(F).
T E 7Tl(AB,AB)
32
I. INTERPOLATION SPACES
On the other hand, if T E '7T 1 (AB, MN), then PT = T E '7Tl(AB, AB), and
therefore
II x ll1-(F) sup IIPTxll F = IlxIIE(F).
TE7Tl(AB,MN)
Thus,
IlxIIE(F) Il x ll1-(F) IIPllllxIlE(F). (4.23)
The interpolation inequality for E(F) in the new norm has the form
II Txll1-(F) II P II II Txll E(F) II P 1111 TII L(AB,AB) II xII E(F)
IIPIIII TIIL(AB,AB)lI x ll1-(F). (4.24)
Now let E be an interpolation space between A and B. By Lemma 4.5 the
subspace F = PE is an interpolation space between M and N. From it we
can construct the space E(F). It turns out that E is imbedded in E(F).
Indeed, if T E '7T 1 (AB, MN), then by (4.19) we have II Txll F CEil xII E and
hence IlxIIE(F) CEllxlIE.
We summarize our results.
THEOREM 4.7. Let M, N be a complemented subcouple of the Banach couple
A, B, and let P be the corresponding projection operator from A, B onto M, N.
The operator P projects every interpolation space between A and B onto a
subspace PE which is an interpolation space between M and N. The space PE is
the smallest among the spaces F for which (A, B, E) is an interpolation triple
relative to the triple (M, N, F) (PE is isomorphic to F(E».
From every interpolation space F between M and N a space E(F) can be
constructed which is an interpulation space between A and B and is the largest
among all interpolation spaces E between A and B for which P E = F.
(A, B, E(F» is the optimal interpolation triple relative to (M, N, F).
The last assertion follows from the corollary of Lemma 4.5 and Theorem
4.4.
We specified the constants in (4.17), (4.18), (4.23), and (4.24) in order to
make the following useful remark.
REMARK. If IIPIIL(AB B) = 1 and the interpolation constant C E is equal to
1, then Fp(E) is isometric to the subspace PE, the interpolation constant in
the inequality of type (4.21) for the space F = PE is equal to 1, the norms
(4.20) and (4.22) coincide, and the interpolation constant in (4.24) is equal
to 1.
We mention still another assertion.
LEMMA 4.6. If the triples (A, B, E) and (M, N, F) are interpolation triples
relative to each other, then F is isomorphic to PEe
. INTERPOLATION SPACES AND TRIPLES
33
PROOF. Since P E L(AB, MN), we have PE c F. The operator j imbed-
ding M + N in A + B belongs to L(MN, AB), and therefore jF C E, i.e.
FeE. If we take into account that PF = F, our assertion follows from the
above inclusions.
5. Right invertible 1IUlppings of Banach couples. Let R be a bounded linear
operator defined on a Banach space E, mapping into a Banach space F and
having a right inverse, i.e. a linear operator II from F to E such that
RII = IF. Since RIIy = y for any y E F, the range of R coincides with the
whole space F. The operator P = IIR is a bounded projection in the space E:
p2 = IIRIIR = IIR = P. We denote by EI the subspace onto which E is
projected by P.
This subspace coincides with the range of II. Indeed, if x = IIy, then
Px = IIRIIy = IIy = x. The complementary projection I - P maps E onto
the kernel N(R) of R. Indeed, every element z of the kernel can be repre-
sented in the form z = z' - IIRz = (I - P)z. Conversely, every element
(1 - P)z belongs to N(R): R(I - IIR)z = (R - R)z = B. Thus, E decom-
poses into the direct sum of the subspaces N(R) and EI. The restriction RI of
R to EI effects an isomorphic mapping of EI onto F, and II = R I - I .
Conversely, if the range of R coincides with the space F, and the kernel
N(R) is a complemented subspace of E, then this operator is right invertible.
Indeed, if E = N(R) + E I , then we may set II = R I - I , where RI is the
restriction of R to E I.
Now let A, Band C, D be two Banach couples, and let R E L(AB, CD).
Suppose that there exists an operator II defined on C + D whose restrictions
to C and D are right inverses of the restrictions of R to A and B, respectively.
Then we say that R is a right invertible mapping of the couple A, B onto the
couple C, D. If the restrictions of R to A and B have right inverses III and II 2 ,
then for the existence of the operator II discussed above, which coincides
with III on C and with II 2 on D, it is necessary and sufficient that the
operators III and II 2 coincide on C n D.
We denote by M and N the images of C and D under the mapping II. As
was explained above, these images are subspaces of A and B isomorphic to C
and D, respectively. The operator P = IIR projects A onto M and B onto N,
and so M, N is a complemented subcouple of the couple A, B. The restriction
RI of R to M + N effects an isomorphism of this space and the space C + D.
To every space F intermediate between M and N this isomorphism assigns a
space G intermediate between C and D. Moreover, a natural correspondence
is thus established between the interpolation properties of (M, N, F) and
(C, D, G). Then Theorem 4.7 immediately implies the following theorem.
34
I. INTERPOLATION SPACES
THEOREM 4.8. Let R be a right invertible mapping of the Banach couple A, B
onto the Banach couple C, D.
To every interpolation space E between A and B corresponds an interpolation
space GR(E) between C and D, with norm
lIy II G (E) = inf Ilxll E.
R Rx=y
The space GR(E) is the smallest among all spaces G for which (A, B, E) is an
interpolation triple relative to the triple (C, D, G).
From every interpolation space G between C and D an interpolation space
E( G) between A and B can be constructed that is the largest among all
interpolation spaces E between A and B for which RE = G. An equivalent norm
can be introduced in E(G) by the formula
IlxIIE(G) = sup IIRTxIIG.
T E 7Tl(AB,AB)
(A, B, E( G» is the optimal interpolation triple relative to (C, D, G).
Similarly, from Lemma 4.6 follows
LEMMA 4.7. If (A, B, E) and (C, D, G) are interpolation triples relative to
each other, then the space G is isomorphic to GR(E).
We note that formally Theorem 4.7 and Lemma 4.6 are special cases of
Theorem 4.8 and Lemma 4.7. The projection P induces a right invertible
mapping of the couple A, B onto the couple M, N, where the role of the right
invertible mapping II is played by the operator imbedding M + N naturally
in A + B.
Another special case can be obtained if we consider the quotients of the
spaces of the couple A, B modulo the spaces of a complemented subcouple
M, N. We consider the quotient spaces (A + B)/(M + N), AI M and BI N.
To every coset x + M (x E A) of A modulo M corresponds the coset
x + M + N of A + B modulo M + N, which contains x + M. Under this
mapping distinct cosets in A / M are mapped into distinct cosets in
(A + B)/(M + N). Indeed, if x + Mandy + M are contained in the same
class z + M + N, then x - y E M + N, and consequently x - y = P(x -
y). Since x - yEA, we then have x - y E M and the classes x + M and
y + M coincide. Finally,
II x + M + NII(A+B)/(M+N) = inf Ilx + uII A + B
uEM+N
inf Ilx + uil A = IIx + MIIA/M.
uEM
g4. INTERPOLATION SPACES AND TRIPLES
35
Thus, AI M and B/ N are imbedded in (A + B)/(M + N), and so they
form a Banach couple. The natural projection
'11": (A + B) (A + B)/ (M + N)
induces a right invertible mapping of the couple A, B onto the couple A I M,
BIN. Its right inverse is the isomorphism which is naturally established
between A I M and BIN and the subspaces (I - P)A and (I - P)B comple-
mentary to M and N.
The reader can easily reformulate Theorem 4.8 and Lemma 4.7 for this
special case.
6. The interpolation ftmctor. Let two categories and l be given. A
covariant functor acting from to l is, by definition, a pair of mappings,
each of which is usually denoted by the same letter . One of the mappings is
defined on objects and to every object of assigns an object ) of l;
the other one is defined on morphisms and to every morphism y E
Mor( , ') assigns the morphism y) E Mor( ), (t'). This pair of
mappings must satisfy the following conditions:
(I ) = 1 ( ) ( E ) and (f3y) = (f3) (y),
where f3y is defined in .
As we take the category of Banach couples A, B in which the morphisms
are the operators from L(AB, CD) (A, B is the first object, and C, D is the
second object). Denote by l the category of Banach spaces E in which the
morphisms are the bounded linear operators from L(E, F).
DEFINITION 4.7. An interpolation functor is a functor acting from the
category of Banach couples into the category of Banach spaces and assigning
to every Banach couple A, B a Banach space (A, B) intermediate between
A and B: A n B c (A, B) c A + B and to every operator T E
L(AB, CD) assigning its restriction 6J(T) to (A, B).
If the above correspondence is a functor, then the operator Cff(T) belongs
to the set of morphisms Mor( (A, B), (C, D»), i.e. is a bounded linear
operator from 6J(A, B) into (C, D). In other words, the triples
(A, B, (A, B» and (C, D, (C, D)) must be interpolation triples for any
Banach couples A, Band C, D.
Simple examples of functors from the category of Banach couples into the
category of Banach spaces are the functors of the sum and intersection of
spaces in a Banach couple. By Lemma 4.1 these are interpolation functors.
An interpolation functor Cff is said to be normalized if the interpolation
constant for any triples (A, B, 6J(A, B» and (C, D, (C, D) is not greater
than I. The functors of sum and intersection are normalized.
36
I. INTERPOLA TION SPACES
We say that an interpolation functor has type (normal type) a if any
triple (A, B, (A, B» is a (normalized) interpolation triple of type a relative
to any triple (C, D, (C, D».
The study in subsection 3 enables us to give some constructions of general
interpolation functors. Let A, B be a fixed Banach couple and E a space
intermediate between A and B. For any other Banach couple C, D (which
may coincide with A, B) a space F(E) intermediate between C and D has
been constructed in subsection 3. We show that the correspondence (C, D)
= F(E) induces an interpolation functor. Co, DO, and cI, D 1 are two
Banach couples and FO(E) and F1(E) are the corresponding intermediate
spaces. We recall that Fi(E), i = 0, 1, is the sum of the spaces F;,.{E), where
T runs over the set 7T I (AB, CiD i ). Let S E 7T I (CoDo, CID I) and z E FO(E).
The latter means that z = LYk, where Yk E F /c(E) and Tk E 7T 1 (AB, COD .
Then Sz = LSYk' since LYk converges in CO + DO, and the operator S acts
continuously from CO + DO to C I + D I. Next, Yk = Tkx k (x k E E), and
consequently, SYk = STkx k , where ST k E 7T 1 (AB, C ID I). To prove that Sz E
F I( E) it is sufficient to verify that
IISYkIIFb-/c(E) < 00.
We choose X k E E so that Yk = Tkx k and IIYkllI1/c(E) + f/2k Ilxkll E .
Then
II Sy k II F T/c (E) II X k II E II Y k II I1/c (E) + f.
Using the arbitrariness of f and then taking the infimum over all represen-
tations z = LYk' we obtain
II Sz II F1(E) II z II pO(E).
Thus we have constructed a normalized interpolation functor, which we
denote by A,B,E).
If we apply A,B,E) to the couple A, B, we obtain a space which is in
general different from E. Now let E be an interpolation space between A and
B. We construct the space A,B,E)(A, B). It consists of elements of the form
z = LYk' where Y k = Tkx k , Tk E 'lTI(AB, AB), X k E E, and LIIYkIlFT/c(E) <
00. If Y = Tx (T E 'lTI(AB, AB», then by the interpolation property of E we
have IIYIIE CEllxII E . Therefore
IIYI\F(E) = inf Ilxil E CE-IIIYIIE.
T Tx=y
Then from the convergence of LIIYkIIFT/c(E) it follows that LI\YkllE con-
verges, and hence z E E. Besides,
IlzllF(E) = inf IIYkIIFT/c(E) C E - I inf IIYkllE = CE-IllzIIE.
94. INTERPOLATION SPACES AND TRIPLES
37
Finally, we consider the representation z = Iz and conclude that E is
contained in F(E) and IlzIIF(E) < IlzIIE. Thus E and A,B,E)(A, B) are iso-
morphic, and if C E = 1, then they coincide.
We have arrived at the following important assertion.
THEOREM 4.9. If a Banach space E is an interpolation space between the
spaces A and B of a Banach couple, then there exists a normalized interpolation
functor 'J such that E is isomorphic to the space 'J(A, B). If the interpolation
constant C E is equal to 1, then E = 'J(A, B).
In subsection 3, from the intermediate space F between C and D in the
fixed Banach couple we have constructed for every other Banach couple A, B
an intermediate space E(F). The correspondence 'J(A, B) = E(F) also in-
duces an interpolation functor, which we denote by f9(C,D,F). If F is an
interpolation space between C and D, then it is isomorphic to f9(C,D,F)(C, D)
and, if C F = 1, then F = f9(C,D,F)(C, D). Thus, we obtain yet another functor
having the properties described in Theorem 4.9.
It turns out that the functors A,B,E) and f9(A,B,E) are extreme cases in a
certain sense: for any functor having the properties indicated in Theorem 4.9
the imbedding
'J(A,B,E)( C, D) C 'J( C, D) C f9 (A,B,E)( C, D)
is valid.
The proof of the above facts is left to the reader (see the Notes on the
Li terature).
Lemma 4.7 immediately implies the following theorem.
THEOREM 4.10. Let R be a right invertible mapping of the Banach couple A, B
onto the Banach couple C, D. For every interpolatio functor 'J the space
'J(C, D) is isomorphic to FR('J(A, B» (the image of 'J(A, B) under the
mapping R).
The hypotheses of Lemma 4.7 are satisfied, since (A, B, 'J(A, B» and
(C, D, 'J(C, D» are interpolation triples relative to each other.
CHAYfER II
INTERPOLATION IN SPACES OF MEASURABLE FUNCI10NS
Introduction
1. Measurable functions on a measure space. Let [Q be a measurable space
and J.L a measure defined on its a-algebra of measurable sets Wl (see [16]). In
what follows we always assume that this measure is a-finite, i.e. the space Wl
is the union of countably many sets of finite measure..
By Xe(t) we always denote the characteristic function of a measurable set
e C [Q. A function x(t) is said to be simple if it assumes only finitely many
nonzero finite values, each on a measurable set, and strictly simple if the
requirement of finiteness of the measure on these sets is imposed... Every
simple function can be written in the fonn x(t) = L7 xiXe;(t), where the e i are
measurable sets.
Every measurable function is the limit of a sequence of simple functions,
and if the function is nonnegative, the sequence can be chosen increasing. If
the function is bounded, the sequence can be chosen uniformly convergent.
EGORov'S THEOREM. If the measure of [Q is finite, and the sequence of almost
everywhere finite measurable functions x n ( t) almost everywhere converges to an
almost everywhere finite function, then for every E there exists a measurable
subset e with measure J.L(e) > J.L(Wl) - E on which the sequence xn(t) converges
to x(t) uniformly.
We say that a sequence xn(t) converges to a function x(t) in measure if the
measure of the set where IXn(t) - x(t)1 E converges to zero as n 00 for
any E > O.
· Editor's note. Actually the measure is totally a-finite in Halmos' terminology ([10], 17).
.. Editor's note. The Russian terms used by the authors are KOBe1lH03Ha1lHU CPyHKUB. (lIter-
ally: finitely valued function) for strictly simple function, and 0606meBBo KOBe1lH03Ba1lllU
4>YBXUB. QiteralIy: generalized finitely valued function) for simple function. The terminology
adopted for the translation is consistent with that of Halmos ([ 10], 120).
39
40
II. SPACES OF MEASURABLE FUNCfIONS
If a sequence converges almost everywhere on a space with a finite
measure, then it converges in measure. If a sequence converges in measure,
then it contains a subsequence which converges almost everywhere.
The space (9)(, J-L) of all real-valued measurable functions (more precisely,
of classes of equivalent functions) on WC is linear. In this space a metric can
be introduced in the following way: Let vet) be a positive function integrable
wi th respect to J-L, and set
J Ix(t) - y(t)1
p(x, y) = 1 + Ix(t) _ y(t)1 ,,(t) dp..
The space (9)(, J-L) becomes a complete linear metric space with this metric.
Convergence in this space is equivalent to convergence in measure on every
set of finite measure.
The Banach space of all functions (more precisely, of all classes of
equivalent functions) integrable on 9)( with respect to J-L is denoted by
L 1 (9)(, J-L).
LEBESGUE'S THEOREM. If a sequence of integrable functions xn(t) converges
almost everywhere to a function x(t) and Ixn(t)1 ly(t)1 almost everywhere,
where yet) is some integrable function, then x(t) is integrable and the sequence
{x n } converges to x in the norm of L 1 (9)(, J-L).
F ATOU'S LEMMA. If xn(t) is a sequence of nonnegative measurable functions,
then
J lim xn(t) dp.";; lim f xn(t) dp..
n oo n oo
B. LEVI'S THEOREM. If an increasing sequence of measurable nonnegative
functions xn(t) converges to a function x(t) almost everywhere, then
lim f xn( t) dJ-L = f x( t) dJ-L.
n oo
2. Ideal &uulch lattices. A Banach space E which is a nonzero linear
manifold of (9)(, J-L) is called a Banach function space.
In (9)(, J-L) there exists a natural partial ordering: x y (x, Y E (WC, J-L))
is equivalent to the inequality x(t) yet) for almost all t E 9R.
DEFINITION 1.1. A Banach function space E is said to be an ideal Banach
lattice, or briefly an ideal lattice, if the condition Ixl Iyl, where x(t) is a
measurable function andy E E, implies that x E E and IIxllE IIYIIE.(I)
e) In the literature such spaces are also called "Banach lattices", and "function lattices".
INTRODUCTION
41
It follows from this definition that II Ixlll E = II xII E for x E E.
We cite the simplest properties of ideal lattices.
1 0. If X n x in E and the sequence xn(t) is increasing, then xn(t) converges
to x(t) almost everywhere.
For the proof we first note that x(t) - xn(t) O. If we had x(t) - X n (t) <
o
-Y (y > 0), on a set e of positive measure, then for all n > no we would also
have x(t) - xn(t) < -y (t E e) and
IIx - xnll E II (x - xn)XeIIE > YIIXeIIE'
which contradicts the convergence of X n to x in E. We write z(t) =
lim n -+ oo xn(t). Then x(t) - xn(t) x(t) - z(t) 0, and therefore the function
x(t) - z(t), and so z(t) as well, belongs to E. Next, Ilx - zilE II x - xnll E
0, and so x( t) = z( t) almost everywhere.
2°. If the sequence {Yn} (Yn E E) is increasing and tends to 00 on a set e of
positive measure, then llYn liE 00.
Indeed, routine arguments show that there exists a set e' C e with J-L( e') > 0
on which the sequence {Yn(t)} uniformly tends to 00. Then the characteristic
function Xe,( t) belongs to E, and
IIXe,(t)IIE min Yn(t) IIYnllE 00.
tEe'
3°. Let J-L(W) < 00, e k E W (k = 1,2, . . . ) and J-L(e k ) > 8. Then IILf XeJI
00 as N 00.
We writeYN(t) = LZ-I Xek(t). The sequenceYN is increasing. If we assume
that its limit is finite almost everywhere, then because of the finiteness of the
measure of W there exists a set e' C Wl with measure arbitrarily close to that
of W (for example, J-L(W - e') < 8/2) on which the sequence YN(t) is uni-
formly bounded. However,
N N
£,YN(t) dp, = k l £, x:e.(t) dp, = k l p,(e k n e') > N13j2,
since J-L(e' n e k ) > 8/2, and this contradicts the boundedness of the sequence
YN(t) on e'. Consequently the limit of YN(t) is infinite on a set of positive
measure, and by 2° we have llYN liE 00 as N 00.
THEOREM 1. An ideal lattice is imbedded in (Wl, J-L).
PROOF. It is sufficient to show that the condition IlxnllE O implies the
convergence of X n to zero in (Wl, J-L), or, which is the same, the convergence
of xn(t) to zero in measure on any set Q of finite measure. Assume the
contrary: Let there exist a set Q, 0 < J-L(r2) < 00, and a positive number € > 0
such that for some sequence x nk (t) the inequality I xlIJc (t)1 > € is satisfied on a
42
II. SPACES OF MEASURABLE FUNCTIONS
set k C with measure J-L( k) > 8, where 8 is some fixed number. Then
€Xnlc(t) Ixnlc(t)l, and so €llx JIE IlxnJIE. Without loss of generality we
may assume that the sequence n k is chosen so that L%'_I IlxIIJcliE < 00. From
this we obtain that
N N N 00
€ XQIc € IIxQlcllE IIxllJcll E IIxlIJcll E < 00,
k=l E k=l k=l k=l
which contradicts property 3°.
COROLLARY 1. Any two ideal lattices (on the same set WC and with the same
measure J-L) forms a Banach couple.
COROLLARY 2. If L Ilxnll E < 00 and Lf X n x in E as N 00, then
L xn(t) converges absolutely to x(t) almost everywhere.
Indeed, L Ixnl converges in E, and by 1 ° it converges almost everywhere.
Therefore xn(t) also converges almost everywhere. Since E is imbedded in
(WC, J-L), the latter series converges to x(t) in (WC, J-L), and consequently
L xn(t) = x(t) almost everywhere.
4°. If 0 x Y and Y n Y in E, then the functions min(x,Yn) = x n
converge to x in E.
Indeed,
o x(t) - xn(t) = [x(t) - Yn(t)] + Iy(t) - Yn(t)l. (2)
Then
lim Ilx - xnll E lim IllY - YnlllE = lim IIY - YnllE = o.
n-+oo n-+oo n-+oo
5°. If x, z E E, then the functions Ix(t)1 9 Iz(t)II-9 also belong to E for
O (} 1.
This property follows from the inequalities
\x(t)\9 Iz (t)\1-8 max(lx(t)l, \z(t)\) Ix(t)1 + Iz(t)\.
6 0. If x, z E E and for almost all t the inequality
\y(t)1 Ix(t)1 8 I z (t)II-9
holds for some () E [0, 1], then
(0.1 )
IIYIIE IIxll lIzl\1-8,
and for every positive functional from E' we have
(0.2)
f(lyl) [f(lxl) ]8[ f(lzl) ] 1-8.
(0.3)
e) z +(t) = max{ z(t), O}.
INTRODUCTION
43
Inequality (0.1) is equivalent to the following:
IY ( t ) I 0 8 - II x( t) I + (1 - 9) f 81 z ( t) I
for all f > O. Therefore,
IIYIIE Oe 8 - l lI x llE + (1 - O) f8 11 z 1lE
and
f( I Y I) Of 8 - (I x I) + (1 - 9) f '1( I z I).
By minimizing the right sides with respect to f, we obtain (0.2) and (0.3).
A special case of (0.2) is the inequality
IIlxI8IzI1-811E IIxll lIzIl1-8,
(0.4)
which becomes the Holder inequality in the case where E = LI.
7°. Let x, Y, X n and Yn be positive elements of an ideal lattice E, and assume
that x n x andYn Y in E. Then
x!:Y; -B X'Y 1-8 in E.
Using (0.4), we obtain
Ilx'YI-8 - x!:y -81IE Ilx 8 (yl-8 - y -8)IIE +lly -8(x8 - x:)IIE
..;; IIxll lllyl-8 _ y _811/(1-8)11 -8
1- 8 111 8 8 1 1 / 8 11 8
+ llYn liE x - X n E.
The function Ix 8 - x:I I / 8 belongs to E, since Ix 8 - x:I I / 8 Ix - xnl.
Similarly, lyl-8 - y -811/(1-8) Iy - Ynl. Consequently,
Ilx'YI-B - x!:y;-81IE IIxll IIY - Ynll -8 + IIYnll -8I1x - xnll O.
We consider some operations which allow us to construct ideal lattices
from other idealla ttices.
Restriction of an ideal lattice. If 9)(1 is a measurable subset of WC of positive
measure and XI(t) is the characteristic function of WC I , then the functions
XI(t)x(t) (x E E) fonn a subspace EI of E. Since Ilxlxll E Ilxll E' multipli-
cation by XI is a projection onto this subspace, bounded by unity. On the
other hand, the elements of the Banach space EI can be identified with
restrictions of functions in E to Wl i in a natural manner. Thus, this space can
be considered as a Banach function space on WC I with measure ILl which is the
restriction of IL to WC I . In what follows we often speak of EI as "the space E
on WC I ".
Weighted ideal lattices. Let pet) be an almost everywhere positive measura-
ble function (weight) on the space WC. The collection of all measurable
44
II. SPACES OF MEASURABLE FUNCfIONS
functions (classes) x(t) for which the functions p(t)x(t) belong to an ideal
lattice E and which is equipped with the norm
II xII Ep(l) = IIpxll E
is obviously an ideal lattice on WC, to be denoted by E p(t).
Sum and intersection of ideal lattices. If Eo and E. are two ideal lattices on
the same space WC with the same measure J.L, their intersection Eo n E. is
obviously also an ideal lattice. We show that the sum Eo + E. also has this
property.
Let Y E Eo + E. and Ixl lyl. Given € > 0, we choose Yo E Eo and
Y. E E. so that IIYollE o + IIY.IIE 1 IIYIIEo+EI + € and Y = Yo + Y.. We set
h(t) = x(t)/y(t) if y(t) =1= 0 and h(t) = 0 if yet) = O. Then x = yJt + y.h,
and since Iy;hl ly;l, i = 0, 1, we havey;h E E;. Therefore x E Eo + E. and
II xII Eo+ EI IIYohll Eo + lIy. hll EI
IIY 0 II Eo + lIy.1I E 1 lIy II Eo + E 1 + €.
Since € is arbitrary, we have IlxllEo+EI lyIEo+EI.
Discretization of an ideal lattice. Let WC = U WCn be a decomposition of
WC into disjoint sets WCn of finite positive measure J.L". Assume that XIDl E E.
II
Such a decomposition always exists if E contains an almost everywhere
positive function. Denote by E C the collection of all functions in E constant
on each set WCn. This collection is a subspace of E. On the set N + of natural
numbers we introduce a measure J.L' by the formula J.L'({n}) = J.Ln. Then the
space E C can be identified with the space dE of all functions x(n) on N+ with
finite norm
00
IIxlidE = x (n)XIDl II .
n=l E
The space dE is an ideal lattice. We shall say that the lattice dE is obtained
by discretization of the lattice E.
From the decomposition U WCn we may construct an averaging operator
00 1
Tx = - L xes) dp. XIDl.c t ).
n = 1 J.L" IDl n
If this operator is bounded in E, then it maps E onto E C . The operator
Tdx = { L xes) dP. }
J.L" [Q II
then maps E onto dE. The imbedding operator dE E C is its right inverse.
3. Special classes of ideal lattices. We say that an ideal lattice has the Fatou
property if the condition that a sequence of functions xn(t) E E converges to a
INTRODUCTION
45
function x(t) almost everywhere and is bounded in E (1lxnll <; M) implies
that x E Eand IIxllE lim llxnIIE.
A norm in an ideal lattice E is said to be absolutely continuous if for any
x E E and any decreasing sequence of measurable sets en with empty
intersection we have IlXe xilE 0 as n 00. A lattice with an absolutely
II
continuous norm is said to be regular.
The property of absolute continuity of the norm is closely related to the
separability of the space E. The measure J.L is said to be separable if there
exists a countable system r of measurable sets in Wl such that for every € > 0
and every measurable set e E Wl there exists a set e' C r for which
J.L( e \ e' U e' \ e) < €.
It turns out that an ideal lattice E is separable if and only if the measure J.L is
separable, and the nonn in E is absolutely continuous.
The support 9C of an ideal lattice E is, by definition, the smallest measurable
set outside which all functions in E are equal to zero. In an ideal lattice there
is always a function which is positive at all points of the support.
The associated space E 1 of an ideal lattice E is, by definition, the collection
of all measurable functions y( t) on [Q whose support is contained in the
support of E and for which
IIYIIEI = sup f x(t)y(t) dt < 00.
II x llE=l
The associated space is itself an ideal lattice. It is contained in the dual E'
of E and is a subspace of it. The space E 1 coincides with the whole dual if
and only if the norm of E is absolutely continuous.
From the space E 1 we construct the second associated space ElI = (E 1) 1.
The space E can be imbedded in E 11 in a natural way; moreover, we have
IlxllEII IlxllE for all x E E. If the space E is separable, then IIxllEII = IlxilE
for x E E; that is, E is isometrically imbedded in ElI. If E has the Fatou
property, then E 11 = E.
The proofs of the above assertions can be found in [29] and [49].
We note still another generalization of the Minkowski inequality. Let
u(t, T) be a function of two variables t, T E we such that for almost every
fixed tit belongs to E as a function of T, the function II u( t, T) II E is
measurable, and
f lIu(t,'T)IIEdt < 00.
Then
f u(t, T) dt f lIu(t, T)IIE dt.
Ell
(0.5)
46
II. SPACES OF MEASURABLE FUNCTIONS
Indeed, for z EEl we obtain
f I u( t, 'T )llz( 'T) I d'T , "u( t, 'T )11 Elizil E"
and therefore, by Fubini's theorem,
f u ( t, 'T) dt
Ell
sup
IIz1lE1<;1
f [f I u(t, 'T)llz( 'T)I dt] d'T
sup f [ f I u(t, 'T)llz( 'T)I d'T ] dt = f II u(t, 'T)IIE dt.
IIzllE I <; 1
4. Lebesgue spaces. Two spaces Wl i and [Q2 with measures ILl and IL2 are
said to be isomorphic if, after deleting sets of measure zero from them, we can
establish a one-to-one correspondence between the remaining parts which
preserves the class of measurable sets and the measure of sets.
The spaces which are isomorphic to an interval (0, a) of the real axis (with
finite or infinite a) with Lebesgue measure are called here Lebesgue spaces..
There is an axiomatic definition of Lebesgue spaces. It requires the separabil-
ity of the measure, its continuity (the measure of any point is equal to zero),
and a certain completeness property which will not be formulated here (see
[324]). An important property of Lebesgue spaces is the fact that every
measurable subset of finite measure is also a Lebesgue space. The space R n
with Lebesgue measure is obviously a Lebesgue space, and consequently any
measurable subset of finite measure in R n is a Lebesgue space. From this it
follows in particular that for two sets in the spaces R n and R m , respectively,
with equal nonzero finite measures there exists a one-to-one mapping of one
set onto the other preserving the measure of all measurable subsets.
1. Positive functions on a semiaxis and their dilation functions
1. Classes of positive functions on the semiaxis (0, (0). A function (t) given
on an interval I of the real line is said to be concave (convex) if
1/1( t 1 ; t 2 ) [ 1/1 ( t 1) + 1/1 ( t 2 ) ];
( 1/1 ( t 1 ; t 2 ) , [ 1/I(t) + 1/I(t 2 )]) t), t 2 E I. (1.1)
The properties of concave (convex) functions are well known, and we only
recall some of them without proof, referring the reader to the books [18] and
[22] .
· Editor's note. Actually these are Lebesgue spaces with continuous measure, in Roblin's
terminology (see [305], 2.4).
I. POSITIVE FUNCTIONS ON A SEMIAXIS
47
If a concave function is measurable, then it is continuous. If it is bounded
below on some interval in the interior of an open interval I, then it is also
continuous on I. In particular, a nonnegative function which is concave on an
interval is continuous there. For a continuous concave function, (1.1) implies
the Jensen inequality
((I - a)t l + at 2 ) (1 - a)1/;(t l ) + a (t2). (1.2)
A continuous concave function is absolutely continuous; it has a derivative
everywhere except at countably many points, and this derivative is a decreas-
ing function.(3)
In what follows, we shall be interested in positive concave functions on the
semiaxis (0, 00). As we mentioned above, such a function is always continu-
ous. Inequality (1.2) implies that (I - a)t l + at0 > a (t0. Letting t l tend
to zero, we obtain that (at2) 00/;(t0. This says that if we set (o) = 0, then
the concavity inequality (1.2) will be satisfied for all points of the closed
semiaxis [0,00). If in (1.2) we write t = (1 - a)t l + at 2 , then it takes the form
t - t t - t
(t) 2 (tl) + I (t2) (t l t t 2 ). (1.3)
t 2 - t I t 2 - t I
From this it is obvious that 1/;(t) is increasing. Indeed, if we assume that
(t) < (tl) (t l < t), then for sufficiently large t 2 (1.3) implies that (t0 < o.
This contradicts the hypothesis.
Inequality (1.3) can be written also in the fonn
( (t) - (tl»/ (t - t l ) ( (t2) - (tl»/ (t 2 - tl). (1.4)
Thus, the concavity inequality (1.3) is equivalent to the condition that the
function ( (t) - (tl»/(t - t l ) is decreasing for any t l E [0, 00). In particu-
lar, if we set t l = 0, we obtain that (t)/ t is decreasing.
It is easy to verify that the infimum of any set of positive concave functions
has the property (1.2), and consequently it will be either a positive concave
function or identically zero. If now cp( t) is a positive function having a
concave majorant o(t): cp(t) o(t) (0 < t < (0), then the infimum of all
concave majorants of cp(t) will be the smallest concave majorant of cp(t). We
denote this majorant by <p(t). We give its constructive description. From (1.2)
we obtain by induction that
1/;C l \t;) i l \1/;( tJ
e) Following Bourbaki, we do not use the term nondecreasing (nonincreasing) function, but
speak of increasing (decreasing) functions and strictly increasing (strictly decreasing) functions.
48
II. SPACES OF MEASURABLE FUNCTIONS
for any n 1 and "A; 0, 'L7= 1 "A; = 1. Therefore, if we write t = 'L. "A;t i , we
have
n n
cp(t) }: "A;cp(t;) A;CP(t i ).
;=1 ;=1
We write
n
o/(t) = sup "A;cp(t;)
;= 1
( n = 1, 2, . . . ; "\ 0; ; "\ = 1; "\t i = t).
It follows from the above that cp(t) o/(t) cp(t). The function 1/;(t) is
concave. Indeed, if t', t" E (0, (0), then there exist the "A;, J.L;, t; and tf' such
that
n'
"A;cp( t;) 0/( t') - E,
1
Then we have
nil
IL;CP( tf') 0/( t") - E.
1
n' nil
(1 - a) A;CP(t;) + a IL;CP(tf') (1 - a)o/(t') + mfI(t") - E.
1 1
Since
nil
nil
(1 - a)\ + alLi = 1
1 1
and
n' nil
(1 - a)"A;t; + alL;t;' = (1 - a)t' + at",
1 1
it follows that
0/((1 - a)t' + at") (1 - a)o/(t') + mJ;(t") - E.
Since E is arbitrary, it follows from this that the function is concave, and so
by the definition of the smallest concave majorant we have o/(t) = ip(t). Thus,
n
cp( t) = sup \cp( ti)
1
( n = 1, 2, . . . ; "\ 0; "\ = 1; "\t i = t). (1.5)
Two positive functions on (0, (0) are said to be equivalent if there exist
positive constants C 1 and C 2 such that
C10/(t) cp(t) C 2 0/(t)
for all t > O.
1. POSITIVE FUNCTIONS ON A SEMIAXIS
49
THEOREM 1.1. A positive function cp( t) is equivalent to a positive concave
function if and only if for any positive to and t l (to < t l )
cp(t o ) kcp(t l ), cp(tl)/t l kcp(to)/t o (1.6)
with some constant k. Moreover, as equivalent concave function we may take the
smallest concave majorant ip( t) of cp( t).
PROOF. Necessity. From the definition of equivalent functions we obtain
cp( to) C 2 o/( to) C 2 o/( t I) (C 2 / C I)CP( t I)
and
cp(tl)/t l C 2 o/(t l )/t. C 2 o/(t O )/t o (C 2 /C I )(CP(t O )/t o ),
i.e., (1.6) is satisfied with k = C 2 / C I.
Sufficiency. By definition, ip( t) > cp( t). Next, by (1.5) and (1.6) we have
n n { t. }
ip(t) = sup A;CP(t;) sup A;k max 1, cp(t)
lIt
{ n n t. }
k sup A; + }: A;...!.. cp(t) = 2kcp(t).
lIt
Hence ip(t)/2k cp(t) <f;(t).
DEFINITION 1.1. A function cp( t) on the semiaxis [0, (0) is said to be
quasiconcave if 1) cp(O) = 0, 2) cp( t) is positive and increasing for t > 0, and 3)
cp(t)/ t is decreasing for t > O.
From the above it follows that a positive concave function on (0, (0),
extended by zero to the origin is quasiconcave on [0, 00). The converse is not
true in general. For example, the function defined by the equalities
0 for t = 0,
cp( t) = 1 for 0 < t 1,
2
1 for t > 1,
2 X
is quasiconcave but not concave.
Theorem 1.1 has the following corollary.
COROLLARY. Every quasiconcave function cp(t) is equivalent to its smallest
concave majorant ip( t), and
<f;(t) cp(t) ip(t).
( 1.7)
Every quasiconcave function is continuous for t > O.
Indeed, if we had cp(t o - 0) < cp(t o + 0) for some to > 0, then for points t'
and t", t' < to < t", sufficiently close to to we would have cp( t') / t'
cp( t") / t", which contradicts Definition 1.1.
50
II. SPACES OF MEASURABLE FUNCTIONS
LEMMA 1.1. Let Ll;( t) = (t;, tt), i = 1, . . . , n, be a system of disjoint inter-
vals on the semiaxis, and let L Ill; I = d. Then, for any quasiconcave function
cp( t),
n
[cp( tf') - cp( t;)] 2cp( d).
;= 1
( 1.8)
In particular, if cp( +0) = 0, then cp(t) is absolutely continuow.
PROOF. We split the sum on the left side of (1.8) into two: ' and ", where
L' is taken over those intervals which lie to the left of d and L" over those
intervals which lie to the right of d (if d belqngs to one of the intervals, then
the right-hand part of it is taken into L", and the left into '). Then by the
monotonicity of cp(t) we have
I [cp( tt) - cp( t;) ] cp( d).
Next, for d t; we have
( t" ) ( t' )
( t!' ) - ( t ) = cp ; t!' - cp ; (
cp I cp 't!' I ('
I I
« cp(t;) ( t!' _ ( ) cp(d) ( t!' _ ( )
( I I d I I
I
and consequently
L" [cp(tf') - cp(t;)] « cp ) " (tf' - t;) « cp(d).
Inequality (1.8) is established, and the second assertion of the lemma
immediately follows from it. The lemma is proved.
We note that (1.8) is a sharp inequality. If we consider the example of the
quasiconcave function on p. 49, then for the intervals (0, ) and (1, 2 - )
with arbitrarily small 8 > 0 we have d = 1 and
cp(8) - cp(O) + cp(2 - ) - cp(l) = + (l - 8) = 2cp(l) - .
If cp( + 0) > 0, then the quasi concave function is absolutely continuous on
every ray [a, 00) (a > 0). In order to see this it is sufficient to consider the
quasiconcave function
( t ) = ( cp(a)
cp\ cp( t)
for 0 t a,
for t > a.
The sum of two quasiconcave functions is obviously quasiconcave. If the
supremum of a family of quasiconcave functions is finite at at least one point
different from zero, then it is a quasi concave function. The infimum has an
analogous property if it is not identically zero.
1. POSITIVE FUNCTIONS ON A SEMIAXIS
51
DEFINITION 1.2. A positive function u(t) given on an interval I of the real
axis is said to be logarithmically convex if the function In u(t) is convex on I.
I t follows from (1.1) that this definition is equivalent to the requirement
that
u 2 «(t} + t 2 )/2) u(t})u(t 2 ). (1.9)
If u(t) is continuous, then the inequality (1.3) for a convex function implies
that
u ( t ) [ u ( t }) ] (t 2 - t) / (t 2 - t I) [ u ( t 2) ] (t - t I) / (t 2 - t I) . ( 1.10)
We note the following important property of logarithmically convex func-
tions: The sum of two logarithmically convex functions is logarithmically
convex.
For the proof we note that (1.9) is equivalent to the condition that the
quadratic form
U(t}) 2 + 2u«(t} + t2)/2) + U(t 2 )17 2
is nonnegative. Since the sum of two nonnegative quadratic fonns is nonnega-
tive, it is clear that the property of logarithmic convexity is preserved by
addition.
Clearly the product of two logarithmically convex functions is also logarith-
mically convex.
DEFINITION 1.3. A function v(t) defined on the axis (- 00, (0) or the
semiaxis (0, (0) is said to be subadditive if
v(t} + t 2 ) v(t}) + v(t 2 ). (1.11)
The properties of subadditivity are expounded in detail in [18].
We note that, for example, every quasiconcave function on (0, (0) is
subadditive on (0, (0). In fact,
( ) v ( t} + t 2) v( t} + t 2)
V t} + t 2 = t} + t 2
t} + t 2 t} + t 2
v( t}) v( t 2 )
t} + t 2 = v(t}) + v(t 2 ).
t} t 2
If a subadditive function is different from + 00 at every point, then it is
bounded from above on any finite interval lying in its domain; if it is also
different from -00, then it is bounded on every such interval. Hence an
everywhere finite subadditive function can become unbounded only when
approaching the endpoints of the interval where it is defined. Its behavior
near the endpoints is described by the following important theorem, which we
give with proof.
52
II. SPACES OF MEASURABLE FUNCTIONS
THEOREM 1.2. If a function v(t) is everywhere finite and subadditive on the
whole axis (- 00, 00), and if
inf pet) = /3 and sup p( t t) = a, (1.12)
1>0 t t<O
then
lim pet) = /3
t + 00 t
and
lim
t - 00
v(t)
-=a.
t
(1.13)
Here - 00 < a /3 < + 00.
PROOF. Since v(t) is everywhere finite, it follows that /3 is either finite or
equal to -00, and a is either finite or equal to + 00. We assume that {3 is
finite. Then there exists to > 0 such that v(to)j to < /3 + e. Now let t > to.
There exists an integer n such that (n + l)t o t (n + 2)t o . We have
/3 v( t) v( nt o ) v( t - nt o ) nv( to) v( t - nt o )
- + +
t t t t t
..;; nt o (/3 + e) + pet - nt o )
(n + I)t o t
The quantity t - nt o varies on the interval [to, 2t o ] on which the function
v(t) is bounded. Therefore, the right side converges to /3 + east 00. From
this it follows that limt oo(v(t)j t) = /3. This fact can be proved similarly in
the case where /3 = -00. The second limit in (1.13) can be considered in the
same way.
N ext, we note that v(O) v(O) + v(O), and so v(O) > O. Using (1.11) again,
we obtain that 0 v(O) v( t) + v( - t). Dividing by t and passing to the
limit as t 00, we obtain the inequality a /3. Then -00 < -v( -1) a
/3 v(l) < 00, and the theorem is completely proved.
The theorem just proved says that an everywhere finite subadditive func-
tion on the whole axis has the same asymptotic behavior at + 00 and -00 as
some linear functions. We note that the behavior of a subadditive function on
the semiaxis can be more complicated (see [18]).
DEFINITION 1.4. A positive everywhere finite function v( t) on (0, (0) is said
to be sub multiplicative if
v ( tIt 2) v( t 1) V ( t 2). ( 1.14)
It is obvious that if we introduce the variable s = In t, then the function
v(s) = In v(e S ) is an everywhere finite sub additive function on the whole axis
-00 < S < 00. An application of Theorem 1.2 to v(s) immediately leads to the
following assertion.
1. POSITIVE FUNCTIONS ON A SEMIAXIS
53
THEOREM 1.3. For every submultiplicative function v(t) there exist numbers a
and /3 such that - 00 < a /3 < 00 and
v(t) > t{3 for t > 1 and v(t) > t a for t < 1. (1.15)
For any € > 0 we have v(t) t/3+£ for sufficiently large t, and v(t) t a -£
for t sufficiently close to zero. Moreover,
I . Inp(t) lnp(t)
a = 1m = sup
I O In t 0<1 < 1 In t '
f3 = Iim In v(t) = inf In v(t) .
I oo In t I> 1 In t
COROLLARY I. If v( t) is a submultiplicative function and v( to) < tCO for some
to > 1 (for some to < 1), then v(t) < tP- for sufficiently large t (for t sufficiently
close to zero), where Il is a number less than JLo (greater than JLo).
PROOF. If v(t o ) < tCO for to > 1, then /3 < In v(to)[ln to]-1 < JLo. Choosing Il
so that {3 < Il < JLo and applying Theorem 1.3 with € = Il - /3, we arrive at
the required assertion.
The second case can be considered similarly.
COROLLARY 2. If v(t) is a submultiplicative function and v(t) = o(tll-o) as
t 00 (as t 0), then there exists a number III < JLo (Ill> Ilo) such that
v(t) = O(tP-l) as t 00 (t 0).
The proof immediately follows from the preceding corollary.
2. Dilation fimctions of positive functions on the semiaxis. Let o/(t) be a
positive everywhere finite function on the semiaxis (0, 00). We introduce the
function
M",(s) = sup
0<1 < 00
0/( st)
0/( t)
(0 < s < 00),
( 1.16)
which we call the dilation function of 0/( t). Obviously
inf 1/1(st) = inf 1/1 ( "1") = f l sup 1/1(s -1"1") ] -1
0<1<00 o/(t) 0<,.<00 o/(S-I".) 0<,.<00 0/(".)
= [ M",(S-I) ] -I M",(s). (1.17)
The function M",(s) satisfies the submultiplicativity relation (1.14). Indeed,
0/(SIS2 t ) 0/(S I S2 t ) 0/(S2 t )
M",(S I S2) = sup sup sup
0<1<00 o/(t) 0<1<00 0/(S2 t ) O<t<oo o/(t)
= M",(SI)M",(S2). (1.18)
From this it follows that M",(s) is everywhere finite if it is bounded in the
neighborhood of one.
54
II. SPACES OF MEASURABLE FUNCTIONS
If the function Mo/(s) is everywhere finite, then it is submultiplicative, and
by Theorem 1.3 there exist numbers 1'", and o/ (-00 < 1'0/ o/ < (0) such
that
Mo/(s) > s for s > 1 and Mo/(s) > sYt/I for s < 1, (1.19)
and, for any € > 0, for sufficiently large s
Mo/(s) s +£, (1.20)
and for sufficiently small s
Mo/(s) sYI/I-£. (1.21)
The numbers 1'0/ and o/ will be called lower and upper dilation exponents of
0/( t).
Combining (1.17), (1.20), and (1.21), we may say that for sufficiently large s
s YI/I - £ [ M 0/ ( s - I) ] - I M", ( s) s + £ . ( 1.22)
If the positive function cp(t) is increasing, then Mrp(s) 1 for s < 1, and
consequently by (1.19) we have SYf(J 1, i.e. I'rp > O. Let the function be
quasiconcave. Then cp(st)j st cp(t)j t for s > 1, and hence cp(st)j cp(t) s.
From this it follows that Mrp(s) s, and thus rp 1. Thus, for a quasicon-
cave function cp(t) we have
1 [Mrp(S-I) ] -I Mrp(s) s
fors>l,
( 1.23)
( 1.24)
o v 8 1.
Irp rp
From the definition of dilation exponents it is clear that they coincide for
equivalent functions, and therefore, if a function is equivalent to a concave
function, then it satisfies (1.24). In testing whether a function is equivalent to
a concave function, the following lemma is useful.
LEMMA 1.2. If for a function cp(t) the inequalities Mrp(so I) 1 and Mrp(sl)
Sl hold for some so' Sl > 1 and the function Mrp(s) is bounded on some interval
[ 1, a], then cp( t) is equivalent to its smallest concave majorant.
PROOF. It is sufficient to verify that for any to < t l the function cp(t)
satisfies (1.6) with some constant k.
We write C = sUPI<s<a Mrp(s). From the submultiplicativity inequality
(1.18) it follows that Mrp(s) does not exceed the number C r on [1, ar]. If we
choose the integer r so that a r > so' then Mrp(s) will not exceed C r on [1, so].
This implies that the inequality cp(t) cp(tl)Mrp(t j t l ) Crcp(t l ) is satisfied on
[tl' sot.]. Let an integer i be chosen so that s6to t l s6+ It O . The condition
Mrp(so I) 1 implies that cp(t) cp(sot). Therefore
cp(t o ) cp(s6+ lt o) Crcp(t l ). (1.25)
1. POSITIVE FUNCTIONS ON A SEMIAXIS
55
We now choose an integer j so that SI- U +l)t l to SI-jtl. Then from
(1.25), with t l replaced by to and to by sl u + l )t l , we obtain
cp(slj-ltl) Crcp(t o ).
The condition Mcp(SI) SI implies that SI-lcp(t) cp(t / SI)' and therefore
cp(Sl j - lt l) > SI- j -lcp(t l ) > SI- 1 (t o /t l )CP(t l ).
Combining this inequality with the previous one, we see that
cp(t l )/t l SlcrCP(tO)/to. (1.26)
Inequalities (1.25) and (1.26) imply that (1.6) is satisfied with k = SI Cr.
COROLLARY 1. If the function o/(t) is increasing and Mo/(SI) SI for some
s 1 > 1, then it is equivalent to its smallest concave majorant.
Indeed, because of monotonicity, Mo/(so- 1) < 1 for any So > 1. Moreover,
for any s SI we have o/(st) o/(SI t) Slo/(t), i.e. Mo/(s) SI on [0, stJ.
In this case we may set a = C = 51' and r = 1, and then (1.26) becomes
cp(t l )/t l sicp(to)/t o .
COROLLARY 2. If 0 < Yo/ 80/ < 1, then o/(t) is equivalent to its smallest
concave majorant.
PROOF. If we choose € so that Yo/ - € > 0 and 80/ + € < 1, then (1.22)
implies that
1 < s{t/I-£ [Mo/(sI1) ] -1 Mo/(SI) s +£ < SI (1.27)
for sufficiently large SI. Next, for fixed a > 1 and 1 S a
Mo/(s) Mo/(SI)Mo/(s/SI) s +£(S/SI)Yt/l-£ s -Yt/l+aaYt/l-e
for sufficiently large SI. Hence, all hypotheses of Lemma 1.2 are satisfied.
REMARK. Under the hypotheses of Corollary 2 the equivalence constant
depends on yo/' 80/' and that SI for which (1.27) is satisfied.
The assertion of Corollary 2 is in a certain sense unimprovable. For
example, the function %(t) = max{t, t In t} is increasing more rapidly than a
linear function at infinity. Therefore it is not equivalent to a concave
function. We have M%(s) = max{s, s(1 + In s)}, and consequently 0 < yy,o =
80/0 = 1. Thus, under the hypothesis of Corollary 2, the inequalities cannot be
replaced by equalities.
LEMMA 1.3. If we have Mo/(SO-I) = 1 (so> 1), for a positive concave function
0/, then Mo/(S-I) = 1 (s > 1). If we have Mo/(SI) = 51 (SI > 1), then Mo/(s) = s
(s > 1).
56
II. SPACES OF MEASURABLE FUNCTIONS
PROOF. In consequence of (1.17), the function [Mo/(S-I)]-I is the infimum
of the concave functions o/(st)/o/(t) as functions of s, and therefore it is
concave. By (1.23) we have [Mo/(S-I)]-I 1 for s > 1, and if this concave
function assumes its minimal value 1 at s = So > 1, then it is a constant:
[Mo/(s -1)]-1 = 1 (s > 1).
The function s-IMo/(s) = sUPo<t<oo (o/(st)/st). (t/o/(t)) is decreasing be-
cause of the concavity of o/(t). Therefore the equality SI-IM (SI) = 1 and
(1.23) imply that s-IMo/(s) = 1 for 1 < s Sl. Now let 1 < Sl < s. Then
s(s - 1) s - s
1/J(st) ;;;. (s l)sl 1/J(Slt) + (s _ 1)I SI 1/J(SSl t ).
Hence
o/(Slt) SI(S - 1) o/(st) s - Sl
-
0/( ss 1 t) s( S 1 - 1) 0/( ss 1 t) s( S 1 - 1) .
Then
[ inf
0<1<00
0/( Sit) ] - 1
0/( ss 1 t)
{ S 1 (s - 1) 0/( st) s - S 1 } ] - I = s.
S(SI - 1) o/(SSlt) S(SI - 1)
Mo/(s) = [ inf
0<1<00
By (1.23) we have M (s) = s. The lemma is proved.
We note some further properties of the lower and upper dilation exponents.
I t is obvious that
M (0/)9 ( s) = [ M 0/ ( s) ] 8.
Therefore Y(o/)' = ()Yo/ and 8(0/)9 = 980/.
Next, Mq;Oq;I(s) Mq;o(s)Mq;I(s), and therefore
In Mq;rlPl( t) . In Mq;o( t) + In Mq;I( t)
Y q;JrJ = lim I hm I = Y q; + Y q; .
UTI t O n t t n t 0 I
( 1.28)
Similarly,
8cprFPI 8q;O + 8q;1.
( 1.29)
3. The study of the integral J 0/(7')7' - 1 d7'. For a positive function 0/( t) on
the semiaxis (0, 00) we wish to compare the behavior of the indicated integral
with that of the function o/(t) itself.
LEMMA 1.4. If Mo/(so I) < 1 for some So > 1, and Mo/(s) is bounded on some
interval [1, a], then the function J 0/(7')7' - 1 d7' is equivalent to t).
1. POSITIVE FUNCTIONS ON A SEMIAXIS
57
PROOF. It follows from the proof of Lemma 1.2 that, under our hypotheses,
\[;(7) Cr\[;(t l ) for any t l and 7' E [tl' sot.]. Setting t l = so-j+l t successively in
this inequality, we obtain
i t 1/1(-r) d7' = f f soi+lt \[;(7') d7'
o 7' j=l So it 7'
00 00
C r L \[;( so- j+ It)ln So C r In So L [M ( SO-I) J j - I \[;( t)
j=l j=l
= C r In so[ 1 - Mo/( SO-I) ] -I \[;( t) = C 2 \[;( t).
On the other hand,
i t \[;(7') d7' > f t 1/1( 7') d7' In a inf 1/1 ( 7')
o 7' ta - I 7' ta - I <;; 7" < t
> In a [ sup Mo/(S) ] -I \[;( t) = C I1/!( t).
l<s<a
Thus,
i t \[;(7')
CI\[;(t) dT C 2 \[;(t).
o 7'
(1.30)
COROLLARY 1. If the function \[;(t) is increasing, and Mo/(SO-I) < 1 and
M",(sl) < 00 for some so' 51 > 1, then (1.30) holds.
COROLLARY 2. If the positive function \[;(t) is concave and Mo/(SO-I) < 1 for
some So > 1, then (1.30) holds.
COROLLARY 3. If 0 < Yo/ 80/ < 00, then (1.30) holds.
Lemma 1.4 immediately implies the following lemma.
LEMMA 1.5. If Mo/(so) < 1 for some So > 1 and Mo/(s) is bounded on an
interval [1/ a, 1] (a > 1), then
f 00 \[;( 7')
C I\[;( 7') d7' " C 2 \[;( t).
t 7'
( 1.31 )
Indeed, the changes 7' = 1/ z, 9 = l/t, \[;(1/9) = \[;1(9) reduce (1.31) for
(t) to (1.30) for 1(9). We also have to take into account that M0/1(s) =
Mo/(s - I).
Of the three corollaries which follow from this lemma similarly to the
preceding one, we mention only one.
COROLLARY. If -00 < Yo/ 80/ < 0, then (1.31) holds.
58
II. SPACES OF MEASURABLE FUNCTIONS
2. Rearrangements of measurable functions
1. Equimeasurable functions. Rearrangements. We shall consider the metric
space 8(0, (0) of Lebesgue measurable functions, finite almost everywhere on
the semiaxis (0, (0). For every nonnegative function x E 8(0, (0) we intro-
duce its distribution function n x ( '1') defined by the formula
n x ( '1') = mes{ t: x(t) > T}.
The distribution function is decreasing, right continuous, and may take on
infinite values. The collection of all functions x(t) for which nlxl(T) X; 00, will
be denoted by 8 0 (0, (0). For every function from 8 0 (0, (0) we have nl xl ( '1') 0
as '1' 00. Indeed, if nlxl( TO) < 00, then forT> TO the measure of the sets {t:
/x(t)/ > 'T} is finite and their intersection has measure zero. Therefore mes{ t:
/ x( t)/ > 'T} 0 as '1' 00. In what follows we consider only functions in
8 0 (0, (0).
If a function x( t) is summable on (0, (0), then
/lXllL, = oo Ix(t)J dt = oo nlxl(r) dr.
(2.1 )
DEFINITION 2.1. Two nonnegative functions x( t) and y( t) in 8 0 (0, (0) are
said to be equimeasurable if
n x ( '1') = ny( '1').
The equimeasurability of x( t) and y( t) does not imply In general the
equali ty
mes { t: x( t) T} = mes { t: y ( t) '1' }.
(2.2)
For example, the functions x(t) = arctg t and y(t) = 1 are equimeasura-
ble. However, mes{ t: x(t) I} = 0, and mes{ t: y(t) I} = 00.
Equality (2.2) will hold for '1' interior to the interval where n x ( '1') < 00.
Indeed, in this case
mes{t: x(t) T} = lim mes{t: x(t) > '1' - e} = lim nx('T - e)
E O E O
and similarly mes{ t: y(t) 'T} = lim ny( '1' - e). The condition n x ( '1') = ny( '1')
(0 '1' < (0), implies (2.2).
We also note that for two equimeasurable functions x(t) and y(t) the
functions x( t)Xe (t) and y( t)Xe' (t), will also be equimeasurable, where Xe (t)
a Q Q
and X; (t) are the characteristic functions of the Lebesgue sets e a = {t:
Q
x(t) > a} and e = {t: y(t) > a}, respectively. If nx(T) < 00 forT> 0, then
the sets e a and e can be replaced by {t: x(t) a} and {t: y(t) a}.
Let t w(t) be a measurable mapping of the semiaxis (0, 00) into itself. We
shall say that w{t) does not decrease the measure of sets if mes w- 1 (e) < mes e,
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
59
and that it preserves the measure of sets if mes w(e) = mes e, for any measura-
ble set e. Since w( w - I( e)) C e, every measure preserving mapping does not
decrease measure.
If w( t) is a mapping which does not decrease measure, then
mes{ t: y(w( t» > T} mes{ s: y(s) > T},
and so for nonnegative functionsy(t) andYw(t) = y(w(t» we have
nyw( T) ny( T).
It is obvious that IIYwllL IlyliL , and (2.1) implies that
00 00
IIY wll LI IIY II LI.
DEFINITION 2.2. The rearrangenrent of a nonnegative function x E 8 0 (0, (0)
is a decreasing left continuous function x*(t) equimeasurable with x(t).
We show that the rearrangement is unique. Let x*(t) and xT(t) be two
different rearrangements of the same function x(t). Assume that xT(t o ) >
x*(t o ). Let e > 0 be such that xr(t o ) - e > x*(t o ). By the left continuity of
x*(t) there exists an interval [tl' to] in which x*(t) < xT(t o ) - E. Then
mes{ t: xT( t) > xr( to) - e} to'
while
mes { t: x * ( t) > x * (to) - e} t I < to.
The functions x*(t) and xT(t) are not equimeasurable, which contradicts their
equirneasurability with x(t).
Consequently, the rearrangement is unique. It can be defined by the
formula
x * ( t) = inf { T: n x ( T) < t}.
It is easy to verify that nx(x*(t» = t, if x*(t) is a point of continuity of
nx(T), and nx(x*(t)) t, if x*(t) is a point of discontinuity of nx(T). Similarly,
x*(n x ( T» = T, if n x ( T) is a point of continuity of x*(t) and x*(n x ( T») ;> T, if
n x ( T) is a point of discontinuity of x*( t).
We denote by x*(t) the rearrangement of the absolute value of an arbitrary
function x( t) in 8 0 (0, (0).
It is easy to calculate the rearrangements of strictly simple functions. Every
strictly simple function can be written in the form
N
x(t) = L XkXek(t),
k=l
where the e k are disjoint sets of finite measure and Xi =f= x j for i =1= j. Then
N
x*( t) = L xj*X('rJ_I.'rj]'
j=l
60
II. SPACES OF MEASURABLE FUNCTIONS
where x j * is the rearrangement of the numbers Ixkl in decreasing order and
Tj = Llxil .x/ mes e; (TO = 0).
It is sometimes convenient to write a nonnegative strictly simple function in
the form
N
x( t) = f1k k'
k=l
where f1 k > 0 and e C e 2 C . . . C e;" are sets of finite measure. For t > 0
we then have
N
x*( t) = f1 k X(O.mese k ].
k=l
(2.3) ,
We introduce the number
x*( (0) = lim x*(t) = inf{ T: n x ( T) < oo}.
1-+00
This number has the property that
{ < 00
nx(r) ..;; :
for T > X * ( 00 ),
for T = X * ( 00 ),
for T < x* ( 00 ).
LEMMA 2.1. Let x( t) and y( t) be two nonnegative equimeasurable functions.
For any e > 0 there exists a measure preserving mapping w( t) of the semiaxis
(0, (0) into itself such that
IIX I - Ywl/LQ)nL I e,
where x1(t) = max{ x(t), x*( oo)} and Y w(t) = y(w(t».
(2.4)
PROOF. We set TO = x*(oo) + e/8 and partition the semiaxis into three
sets:
E; = {t: TO < x(t)}, E; = {t: x*( (0) < x(t) < TO}'
E; = {t: x(t) x*(oo)}.
The mapping w(t) will be defined separately on each of these sets. For the
function y(t) we introduce three similar sets E), i = 1, 2, 3, having in mind
that x*( (0) = y*( (0).
The definition of w(t) on Ex l . The set E; has finite measure. We choose
8 = min{e, e/(2 mes E;)} and consider the sets e .x = {t: TO + k8 < x(t)
TO + (k + 1)8} and e .y = {t: TO + k8 <y(t) TO + (k + 1)8} (k =
0, I, . . . ). They have equal measures for the same k, and Ex I = U k el. x and
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
61
Eyl = U k e ,y. On every set e ,x we define the measure preserving mapping
w(t) onto el,y. Then, by construction,
Ix(t) - y(w(t»1 8 £
and
f Jx(t) - y(w(t))l dt ..;; 8 mes E; < e/2. (2.5)
E1
The definition of w(t) on E;. If the measure of the set E; is finite, then w(t)
is defined on it in the same way as on Ex l with TO replaced by x*(oo) and £ by
£ /2. Then (2.5) will be satisfied. The case where mes E; = mes Ey2 = 00 is
more complicated. In this case the sets E; and Ey2 can be partitioned into
disjoint sets e1,x and ef,y, k = 0, 1, . . ., respectively, so that mes e:,x =
mes e:,y = 1,
Tk+1 x(t) Tk
(t E e:,x)
and
Tk+1 y(t) Tk (t E e:,y),
where Tk!X*(oo) as k 00. We choose an infinite sequence of natural
numbers m l , m 2 , . .. so that
00
(T +I - x*(oo») < 00/8.
;= I
Take the union F;, of the sets .Y" Denote. the remaining s.ets by e y
(k = 0, 1, . . . ; n k < nk+ I). We defIne the functIon w(t) so that It maps e k x
,
onto e; Y ' preserving measure.
k'
Then
00
f 2 Ix( t) - y,.,( t)1 dt";; f2 Ix( t) - y..,( t)1 dt
Ex k=O ek.x
00
(Tk - T 1Itc + 1 ).
k=O
We estimate the last sum. For any N > 0 we have
N N
(T k - T 1Ztc + I ) = Tk -
k = 0 k = 0 nk + I <. N
T1Ztc+ 1 -
nk + I > N, k <. N
T1Ztc+ 1
= TO +
m;+I<;.N
T -
nlt+ I
T",,+I
k<.N
nk + I > N
00
TO - X*(oo) + (Tm,+1 - x*(oo») < £/4.
;= I
62
II. SPACES OF MEASURABLE FUNCTIONS
Thus
Ix(t) - yw(t)1 Tk - T1Itc+ 1 TO--X*(oo) e
(t E E;)
(2.6)
and
L2Ix(t) - y.,(t)1 dt < : .
x
The definition of w(t) on E;. On the set E; we have x 1 (t) = x*( (0), and so
the mapping w{t) has to send E; into parts of the semiaxis where y(t) is
sufficiently close to x*( (0) = y*( (0).
If mes E; = 00, then for this purpose we use the set F;, chosen above. If
mes E; < 00, then in F;, we choose a set e y of the same measure so that
x*(oo) y*(t) x*(oo) + min{ e, ej (4 mes E;)}
on e y , and we map E; onto e y in a measure preserving way. Then
Ix*(oo) - y(w(t»1 e
on E; and
JE3IX*(OO) - y(w(t»1 dt " : .
x
(2.7)
If, however, mes E; = 00, then we partition E; into sets e x (k =
,
1, 2, . . . ) of measure one in an arbitrary way, and map them with preserva-
tion of measure onto sets e 1?' for which T",Jc x*(oo) + ej2k+2. Then (2.7)
will also be satisfied.
It remains to consider the; case where mes E; = mes Ey2 < 00. But then
mes E; = mes E} = 00, and for any T < x*(oo) we have mes{t: T <y(t) <
x*( oo)} = 00. Therefore the set E} may now play the same role as F;, did
above. Repeating the preceding constructions, we obtain a mapping with the
properties (2.7).
Thus, we have defined a measure preserving mapping w(t) on the whole
semiaxis, and by (2.5)-(2.7) we have
IXJ(t) - y(w(t»1 ,,£ and {X> IXJ(t) - y(w(t»1 dt < £.
The lemma implies the following important theorem.
THEOREM 2.1. If for two measurable functions x( t) and y( t) the inequality
x*(t) y*(t) holds for all t > 0, then for any e > 0 there exist a measure
preserving mapping w(t) of the semiaxis (0, (0) into itself and a measurable
function {3( t) with I {3( t)1 1 such that
IIx - {3YwIILQ)nL I e (Yw(t) = y(w(t»). (2.8)
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
63
PROOF. The function y*(t) is equimeasurable with ly(t)l, and therefore by
Lemma 2.1 and the monotonicity of y*(t) there exists a measure preserving
mapping wl(t) such that Ily* - lylwlliLoonL I < £/2. We write al(t) =
x*(t)[y*(t)]-l (assuming that a(t) = 0 if y*(t) = 0); then 0 al(t) 1 and
IIx* - allYlwlliLoonL I =llal(y* -IYlwl)IILoonL I < £/2. (2.9)
Similarly, by the equimeasurability of x* and lxi, there exists a measure
preserving mapping w 2 ( t) such that
IIx I - (x*)w 2 11 L oonLI < £/2,
where xl(t) = max{lx(t)l, x*(oo)}. Write a2(t) = Ix(t)l[xl(t)]-I. Then 0 <;
a 2 (t) 1 and
" Ixl- a 2 (x*)w 2 11L oo n LI < £/2.
We choose w(t) = W I (W2(t» and
{3(t) = a 2 (t)a l (w2(t»sgn x(t)sgnyw(t).
Then (2.9) and (2.10) imply that
Ilx - {3YwiILoonLI = II Ixl- a2(al) IYlwl 0 w211LoonLI
II Ixl- a2(x*) IILoonLI +lla2[(x*) - (al) IYlwl oW2JIIL oo nL I
< ; + Ilx* - adYlw,IIL..nL, < E.
(2.10)
2. The properties of rearrangements.
1 0. The equimeasurability of I x( t)1 and x*( t) implies that
£'>0 Ix(t)1 dt = £'>0 x*(t) dt = oo nlxl(-T) dr, (2.11)
where both sides may be infinite.
2°. If Ix(t)1 ly(t)l, then x*(t) y*(t).
3 0. If x( t) = y( t) everywhere except for a set of measure not exceeding a
number y < 00, then x*(t) y*(t - y) andy*(t) x*(t - y) (t > y).
Indeed, let x*(t o ) = TO (to> y). For any £ > 0 there exists a set e of
measure to on which Ix(s)1 > TO - £. Then we can construct a set e' with
mes e' to - Y on which ly(s)1 > TO - £. From the definition of y* it follows
that y*(t o - y) > TO - £ = x*(t o ) - £. Due to the arbitrariness of £ and the
equimeasurability of x and y the assertion is proved.
4°. If x(t) 0 and £ > 0, then (x + £)*(t) = x*(t) + £.
5°. If Ix(t) - y(t)1 £ for almost all t, then Ix*(t) - y*(t)1 £.
Indeed, Ix(t)1 ly(t)1 + £, and hence x*(t) < (Iyl + e)*(t) = y*(t) + £.
Similarly, y *( t) x*( t) + £.
64
II. SPACES OF MEASURABLE FUNCTIONS
6°. The inequality
f Ix(s)1 ds i mese x*(s) ds
e 0
(2.12)
holds .
We denote by Xe(s) the characteristic function of the measurable set e.
Then n1xIXe(T) mes e and nlxlk(T) nlxl(T); therefore nlxlk(T)
min{ mes e, nlxl( T)}. Integrating this inequality and using (2.11), we obtain
f Ix(s)1 ds = LX) nlxlx.(T) dr
e 0
r 00 r mes e
J o min{ mes e, nlxl( T)} dT = J o x*(s) ds.
7°. If x*(O) > x*(oo), then there exists a set e8(x) with mes e8(x) = 0 for
which
1 9 x*(s) ds = f Ix(s)1 ds. (2.13)
o ee( x)
The condition x*(O) > x*( (0) implies that in the neighborhood of x*(O) we
have nlxl(T) < 00, and therefore the sets of those s at which x*(s) > x*(O) and
/x(s)1 > x*(O) have the same measure, which is at least O. If we denote by e 8
the set of those s where /x(s)1 > x*(O) and bye; the set of those s where
/x(s)1 > x*(O), then mes e8 0 mes e;. The set e 8 (x) can be chosen so that
e9 C e8(x) C e 9 ' and mes e8(x) = O. The restrictions of Ix(t)1 and x*(t) to
e8(x) and (0, 0], respectively are equimeasurable, and so (2.13) is satisfied.
If x*(O) = x*( (0), then there is no set e 8 (x) on which equality (2.13) is
attained. The function x(t) = arctg t considered above serves as such an
example. However, we have the following assertion.
8°. The equality
i 9 x*(s) ds = sup f Ix(s)1 ds
o mes e = fJ e
(2.14)
holds .
From (2.12) we immediately obtain the inequality
fo9 x*(s)ds m:su 9 i Ix(s)1 ds. (2.15)
Therefore, in the case where x*(O) > x*(oo), (2.13) and (2.15) imply (2.14).
If x*( 0) = x*( (0), then by e8 we again denote the set of those s for which
/x(s)/ > x*(oo), and by e 9 ' the set of those s for which Ix(s)1 >x*(O) - elO,
where e > 0 is an arbitrarily small number. Then by the equimeasurability of
x*(s) and Ix(s)1 we have mes e8 0 mes e;; = 00. Therefore there exists a
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
65
set e; such that e 8 C e; C ef/ and mes e; = O. The restrictions of Ix(t)1 and
x*(t) to the sets e9 and (0, mes e8] are equimeasurable. Therefore
f Ix(s)1 ds = I mes ee x*(s) ds.
e9 0
Then
i3 Ix(s)1 ds = iiJ Ix(s)1 ds + i 3 - e iJ Ix(s)1 ds
;;;. imeseiJ x*(s) ds + f [x*(oo) - £/0] ds
o e£ - e9
1 mes eo
;;;. 0 x*(s) ds + x*( 00)( 0 - mes es) - £.
On the interval (mes e9, 0) the function x*(s) is equal to x*( (0), and so
f Ix(s)1 ds i 9 x*(s) ds - f.
el 0
(2.16)
Inequalities (2.15) and (2.16) imply (2.14).
We note that (2.14) implies the important inequality
8 (x + y)*(s) ds '" 8 x*(s) ds + 8 y*(s) ds. (2.17)
REMARK. If nlxl( T) < 00 for all 7" > 0, i.e. if x*( (0) = 0, then the sets e8(x)
constructed in 7° are defined for all 0 for which x*(O) > O. We write
e(x) = U e 8 (x).
x*(9»O
(2.18)
Then e(x) coincides with the support of x(t).
9°. For two locally summable functions x(t) and y(t) with nx(T), ny(T) < 00
the equality
(x + y)*(t) = x*(t) + y*(t) (2.19)
holds for all 7" > 0 if and only if x(t) and y(t) are of constant sign almost
everywhere and have a common system of sets e 8 : e 8 (x) = e 8 (y) (0 < () < (0).
PROOF. Necessity. Let (x + y)* = x* + y*. By (2.14) we have
f Ix + yl dt = i 8 (x + y)* dt = 1 8 x* dt + 1 8 y* dt
ee(x + y) 0 0 0
sup f Ixl dt + sup f Iyl dt.
mes e= 9 e mes e=9 e
(2.20)
66
II. SPACES OF MEASURABLE FUNCTIONS
If we assume that
f Ixl dt < sup f Ixl dt
ee( x + y) mes e = fJ e
or
f IYI dt < sup f IYI dt,
ee( x + y) mes e = fJ e
then
f Ix + YI dt f Ixl dt + f IYI dt
ee( x + y) ee( x + y) ee( x + y)
< sup f Ixl dt + sup f iYl dt,
mes e=fJ e mes e=fJ e
which contradicts (2.20). Therefore,
f IYI dt,
e
(2.21 )
and consequently the system of the sets e8(x + y) can serve as the system
e 8 (x) and e8(Y). From (2.20) and (2.21) it follows that
f Ix + YI dt = f Ixl dt + f IYI dt,
ee(X+y) ee(X+y) ee(X+y)
which implies that
f Ixl dt = sup f Ixl dt and f lyl dt = sup
ee(x + y) mes e = fJ e ee(x + y) mes e= fJ
Ix(t) + y(t)1 = Ix(t)1 + ly(t)1
(2.22)
at almost all points of e 8 (x + y).
By the remark following (2.17) this equality holds almost everywhere on the
whole semiaxis. The necessity is proved.
Sufficiency. If e8 is a common system of sets for the functions x( t) and y( t),
and these functions satisfy (2.22) almost everywhere, then
i 8 x* dt + i 8 y* dt = f Ixl dt + f lyl dt
o 0 ee ee
= f Ix + YI dt i 8 (x + y)* dt.
ee 0
Then it follows from (2.17) that
i 8 x* dt + (8 y* dt = (8 (x + y)* dt.
o J o J o
Differentiating with respect to 0, we obtain (2.19).
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
67
10°. The inequality
(xy )*( t I + t 2 ) x*( t I)Y*( t 2 )
is true.
For the proof we write x*(t l - e) = a andy*(t 2 - e) = b, where e > 0 is
sufficiently small. We have the inclusion
{ t: I x( t)y ( t) I > ab} c {t: I x( t) I > a} U {t: I y ( t) I > b },
which implies that
nlxy,(ab) n\xl(a) + n1yl(b) t l + t 2 - 2e < t l + t 2 .
Hence
(xy)*(t l + t 2 ) ab = x*(t l - e)y*(t 2 - e).
This inequality implies the required one by the left continuity of rearrange-
ments.
COROLLARY. The following inequality holds:
(x + y) * ( t I + t 2) x * ( t I) + y * ( t 2).
(2.23 )
Indeed,
e(x+ y )*{tI+t 2 ) = (e\x+Y\)*(t l + t 2 ) (elxleIYI)*(tl + t 2 )
(elxl)*(tl)(eIYI)*(t2) eX*{tI)ey*(t2) = e X *{tI)+y*(t 2 ),
which implies (2.23).
11°. If a sequence xn(t) converges to a function x(t) in measure, then
x:( t) x*( t) at all points of continuity of x*( t).
If mes(t: Ix(t) - xn(t)1 > e) < T for n no(e, T), then (x - x n )*( T) e, as
n 00. Then by the corollary of property 10° we have
x*(t + 11) x:(t) + (x - X n )*(11)
and
x:( t) x*( t - 11) + (x - x n )*( 11).
From this and the preceding it follows that for any 8 > 0 and 11 > 0 for
sufficien tly large n
x*(t + 11) - 8 x:(t) x*(t - 11) + 8
holds.
If x*(t) is continuous at t, this inequality implies that lim n -+ oo x:(t) =
x*( t).
12°. Let nIYI(T) < 00 for all 7" > 0, and Ixm(t)1 y(t). If xm(t) x(t)
almost everywhere, then (x - x m )*( 7") 0 for all T > O.
68
II. SPACES OF MEASURABLE FUNCTIONS
For every E > 0 the semiaxis can be divided into two sets e and e' so that
/y(t)/ < EI2 on e and e' has finite measure. On e we then have Ix(t) - xm(t)1
< E, and on e' the sequence xm(t) converges to x(t) in measure. Therefore
mes{ t: /x(t) - xm(t)/ > E} 0 as m 00. Thus, on the whole semiaxis we
have X m - x 0 in measure. Property 11 0 implies that (x - x m )*( T) 0 for
all T > O.
We note that the restrictions imposed on the sequence of the functions
x m ( t) are essential. For example, for the sequence of the functions X[m,m + 1]( t)
the condition n x (T) 1 is satisfied; it converges to the function x(t) = 0
[m.m+ I)
everywhere, but X[m,m+ 1]( T) = X[O,l]( T).
COROLLARY. If under the preceding hypotheses the function y(t) is summable
on the whole semiaxis, then, by Lebesgue's theorem,
lim f x:(t) dt = f x*(t) dt
n oo e e
(2.24)
for any measurable set e.
13 0 . The following inequality holds:
oo x(s)y(s) ds ..;; oo x*(s)y*(s) ds.
(2.25)
PROOF. Without loss of generality we may assume that the functions x and
yare nonnegative. We construct an increasing sequence of strictly simple
functions Yn(t) converging to y(t) almost everywhere. We represent the
functionsyn(t) in the form
N n
Yn( t) = Yn,kXe n . k (t),
k=l
where the Yn,k are positive numbers, and en, I C e n ,2 C . . . C en,N n . Then 6 0
implies that
N N
00 n n mes e k
1 X(S)Yn(S) ds = Yn,k f x(s) ds..;; Yn,k 1 n. x*(s) ds
o k = I en.k k = I 0
N n
= 1 00 x*(s) Yn,kX[O,mesen.kj(S) ds.
o k=l
The function Z 1 Y n,kX[O,mesen.k]( t) coincides with Y n( t) by (2.3). Thus
oo x(s)Yn(s) ds ..;; oo x*(s)y:(s) ds ..;; oo x*(s)y*(s) ds.
Passing to the limit as n 00, by B. Levi's theorem we obtain (2.25).
Inequality (2.25) is sharp in the following sense.
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
69
14°. The equality
s p oo u(s)y(s) ds = oo x*(s)y*(s) ds
(2.26)
holds, where the supremum is taken over all functions u(s) for with lu(s)1 is
equimeasurable with x*(s).
By virtue of (2.25) it is sufficient to establish the inequality (2.27).
Without loss of generality we may assume thaty(t) 0, and then u(t) o.
We also assume that y*(t) > y*(oo) for t E (0, 0) and y*(t) = y*(oo) for
t > 0 (0 = 00 is allowed). We partition the semiaxis into intervals of length 8
so that N8 = 0 (N (0). Reasoning in the same way as in 7° and 8°, we can
construct sets e l , . . . , eN of measure 8 such thaty*«i - 1)8) > y(t) y*(i8)
for t E e j , and sets e N + I' e N + 2 , . .. of measure 8 such that y*( (0) > y(t)
y*( (0) - E for t E e j , where E > 0 is arbitrarily small. We map every set e;
one-to-one onto the interval «i - 1)8, i8] in a measure preserving manner by
means of the functions wj(t), and set
00
u 8 (t) = x*(wj(t)Xe(t).
,
;= I
The function u 8 (t) is equimeasurable with x*(t). Next,
00 00 00
sup 1 u(s)y(s) ds > 1 u 8 (s)y(s) ds = f x*(w;(s»y(s) ds
u 0 0 ; = I e;
N 00
> 8. x*(i8)y*(i8) + . f x*(w;(s»[y*(oo) - e] ds.
, = I , = N + I e,
We have
00
f x*(w;(s» ds = loo x*(s) ds.
e, fJ
i=N+ I
If this integral is infinite, then (2.27) is true, since its left side is infinite. If,
however, the last integral is finite, then
00 00 00
sup 1 u(s)y(s) ds > 8 x*(i8)y*(i8) - e 1 x*(s) ds.
u 0 i=l fJ
The function x*(t)y*(t) is decreasing. Therefore, it is Riemann integrable
on any interval [1/ a, a]. From this it follows that the limit as 8 0 of the
first term on the right side is equal to the integral of x*(t)y*(t) over (0, (0).
Thus,
sup 1 00 u(s)y(s) ds 1 00 x*(s)y*(s) ds - E l oo x*(s) ds,
u 0 0 fJ
which implies (2.27) in view of the arbitrariness of E, and hence (2.26).
70
II. SPACES OF MEASURABLE FUNCTIONS
REMARK. It can be seen from the proof that for y(t) 0 the supremum can
be taken over functions u(t) of the form u(t) = x*(w(t)), where w(t) is a
one-to-one measure preserving mapping of (0, (0) onto itself.
COROLLARY. The following inequality holds:
£00 x*(s)[y(s) + z(s)]* ds ..;; {;c x*(s)y*(s) ds + £00 x*(s)z*(s) ds.
15°. l'he equality
(00 x*(s) ds = inf f \x(s)\ ds
.19 e'
(2.28)
holds, where the infimum is taken over all measurable sets e' whose complement
has measure O.
First we establish that
(00 x*(s) ds f "\x(s)\ ds. (2.29)
.18 e'
If x(s) is summable on (0, (0), this inequality follows from (2.11) and
(2.12). Let x(s) not be summable on (0, (0). For the proof of the inequality it
is of interest to consider only those sets e' for which fe' Ix(s)1 ds < 00. The
function Ye(s) = min{lx(s)l, x*(O)} is bounded on the complement of e',
which has finite measure 0, and does not exceed Ix(s)1 on e'. Therefore it is
summable on (0, (0). Next, Yl(s) = rnin{ x*(s), x*( O)}; therefore
(00 x*(s) ds = (00 Yl(s) ds f Ye(s) ds f \x(s)\ ds.
.I 9 .I 9 e' e'
The required inequality (2.29) is proved.
In the proof of (2.28) we may assume that f x*(s) ds < 00. Then nlxl( T) <
00 for all 'T > 0, and therefore the set ee(x) of measure 0 constructed in 7°
exists. We write ee(x) = (0, (0) \ ee(x). The functions x*(s) and Ix(s)1 are
equimeasurable on the sets (0, (0) and ee(x), respectively, and therefore
1 00 x*(s) ds = f Ix(s)1 ds.
8 (x)
COROLLARY. If x(s) = 0 for s > a, then
a x*(s) ds ..;; a Ix(s)1 ds.
16°. If x(t) = y(t) = 0 for t > a, then
La x*(s)y*(a - s) ds ..;; L a Ix(s)Y(s)1 ds.
o 0
(2.30)
(2.31 )
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
71
PROOF. Assuming that x and yare nonnegative, we construct an increasing
sequence of strictly simple functions Y n( t) converging to y( t) almost every-
where. The functions Yn(t) can be written in the form
N n
Yn(t) = Yn.kXen.k(t),
k=l
where the Yn.k are positive numbers and e n . l C e n . 2 C . . . C e n . Nn . By (2.29)
we then have
(a x(s)y(s) ds La X(S)Yn(S) ds
J o 0
N n N n
= Yn,k f x(s) ds Yn,k fax*(s) ds
k = 1 en.k k = 1 a - mes en.k
= a x*(s) Yn,kX[a-mese., a](s) ds
= a x*(s)y:(a - s) ds. (2.32)
The sequence y:(a - s) is increasing, and by property 11 ° it converges to
y*(a - s) almost everywhere. B. Levi's theorem allows us to obtain (2.31)
from (2.32) by passing to the limit.
17°. If x(t) = y(t) = Ofor t > a, then
L a x*(s)y*(a - s) ds = inf L a lu(s)Y(s)1 ds, (2.33)
o u 0
where the infimum is taken over all functions u(t) such that lu(t)1 is equimeas-
urable with x*(t).
It is sufficient to establish the inequality
L a x*(s)y*(a - s) ds inf (a lu(s)y(s)J ds.
o u )0
(2.34 )
On the left there is a Riemann integral. For an arbitrary e > 0 we construct
a sequence {s;} oooo of points having the following properties:
l)s_; + s; = a,
2) y*(a - s; + 0) (1 + e)y*(a - S;-l)' and
3)
00
X*(S;_1 + O)y*(a - S;_I)(S; - S;-l)
; =-00
(1 + e) La x*(s)y*(a - s) ds.
o
72
II. SPACES OF MEASURABLE FUNCTIONS
Properties 2) and 3) imply the inequality
00
L X*(S;_1 + O)y*(a - s; + O)(s; - S;-I)
; =-00
2 1 a
..;; (1 + e) 0 x*(s)y*(a - s) ds.
We construct disjoint sets e; of measure s; - S;-1 on which
y*(a - S;-I) \y(t)\ y*(a - s; + 0)
(t E eJ.
We map every set e; onto [S;-I' s;] with preservation of measure by means
of the function w;(t), and set
00
v(t) = L
x*( w;( t)Xe (t).
,
t;=-oo
The function v(t) is equimeasurable with x*(t). We have
a a 00
inf 1 lu(S)Y(s)1 ds ..;; 1 v(s)IY(s)1 ds = . L f x*(w;(s))IY(s)1 ds
o 0 I = - 00 e;
00
L X*(S;_1 + O)y*( a - s; + O)(s; - S;-I)
;=-00
2 (a
..;; (1 + e))o x*(s)y*(a - s) ds.
In view of the arbitrariness of E, the last inequality implies (2.34), and with
it (2.33).
18 0 . Assume that for the locally summable functions x( t) and y( t)
8 x(s) ds ..;; 8 y(s) ds
(2.35)
holds and b(t) is a decreasing nonnegative function on (0, (0) such that the
functions x(t)b(t) andy(t)b(t) are locally summable. Then
LX) x(s)b(s) ds ..;; 1 00 y(s)b(s) ds. (2.36)
o 0
If x(t) and y(t) are nonnegative, then the condition of summability of xb and
yb can be omitted.
PROOF. First consider the case where b( t) is strictly simple:
N
b( t) = L kX(O. 8 k ]( t),
k=l
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
73
where d k > 0 and 0 1 < O 2 < . . . < ON. Then
00 N f}
i x(s)b(s) ds = L l1 k i k x(s) ds
o k=l 0
N f} 00
< L l1 k i k y(s) ds = 1 y(s)b(s) ds,
k=l 0 0
i.e. (2.36) is satisfied. If b(t) is a bounded function, then it can be approxi-
mated uniformly by a sequence of decreasing strictly simple functions bN(s)
on every finite interval [0, a], and then
i a x(s)b(s) ds = lim i a x(S)bN(S) ds
o N-+oo 0
lim i a Y(S)bN(S) ds = i a y(s)b(s) ds,
N-+oo 0 0
from which we also obtain (2.36).
Finally, for any b(t) we write b(n)(t) = min{ b(t), n}. Then in the inequality
1"00 x(s)b<n>(s) ds < 1"00 y(s)b<n>(s) ds, (2.37)
already proved, we can pass to the limit in view of the absolute continuity of
the integrals of surnmable functions. Thus, we again arrive at (2.36).
If x(t) and y(t) are nonnegative, then in the case where y(t)b(t) is not
summable, (2.36) is obvious, and in the case where it is summable, the
passage to the limit in (2.37) is possible in view of Fatou's lemma.
REMARK 1. If the support of b(t) is contained in (0, N], it is sufficient to
require that (2.35) be satisfied on (0, N].
REMARK 2. If b(t) is increasing, then the inequality
1"0 00 Ix(s)lb(s) ds < 1"0 00 ly(s)lb(s) ds
follows from the inequality
oo Ix(s)1 ds < oo lY(s)1 ds.
19°. The inequality
i 9 x*(s) ds < i f} y*(s) ds
o 0
for all ° > 0 implies the inequality
1"0 00 x(s)a(s) ds < 1"0 00 y*(s)a*(s) ds
for any measurable function a(s).
(2.38)
74
II. SPACES OF MEASURABLE FUNCTIONS
Indeed, this follows from properties 18° and 13° of rearrangements.
20°. If b( t) is an increasing nonnegative function, then
( 00 x*( t)b( t) dt = inf (00 x( w( t) )b( t) dt, (2.39)
J o w J o
where w(t) is a one-to-one measure preserving mapping of (0, (0) onto itself.
PROOF. The functions Ix(w(t))1 and x*(t) are equimeasurable. Therefore
(2.29) applied to x(w(t)) and Remark 2 after property 18° imply the inequality
fooo x*(s)b(s) ds " fooo Jx(w(s))Jb(s) ds.
Thus it is sufficient to prove the reverse inequality under the assumption
that the left side is finite. Given e > 0, consider the sequence of numbers
(1 + e)k (- 00 < k < (0). Assuming that b( -0) = 0, we denote by T k a point
on the semiaxis at which b(T k - 0) (1 + e)k b(Tk + 0). The limit of Tk as
k -00 will be denoted by T -00. If b(t) is bounded, then among the Tk there
exists a last one Tko-I' and we set Tko = 00. The numbers Tk have the
properties
b(T k - 0) (1 + e)b(Tk_1 + 0) (-00 <k < (0).
The finiteness of the integral on the left side and the increasing property of
b(t) imply that x*( (0) = 0, and therefore n1xl( T) < 00 for all T > o. Hence,
reasoning in the same way as in 7° and 8°, we can construct sets e k with
measure Tk - T k - I so that the functions x*(t)X(Tk_I,Tk](t) are equimeasurable
with the functions Ix(t)IXe k (t) for all numbers Tk 00 (k > -(0). If T -00 > 0,
then we also construct a set e _ 00 so that the function x*(t)X(O,T _oo](t) is
equimeasurable with the function Ix(t)IXe (t). Next, we define a one-to-one
-00
measure preserving mapping w(t) of (0, (0) onto itself which carries the
intervals (T k _ I' T k] onto the sets e k and the interval (0, T _ cxJ onto e _ 00. Then
the functions x*(t)X(Tk_I,Tk](t) and Ix(w(t))lx(Tk_I,Tk](t), and also x*(t)X(O,T_oo](t)
and Ix(w(t))lx(O,T _oo](t), respectively, are equimeasurable, and therefore
1 00 x(w(t))b(t) dt = L L r * x(w(t))b(t) dt
o k Tk _ I
" L b('rk - 0) L r * x(w(t)) dt = L b('r k - 0) L r * x*(t) dt
k Tk - I Tk - I
" (1 + e) L b( 'r k - 1 + 0) L r * x*(t) dt " (1 + e) 1 00 x*( t)b( t) dt.
Tk _ I 0
By the arbitrariness of e the required inequality, and with it (2.39) as well,
are proved.
2. REARRANGEMENTS OF MEASURABLE FUNCTIONS
75
CoROLLARY. If b( t) is an increasing nonnegative function, then
{YJ (x + y)*(t)b(t) dt) faoo x*(t)b(t) dt + faoo y*(t)b(t) dt.
21 o. If a > -1 and x( t) is a decreasing nonnegative function, then
C I fa 00 x( t)t y + a dt « fa 00 (x( t)t Y )* t a dt « C 2 fa 00 x( t)t y + a dt, (2.40)
where C I and C 2 do not depend on the choice of x(t).
For y 0 the integrals on all sides of the inequalities coincide. Therefore
y > 0 is the only interesting case.
For X(O.h t) we have
1 00 (X(O.hj(t)tY)*t a dt = l h (X(O.hj(t)tY)*ta dt = l h (h - t)"Yta dt
000
= ha+Y+lfal (1 - sf sa ds
= (0: + Y + 1) II (1 - sf sa ds l h ty+a dt
o 0
= carfaoo X(O.hj(t)t y + a dt. (2.41)
1) Let a > O. We represent the decreasing nonnegative strictly simple
function x(t) in the form
N
x( t) = L kX(O.hk]( t)
k=l
( k > 0; hI > h 2 > . . . > h n ). (2.42)
The function t a is increasing, and so in view of 20 0
1 00 (x(t)tY)*ta dt = inf 1 00 x(w(t))[ w(t)pa dt
o w 0
N 00
L kinf 1 X(O.hk](W(t)) [ w(t)]Yt a dt
k=l w 0
N 00
= L Ll k 1 (X(O.hkj(t)tY)*t a dt
k= I 0
N 00
= Cay L Ll k l x(o.hit)t y + a dt
k= I 0
= Car OO X(t)t y + a dt.
(Here w(t) is a one-to-one measure preserving mapping of (0, (0) onto itself.)
We represent an arbitrary decreasing nonnegative function x(t) as the limit of
76
II. SPACES OF MEASURABLE FUNCTIONS
an increasing sequence of nonnegative step functions xn(t). Then
LX) (x(t)tY)*t a dt ;;. (00 (xn(t)tY)*ta dt ;;. Cay (00 xn(t)t y + a dt.
o J o )0
Passing to the limit as n 00 on the right side, we obtain
00 (x( t)tY)* t a dt ;;. Cay oo x( t)t y + a dt.
The left side of (2.40) is proved. The right side is satisfied with C 2 = 1 in
view of (2.39).
2) Let - 1 < a < O. The function t a is decreasing and the left side of (2.40)
is satisfied with C 1 = 1 in view of 13 0 .
For a function of the form (2.42), we obtain from (2.26) and (2.41)
1 00 (x(t)tY)*t a dt = sup (00 x(t)tYua(t) dt
o )0
N 00
L kSUP 1 X(O,h.J(t)tYua(t) dt
k=l 0
N 00
= L k 1 (X(O,hkl( t)t Y )* t a dt
k= I 0
N 00 00
= Cay L k 1 X(O,hkJ(t)t y + a dt = Cay ( x(t)t y + a dt,
k = I 0 )0
where the ua(t) are functions equimeasurable with ta.
We represent any decreasing nonnegative function x(t) as the limit of an
increasing sequence xn(t) of the form (2.42). Then
1 00 xn(t)tYua(t) dt Cay (00 xn(t)ty+a dt Cay (00 x(t)t y + a dt.
o J o J o
Passing to the limit as n 00, we obtain
oo x(t)tYua(t) dt Cay oo x(t)t y + a dt.
Taking the supremum over all functions ua(t) equimeasurable with ta, by
virtue of (2.26) we arrive at the inequality
1 00 (x( t)t Y )* t a dt Cay (00 x(t)t y + a dt.
o J o
22 0 . Let a > -1 and a + y < -1, and let x( t) be a nonnegative increasing
function. Then (2.40) is satisfied.
3. OPERATORS IN THE BANACH COUPLE L}(O, (0), Loo(O, (0) 77
Let x( t) = X(h,oo)( t). Then
L oo (X(h,oo)(t)tY)*t a dt = 1 00 (t + h)Yta dt = h y + a +} l OO (1 + s)Ysa ds
000
=I'Y + a + 11 00 (1 + sfsa ds oo t y + a dt
= Cay oo X(h,oo)( t)t y + a dt.
(Here f (1 + s)Ysa ds < 00 in view of the conditions imposed on a and y.)
The remaining reasoning is analogous to that carried out in the proof of
21 0 . The increasing function x( t) is approximated by a sequence of simple
functions of the form
N
x(t) = L kX(hk,OO)(t).
k=l
3. Operators in the Banach couple L}(O, (0), Loo(O, (0)
1. The spaces L} n Loo and Ll + Loo. The space Ll n Loo consists of all
bounded summable functions on (0, (0) with norm
IIXIILlnL = max{ ess suplx(t)l, oo Ix(t)1 dt}.
The set of strictly simple functions is dense in Ll n Loo. Indeed, let
x E Ll n Loo and x(t) O. We partition the interval [0, Ilxil L ] into N equal
00
parts and set
N-I k
x N ( t) = L N II xII LooXe k (t),
k=O
where e k = {t: (k/ N)llxllL x(t) < «k + 1)/ N)llxllL }. The condition
00 00
x E L} implies that mes e k < 00 for 1 k N - 1, and consequently the
functions XN(t) are strictly simple. It is obvious that Ilx - xNIIL oo <; llxllLoo
O as N 00. Next,
IIx - xNIIL = 1 00 [x(t) - xN(t)] dt 0
I 0
by Lebesgue's theorem, since x(t) - xN(t) x(t).
For any function x E Ll n Loo the assertion follows from the representa-
tionx = x+ - X_, wherex+ = max{x(t), O}.
The space Ll + Loo consists of functions which are sums of bounded
measurable and surnmable functions on (0, (0). The norm in this space is
defined by
/lX\\LI+Loo = inf (11u11 L + Ilvll L ).
x( t) = u(t) + v(t) I 00
uEL., vEL oo
(3.1 )
78
II. SPACES OF MEASURABLE FUNCTIONS
It is easy to see that L} + Loo consists of all locally summable functions
x(t) for which nlxl(T) 00. Indeed, if x E L} + Loo, then x(t) = u(t) + v(t),
where u E L} and v E Loo, and so x(t) is locally summable. Next, if T >
ess suplv(t)1 = c, then nlxl( T) nlul( T - c) < 00. Conversely, let x(t) be lo-
cally summable and let nlxl(T o ) < 00. Then for the set eo = {t: Ix(t)1 > TO} we
have mes eo < 00. Setting x(t) = x(t) o(t) + x(t)(1 - Xeo(t», we obtain that
the first term on the right belongs to L}, and the second to Loo, i.e.
x E L} + Loo.
An explicit fonnula can be given for the calculation of the nonn of a
function x(t) in the space L} + Loo. Let mes e = I. We have
IIxIlLl+L IIxXeIlLl+L = inf (lIull LI + IIvll L ).
u( t) + v(t) = x( t)Xe(t)
It is obvious that for the calculation of the infimum it is sufficient to take
functions u and v with supports on the set e, and then Ilvll L IIvll L .
I CIO
Therefore
IlxIILl+L inf(lIull LI + IIvII LI ) IlxXeIIL I .
By (2.14) we have
IIXIIL,+Loo sup II xx.,11 L, = 1 1 x*(s) ds.
me= I 0
(3.2)
For the proof of the reverse inequality we introduce the functions
v(t) = sgn x( t)min(lx(t)l, x*(I))
and u(t) = x(t) - v(t). Then lu(t)1 = [lx(t)1 - x*(I)]+, and since the func-
tions [lx(t)1 - x*(I)]+ and [x*(t) - x*(I)]+ are equimeasurable, we have
IlullL, = { )llx(s)l- x*(l)I+ ds = £1 [x*(s) - x*(l)] ds.
On the other hand, Ilvil L = x*(I), and consequently
IlullL, + IlvllLoo = I x*(s) ds. (3.3)
From (3.1)-(3.3) we obtain the basic fonnula
IIXIIL,+Loo = £1 x*(s) ds. (3.4)
This formula in particular implies that x*(t) E L} + Loo for any x E L} +
Loo.
In what follows we will be interested in the functional defined on L} + Loo
by the formula
K(t, x) = inf (lIullL + tllvll L ) (x E L} + Loo, L > 0).
x(s) = u(s) + v(s) 1
3. OPERATORS IN THE BANACH COUPLE LI(O, 00), Loo(O, (0) 79
For fixed t the functional K(t, x) is equivalent to the norm of the space
LI + Loo.
The expression for K(t, x) can be written in the fonn
K(t, x) = t inf [ (00 lu(to)ldo + sup I v(to)l ]
x(to)=u(to)+v(to) )0 0<0<00
and, by the above (see (3.4», we have
K(t,x) = tfo1 [x(ta)]*da = for x*(s)ds. (3.5)
Here we have used the obvious equality [x(to)]* = x*(to).
We note that the spaces LI n Loo and LI + Loo are not separable. Indeed,
in LI n Loo for the continuous family of functions X[O,'T] the distance between
any two different functions is equal to one. In LI + Loo the same property is
possessed by the continuous family of functions X u [1Itc,nk + I]' where {n k } is an
arbitrary increasing sequence of natural numbers.
We note that the set LI is not dense in LI + Loo. The closure LI in
LI n Loo consists of all locally summable functions for which nlxl( T) < 00 for
all T > O. Indeed, if x(t) has these properties, we write e k = {t: Ixl > 1/ k}.
Then mes e k < 00, and so the function x(t)Xek(t) belongs to LI. Next,
II x - xXek11L1+Loo Ilx - xXekllLoo Ilk
and hence x ELI. Conversely, if nlxl(T o ) = 00 (TO> 0) and y(t) is any
function in L I , then ly(t)1 > E on a set of finite measure, and so Ix(t) - y(t)1
TO - E on a set of infinite measure. By (3.4) we have Ilx - yilL +L TO -
_ I 00
E, i.e. x fl LI.
It is easy to see that Loo is dense in LI + Loo.
LEMMA 3.1. For every function x(t) from LI + Loo there exists a sequence of
sets en of finite measure such that the functions [x( t)Xe,. (t)]* converge to x*( t)
almost everywhere.
PROOF. We consider the sets e = {t: Ix(t)1 > x*(oo) + I/n}. Then
[x(t)Xe (t)]* = x*(t)X[O,mese ](t). If the measures of the sets e tend to 00, we
use them as sets en' and the assertion is obvious.
If mes e M < 00, then the set e = U e is of finite measure. The
function x*(t) is equal to x*(oo) on the interval (mes e, 00). We denote by fn a
set of measure n on which x*(oo) Ix(t)1 > x*(oo) - I/n holds, and write
en = e U In. Then the function [x(t) (t)]* coincides with x*(t) on [0, mes e],
and differs from it by less than I/n on the interval [mes e, mes e + n].
REMARK. If we replace the system {en} of sets by a larger system {e }
(e ::) en)' then the lemma remains valid for the system {e }.
80
II. SPACES OF MEASURABLE FUNCTIONS
The spaces Ll n Loo and Ll + Loo are dual in the sense that for x E Ll +
Loo andy E Ll n Loo we have
It 00 x( t )y( t) dtl II xii L, H..,IIyll L, n L.,.
Indeed, if x = u + v, where u E Ll and v E Loo, then
(3.6)
I{>O x(t)y(t) dtl IlulIL,llyllL., + IIVIIL.,IIYIIL,
IIYIILlnL (lIuIILI + IlvIlL ).
Inequality (3.6) then follows from the definition of the norm in Ll + Loo.
Next,
sup 1 1 0 00 x(r)y(t) dt l = IIYIILlnL .
IIxllLl+Leo <;; I
(3.7)
Indeed,
sup 1 1 0 00 x(t)y(t) dr l
Ilxll L. +Leo <;; I
max [ sup I L x(t)y(t) dt J , sup 1 1 0 00 x(t)y(t) dt l]
\lxll L. <: I 0 Ilxll Leo <;; I
= max[ IIYIIL 1 ' IIYIIL J = IIYIILlnL . (3.8)
Formulas (3.6) and (3.8) imply (3.7).
Similarly,
sup I l oo x(t)y(t) dt l = IlxIILI+Loo.
IIYIILlnL eo <; I 0
Indeed, (3.4) and (2.14) imply that
IIXIIL,H., = 1 1 Ix(t)l* dt f Ix(t)1 dt + f,
o ef
where ef is a set of measure 1. Then
(3.9)
sup 1 1 00 x(t)y(t) dt l loo x(th ef(t)sgn x(t) dt IIxIIL,H., - f.
IIYIIL.nLeo<;1 0 0
From what was said on p. 45 and from (3.7) and (3.9) it follows that the
spaces Ll n Loo and Ll + Loo are associated with each other.
2. Averaging operator. Let the semi axis (0, 00) be partitioned into a counta-
ble number of measurable sets e; (i = 1, 2, . . . ) of finite measure. From the
3. OPERATORS IN THE BANACH COUPLE LI(O, 00), LOt:)(O, 00) 81
system d = {e i } we construct an averaging operator defined for every
locally summable function by
00 1 00
T,6.x(t) = L f x(-r) d-rXe,( t) = L xjXe, (t). (3.10)
; = I mes e i e; ; = I
It is obvious that T (t) = Xe(t). The operator T is bounded in Loo and
, ,
has norm equal to one. Next, for x E LI we have
oc
II T ,6.XIiL. =.L f x( 7") d-r "q xii L,.
z = I e,
This inequality becomes equality for the functions Xe (t). Therefore II T II L
, I
= 1. From what was said it follows that T is bounded in LI + Loo, and
II T IIL +L 1. By (3.4) for the norm in LI + Loo, this is equivalent to the
1 00
inequality
r 1 (T,6. u ) * ( 7") d7" ..; L 1 U * ( 7") d7".
)0 0
Applying the same reasoning to the norm in LI + Loo determined by the
functional K( t, x), from (3.5) we obtain
It (T,6.u )*( 7") d7" ..; l t u*( 7") d7". (3.11)
3. An additional property of rearrangements. We consider the finite-dimen-
sional Banach space En of vectors x = ( I' . . . , n). We shall say that this
space is symmetric if the norm of a vector does not change under arbitrary
permutations and changes of sign of its components.
I f x = ( I' . . . , n) and y = (11 I' . . . , 11n)' and 1 i 1 111 i 1 (i = 1, . . . , n),
then the point y belongs to the parallelepiped with vertices at the points
( + I' . . . , + n). The radius vectors of these vertices have the same norm
equal to IIxll. Therefore, all points of the cube have norm not exceeding II xii,
and, in particular, lIyll Ilxll.
By x* we denote the vector with components * I = I i I, . . . , : = I i I,
1 n
where (i l ,..., in) is a permutation of the indices (1,..., n) such that
1 i 11 1 i21 . . . I iJ .
LEMMA 3.2. In a space with symmetric norm
Ilx* - Y*II II x - yll.
(3.12)
PROOF. The left and right sides of the inequality will not change if we carry
out the same permutations and changes of sign of the components in both x
and y. Therefore in the proof we may assume that x = x*. Under this
assumption, by the mono tonicity property of the norm described above, we
82
II. SPACES OF MEASURABLE FUNCTIONS
have Ilx - Iylll IIx - yll, Iyl = (11]11, . . . , l1]nl). Consequently, (3.12) will
be proved if we establish it for vectors y with nonnegative components. Thus,
let x = x* andy O. Let 1]; < 1lj for some i and} with i <i. This means that
the points x andy lie on different sides of the plane I with equation ; = (x
can lie on this plane). We consider the pointy symmetric to y with respect to I
and the point z symmetric to y with respect to the plane parallel to I and
passing through x. It is obvious that y lies on the line segment connecting y
and z. Next, II x - y II = II x - z II in view of the symmetrici ty of the space,
and consequently Ilx - yll Ilx - yll. Thus, the transposition of a pair of
components of y in decreasing order does not increase Ilx - yll. Applying
this property successively, we arrive at (3.12).
'We leave it to the reader to prove that the quantity
I
Ilxll = ;*
;= I
has the properties of the norm of a symmetric Banach space for every fixed
I n (see (2.16)). Then the lemma will imply the inequality
I I
( ;* - 1];*)* ( ; - 1];)*
;=1 ;=1
(1 I n).
(3.13)
We now return to rearrangements of functions.
THEOREM 3.1. For any functions x, yELl + Leo
{r [x*(t) - y*(t)]* dt < t T [x(t) - y(t)]* dt.
(3.14)
PROOF. First we establish the following auxiliary assertion: Assume that for
the sequences of pairs of functions xn(t), Yn(t) E LI + Leo we have already
shown the validity of (3.14). If a) x:(t) x*(t) and y:(t) y*(t) almost
everywhere, and b)
t T [xn(t) - Yn(t)]* dt < t T [x(t) - y(t)]* dt,
then (3.14) will be true for the two functions x and y, as well. Indeed, (3.14)
and condition b) imply that
t T [x:(t) - y:(t)]* dt < t T [x(t) - y(t)]* dt.
From property 7 0 it follows that for any e with mes e = T we have
f Jx:(t) - y:(t)1 dt < iT [x(t) - y(t)]* dt.
e 0
3. OPERATORS IN THE BANACH COUPLE L1(0, 00), Loo(O, 00) 83
By Fatou's lemma, this inequality and property a) imply that
f Ix*(t) - y*(t)1 dt " i T [x(t) - y(t)]* dt.
e 0
Taking the supremum over e on the left side, from 7 0 we obtain (3.14).
The above assertion enables us to enlarge the class of functions for which
(3.14) holds.
We first consider step functions of the form
n n
x( t) = XkX[(k -l)a, ka)( t) and y( t) = YkX[(k -l)a,ka)( t).
k=l k=l
For such functions we have
n n
x*(t) = X:X[(k-l)a,ka)(t) and y*(t) = Y:X[(k-l)a,ka](t).
k=l k=l
Inequality (3.13) immediately leads to the inequality
(W l w
)n [ x*( t) - y*( t) ] * dt [ x( t) - y( t) ] * dt
o 0
The functions of T on the left and right sides of (3.14) are piecewise linear
in our case, and so (3.15) implies (3.14).
Now let x,y E L1(0, 00). We introduce the step functions
(0 I n). (3.15)
2
n k/n
xn(t) = n f X(S) dsX[(k-I)/n, k/n)(t) = Tnx(t),
k=l (k-l)/n
2
n k/n
Yn( t) = n f y(S) dsX[(k-I)/n. k/n)( t) = T,.Y( t).
k= I (k-l)/n
By the above, (3.14) holds for these functions. We verify that the sequences
X n and Yn satisfy conditions a) and b). By the properties of the integral of a
summable function the sequences Tnx(t) and T ,.y(t) converge to x(t) and y(t)
almost everywhere. Since x and y belong to L 1 , the indicated sequences
converge in measure. By property 11 0 of rearrangements this implies that
x:(t) x*(t) and y:(t) y*(t) at all points of continuity of x*(t) and y*(t),
and consequently, almost everywhere. Property a) is satisfied. Property b)
follows from (3.11):
L r [xn(t) - Yn(t)]* dt = I T [Tn(x - y) ]*(t) dt" (T (x - y)*(t) dt.
o 0 )0
Now let x, yELl + Loo. According to Lemma 3.1 we construct the sets en
andh for x(t) andy(t), respectively, and then take their unions en = en U In.
The functions xn(t) = x(t)x:e (t) and Yn(t) = y(t)Xe (t) belong to L1(0, 00).
n n
84
II. SPACES OF MEASURABLE FUNCTIONS
Next, by the remark to Lemma 3.1, we have x:(t) -=) x*(t) andy:(t) -=) y*(t)
almost everywhere. Finally, by property 2 0 of rearrangements we have
T [xn(t) _ yn(t)] * dt = T [(x - y)(t)xe.(t)]* dt " T [ x(t) - y(t)] * dt.
Properties a) and b) are satisfied. The theorem is proved.
4. The contraction semigroup operators. We consider the set L of linear
operators in L) + Loo for which the restrictions to Ll and Loo act in each of
these spaces and are con tractions:
L = {T: II TII Ll Ll 1, 11111 L --+L I}.
In accordance with the notation of the first chapter, L = 'lT1(LIL 1 , LooLoo);
the set L is convex and forms a semigroup with respect to the composition of
operators.
The operators in L have the following important property:
9 (Tx)*(s) ds " 9 x*(s) ds
(0 < 0 < 00).
(3.16)
Indeed, by (3.5) this inequality is equivalent to K(O, Tx) K(O, x). The
functional K(O, x) is the norm in the sum of the spaces Ll and OL oo (with
norm 0 II xii L ), and the above inequality follows from the fact that this sum is
00
an interpolation space between L) and OL oo with interpolation constant 1 (see
Lemma 4.2 in Chapter I).
It is easy to see that, conversely, (3.16) implies that TEL.
We give examples of operators in L. It is obvious that L contains the
operators of multiplication by a measurable function not exceeding one in
absolute value.
\
By (3.11) the set L contains all averaging operators.
Finally, L contains the operators of the form Tx(t) = x(w(t), where w(t) is
a transformation of (0, 00) into itself that does not decrease measure.
We introduce an operator V which. s the adjoint of the restriction of
TEL to L): V = (TIL I )*. The operator V acts in Loo and has norm 1.
We show that V can be continuously extended from LI n Loo to a contrac-
tion in L 1. By definition,
oo Tx(t)y(t) dt = oo x(t) Vy(t) dt
(3.17)
for all x E L) andy E Loo.
3. OPERATORS IN THE BANACH COUPLE L1(O, 00), Loo(O, (0) 85
For yELl n Loo we have
II VYIIL 1 = loo I Vy(t)1 dt
- sup L o oo X[O,Nj(t) sgn Vy(t) Vy(t) dt
O<N<oo
- sup L o oo T[Xio,Nj(t)Sgn Vy ] (t)y(t) dt
O<N<oo
II TIIL --+L I IILI IlyIIL J .
Since Ll n Loo is dense in L 1 , the last inequality implies our assertion.
The operator V can be extended to Ll + Loo by linearity. Then it becomes
a member of L and is denoted by TO.
If x E Ll n Loo, on passing to the limit we obtain from (3.17) that
(00 Tx( t)y( t) dt = L 00 x(t) T«y(t) dt
)0 0
(x E Ll n Loo,y E Ll + Loo).
(3.18)
Any operator of multiplication is selfadjoint in the sense that T = TO. The
averaging operators have the same property, which follows from the equality
1 00 Tx(t)y(t) dt = L ] f x(s) ds f y(s) ds.
o mes e J " e. e.
J J
We can give an example of a nonzero operator Tl E L for which T = O.
On Loo we construct a linear functional fo(x) with norm equal to one,
vanishing on Ll n Loo and not identically zero. Then the operator T1x =
fo(x)x(O,OO) belongs to L (on Ll the operator Tl is identically zero). Then it
follows from (3.18) that
loo x(t) TPy(t) dt = 0
for all x E LI andy E Loo, and so T y = O.
From this example it follows in particular that Too =1= T in general.
By LO we denote the collection of all operators TEL for which Too = T.
For the operators T E LO it follows from (3.18) that
(00 x(t) T«y(t) dt = (00 TOOx(t)y(t) dt = L oo Tx(t)y(t) dt
)0 )0 0
(x E Ll + Loo'Y E Ll n Loo).
86
II. SPACES OF MEASURABLE FUNCTIONS
By (3.18) the equality
oo Tx(t)y(t) dt = oo x(t) T«y(t) dt
(x E Ll + Loo,Y E Ll n Loo) (3.19)
is a characteristic property distinguishing LO in L.
In Ll + Loo we may introduce the weak topology O(LI + Loo, Ll n Loo).
An operator TEL belongs to LO if and only if it is continuous in this
topology. Indeed, if T E Lo and X n converges to x weakly, then for yELl n
Loo we have
lim oo Txn(t)y(t) dt = lim oo xn(t)T«y(t) dt
= oo x(t)T«y(t) dt = oo Tx(t)y(t) dt.
Conversely, if TEL is continuous, then, using the fact that for any
x E Ll + Loo we have x(t)X[O,N](t) E Ll and this sequence converges to x(t)
in the topology O(LI + Loo, Ll n Loo) as N 00, from (3.17) for yELl n
Loo we obtain
oo Tx(t)y(t) dt = lim oo TXN(t)y(t) dt
= lim oo xN(t)T«y(t) dt = oo x(t)T).(t) dt,
which proves that T belongs to LO.
From the last property it follows in particular that LO is a semigroup with
respect to composition of operators.
Let w(t) be a one-to-one measure preserving mapping of the semiaxis into
itself. We show that the operator Twx = x(w(t» belongs to Lo. For this we
denote by S the image of (0, 00) under the mapping w. It is easy to verify that
for any functions x E Ll + Loo andy E Ll n Loo we have
1 00 x(w(t»y(t) dt = f X(T)y(w-l(T)) dT.
o S
If we introduce the function
z( 1') = { (w -1( 1'»)
for T E S,
for T fl S,
we obtain
oo x(w(t»y( t) dt = 00 x( 1')z( 1') dr.
3. OPERATORS IN THE BANACH COUPLE Ll (0, 00), Loo(O, 00) 87
It is obvious that z E Ll n Loo. Therefore from the weak convergence of
X n to x it follows that xn(w(t» converges to x(w(t» weakly, and consequently
Tw E LO.
THEOREM 3.2. If u E Ll n Loo and v E Ll + Loo, then
sup L oo Tu(t)v(t) dt = sup loo u(t) Tv(t) dt = L oo u*(t)v*(t) dt. (3.20)
TELo 0 TELo 0 0
PROOF. By property 13 0 , (3.16), and property 18 0 of rearrangements we
have
L oo Tu(t)v(t) dt (00 (Tu)*(t)v*(t) dt (00 u*(t)v*(t) dt. (3.21)
o )0 )0
Similarly,
tOO u(t)Tv(t) dt tOO u*(t)v*(t) dt.
(3.22 )
We prove that
sup 1 00 Tu(t)v(t) dt L oo u*(t)v*(t) dt. (3.23)
TELo 0 0
Since LO contains the operators of multiplication by a function not exceed-
ing one in modulus, we may assume that u(t) 0, v(t) 0 and Tu(t) O.
According to Lemma 2.1, we construct operators TWI and T W2 belonging to
LO such that
Tw 1 U ( t) = u*(t) + a(t), T W2 v(t) = v*( t) + {3(t)
and II a II Ll n L«J < E, 11,811 Ll n L«J < E. Then
(00 T2 Tw u(t)v(t) dt = (00 Tw u(t)Tw v(t) dt = (00 u*(t)v*(t) dt + y,
)0 2 1 )0 1 2 )0
where
Iyl IlaIIL1nL«Jllv* + ,8II L I+L«J +lluIILlnL«JII,8IILI+L«J
E( II ull Ll n L«J + II vii Ll + L«J + e) 0 (3.24)
as E O.
Thus
sup i oo Tu(t)v(t) dt i oo T22T""u(t)v(t) dt
TELo 0 0
tOO u*(t)v*(t) dt +Irl,
from which (3.23) follows by (3.24).
88
II. SPACES OF MEASURABLE FUNCTIONS
Similarly, by means of the operator T21T"'2 we may prove the inequality
sup 1. 00 u(t) Tv(t) dt 1. 00 u*(t)v*(t) dt,
TE 0 0
which, together with (3.21) and (3.22), gives (3.20).
With every operator TEL we associate the function
Pr(x,y) = oo Tx(t)y(t) dt
on the product Loo X (L 1 n Loo). It is obvious that
\Pr(x, y)1 II TxIIL llyIILI IlxIIL llyIILI.
(3.25)
It can be verified easily that the mapping T Pr is one-to-one.
By means of the functions Pr we introduce a topology on L in which a
basis of neighborhoods of the point To is determined by all possible finite
collections of functions {Xi} E Loo and {y;} E Ll n Loo. A neighborhood
consists of all those T for which IPr(x;, Yj) - Pro(x;, Yj)1 < 1. In this topology
the mapping T Pr is a continuous imbedding of L in the compactum Q
which is the product of the intervals Ixy = {a: lal IlxIIL IIYIILI} (x E Loo,
yELl n Loo).
We show that the image of L is closed, whence it will follow that it is
compact. Let p(x,y) be a limit point of the image of L. Then p(x,y) is a
bilinear functional of x and y. From (3.25) for all TEL we obtain the
inequality
Ip(x,Y)1 \llxIIL \llyIILI.
(3.26)
Thus, p(x, y) is bounded in Ll for fixed x E Loo and so it can be written in
the form
p(X,y) = oo i(t)y(t)dt,
where z E Loo. We set Tx = z. From the linearity of p(x, y) in x and the
-
density of Ll n Loo in Ll it follows that the operator T is linear. The set
Ll n Loo as a subset of L'oo is normative (see Chapter I, 2) for Loo. Therefore
(3.26) implies that II TilL 1.
eX)
Since for the functionals pr<x, y) for x E Loo n Ll we have
IPr(x,y)1 IlxIILlllyIIL ,
the same inequality holds for p(x, y) as well, i.e.
I oo Tx(t)y(t) dtl "llxIIL,IIYIIL", (x E LOC) n L),
whence II Txll LI IlxlI LI . Since Ll n Loo is dense in Lt, the operator f
admits a continuous extension to the whole space L 1 , and then to Ll + Loo
3. OPERATORS IN THE BANACH COUPLE L}(O, 00), Loo(O, 00) 89
by linearity; here IITIILI 1. Thus, T E . Consequently p(x,y) = pr<x,y),
and the image of L is closed in Q.
We denote by Ox the orbit of the point x E L} + Loo with respect to tile
semigroup L: Ox = {y: y = Tx, TEL}.
THEOREM 3.3. The orbit of any point x E L} + Loo is closed in the weak
topology a(L} + Loo, L} n Loo).
PROOF. Let x E L} + Loo. For the proof of the assertion it suffices to show
that the mapping T Tx of the compactum L into L} + Loo is continuous.
Let Ta To in L. Then
Pr (z, y) = 1 00 Taz(t)y(t) dt l oo T oZ(t)y(t) dt
a 0 0
for any z E Loo and y E L} n Loo. We choose a sequence Zn E Loo so that
Zn x in L} + Loo. Then
I{'" Tax(t)y(t) dt - oo T oX(t)y(t) dtl
..;; I oo Ta(x _ zn)y dtl +I oo To(x - zn)y dtl
+ I oo (Tazn - T oZn)Y dtl
..;; 211 x - ZnIlLt+LooIIYIIL,nLoo +I oo (Tazn - ToZn)Y dtl.
Choosing n and then a sufficiently large, we obtain
1 00 Tax(t)y(t) dt 1 00 ToX(t)y(t) dt
o 0
for any y E L} n Loo.
In what follows, the next theorem plays an important role.
THEOREM 3.4. Let x E L} + Loo. A measurable function y belongs to the
orbit Ox if and only if
9 y*(t) dt ..;; 9 x*(t) dt (0 < () < 00). (3.27)
PROOF. The necessity of (3.27) immediately follows from (3.16).
We prove the sufficiency indirectly. For the function y(x) let (3.27) be
satisfied and assume that y fl Ox. We observe that (3.27) implies that y E L}
+ Loo. The orbit Ox is convex and closed in the weak. topology
a(L} + Loo, L} n Loo).
90
II. SPACES OF MEASURABLE FUNCTIONS
By general duality theorems (see [6], Chapter II) the space LJ n Loo
is dual to the locally convex linear space LJ + Loo with the topology
a(LI + Loo, Ll n Loo). Using the separability theorem ([6], loc. cit.), we may
assert that there exists a function a(t) E LJ n Loo such that
(00 a(t)y(t) dt > sup 1 00 a(t) Tx(t) dt.
J o TEL 0
From (2.25) and Theorem 3.2 we obtain
oo a*(t)y*(t) dt oo a(t)y(t) dt > oo a*(t)x*(t) dt.
(3.28)
By property 18 0 of rearrangements, inequality (3.28) contradicts (3.27).
4. Symmetric spaces. Interpolation between Ll and Loo
1. Definition of symmetric spaces. A Banach function space on the semiaxis
(0, 00) with Lebesgue measure is said to be symmetric if it satisfies the
following conditions:
1) If y E E and Ix(t)1 ly(t)1 almost everywhere on (0, 00), then x E E
and IlxilE IlyliE (i.e., E is an ideal lattice).
2) If y E E and Ix(t)1 is equimeasurable with ly(t)l, then x E E and
Ilxil E = IlylIE.
Conditions 1) and 2) are equivalent to the following single condition.
I') Ify E E and x*(t) y*(t) for all t E (0,00), then x E E and IlxllE
II y II E.
Let the Banach function space E satisfy condition 1). It contains at least
one function xo(t) O. Then for some e > 0 there exists a set eo of positive
measure mes eo = J-Lo, such that Ixo(t)1 e for t E eo. This implies that e o(t),
and hence o(t) as well, belongs to E.
LEMMA 4.1. If an ideal lattice E (condition 1» contains all functions (t) with
mes e = JLo and IIXeII E C, where C does not depend on the choice of e, then
the space Ll n Loo is imbedded in E.
PROOF. Let x E Ll n Loo. Without loss of generality we may assume that
x( t) O. Write "\ = x*UJLo) U = 0, 1, . . . ). There exist disjoint sets e j such
thatmes = JLo and "\+1 x(t) \ U = 0,1,...). If
00 00
x(t) = .L "\ (t) and (t) = L "\+J (t),
J=O j=O
then (t) x( t) x( t). Hence
00
ILo L Aj+ I = II IILI IlxllLI < 00.
j=O
( 4.1 )
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN Ll AND Loo 91
We consider the functions XN(t) = (t). The sequence X N is funda-
mental in E. Indeed,
N+p N+p
Ilx N + p - xNll E \II II c >y O
j=N+l E j=N+l
as N 00, because the series (4.1) converges. Since X N x in 8(0, 00), we
have X N x in E, and consequently x E E. By 1) we also have x E E and
IlxilE (; IlxilE (; Cj O l} = c( Ao + j O A j + 1 )
(; C(IIX11Loo + IIXIIL.) (; c( 1 + )llxllL,nLooo
REMARK. The hypothesis of the lemma and 1) imply that the space E
contains the characteristic function of any measurable set of finite measure.
Therefore we may set JLo = 1, and the preceding inequality takes the form
IlxIIE 2 sup IIXeIIIIXIILlnL .
roes e = 1
THEOREM 4.1. Any symmetric space is intermediate between the spaces
L1(0, 00) and Loo(O, 00), i.e. Ll n Loo C E C Ll + Loo.
PROOF. The space E obviously satisfies the hypotheses of Lemma 4.1, and
IlXe II E = Ilx(o,I)11 E if mes e = 1. Therefore Ll n Loo is imbedded in E, and
IlxIIE 21Ix(O,I)IIEllxIILlnL . (4.2)
We pass to the proof of the second imbedding. Let x E E and x( t) =
x*(t). By (3.4) we have
IIXIILooH. = l x(t) dt,
(4.3)
and therefore it is sufficient to estimate the norm in Ll of functions belonging
to E with support in [0, 1]. First we consider a step function yo(t) on [0, 1],
assuming the values Xk 0 on sets ek (k = 0, 1, . . . , N - 1) in [0, 1] of the
same measure mes e k = 1/ N. Then
N-l
Yo(t) = xkXe k (t).
k=O
We set
N-l
Yj(t) = Xk+j(mod N) Xe k (t).
k=O
92
II. SPACES OF MEASURABLE FUNCTIONS
It is obvious that the functions Yo(t) and Yj(t) are equimeasurable. There-
fore IIYjllE = IIYoIIE. Next,
N-I N-I
Yj(t) = xkX[O,l]'
)=0 k=O
and hence
( N-I )
k O x k Ilx[o,IJIIE =
N-I
Yj
)=0
N-I
IIYjllE = NIIYoIIE.
E )=0
From this we obtain
1 N-I
IlYoIIE Ilx[o,l]IIE N L x k = IIx[o,IIIIEIIYoIIL I "
k=O
An arbitrary decreasing nonnegative function z E E which is equal to zero
for t > 1 can be approximated by an increasing sequence of step functions
( t ) N l ( k+l ) ( t ) ( N=2n ),
ZN = k O Z N X[k/N,(k+I)/N]
which converges to z(t) almost everywhere. Since ZN(t) z(t), we have
lim N -+ oo IlzNllE IlzilE. On the other hand, by B. Levi's theorem we have
IlzllL I = lim N -+ oo IlzNIILI. Applying (4.4) to the functions ZN and using the last
relations, we obtain
(4.4)
1
IlzIIL , ' II II Ilzlk
X[O,I] E
Combining this with (4.3), we arrive at the final inequality
1 1
II xii L", HI ' II II II XX[O, IJII E' II II II xii E'
X[O,I] E X[O,l] E
Thus, E C Loo + L I . The theorem is proved.
REMARK. In the sequel we need the inequality
Ilx*x[O,hIIIL I ' IIx h II Ilx*x[O,hIIIE"
[O,h] E
( 4.5)
( 4.6)
LEMMA 4.2. The set of elementary * functions is dense in any symmetric space
E.
PROOF. By Theorem 4.1 every function x E E can be represented as the
sum of a summable and a bounded function, and so its restriction to any
· Editor's note. The Russian term used by the author is CQeTH03Ba'IBa.Sl (literally: countably
valued). The term adopted for the translation is that used by Halmos in [10].
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN LI AND Loo 93
finite interval belongs to LI. Then the function xX(n-l,n] E L(n - 1, n) can be
uniformly approximated by an elementary function zn(t) (n - 1 t n) of
smaller modulus with accuracy up to e/2n. This means that if zn(t) is
extended to be equal to zero outside (n - 1, n], then
IlxX(n-l,n] - znllLlnL < e/2 n . (4.7)
Then z(t) = zn(t) is elementary, and Iz(t)1 Ix(t)l. Therefore z E E.
Next, by (4.7) we have
00
Ilx - zlILlnL IlxX(n-l,n] - znllLlnL e. (4.8)
1
By (4.2) we have Ilx - zilE 21Ix(O,I]IIE e .
LEMMA 4.3. Under condition 1) of the definition of a symmetric space,
condition 2) is equivalent to the following: For any mapping t w( t) of (0, 00)
into itself which does not increase measure and for any y E E we have
y w(t) = y(w(t» E E and
IIYwllE IlylIE.
( 4.8')
PROOF. As was shown above, condition 1) implies that E contains a
function of the form ko(t) with mes eo = JLo > O. If e l is another measurable
set of the same measure, then we can construct a mapping w(t) of (0, 00) onto
itself which maps e l onto eo and preserves measure. Then lt) = o(w(t»,
and therefore k l E E and IIklll E IIXeol1 E. Thus, the hypotheses of Lemma
4.1 are satisfied, and LI n Loo is imbedded in E.
Now let y E E and let Ix(t)1 be equimeasurable with ly(t)l. Then by
Theorem 2.1 for any e > 0 there exist a measure preserving mapping w(t)
of (0, 00) into itself and a function {3( t) with I {3( t)1 1 such that
Ilx - 13YwIILlnL < e. By Lemma 4.1 and (4.2) we have x - 13Y w E E and
II x - ,By wll E 21Ix[O,I]11 Ef.
By property 1) and the hypothesis of the lemma we have f3y wEE, and
hence x E E and
IlxilE II{3YwIlE + 21Ix[o,1]IIEf
I wilE + 21Ix[O,I]IIEf IlyliE + 21Ix[o,1]11 E e .
Since e is arbitrary and x and y occur symmetrically, we obtain that
Ilxil E = IlylIE.
Conversely, if w(t) is a measure nondecreasing mapping of (0, 00) into
itself, then y:(t) y*(t). In a symmetric space the condition y E E then
implies that y wEE, and (4.8') is satisfied. The lemma is proved.
94
II. SPACES OF MEASURABLE FUNCTIONS
A linear subset F of a symmetric space E is said to be symmetric if the
condition that Y belongs to F implies that x belongs to F whenever x*(t)
y*(t).
LEMMA 4.4. The closure in E of a symmetric linear subset F is a symmetric
space with respect to the norm of E.
PROOF. Let Y E F, and let the sequence Y n E F converge to y in E. If
Ixl < Iy!, then the sequence min(lxl, IYnD < IYnl belongs to F, and by prop-
erty 4° of ideal lattices (see p. (0) it converges to Ixl in E; consequently
sgn x . min(lxl, IYnl) E F, and this sequence converges to x in E, i.e. x E F.
Thus, property 1) is satisfied. If now wet) is a measure nondecreasing
transformation of the semiaxis into itself, then II(Yn)w - YwllE llYn - yilE
O. Since (Yn): < Y: E F, we have (Yn)w E F, and thus y w E F. On the
basis of Lemma 4.3 this implies property 2) for the space F.
LEMMA 4.5. The intersection and the sum of two symmetric spaces are
symmetric spaces.
PROOF. Let EI and E 2 be symmetric spaces. The fact that El n E 2 is
symmetric is obvious. Consider the sum E = E 1 + E 2 . Let Y E E and x*( t)
< y*(t). By Theorem 2.1, for every € > 0 there exist a measure preserving
transformation of [0, (0) into itself and a measurable function {3( t) with
1 {3(t)1 < 1 such that
x(t) = {3(t)y(w(t» + z(t),
where IlzlILlnLoo < €. If now yet) = u(t) + vet), u EEl' V E E 2 , then x(t) =
{3(t)u(w(t» + l3(t)v(w(t» + z(t), where by Lemma 4.3 we have l3u w E El and
I3v w E E 2 . From this it follows that x E E and
IlxilE < II{3u + zllE I + II{3vIIEI < II ullEI + II VllE2 + 21Ix[o,IJIIE 1 €.
Taking the infimum over all representations Y = u + v, u EEl' V E E 2 ,
we obtain that IlxilE < IlyIIE + 211X[0,I]IIE€. Since € is arbitrary, this means
that II x liE < II Y II E' and so property I') is satisfied in E 1 + E 2 .
THEOREM 4.2. Let Eo and EI be two symmetric spaces, and let E be an
interpolation space between them with interpolation constant one, i. e.
II TJIE E < max(lll1IEo Eo' 111lIEI E)
for all T E 1T( Eo, E I; Eo, E I). Then E is a symmetric space.
( 4.9)
PROOF. Let a(t) be a function with la(t)1 < 1 and w(t) a measure nonde-
creasing mapping of (0, (0) into itself. Property 1) and Lemma 4.3 imply that
the operator aT wY(t) = a(t)y(w(t» acts in Eo and E 1 , and its norm there does
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN Ll AND Loo 95
not exceed one. Then it acts in E as well, and IlaT cYllE lIyllE. For w(t) = t
this implies that condition 1) is satisfied in E, while for a = 1 condition 2) is
satisfied by Lemma 4.3.
REMARK. If instead of (4.9) we have
111l1E E c max(lll1IEo Eo' 1111IEI EI)' (4.10)
then from what was said on p. 20 it follows that we may introduce an
equivalent norm in E with respect to which E becomes symmetric.
2. The nuun interpolation theorem. If E is an interpolation space between Ll
and L 00 and
11111E E max(lll1ILI LI' "l1'L L ),
( 4.11 )
then E is symmetric by Theorem 4.2.
In what follows we need
LEMMA 4.6. In every interpolation space E between Ll and Loo for which
(4.11) holds',
Ilx* - y*IIE Ilx - YIIE
(x,yEE).
Indeed, (3.14) for any functions belonging to Ll + Loo and Theorem 3.4
imply that x* - y* = T(x - y) for some Tin L. By (4.11) we then have
Ilx* - y*IIE = II T(x - y)IIE 11111E Ellx - yilE Ilx - yiIE.
The lemma is proved.
There arises the problem of describing all symmetric spaces that are
interpolation spaces between Ll and Loo and have the property (4.11). The
answer to this question is given by the following theorem.
THEOREM 4.3. A Banach space E intermediate between Ll and Loo is an
interpolation space with interpolation constant one if and only if the following
condition is satisfied: If y E E, x E La + Ll and
t X*(T) dT " t Y*(T) dT (4.12)
for all t 0, then x E E and IIxll E IlylIE.
PROOF. Necessity. If the functions x and yare connected by the relation
(4.12), then by Theorem 3.4 there exists an operator TEL such that x = Ty.
Then the interpolation property of E implies that x E E, and (4.11) implies
that
IlxilE = II TyllE 11111E Elly11E IIYIIE.
96
II. SPACES OF MEASURABLE FUNCTIONS
Sufficiency. Let T be an arbitrary operator in Land lety E E. By (3.16) we
have
t (Ty) *(-r) dr " t Y * ( r) dr.
From the property of E we have assumed to be satisfied it follows that
Ty E E and II TyllE IlylIE.
In connection with Theorem 4.3 it is natural to ask whether there exist
symmetric spaces which are not interpolation spaces between Ll and Loo. It
turns out that such spaces exist; they will be given in 5.7.
REMARK. If E is such that (4.12) implies that IlxilE CllyllE' then an
equivalent norm can be introduced in E by the formula Ilxlik = supllullE
where the supremum is taken over all u E Ll + Loo for which
1 t U * ( r) dr " 1 t X * ( 'T) d'T.
o 0
The space E with the new norm satisfies the hypotheses of Theorem 4.3,
and so it is an interpolation space between L I and Loo with property (4.11).
Since IlxllE Ilxlll- Cllxll E , inequality (4.10) is satisfied for the original
norm in E with the same constant C.
3. Dilation operators in a symmetric space. In the space S(O, 00) we consider
dilation operators defined by
<11"x(t) = x('T-It) ('T > 0).
It is obvious that the operators <11" are continuous in S(O, 00). These operators
form a group with unit element equal to <1 1 = I. It is easy to see that the
operators <11" commute with the operation of rearrangement: (<11"x)*(t) =
<11"( x*)( t) = x*( 'T - It).
THEOREM 4.4. The operators <11" are bounded in any symmetric space E.
PROOF. If 'T I < 'T 2' then x*( 'T 1- It) x*( 'T 2- It), and consequently (<11" IX )*( t)
( <11"2 X )*( t). This implies that for x E E the function II <11" X II E is increasing in 'T.
In particular, for 'T 1 the operator <11" is bounded in E and II <11"11 E 1
('T 1).
Now let T > 1 and x E E. By what was said earlier, to prove the bounded-
ness of the operators <11" it suffices to prove the boundedness of the operators
<1m' m = 2, 3, . . . . We first assume that the function x E E is elementary,
and
00
x(t) = L xjXe.(t),
1 :I
( 4.13)
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN L} AND Loo 97
where e; n = 0 for i =1= j, and mes < 00. We divide every set into m
disjoint parts l' . . . , e jm of equal measure and set
00
ys(t) = L Xj s(t).
j=l
Then
m
x( t) = L Ys( t) and (<1mYs)*( t) = x*( t).
1
Therefore
m
II<1 m xII E L lI<1mYsllE = mllxllE.
1
( 4.14)
Since the elementary functions are dense in E and the operators <1" are
continuous in S(O, 00), inequality (4.14) can be established for any x E E by
a passage to the limit.
THEOREM 4.5. The function II <1,,11 E is submultiplicative and quasiconcave.
PROOF. The first assertion is obvious:
II <1"1"211 E II <1"111 Ell <1"211 E.
To prove quasiconcavity we again consider a function of the form (4.13) in
E and construct the functions
00
zs( t) = L X } "Xe.\e. (t).
"J "JS
j=l
Then zs(t) is equimeasurable with <1(n-I)/nX(t), and
n
L zs(t) = (n - l)x(t).
1
Hence
n
(n - 1)ll x llE = L Zs
1
n
L IlzsllE = nll<1(n-I)/nXIIE.
E 1
( 4.15)
The inequality can be established for all x E E by a passage to the limit.
Let y be any function in E. We set x = <1 n /mY. Then from (4.15) we obtain
that
m m
-;11<1 n /mYIIE n _ 1 11 <1(n-I)/mYIIE.
This implies that for any rational numbers
1 1
-;; II <1 r "" E II <1r II E'
r r
98
II. SPACES OF MEASURABLE FUNCTIONS
if r' < r". For arbitrary TI and T2 with TI < T 2 we construct sequences of
rational numbers r iTI and r'; J..T2. Then
1 r" 1
II oT211 £ < Tn r" II Or '11 £
2 2 n
r ' 1 r; T I 1
< 711 0 11£ < -;:' -:;:-11 °T.II £.
2 n 2 n I
The passage to the limit as n 00 gives the inequality
1 1
II oT211 £ < -:;:-11 °T.II £.
2 I
COROLLARY 1.
IIOTII£ < max{ 1, T}
(T > 0).
( 4.16)
This follows from the fact that 110 1 11£ = 1.
COROLLARY 2. A symmetric space E has the following property: If Y E E and
nlxl(T) < Cnlyl(T), (4.17)
then x E E and IIxll£ < max{l, C}IIYII£.
Indeed, (4.17) is equivalent to the inequality
x*(t) < y*( C -It) = 0cy*(t),
and the required inequality immediately follows from (4.16).
REMARK. It can be seen from the proof of Theorem 4.5 that the function
1I0TYII£ is quasiconcave for any Y E E.
It can be verified immediately that IioTI14 = T llp . In 5 we shall study the
Lorentz spaces and Marcinkiewicz spaces M1f. If \f;(t) is a concave
function and \f;(0) = 0, then
Ilxll", = oo x*(t) tA{I(t).
We calculate
lIa,-xll", = oo x*('T-1t) tA{I(t) = oo x*(s) tA{I('Ts).
By property 18° of rearrangements we have
Ila,-xll", 71 oo x*(s) tA{I(s) = M1f('T)llxll",. (4.18)
On the other hand,
II oTx(o,t)II \f;( Tt)
II a,-II", ;;;. sup II II = sup () = M1f( 'T). (4.19)
t >0 X(O,t] t >0 \f; t
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN L} AND Loo 99
Inequalities (4.18) and (4.19) show that
\I 0TIIA", = M1/1( 'T). (4.20)
If o/(t) is a quasiconcave function and 0/(0) = 0, then
Ilxll = sup _ ( 1 ) (I x*(s) ds.
1>0 0/ t )0
A similar calculation shows that
II ° I I = 7' sup t/1( t / 7') = 7' M ( ! ) = M ('T ) ( 4.21)
T \[;(t) 1/1 'T 1/1.'
where 0/*( t) = t / \[;( t).
DEFINITION. The upper and lower dilation exponents of the function II0TII£
are called upper and lower dilation exponents of the space E and will be
denoted by {3E and CX E .
If we write s( 'T) = II 0T II E' then by the submultiplicativity of this function we
have Ms( 'T) = s( 'T). Indeed,
Ms( 7') = sup sY7'i s( 7') and Ms( 7') S(l(.)7') = s(-r).
1 >0 s t s 1
Therefore the formulas for determining the upper and lower dilation expo-
nents take the form
. In II ° T II E
cx E = hm In
T O 'T
d {3 - 1 . In II ° TilE
an E- 1m In .
'T OO 'T
( 4.22)
REMARK. If s('T), 'T > 0, is submultiplicative and quasiconcave and s(l) = 1,
then there exists a symmetric space E for which s( 'T) = II 0T II E.
Indeed, as E we may take the Marcinkiewicz space Mt/s(/). By (4.21) we
have
s(t)
Ila,-IIE = 7' sup (/) = Ms( 7') = s( 7').
1 > 0 'TS t 'T
Now we are in a position to improve inequality (4.11) under the hypotheses
of Theorem 4.3. Let T be an operator acting in Ll and Loo. Then for any
'T > 0 we have
II TOT II LI-+L. < 'T11711 L.-+L. and II TOT II Loo-+Loo 11711 Loo-+Loo.
Using the representation T = TOTOl/T' we obtain the reverse inequalities, i.e.
II TOTIILI-+L. = 'T117lIL.-+L. and II TOTIILoo-+Loo = II TtILoo-+Loo.
We choose
'To = 11711 Loo-+Loolllll LII-+L..
( 4.23)
100
II. SPACES OF MEASURABLE FUNCTIONS
Then by Theorem 4.3 we have
II TOToxll£ < 1111IL -.LQ)llxll£.
Replacing 0TOX by y, we obtain
II TYII£ < II TJIL -.L llol/ToYII£,
and hence
11711£ £ < 1171ILQ) LQ)11 01/Toll£. (4.24)
By (4.16) this inequality implies (4.11). Inequality (4.24) is sharp in the
sense that it becomes an equality for the operator 0l/To.
THEOREM 4.6. A space E intermediate between LI and Loo is a normalized
interpolation space of type a between LI and Loo if and only if it satisfies the
hypothesis of Theorem 4.3 and the equality IloTII£ = 'T a .
PROOF. If the last equality is satisfied, then by (4.23) and (4.24) we have
1171/ £ £ < II TJI :Lllllll -.L . (4.25)
Conversely, (4.25) applied to 0T gives II 0T II £ < 'T a. On the other hand, II 0T II ;>
Iiol/TII- I (1/'T)-a = 'T a , i.e. IloTII£ = 'T a .
In what follows we shall frequently use
LEMMA 4.7. Let \fI(t) be a continuous increasing function. Then for any
function x E E, where E is a symmetric space,
11 00 aTx*d\¥( T)t < 00 IlaTxliE d./J( T).
( 4.26)
PROOF. The function OTX*(t) is increasing in 'T, and so for any A > 1 we
have
1 00 00 ('Ak+l
OT X *( t) d\fl( 'T) = J). k OT X *( t) tAJ;( 'T)
o k=-oo 'A
00
< Oi\k+IX*( t) [ \fI(A k+ I) - \fI(A k) ].
-00
Since E is complete, we can apply the triangle inequality to infinite sums in
i t. Therefore
00
11 00 aTx*d./J(T)t < IlaAk+lXIIA \¥(Ak+l) -\¥(A k )]
00
< A 110i\kXII£[ \fI(A k+ I) - \fI(A k) ]
-00
(we have used (4.16».
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN LI AND Loo 101
On the other hand, by the monotonicity of the function 110'Txll£ we have
1 00 II a,-xil E t/1fICr) = f (A k+ I II a,-xil E t/1fI( 'T)
o - 00 J'A k
00
IIOi\kXll£[ "-'(A k+l) - "-'(A k)].
-00
Thus,
1 11 00 a,-x. d\[;( 'T ) 11 A 1 00 II a,-xll E t/1fI( 'T).
I 0 £ 0
By the arbitrariness of A > 1 this implies (4.26).
REMARK. The integral on the left side of (4.26) is understood as a Stieltjes
integral for every t. We could not use the theory of abstract integration in
obtaining (4.26), since the range of O'Tx*( t) as an abstract function of T with
values in E is in general not separable, and consequently the function is not
measurable.
4. The fundamental function. From the definition of a symmetric space it
follows that the norm of the characteristic function Xe(t) of a measurable set
e C (0, (0) depends only on the measure of e. Hence, for a symmetric space E
the function CfJ£( t) (0 < t < (0) is defined by
IIXeII£ = CfJ£(mes e).
I t is obvious that CfJ £(0) = O.
The function CfJ£( t) is called the fundamental function of the space E.
I t is easy to verify that
CfJlp(t) = t llp ,
1
fPL:,(t) = M -I( 1 / t)
(see [23]).
( 4.27)
For the Lorentz and Marcinkiewicz spaces mentioned in the preVIOUS
section we have
t
fP A/ t) = \[;( t) and fP (t) = \[;( t) = \[;.( t).
( 4.28)
THEOREM 4.7. A function CfJ(t) is the fundamental function of some symmetric
space if and only if it is quasiconcave.
PROOF. If E is a symmetric space, then
CfJ£( T) = Ilx(o,'T] 11£ = II °'TX(O,I] 11£,
and therefore the quasiconcavity of CfJ£(T) follows from the remark to Theo-
rem 4.5.
102
II. SPACES OF MEASURABLE FUNCTIONS
If C(J( t) is a quasiconcave function, then by (4.28) it is fundamental for the
space Mt/ t). The theorem is proved.
Between the functions II 0'T11 E and CPE( T) the following relation holds:
II 0'TX(O.t] II E _ C(JE( Tt) _ (
II °TII E ? sup II II - sup () - M'PE 'T).
t X(o.t] E t C(JE t
For the spaces 4 we have 110'Tlllp = C(Jlp(T) = T llp . As can be seen froJn
(4.20) and (4.28), for Lorentz and Marcinkiewicz spaces (4.29) turns into an
equality. There are examples of spaces for which strict inequality holds in
( 4.29).
Denote by YE and 8 E the lower and upper dilation exponents of McpE. From
(4.29) it follows that
( 4.29)
CX E YE and {3E 8 E . (4.30)
5. Separable symmetric spaces. Since every symmetric space E is an ideal
lattice, from what was said in the Introduction (p. 45) it follows that E is
separable if and only if it is regular, i.e. the norm of any element in E is
absolutely continuous. The separability of E is also a necessary and sufficient
condition for its associated space Eland the entire conjugate space E' to
coincide. In this case E can be isometrically imbedded in a natural way into
ElI.
We note some further properties of separable symmetric spaces.
The function X(O.a) belongs to E for any a > O. Therefore, for T < a we
have
C(J( 'T) = Ilx(o.'T)11 E = IIX(O.a)X(O.'T)II E 0 as T 0
by the absolute continuity of the norm of X(O.a). Thus, in a separable
symmetric space we have C(JE( + 0) = O.
For any measurable function x(t) the function xN(t) defined by xN(t) =
x(t) if Ix(t)1 Nand xN(t) = N sgn x(t), if Ix(t)1 > N is called the truncation
of x(t). The right truncation of x(t) is defined to be the function xN(t) =
x(t)X(O.N)(t). If the norm of x(t) E E is absolutely continuous, then, since
meseN O,whereeN = {t: Ix(t)1 >N},andduetothefactthatxbelongsto
Ll + Loo' we obtain
II x - xNllE II xXe",11 £ 0 as N 00.
Under the same conditions, we have
Ilx - xNllE = Ilxx(N.oo)IIE 0 as N 00.
Thus, in a separable space E every function is the limit of its truncations
and right truncations.
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN Ll AND Loo 103
Every bounded function with bounded support can be uniformly ap-
proximated by a sequence of strictly simple functions. This sequence will
converge in the norm of Ll n Loo, and hence in the norm of E. Since the
two-fold truncations. x; of x E E are bounded functions with bounded
support, from the preceding it follows that the strictly simple functions are
dense in a separable. symmetric space.
Conversely, if th strictly simple functions are dense in E, then every
function x E E is the limit of its truncations and right truncations. Indeed, let
y(t) be a strictly simple function such that Ilx - yIIE < f. For N > IIYllLoo
we have Ilx N - yIIE < Ilx - YIIE. Therefore Ilx - xNllE < 2f. Next, if
(0, N) contains the support of y(t), then Ilx N - YIIE = II(x - y)x(o,N)IIE < f,
and consequently Ilx - xNllE < 2f.
Finally, if C(JE( + 0) = 0 and the strictly simple functions are dense in E,
then the space is separable. Indeed, for every strictly simple functiony(t) with
support in (0, N) (N is an integer), given T > 0, by Luzin's theorem one can
construct a continuous function z-r(t) such that IZ-r(t)1 < ly(t)1 and mes e-r < T,
where e-r = {t: y(t) =1= z-r(t)}. Then
IIY - z-rIIE < 2 sup ly(t)IIIXeJI E = 21IYIILQ)C(J( T) 0 as T o.
The continuous function z-r(t) can be approximated by a polynomial with
rational coefficients uniformly on [0, N]. Thus, the countable set of functions
p(t)X(O,N)' where p(t) is a polynomial with rational coefficients and N =
1, 2, . . ., is dense in E.
We have established the following assertion.
THEOREM 4.8. If E is a separable symmetric space, then the following
assertions are true:
1) C(JE( +0) = O.
2) Every function zn E is the limit in E of its truncations and right
truncations.
3) The strictly simple functions are dense in E.
If conditions 1) and 2) or 1) and 3) are satisfied, then E is separable.
6. Spaces associated with symmetric spaces. Let E be a symmetric space and
let E 1 be the space associated with it. The space E 1 consists of all measurable
functions u(t) for which
IlulIE' = sup i oo x(t)u(t) dt < 00. (4.31)
IIxIIE<1 0
· Editor's note. The meaning is that both the range and the domain of x are truncated (by N
and M respectively).
104
II. SPACES OF MEASURABLE FUNCTIONS
By means of B. Levi's theorem it is easy to see that the supremum can be
taken only over strictly simple functions x(t) in E of compact support.
Let u*(t) be the rearrangement of u(t). For any strictly simple function x(t)
of compact support we have
00 00 N 8
L u*(t)x(t) dt ..;; L u*(t)x*(t) dt = a k L k u*(t) dt.
o 0 k= I 0
Using the arguments in the proof of property 7° of rearrangements, we
may construct a system of measurable sets e such that
a) mes e = Ok'
b) e 8 , C e 82 C . . . C e 8N ,
c) f k u*(t) dt fe lu(t)1 dt + f.
9k
Then
00 N N
L u*(t)x(t) dt ..;; L a k f u(t) dt + E L ak
o k=l elk k=l
00 N
= L u(t)x.(t) dt + E ak-
o k=l
where x£(t) = 2 ak (t) is a function equimeasurable with x(t). Since f is
9k
arbitrary and Ilx£IIE = IlxilE 1, we have
sup L oo u*(t)x(t) dt";; sup L oo u(t)x(t) dt =llulk.
Ilxll£<1 0 Ilxll£<1 0
This inequality shows that u*(t) E E I and Ilu*IIE' lIuIIE,. Conversely,
since II x* II E = II xII E, we have
IlulIE' = sup [ u(t)x(t) dt";; sup L oo u*(t)x*(t) dt
IIxll£<1 0 Ilxll£<1 0
..;; sup L o oo u*(t)x(t) dt = Ilu*IIE I .
Ilxll£< I
This implies that all functions equimeasurable with a u( t) belonging to E I
belong to E I and have the same norm.
It is obvious that the space E I has property 1) of the definition of a
symmetric space.
We have arrived at the following assertion: A space associated with a
symmetric space is itself symmetric.
A symmetric space is said to be maximal symmetric if E is isometric to E II .
Since we always have E 1 = E III, the class of maximal symmetric spaces
coincides with the class of spaces associated with symmetric spaces.
Examples of maximal symmetric spaces are the Orlicz, Marcinkiewicz, and
Lorentz spaces. An example of a nonmaximal symmetric space is the space
Mo( \f;) (see 5.4).
4. SYMMETRIC SPACES. INTERPOLATION BETWEEN Ll AND Loo 105
THEOREM 4.9. Every maximal symmetric space E I is an interpolation space
between Ll and Loo with interpolation constant one.
PROOF. Let v EEl, and for u E Loo + Ll let
t u* ds ..;; t v* ds.
By property 18° of rearrangements, for any measurable functiony(t) we have
oo u*(t)y*(t) dt ..;; oo v*(t)y*(t) dt.
In this inequality we take the supremum over all Y E E such that IIYIIE < 1.
We obtain IlullEI < IlvilEI. By Theorem 4.3 the space E 1 is an interpolation
space between Ll and Loo, and (4.11) is satisfied.
Theorem 4.5 implies the following important theorem.
THEOREM 4.10. If Ll n Loo is dense in a symmetric space E, then E is an
interpolation space between Ll and Loo.
PROOF. Let x andy be measurable functions,y E E, and let
.,. x*( t) dt ..;; .,. y*( t) dt.
In view of Theorem 4.3, to prove the theorem we need to show that x E E
and IlxilE < IIYIIE. From the preceding theorem it follows that IlxllEII <
IIYIIEII = IIYIIE. It only remains to verify that x E E. By Theorem 3.4 there
exists an operator To E L such that x = ToY. We consider a sequence
Yn E Ll n Loo converging to y in E. Then by the interpolation property of
Ell we have ToYn ToY = x in Ell. The operator To acts in Ll n Loo.
Therefore ToYn E Ll n Loo C E. Since the imbedding of E in Ellis isomet-
ric, the sequence ToYn is Cauchy in E, and so x E E.
COROLLARY. If a symmetric space is separable, then it is an interpolation
space between Ll and Loo.
We consider the dilation operators a" in an associated space. From (4.31)
we obtain
= sup
IIxll£< I
00 x( t)a.,.u( t) dt
7' oo a1j.,.x(t)u(t) dt ..;; 7'lla 1 j.,.II E ll u IIE'.
/la"uIIEI = sup
IIxIIE<1
Thus,
Ilo"I/E I < 'Tllal/"IIE.
( 4.32)
106
II. SPACES OF MEASURABLE FUNCTIONS
Since E I = E 111
,
IIOTII£III =lloTII£1 Tll o l/TII£II.
Applying (4.32) to E I, we obtain
II0TII£1 = Tiiol/TII£II.
( 4.33)
( 4.34)
We also need the quantity IloTII£ £II. We have
IloTII£ £1I = sup Il o T x ll£1I sup Il o T x ll£1I = Il o TII£1I = Tllol/TIIEI.
IIxIlE<1 II x IIE II <1
On the other hand,
'Tlla1j.,.vII E , = sup 'T {)to x(t)a1j.,.v(t) dt
IIxliE < 1 0
- sup 1 00 a.,.x(t)v(t) dt .;;;; II a.,. II E-+E" II vii E 1 .
IIxIIE<1 0
Thus,
IloTII£ £1I = TIIOI/TII£1 = 1\0TII£II. (4.35)
If E is isometrically imbedded in Ell, this equality implies that
IloTII£ = Tiiol/TII£I = IloTII£II. (4.36)
THEOREM 4.11. The associated space E I of a symmetric space E satisfies
(4.32) and the inequalities
lX£1 1 - {3£, {3£1 1 - lX£. (4.37)
If the natural imbedding of E in E II is isometric, then (4.36) holds and
lX£1 = 1 - {3£, {3£1 = 1 - lXE. (4.38)
Inequalities (4.37) and equalities (4.38) immediately follow from (4.32),
(4.36), and formula (4.22) for the calculation of lower and upper dilation
exponents.
Now we pass to the calculation of the fundamental function of the
associated space. We have
'PE,(t) = Ilx(o,tlllEI = sup 1 00 X(S)X(O,t](S) th
IlxllE<1 0
.;;;; sup i oo X*(S)X(O,t](S) th = sup it x*(s) th.
IlxllE<1 0 II x IlE<1 0
We use (4.6). Then we have
'PEl(t) .;;;; II I IIX(o t]IIE Ilx*x(O,tIIIE .;;;; 'PE t) '
5. LORENTZ AND MARCINKIEWICZ SPACES
107
On the other hand,
1 00 X(O.t]( s) t
'PE,(t) 0 lIX(o,tjIlE X(O,tj(S) ds = 'PAt) .
Thus,
CPE I ( t) = t / CfJE( t).
( 4.39)
Next we compute
CfJE I ( Tt) TCfJE( t) ( 1 )
M"'E,(T) = sup () = sup () = TM", -.
1>0 CfJE I t 1>0 CfJE Tt E T
( 4.40)
THEOREM 4.12. For the associated space E 1 of a symmetric space the
equalities (4.39) and (4.40) hold, and
-rEI = 1 - 8 E , 8 E I = I - -rE.
We also compare (4.40) with (4.29) and (4.32). We have
Tlloli'rllE II0.,.IIE' M"'Eb) = TM"'E( ).
From this it is clear that if (4.29) becomes an equality, then we have equality
in (4.32) as well.
5. Lorentz and Marcinkiewicz spaces
1. Lorentz space. Let o/(t) ( O) be an increasing concave function on
[0, (0) such that 0/(0) = 0, and let b( t) be a nonnegative decreasing function
on (0, (0). We shall consider the integral
1'000 b( t) t#( t),
understanding by it the following: If 0/( +0) =1= 0, then
1 00 b(t) do/( t) = 0/( + O)b( + 0) + f oo b(t) tAJ;( t),
o +0
where the second term is an improper Stieltjes integral or, which is the same,
1'000 b(t) t#(t) = t/J( +O)b( +0) + 1'000 b(t)t/J'(t) dt,
where the second term is a Lebesgue integral. If 0/( + 0) = 0, then
1 00 b(t) #(t) = f oo b(t) tAJ;(t) = 1 00 b(t)\[;'(t) dt.
o +0 0
We now consider the collection of all measurable functions on (0, (0)
for which
1'000 x*(t) t#(t) < 00.
(5.1)
is called a Lorentz space.
108
II. SPACES OF MEASURABLE FUNCfIONS
If \fI( + 0) =t= 0, then by the preceding it follows from (5.1) that x*( + 0) =
II xII L < 00 and
00
oo x*(t) #(t) = t/J( +O)llxIlL oo + oo x*(t)tfI'(t) dt. (5.2)
Inequality (5.1) implies that x E Ll + Loo. Indeed, if \fI'(t) = 0, then A-JI =
Loo C Ll + Loo. If \fI'(t o ) > 0 for some to, then
1 to 1 1 to
x*(t) dt"; '() x*(t)t/J'(t) dt < 00.
o \fI to 0
Since the function J o x*(s) ds is concave, we have
IIXIIL,+Loo = I x*(t) dt ..; max{ t ,I} to x*(t) dt < 00,
which implies our assertion.
The behavior of (t) at zero and at infinity manifests itself in the properties
of the space A-JI in an essential way. From the preceding it follows that
C Loo if \fI( +0) =t= O. Next, if (oo) = lim t -+ oo (t) < 00, then every
bounded function belongs to , i.e. A-JI ::> Loo. Thus, if #. +0) =F 0 and
\fI( 00) < 00, then A-JI = Loo.
If \fI( (0) = 00, then for any function x E A-JI we have nlxl(-r) < 00 for all
'T > O. Indeed, if for some 'To the set e = {t: Ix(t)1 > 'To} has infinite measure,
then x*( t) > 'To for all t > 0, and
1 00 x*( t)tfI'( t) dt ;;;. 'To 1 00 tfI'( t) dt = 00.
o 0
We obtain still another expression for the quantity (5.2). The function '(t)
is decreasing, and as the derivative of a concave function it is locally
summable. Therefore \fI' E Ll + Loo. By Theorem 3.2 we have
1 00 x*(t)tfI'(t) dt = sup 1 00 x(t)Tt/J'(t) dt.
o TE 0
From this and (5.2) it follows that the quantity (5.2) has the properties of a
norm. Thus, the space A-JI is normed by
IIxllA.t, = 00 x*( t) #( t).
It is obvious that the relations ly(t)1 Ix(t)l, x E A-JI' imply y E A-JI and
" y "A II x II .
We prove the completeness of the space . For this it is sufficient to prove
that the inequality L:f=l IIxkll = c < 00 (x k E A-JI) implies that L Ixk(t)1
belongs to . Since A-JI is imbedded in Ll + Loo, we have
00
L IIxkllLl+Loo < 00,
k=l
5. LORENTZ AND MARCINKIEWICZ SPACES
109
and so Lf" Ixk(t)1 converges to a function x(t) E Ll + Loo almost every-
where. Next, for T E Lo Theorem 3.2 and (5.2) imply that
00 N N 00
1 IXk(t)ITt/J'(t) dt ..;; 1 x;(t)t/J'(t) dt ..;; C.
o k=l k=l 0
Then by B. Levi's theorem we have
oo x(t) Tt/J'(t) dt ..;; C.
Taking the supremum over all T E Lo, we obtain
00 x*( t)t/J'( t) dt ..;; C.
If (+o) =t= 0, then the finiteness of L Ilxkll implies the finiteness of
Ilxkll Loo and the inequality IlxllLoo < 00. Thus, x E A , which proves that
A is a Banach space.
It is clear from the definition that is symmetric and its fundamental
function is equal to (t). By Theorem 4.1 we obtain that
Ll n Loo C A C Ll + Loo
(both imbeddings can be verified immediately).
A is an interpolation space between Ll and Loo. Indeed,
II Txll = t/J( +0)11 TxllLoo + oo (Tx)*(t)t/J'(t) dt,
and if TEL, then by (3.16) and property 18° (see p. 72) we have
II Txll ..;; t/J( +O)lIxIILoo + oo x*(t)t/J'(t) dt = IIxll .
In what follows we need an assertion on passage to the limit in the space
A .
THEOREM 5.1 (on passage to the limit). /1t y E , nlyl(T) < 00 for all
T > 0, and I x n ( t)1 < I y( t)l. Let one of the following conditions be satisfied:
a) \fI( +0) = 0 and lim n --+ oo xn(t) = x(t) almost everywhere.
b) ( + 0) 0 and lim n --+ oo x n ( t) = x( t) uniformly.
Then Ilx - xnll A o.
IJ-
PROOF. It is obvious that x E and nlxl(T) < 00 for T > O. From property
12° of rearrangements it follows that (x - xn)*(t) 0 for any t > 0 as
n 00. If \fI( +0) = 0, then
Ilx - xnllA.; = oo (x - xn)*(t)t/J'(t) dt 0
by the inequality (x - xn)*(t) '(t) 2y*(t) '(t) and Lebesgue's theorem.
110
II. SPACES OF MEASURABLE FUNCfIONS
If \f;( + 0) > 0, then
IIx - xnllA.; =llx - x n lk",1¥(+O) + oo (x - xn)*(t)1¥/(t)dt O
by the preceding and the uniform convergence of xn(t) to x(t).
COROLLARY 1. If \f;( + 0) = 0 and (oo) = 00, then the relations Ixn(t)1 <
y(t) (y E A1jI) and xn(t) x(t) almost everywhere imply that Ilx - x n l11jl o.
Under the hypotheses of Corollary 1 every function in A1jI is the limit of its
two-fold truncations. These hypotheses are necessary for this property. In-
deed, if \f;( (0) < 00, then X[o.OO)(t) E , and the truncations X[O.N](t) are at a
constant distance (oo) from X[o.oo). If (+ 0) > 0, then for the function
L =-l X[n.n+ 1/n2](t) the truncations L = 1 X[n,n+ 1/n 2 ](t) are at a distance not less
than \f;( + 0) from this function.
COROLLARY 2. Every function x E is the limit, in , of its truncations
x N .
Indeed, if \f;( +0) > 0, then C Loo and xN(t) = x(t) for sufficiently large
N.
Now let \f;( +0) = O. There exists TO > 0 such that n1xl(T o ) < 00. Then for
N > TO we have Ix(t) - xN(t)1 Ix(t) - x'T<>(t)l, and n 1x _ x 1"oj(T) < 00 for all
T > O. Theorem 5.1 implies that Ilx - xNII 0 as N 00.
COROLLARY 3. If (oo) = 00 « (0), then the strictly simple (simple) func-
tions are dense in A1jI.
By Corollary 2 any function x(t) E can be approximated by a bounded
function with arbitrary accuracy, and if (oo) = 00, then the approximating
function can be chosen of compact support. Any bounded function of
compact support can be represented as the limit of a uniformly convergent
sequence of strictly simple functions not exceeding it. Therefore, in the case
( (0) = 00 our assertion follows from part b) of the theorem on the passage
to the limit. Every bounded function can be represented as the limit of a
uniformly convergent sequence of simple functions. In the case (oo) < 00
this implies the convergence of the sequence in A1jI.
LEMMA 5.1. The space is separable if and only if
\f;(+0) = 0 and (oo) = 00.
( 5.3)
The proof immediately follows from Theorem 4.8 on separable symmetric
spaces and the discussion above .
5. LORENTZ AND MARCINKIEWICZ SPACES
111
In what follows we need still another expression for the norm in a Lorentz
space. We have
IlxlIA = 10 00 I/t(nlxl(T») dr.
(5.4)
The left and right sides of this equality do not change if from the function
we pass to its absolute value and to a function equimeasurable to it.
Therefore we may assume that x(t) = x*(t). If the function is elementary, say
00
x( t) = LlkX(O"k)
1
(00 = t} > t 2 > . . . > 0, Ll k 0),
( 5.5)
then a direct calculation shows that both sides in (5.4) are equal to
Lr Llk (tk). Any function x E (x = x*) can be uniformly approximated
by an increasing sequence of functions xn(t) of the form (5.5). By Theorem
5.1 we have Ilxll = lim n -+ oo Ilxn" . For every T > 0 the functions n (T)
converge to nx(T), and consequently (n (T)) (nx(T)) for all T > O. By B.
Levi's theorem we have
(00 t/;( n x ( T» dT = lim 1 00 1/t( nx" (T») dT.
)0 n-+oo 0
This and the preceding imply (5.4).
Convex functionals on a Lorentz space. Let (t) be a convex functional on
A , i.e. a functional with the properties
(x + y) (x) + <l>(y), (Ax) = IAI (x).
We assume that <l>(x) can take on infinite values.
LEMMA 5.2. If a convex functional is bounded on all characteristic functions of
measurable sets, then it is bounded on all simple functions in A .
PROOF. Let
(Xe) CIIXeII = Ct/;(mes e).
If x(t) is a nonnegative simple function belonging to A , then it can be
represented in the form
N
X ( t) = LlkXe k ( t),
k=l
where e} C e 2 C . . . C eN and Ll k > O. Then
N N
(x) Llk<l>(Xe k ) C Llk (mes e k ).
k=l k=l
( 5.6)
112
II. SPACES OF MEASURABLE FUNCfIONS
On the other hand, as was shown on p. 60, we have
N
x* ( t ) = '" Ll X ( t )
£.J k [0, mes ek] ,
k=l
and consequently
00 N N
Ilxll = 1 dkX[O.mese.l(t) d1[;(t) = dkl/t(mes ek)' (5.7)
o k=l k=l
Combining (5.6) with (5.7), we obtain <I>(x) ell x II A", .
If now the simple function x(t) from A has values of arbitrary sign, then
<1>( x) = <1>( x + - X _) cI>( X +) + <1>( x _ )
C(IIx+ll + Ilx_II ) 2Cllxll .
COROLLARY 1. Let the functional <I>(x) have the form IIAxIIE, where E is
some Banach space imbedded in a separated topological linear space 21, and let
A be a linear operator on continuous in 21 ( if Ax fl E, we set <1>( x) = (0). If
the functional <I>(x) is bounded on all characteristic functions from A , then A is
bounded in E.
Indeed, let x E and let X n be a sequence of simple functions approxi-
mating x. Then by Lemma 5.2 we have IIA(x n - xm)11 < 2Cllx n - xmllA",'
and so the sequence AX n is Cauchy in E. Since it converges to Ax in 21, its
limit will be the same Ax in E, i.e. Ax E E and IIAxllE 2CllxllA",.
2. Marcinkiew;cz spaces. We study the structure of the dual space of a
separable space A .
THEOREM 5.2. Every linear functional f on a separable space has the form
f(x) = oo x(t)y(t) dt, (5.8)
where y( t) is a measurable function on [0, 00) and
Ilfll = sup [ I/t(h) r I .£h y*(s) dY < 00.
O<h<oo 0
( 5.9)
PROOF. We show that if (5.9) is satisfied, then (5.8) defines a linear
functional on . We note that (5.9) and the condition \fI( + 0) = 0 imply that
J3 y*(s) ds C\fI(h) 0 as h O. Let x(t) be a step function. Then
N
x*( t) = LlkX[O,hd'
k=l
Ll k > O.
5. LORENTZ AND MARCINKIEWICZ SPACES
113
We estimate
00 00 N h
If(x)1 = I i x(t)y(t) dt l ' i x*(t)y*(t) dt = L !1 k i' y*(t) dt
o 0 k=l 0
sup { [ 1/1(h) r l i o h y*(t) dt } L Llko/(hk)
O<h< 00
= sup { [ 1/1(h) r l i o h y*(t) dt } IlxllA",.
O<h< 00
As we saw in the proof of Lemma 5.1, the step functions are dense in the
norm of in the set of bounded functions of compact support. Therefore
the inequality
If(x) I , [YO Ix(t)1 iY(t)1 dt, sup { [1/1(h)r 1 i h y*(t) dt } llxll (5.10)
o O<h<oo 0
can be established for any bounded function of compact support by passage
to the limit. By the corollaries to Theorem 5.1, for any function x E AtJI the
absolute value I x( t)1 is the limit, in , of its two-fold truncations, and so by
Fatou's lemma inequality (5.10) can be carried over to any functions in AtJI.
This implies thatf E AtJI and
If( x ) I sup { [ 1/1 ( h) r I 1 h Y * ( t) dt } II x II A",. ( 5 .11 )
O<h<oo 0
Now let f be an arbitrary linear functional on . Then /f(x)/
Ilfll Ilxii A . If x = Xe is the characteristic function of a set e of finite measure,
IJ-
then for a(e) = f(Xe) we have
la( e)1 IIJlIIIXeII = Ilfllo/(mes e).
Since 0/( +0) = 0, the function a(e) is absolutely continuous, and by the
Radon-Nikodym theorem we have
a(e) = AXe) = oo x.,(t)y(t) dt, (5.12)
where y(t) is a locally summable function. Formula (5.12) implies that
f(x) = oo x(t)y(t) dt (5.13)
for any strictly simple function x( t).
Next,
Xe( t)
sgny(t) ( )
0/ mes e
=1
,
and therefore
[o/(mes e) ] -I f ly(t)1 dt = f ( sgny (x., » ) Ilfll.
e 0/ mes e
114
II. SPACES OF MEASURABLE FUNCfIONS
By property 7° of rearrangements we then have
I l h
sup [ (h)] - y*(t) dt Ilili.
O<.h<oo 0
The extension of (5.13) to all functions in A can be carried out by the
same arguments as in the proof of the first part of the theorem. Inequalities
(5.11) and (5.14) imply (5.9). The theorem is proved.
The space of all measurable functions y(t) for which (5.9) is satisfied is
called the Marcinkiewicz space . We have proved that M is isometric to
the space dual to the separable Lorentz space . Similar arguments show that
for an arbitrary Lorentz space the space coincides with the space
associated with . This implies that is a Banach space. It is obviously
symmetric.
M is an interpolation space between LI and Loo with interpolation
constant 1. Indeed, if TEL, then by (3.16) we have
(5.14)
I l h
II TxllM.. = SUp [ 1¥(h) r 0 (Tx)*(t) dt
- I l h
, SUp [ 1¥(h) ] 0 x*(t) dt = IlxllM+o
It is sometimes convenient to consider spaces M constructed from a
quasi concave function (t). Then for the fundamental function the equality
CfJM (t) = t 1 (t) holds.
v-
Of course, the norm (5.9) makes sense for any function o/(t) positive for
t > O. However, it can be shown that if the space thus obtained has nonzero
elements, then the norm coincides with the norm calculated according to (5.9)
with some quasiconcave function (the largest quasiconcave minorant of (t)).
LEMMA 5.3. Let (t) be an increasing concave function. In order that
(t)/t E it is necessary and sufficient that M (SOI) < 1 for any So > 1.
PROOF. If M (SOI) = 1, then by Lemma 1.3 we have M (S-I) = 1 for
s 1. Then for any integer n > 1 there exists a point t n such that (ntn)
(1 + f) (tn). The function (t)1 t is decreasing, and therefore
1¥(t) > [1¥(ntn)rl l ntn 1¥('T) d'T > [1¥(nt n )r 1 j nt n (T) dT
t Mv- 0 T t n T
1 j ntn dT 1
[ (ntn) ] - (tn) - 1 In n.
t T + f
n
Since n is arbitrary, we obtain that (t)1 t fl M . Thus, for o/(t)1 t to belong
to it is necessary hat M (so I) < 1 (so> 1). Now let this condition be
5. LORENTZ AND MARCINKIEWICZ SPACES
115
satisfied. Then by Corollary 2 from Lemma 1.4 we obtain
( t) 1 1 1 \fI( T )
t M", = sup 1/1(t) 0 'T d'T';;; C.
The lemma is proved.
We introduce the functional
F(x) = sup ..,, ( t t) x*(t).
0<1<00 't'
It is obvious that F(x) > 0 if x(t) 0, and F("Ax) = IAIF(x). However, this
functional does not in general have the properties of a norm.
We have
J x*(t) dt hx*(h) = F1 ( )
IlxllM", = o: 00 1/1(h) ,? o: 00 1/1(h) x "
This and Lemma 5.3 imply the following theorem.
( 5.15)
THEOREM 5.3. The functional F(x) is equivalent to the norm Ilxll if and
only if Mt/J(S-I) < 1 for s > 1.
PROOF. Necessity. Since (t)/ t is decreasing, F(\fI(t)/ t) = 1. If the func-
tional F(x) is equivalent to the norm of the space Mt/J' then this equality
implies that (t)/t E Mt/J' and by Lemma 5.3 we have Mt/J(S-I) < 1 for s > 1.
Sufficiency. Let F(x) 1. Then x*(t) \fI(t)/t and consequently IlxilM
IJ-
11 (t)/tIIM = C < 00. From the homogeneity of both functionals it follows
IJ-
that IlxilM CF(x). This, together with (5.15), proves the theorem.
IJ-
3. The space . Assuming that (+o) = 0 and (oo) = 00, we denote by
M the set of all x E Mt/J for which
lim [ (h) ] -I l h x*(t) dt = O. (5.16)
h O,oo 0
It turns out that is a subspace of Mt/J. The linearity of follows from
(2.17). Let X n Xo in Mt/J and let X n E . Then the sequence of the
continuous functions a,,(h) = [ (h)]-IJ x:(t) dt uniformly converges to the
function ao[h] = [ (h)]-IJ3 xd(t) dt.
Indeed, by (3.14) we have
I [1/1(h) r I t h x;(t) dt - [ 1/1(h) r I t h x6(t) dtl
.;;; [ 1/1(h) r 1 t h (X n - XO)*(t) dt .;;; IIX n - xollM","
Since all functions a,,(h) have the property that limh-+O,oo lXn(h) = 0, so does
the limit function lXo(h). Thus, Xo E .
116
II. SPACES OF MEASURABLE FUNCfIONS
We note that for x E we have nlxl(T) < 00 for all T > O. Indeed, if
n 1xl ( TO) = 00, then x*(t) TO and
-l l h h TO
[!/t(h)] 0 x*(t) dt 'To !/t(h) !/t(1) > 0
for h 1, i.e. x fl M .
If lim,-+o \fI( t) / t = k < 00, then consists only of the element zero.
Indeed, if x E M", and x 0, then for some to > 0 we have x*(t o ) > O. Then
J h x* ( t ) dt h 1
l 0 !/t(h) x*(to) l !/t(h) = kx*(to),
. M o
I.e. x fl "'.
LEMMA 5.4. Under the condition
lim \fI( t) / t = 00
t O
the subspace coincides with the closure of the set of all bounded functions of
compact support in M",.
( 5. 1 7)
PROOF. It is obvious that under the condition (5.17) any bounded function
of compact support belongs to (since (oo) = 00). Let Xo E M . For any
f > 0 there is a 0 > 0 such that
[ !/t(h) r I h x ( t) dt " f
for h '0 or h 1/0. Since n1xol( T) < 00 for all T, by property 12° of
rearrangements (x o - xci')*(t) converges to zero for all t as N 00. Hence,
for sufficiently large N,
l) - I (x 0 - x 0") * ( t) dt " af;( 8).
For these N we have
II Xo - xci'11 M
{ J (xo - xt')*( t) dt J (xo - xt)*( t) dt }
= max sup ..1' ( h ) sup .1' ( h )
h <: 8. h ;> 8 - 1 't' 8 <: h <: 8 - I 't"
( J X6(t) dt Jg-l (xo - xt)*(t) dt )
, max sup, , f.
h <.8. h ;> 8 - 1 \fI( h ) \fI( 0 )
Obviously the space M is symmetric. From the imbedding Theorem 4.1 it
follows that
Ll n Loo C .
( 5.18)
5. LORENTZ AND MARCINKIEWICZ SPACES
117
Any operator TEL maps the set of bounded functions of compact
support into LI n Loo, and consequently into M . Since L I n Loo is dense in
and is an interpolation space between LI and Loo, it follows that
T C , i.e. M is also an interpolation space between LI and Loo, with
interpolation constant 1.
THEOREM 5.4. Under the condition (5.17) every linear functional f on M can
be represented in the form
f(x) = 10 00 x(t)y(t) dt,
where y E A and Ilfll = IIYII .
( 5.19)
PROOF. The fact that (5.19) defines a linear functional on , and conse-
quently on , follows from (5.11) if x and yare interchanged there. This
also implies that
Ilill IIYII . (5.20)
Now let f be an arbitrary linear functional on . The function a(e) =
f( ) defined on all sets of finite measure is additive and absolutely continu-
ous:
mes e
la( e) I = lJ(x.,) I ..; 1111111XeII = 11J11 1/t(mes e) 0
when mes e 0 (by virtue of (5.17)). It follows from the Radon-NikodYm
theorem that there exists a locally integrable functiony such that
f(x) = 10 00 x(t)y(t) dt (5.21)
for any strictly simple function x( t).
We choose xe(t) = sgny(t) (t) with mes e = 1. Then II II = 1/ (1).
tJ-
Therefore
sup
mes e = 1
f ly(t)1 dt IIfll
e (1) ,
i.e. yELl + Loo.
Any function x in LI n Loo can be represented as the limit of a sequence of
strictly simple functions converging in LI n Loo, and consequently, by virtue
of the imbedding (5.18), in M as well. This allows us to extend (5.21) to all
functions x in LI n Loo by passing to the limit.
We introduce the functions f.le E LI n Loo (0 < € < 1), defined by
f.le(t) = '(t)X[e,l/e](t).
118
II. SPACES OF MEASURABLE FUNCTIONS
We have II II 111/;'II = 1. For any operator TEL the functions TJ.Le
belong to Ll n Loo, and by the interpolation property of the space M we
have II TJ.LeIl 1. Formula (5.21) implies that
'"
V( TJL.)I = It oo TJL.( t)y( t) dtl <; 11111.
From Theorem 3.2 it follows that
tOO p.:( t)y*( t) dt = t 1/ e-e 1[;'( t + £ )y*( t) dt <; 11.1]1.
Letting E tend to zero, we obtain from B. Levi's theorem that
I II", = tOO 1[;'( t)y*( t) dt <; 11111. (5.22)
Thus,y E , and by (5.20) and (5.22) we have IIYIIA", = IIfli.
The validity of (5.19) for any x E can now be established by standard
arguments.
4. Imbedding of synunetric spaces. The space is the smallest among
symmetric spaces with the same fundamental function.
THEOREM 5.5. Let E be a symmetric space with fundam£ntal function cp(t).
The space , where 1/;( t) is the smallest concave majorant of cp( t), is imbedded
in E, and IIxllE IIxll .
PROOF. For any characteristic function Xe we have
r mes e
IIXeIIE = <p(mes e) <; 1[;(mes e) = )0 d1f;(t) = IIXeII",.
Let x(t) be an arbitrary nonnegative strictly simple function:
N
x( t) = LlkXe k (t),
k=l
where Ll k > 0 and e 1 C e 2 C . . . C eN. Then
N N N
IlxilE IILlkXeJIE LlkllXekllA,y = Ll k 1/;(mes e k ) = IIxll (5.23)
k=l k=l k=l
(see (5.7)). Obviously this inequality holds for an arbitrary simple function, as
well. By Corollary 3 of Theorem 5.1 any function in can be represented as
the limit of a sequence of simple functions. By (5.23) this sequence is Cauchy
in E, and so it has a limit in E. Since E and AlJI are imbedded in Ll + Loo,
this limit coincides with the initial function. Thus, C E. Inequality (5.23)
can be extended to any function in AlJI by passing to the limit.
THEOREM 5.6. Let E be a symmetric space. If the function IIOt-lIlE E belongs
to the Lorentz space At/1, then E c A .
5. LORENlZ AND MARCI naEWICZ SPACES
119
PROOF. For any 'T E (0, I] we have
oo x*(t) do/(t) " oo x*( 'Tt) fiI¥{t).
We multiply both sides of this inequality by the characteristic function
X[O,l]( 'T) and compare the norms of the functions of 'T thus obtained. Then, by
Lemma 4.7,
<pAl) oo x*(t) do/(t) "IIX[O,lJ( 'T) oo x*( 'Tt) do/(t)t "11 00 x*( 'Tt) do/(t)t
" oo lIal/tx*IIE fiI¥(t) " oo Ilal/tll E fiI¥(t)llxli£.
Thus, for x E E we have
IlxllAv- [<PE(I) ] -llillol/tIIEIIAv-llxIIE.
The largest space among symmetric spaces with the same fundamental
function is the Marcinkiewicz space.
THEOREM 5.7. Let E be a symmetric space with fundam£ntal functllJn \[I(t).
Then E c M ., and
IlxIIM . IlxiIE'
(5.24)
where \[1*( t) = t / \[I( t).
PROOF. Let x be an arbitrary function in E. By (4.6) we have
II x*X[o,h]11 Ll h
.
II X *X[O,h] II E \[I ( h)
Hence
* * Ilx*x[O.h]IIL 1 1 (h *
IlxilE = Ilx liE Ilx X[O,hIIIE h/t[;(h) = t[;*(Jl) J o x (t) dt,
If we take the supremum over h on the right side, then we obtain (5.24).
The theorem is proved.
It is worth noting that every interpolation space E between Ll and Loo is a
union of Marcinkiewicz spaces. Indeed, let x E E. Consider the concave
function
t[;A 'T) = T x*(s) dS',
and the Marcinkiewicz space x constructed from it. We have
f x*(s) ds = 1
IIxlIM x = o: 00 t[;A'T) ,
120
II. SPACES OF MEASURABLE FUNCTIONS
and consequently x E M x. Next, if y E M¥,x' then
i T y*(s) tis ..; IIYIIM i T x*(s) ds.
o x 0
It then follows from Theorem 4.3 thaty E E. Thus, M¥,x is imbedded in E.
If the interpolation constant corresponding to E does not exceed one, then
the last inequality implies that
IIYIIE IIYIIM xllxIIE.
If we set x = y IIIYIIE' the inequality becomes an equality. Therefore we
may write
lIy II E = inf lIy II M""
where the infimum is taken over all 0/ of the form o/x with Ilxll E = lor, in
other words, over all 0/ for which the Marcinkiewicz spaces are imbedded in E
with imbedding constant not exceeding 1.
5. The renonning of a synunetric space. Let E be a symmetric space with
fundamental function 0/( t) and let 0/1 (t) be a quasiconcave function equivalent
to 0/( t) :
Clo/(t) o/I(t) C 2 0/(t).
We introduce the new norm
(5.25)
II x II = max { C III x II E' II x II M",I. }
in E. It is obvious that IIxlik ClllxllE. Next, by (5.25) and Theorem 5.7 we
Mve
IIxllM",l. C 2 I1 x llM",. C 2 I1 x IlE.
Thus, IIxllk is equivalent to the original norm of E. For t = mes e we have
II Xe II = max { C III Xe II E' II Xe II M", I.} = max { C 10/ ( t), 0/1 ( t)} = 0/1 ( t).
We have arrived at the following assertion.
THEOREM 5.8. If the function o/I(t) is quasiconcave and equivalent to the
fundamental function o/(t) of a syrrunetric space E, then an equioolent norm can
be introduced in E so that it remains symmetric and has fundam£ntal function
o/I(t).
From Corollary 1 of Theorem 1.1 we obtain the following corollary.
COROLLARY. In any symm£tric space an equivalent norm can be introduced so
that the space remains symm£tric and its fundam£ntal function becomes concave.
5. LORENlZ AND MARCINKIEWICZ SPACES
121
6. Sum and intersection of two Lorentz spaces. We study the sum of two
Lorentz spaces in the case where the quotient of their fundamental functions
is monotone.
THEOREM 5.9. Let <Po(s) and <Pl(S) be two increasing concave functions whose
quotient <Po(s)[ <PI(S)]-I is decreasing. The space Aq>o + I is isometric to
A min ( q>Otq> I).
PROOF. Let x E Aq>o + Aq>I. By property 18° of rearrangements we have
IlxIIA +A 1 = inf {II uli A + IIvll A }
u+v=x I
ul x {liuIiAuun(cpo.CPl) + IIvIIAmin(cpo.CPl)} IlxIIAuun(cp()CPl).
Thus, Aq>o + AfPI C Amn(q>OtfPl). (Here we have not used the monotonicity of
<Po( s)[ <P I ( s) ] - 1 . )
If CPo(s)[ <PI(S)]-I is greater than 1 for all s or smaller than 1 for all s, then
one of the functions <Po(s) or <PI(S) is smaller than the other, and the space
corresponding to it contains the other space. The sum of the spaces then
coincides with the larger space, and the theorem is proved.
Now let
<Po(so) [ <PI (so) ] -I = 1.
( 5.26)
Then
<Po(s) { <PI(S) for s s(p
<Pl(S) fors so.
Let x E Amn(q>Otq> I). We write Xo = sgn x(s)min{x*(so), Ix(s)l} and XI =
x(s) - xo(s).
Then x6(s) = x*(so) for s So and xT(s) = 0 for s > so. Besides, x*(s) =
x6(s) + xT(s). Therefore
IIXIIAmin( ' I) = IIX611 ' I) + IIxTIlAuun( I)
= x * (so)min( q:>o( so), q:> I (so)) + LX> xo ( s) dq:>o( s) + II x r II A",
So
= IIX6I1A + IlxTII 1 IIx*II + I.
Since the sum of symmetric spaces is symmetric (Lemma 4.5), we have
IIx*IIA +A = IIxli A +A , and consequently Ilxll A . . ) II xii A +A .
'PO I 'PO 4i'1 " I u"'O I
From Theorem 5.9 we obtain the following corollary.
COROLLARY. If under the hypotheses of Theorem 5.9 a function x in Aq>o +
Aq>1 is of compact support, then it belongs to Aq>I; if it is bounded, then it belongs
to AfPo.
122
II. SPACES OF MEASURABLE FUNCTIONS
The case where (5.26) holds is of interest. In this case, by Theorem 5.9 we
have
l so f oo
IIXIIACFO+A'PJ = X*(S) dcpI(s) + x*(s) dcpo(s),
o So
which. mmediately implies our assertion.
REMARK. Instead of the monotonicity of CPo(s)[ CPI(S)]-I it is sufficient to
require generalized monotonicity:
CPo(s) [ CPI(S)]-I Ccpo(t) [ CPI(t)]-1 for t < s,
where C does not depend on s or t. Then in Theorem 5.9 isometry must be
changed to isomorphism. The assertion of the corollary remains valid.
We now consider the intersection of Lorentz spaces.
THEOREM 5.10. Let CPo(s) and CPI(S) be two increasing concave junctions. The
space Aq>o n Aq>J is isomorphic to max(q> q>J»- (where, as above, (max( cPep CPl))-
denotes the smallest concave majorant of max( CPo' CPI)).
PROOF. Let x E Aq>o n Aq>J. Since (max( CPo' CPI))- CPo + CPI' by property
18° we have
II XII1\max(...,.",,»- = 00 x*( s) d(max( CPo, CPI»-
1 00 x*(s) dcpo + 1 00 x*(s) dcpI 2 ax Ilxll ;= 21IXII ni\pJ.
o 0 I = 0, 1
Thus, Aq;O n Aq>J C max(q> q>J»-. Conversely, if x E A(max(q> q>J»-' then
IlxiiA IlxlI A _ _, which proves the theorem.
'P, ... "'(max( 'PJ»
7. Symmetric spaces which are not interpolation spaces between LI and Loo.
We give an answer to the question raised in 4.2 on the existence of such
spaces.
LEMMA 5.5. Let \f;(t) be an increasing concave continuously differentiable
function on [0, (0) having the properties \f;(0) = 0, \f;( (0) = 00 and
lim t[;(2t) = 1. (5.27)
1-+0 \f;( t)
The Marcinkiewicz space M¥, has a proper subspace G which is a symmetric
space and contains the function \f;'(t)X(O,I](t).
-
PROOF. Denote by G the collection of functions in M¥, different from zero
only on a set of finite measure (depending on the function) and having the
property that
x*( t)
t[;'( t) < 00.
5. LORENlZ AND MARCINKIEWICZ SPACES
123
We note that for x E G the function x*(t) is of compact support.
-
We show that G is linear. From (5.27) it follows that
lim \f;'(2t) = l.
1-+0 \f;'( t) 2 '
and theref re on every interval [0, N] we have \f;'(2t) aN\[/(t). If XI and X 2
belong to G, then x;*(t) Cj\f;'(t) (i = 1, 2), and by (2.23) we have
(XI + x 2 )*(t) xr( ) + x;( ) (C I + C 2 )1/1'( )
C + C
1 2 \f;' ( t)
aN
for 0 t N. If N is chosen so that the support of the function (Xl + x:J*(t)
is contained in [0, N], then the last inequality will be valid for all t > O. Thus,
-
Xl + X 2 E G.
As the subspace G we take the closure of G in M¥'. Since G is obviously
symmetric, by Lemma 4.4 G is symmetric. It is clear that \f;'X(O,I] E G c G.
It remains to show that G does not coincide with M¥'. For this we construct
a sequence 'Tn in the following way: We set 'To = 1 and determine 'Tn + 1 from 'Tn
by the two conditions
1
o < 'Tn + 1 2 'Tn and
1/1 ( Tn) - 1/1 ( Tn+ I) 2 .
\[I ( 'Tn - 'Tn + I) 3
We can satisfy the second condition, because (\f;('Tn) - \f;('T))j\f;('T n - 'T) is
continuous and converges to 1 as 'T O. We show that the function
u(t) = 1/1(T n ) -1/1(T n + l ) ( t )
'T - 'T X( 'Tn + 1,'T n ]
n=O n n+I
belongs to \ G. Since \f;(t) is increasing and concave, the function u(t) is
decreasing; thus u( t) = u*( t). Then f u*( t) dt is linear on the intervals
['Tn + l' 'Tn], and f n u*( t) dt = \f;( 'Tn). The concavity of \f;( 'T) implies that
.,. u * ( t) dt 1/1( T) (T > 0), ( 5.28)
and consequ ntly u EM¥'.
Let Y E G. Then y*(t) C\f;'(t). By Lemma 4.6 we have lIu - YIIM",
II u - y* 11M",. Next,
lIu - y*lIM", II(u - Y*)X(T p /2,'T p ]lIM",
II UX( 'Tp/2,'Tp] II M", - CII \f;' x( Tp/2.'Tp] II .
124
II. SPACES OF MEASURABLE FUNCTIONS
By the definition of the norm in and the choice of the numbers 'Tp we
have
'Tp - 'T p + 1
1/;( 7: - 7: ) = IIX(Tp+I,TplllM,;
p p+1
IIX(T p +I,Tp/2] 11M", + IIX(T p /2,T p ] 11M", < 2I1X('-p/ Tp]IIM",.
Hence
\f;( 'T p ) - \f;( 'Tp + I)
IIUX(T p /2,T p ]lIM", = 'T - 'T IIX(Tp/2,Tp]lI
p p+1
1 \f;( 'T p ) - \f;( 'T p + I) 1
- -.
2 \f;( 'Tp - 'T p + I) 3
Thus,
lIu - yllM", 1/3 - CII\f;'X('-p/2,T p ]/lM",.
If we show that the last term on the right tends to zero as p 00, then this
inequality will imply that u fl G. We have
, f \f;' ('T + s) ds
II 1/; X(T,2TIIIM,; = sup 1/;( h)
O<h <,-
- sup
o <h <,-
\f;( 'T + h) - \f;( 'T )
\f;(h)
Since \f;( t) is concave, the function \f;( 'T + h) - \f;( 'T) is decreasing in 'T, and
therefore \f;('T + h) - \f;('T) \f;(2h) - \f;(h). Then
I . II ' II 1 . \f;(2h) - \f;( h) lim ( \f;(2h) 1 ) 0
1m \f; X( T,2T] M", 1m sup (h) = (h) - =.
,- O T O O<h<,- \f; ,- \f;
THEOREM 5.11. The symmetric space G constructed in Lemma 5.5 is not an
interpolation space between LI and Loo.
PROOF. We rewrite (5.28) in the form
£T u*(t) dt .;;;; £T [1/;'X(O,I]]*(t) dt.
The function \f;'X(O,I] belongs to G, while u fl G. Thus, G does not satisfy
the necessary condition of Theorem 4.3. G is not an interpolation space.
6. Operators of weakened and weak type
1. The operation **. For x E LI + Loo we introduce the notation
l l t
x**(t) = - x*(s) ds.
t 0
6. WEAKENED AND WEAK TYPE OPERATORS
125
Obviously x**(t) x*(t). Using property 7° of rearrangements (see (2.13)),
we may write
x**(t) =! sup f Ix(s)1 ds.
t mes e = t e
This implies that the operation x** satisfies the triangle inequality
[Xl + X2J**(t) xT*(t) + xi*(t). (6.1)
We note that the operation x* does not have this property. We show that
x** satisfies the triangle inequality for an infinite number of terms, as well.
Let the series L Xt*(T) (xk E LI + L(; ::) be convergent. By B. Levi's theorem,
for a measurable set e with mes e = T the series L xk(s) converges almost
everywhere on e, and
! f I L xk(s)1 tb ,,! L f IXk(S)1 ds L xt*( T).
T e T e
On the left we take the supremum over all e with mes e = T. This gives us
(L Xk)**(T) L X:*(T).
The last property allows us to construct new symmetric spaces from old ones
by means of the operation **. Let 8(t) and K(t) be two positive measurable
functions on (0, (0). For a symmetric space E we denote by E x ,8 the space of
functions x(t) in LI + Loo for which the functions K(t)x**(8(t)) belong to E.
(The function x**(8(t)) is measurable, since x** is monotone and 8 is
measurable.) With the norm
II xii E IC ,6 = II K( t)x**( 8( t)) II E
this will be a Banach space. To prove this it is sufficient to verify that for any
sequence X k E EIC 8 (k = 1, 2, . . .) with L IIxkll E < 00 the series L X k is
, 6
convergent in EIC 8. By definition
,
L IIx k 1lE IC ,6 = L IIK(t)X:*(8(t))IIE'
and, since E is complete, L K(t)xt*(8(t)) E E. It then follows from the
triangle inequality that the function
K(t)(L x k )**(8(t)) K(t)L x:*(8(t))
also belongs to E. This is equivalent to the condition that L x k ( t) E E IC ,8;
moreover,
II L xkll E IC ,6 L II xkll E IC ,6.
From this it follows by standard arguments that L X k is the limit in E IC ,8 of
the partial sums L X k .
126
II. SPACES OF MEASURABLE FUNCTIONS
We establish some connections between the operation ** and the dilation
operators. In the case of characteristic functions
X (t) = { I forO <t O,
( , ] 0 / t for t > 0
and we consider the averaging operator
1 (h
Th X ( t) = It)o x(s) tis' X(O.h]( t).
For T > 1 we compute
1 (-rtJ ** ( ) 1 [ (f)
TO )O X(O./J] s tis' = TO )0 X(o./J]
1
= - (1 + In 'T).
'T
(s) tis' + £T6 X(O,/J] (s) tis' ]
Hence
(T T6X( ,Z] )( t) = (1 + In T ) X(O,T6]( t) = 1 + TIn T a..X(O./J]( t).
The operator T,,(J is of norm one in the spaces Ll and Loo. Therefore it
satisfies (3.16). Hence
$ a.,X(O./J] ( t) dt ..;; 1 + TInT $ X(O./J] (t) dt (s > 0).
For a strictly simple function of the form
N
.:(t) = Ll i X(O,8;](t),
;= 1
where Ll i > 0 and 0 1 > O 2 > . . . > 0, it follows immediately that
s s N s
1 a.,z(t) dt ..;; 1 Tl ( d;X(;';] (t) dt = 1 Tl (z**(t) dt.
o + n'T)o ;=1' + n'T )0
Any decreasing nonnegative function x(t) can be approximated uniformly by
an increasing sequence of elementary functions of the form
00
y( t) = Ll i X(O,8;] ( t) + x( 00),
;= 1
where Ll j > 0 and 0 1 > (}2 > . . . > O. The preceding inequality implies that
$ a.,[ ; l d;X(o,d t) + x( 00) ] dt
..;; 1 Tl ($ [ d;X(;';] (t) + x( 00) ] dt ..;; 1 Tl ($ x**( t) dt.
+ n T )0 ;= l' + n 'T )0
6. WEAKENED AND WEAK TYPE OPERATORS
127
Passing to the limit twice we arrive at an inequality which can finally be
written in the form
i s T i s
O'T x *( t) dt 1 I x**( t) dt
o + nT 0
2. Operators from Lorentz spaces to Marcinkiewicz spaces. Let \fI(t) be a
quasiconcave function and let \[1*( t) = t / \fI( t). The norm in the Marcinkie-
wicz space M can be written in the form
(T 1, s > 0).
(6.1 ')
I II . = sup tf;(t) it y*(s) ds = sup [ tf;(t)y**(t) J.
0<1<00 t 0 0<1<00
Now if an operator A acts boundedly from Aq> to M¥'., this means that
tf;(t)(Ax)**(t) ,.; C oo x*(s) dqJ(s). (6.2)
If ep(t) is a quasi concave function, we shall consider the space A , where
cjJ(t) is the smallest concave majorant of ep(t). By (1.7) the condition that the
operator is bounded from . to Aq> can again be written in the form (6.2).
The main interpolation theorem to which the present subsection is devoted
concerns operators that act boundedly from the Banach couple (AcPo' A ) to
the Banach couple (M." , M m ). Before formulating this theorem, we intro-
oro. ..,.. I .
duce a class of spaces which are intermediate between Lorentz spaces.
Let epo(t) and epl(t) be two quasiconcave functions for which the quotient
epo(t)[ epl(t)]-I is decreasing. In this case, for T > 1 we have the inequality
epo( Tt) [ ep I ( Tt) J - I (; epo{ t) [ ep I ( t) J - I
or
epO(Tt) [ epO{t)J-I epl(Tt)[ epl(t)J-I,
which implies that Mq>o( T) Mq>l( T) for T > 1. For T < 1 the reverse inequal-
ity holds.
We consider a symmetric space E having the property that for the norm of
the dilation operator at the inequality
I 11 0 I/TII £ dMrp, (-r) + 00 II 0 I/TII £ dMrpo( 'T) < 00 (6.3)
is satisfied. In view of the above inequalities, this condition can be written in
the form
oo II 0l/TII£ d min{ Mrp,(-r)} < 00,
i = 0, 1.
( 6.4)
128
II. SPACES OF MEASURABLE FUNCTIONS
As usual, we denote by Yj and j the lower and upper dilation exponents of
cPj(t) and bya E and (3E the lower and upper dilation exponents of E. Then
N 1I01/TIIE dMrpo(r) ;;> N r- aE dMrpo(r)
= r-aEMrpo(r)1f + aE N Mrpo(r)r- arl dr
o T 8 0-a E I N
-? I.
Uo - a E
Condition (6.3) implies that a E > o. Similarly, from the finiteness of the
first term in (6.3) it follows that YI > {3E. Conversely, if a E > o and YI > (3E'
then lIal/rli E CT-aE+ E and Mrpo(T) CT 8 o+E for T > 1, and lIal/rll E
CT-(PE+E) and Mrpo(T) CT Y1 - E for T < 1. For sufficiently small E this implies
the convergence of the integrals in (6.3). Thus, (6.3) is equivalent to the
inequali ties
YI > /3E a E > o.
These inequalities are in turn equivalent to the relations
( 6.5)
" at II E = 0 ( t 8 0 ) as t 0
and
/latll E = 0(t T1 ) as t O. (6.6)
LEMMA 6.1. If one of the equivalent conditions (6.3)-(6.6) is satisfied, then the
space E is intermediate between the Lorentz spaces A o and A I.
PROOF. The function II al/TII E is decreasing, and therefore, by property 18°
of rearrangements, from (6. ) we obtain that
oo II ° I/TIIE d min { <pl r)} .;;; 2 00 II ° I/TII E d min { epj( r)}
.;;; 2 max{ epj(l)} oo Iiol/TIIE d min{ Mepj( r)} < 00.
Theorem 5.6 implies that E c in{cP;}. The quotient if;o(t)[ if;1(t)]-1 is gener-
alized monotone, and so from the remark to Theorem 5.9 it follows that
Amin{ i} is isomorphic to A o + A I. Thus, E c A o + A I.
To obtain the second imbedding, we note that
A = tOO Ilol/TIIEdmin{M",,(r)}
;;> tt II 0l/TII E d min{ Mrpi( r)} ;;> lIol/IIIE min{ Mrp,(t)}
"llal/tX(O,I)IIE . { cpj(l) }
min
Ilx(o,1)IIE cpj(l/t)
CPE( 1 / t) . { . { I }
p epAl) mm epj(l)}mm epj(1/t) .
6. WEAKENED AND WEAK TYPE OPERATORS
129
Hence
ACPE( 1)
'PE( 'T) = min { 'P;( I)} max 'Pi ( 'T).
An analogous inequality holds for <PE( T) and <p;( T). By Theorem 5.5 we have
E ::> A E ::> AC(max{cp;})-' and by Theorem 5.10 the space Ac(max{cp;})- is isomor-
phic to A o n A I. Thus, E ::> A o n A I.
THEOREM 6.1 (main interpolation theorem). Let four quasiconcave functions
CPo(t), CPI(t), 1/;o(t) and 1/;1(t) satisfy the following conditions:
1) The function CPo( s)[ cP I (s)] - I is decreasing.
2) The range of 1/;o(s)[ 1/;1(S)]-1 for 0 < s < 00 contains the range of
CPo(s)[ CPI(S)]-I (0 < s < 00).
If a linear operator A acts boundedly from the couple (A o' A ) to the couple
(M. , M. , . ), then it acts boundedly from every space E satisfying conditions
H O. ';'1.
(6.3)-(6.6) into the space EI(,6' where (t) is a measurable positive solution of the
equation
1/;o( (t»[ 1/;1( (t» J -I = CPo(t) [ CPI(t) J -I,
(6.7)
and
K(t) = 1/;o( (t»[ CPo(t)J -I = 1/;1( (t»[ CPI(t)] -I. (6.8)
PROOF. By Lemma 6.1 the space E is imbedded in A o + A I' and therefore
A is uniquely defined on it. Let x E E c A o+ I. For fixed t we set xo(s) =
sgn x(s)min{ x*(t), Ix(s)l} and xl(s) = x(s) - xo(s). Then x6(s) = x*(t) for
s t and xi(s) = 0 for s > t. Besides, x*(s) = x6(s) + xi(s), and so the
functions x6(S) and xi(s) belong to A o+ I. From the corollary of Theorem
5.9 we then obtain that x6 E A o and xi E A I.
By (6.1) we have
(Ax )**( (t» (Ax o )**( (t» + (Ax l )**( (t». (6.9)
According to (6.2), from the conditions imposed on A we obtain
1/1o(8(t»(Axo)**(8(t» ..; Co oo x6(S) d'Po(s)
= Cor t x*(t) d'Po(s) + foo x6(S) d'Po(s) l
By (6.7), for the first integral on the right we have
CoX*(t) t d'Po(s) = CoX*(t)'Po(t)
CPI (t )1/;0( (t» 1/;0 ( (t» (t
= Cox*( t) 1/1\( 8( t» = Co 1/1\( 8( t» )o x6(S) d'P\(s),
130
II. SPACES OF MEASURABLE FUNCTIONS
Next,
1/J)( 8(t»(Ax)**( 8( t» ..;; C) oo xHs) dcp)(s) = C) t xr(s) dcp)(s).
From the above inequalities it follows that
(Ax)**(8(t» ..;; c[ t x*(s) d 1/J ;» + Joo x*(s) d 1/J g» l
where C = max{ Co, C 1 }, and by (6.8)
K(t)(Ax)**(8(t» ..;; c[ t x*(s) d :: + foo x*(s) d :: l
By condition 1) we then have
K(t)(Ax)**(8(t» ..;; C (00 x*(-rt) d min Cp; 7't;
)0 ; t
and by property 18° of rearrangements we have
K(t)(Ax)**(8(t» ..;; C oo x*(7't) d min M<p,( 7').
Lemma 4.7 implies that
IIK(t)(Ax)**(8(t»IIE ..;; C oo 110)/TIIE d min M<p,(7')ll x lk
( 6.10)
The theorem is proved.
If (oo) = 00 « 00), then the strictly simple ( simple) functions are dense
in Arp' and therefore, if the operator A is defined on the strictly simple
functions (simple functions) and satisfies (6.2), then it admits an extension by
continuity to a bounded operator from Arp to Mo/.. Moreover, by Lemma 5.2
it is sufficient to verify condition 5.2 only on characteristic functions from .
3. Operators of strong, weakened, and weak types {E, F}. Let E and F be
two symmetric spaces. The bounded linear operators from E to F are often
called operators of strong type {E, F}.
DEFINITION 6.1. A linear operator A, defined on all strictly simple functions
if E( 00) = 00, or on all simple functions if E( 00) < 00, is called an operator
of weakened type {E, F} if it is bounded as an operator from A E to rpF)..
In 5.5 it was shown that for a symmetric space E we have the imbedding
A E C E C M(rpE).'
from which it follows that an operator of strong type {E, F} is an operator of
weakened type {E, F}.
If an operator belongs to the semigroup L (see 4.4), then, as we pointed
out on pp. 109 and 114, it acts boundedly in all Lorentz and Marcinkiewicz
6. WEAKENED AND WEAK TYPE OPERATORS
131
spaces. Consequently, if an operator A is of weakened type {E, F}, then so
are the operators M T and AT A for any T E and A E R I.
DEFINITION 6.2. A linear operator A, defined on all strictly simple (simple)
functions if CPE( 00) = 00 « 00), is called an operator of weak type {E, F} if
(Ax)*(t)fPF(t) C oo x*(s) dfPE(S).
( 6.11 )
Since (Ax)** (Ax)*, every operator of weakened type is an operator of
weak type. Consequently, every operator of strong type is an operator of
weak type.
LEMMA 6.2. If MrpF(SO) < So for some So > 1, then any operator of weak type
{E, F} is an operator of weakened type.
PROOF. The condition MrpF(SO) < So implies that the function cpp(t) =
t / cP t) satisfies the condition MrpF.(SO I) < 1. Therefore by Corollary 2 of
Lemma 1.4 we have
.!.. (T dt =.!.. (T fP (t) dt C fP (r) = C .
If )0 cP F( t) If )0 t If cP F( If)
We then obtain from (6.11) that
1 1 T dt 1 00
(Ax)**(r) c- ( ) x*(s) dfPAs)
If 0 CPF t 0
c 1 00
() x*(s) dfPE(S).
CPF T 0
( 6.12)
The lemma is proved.
As an example of an operator of weak type which is not an operator of
weakened type we may take the Hardy-Littlewood operator
1 1 t
Hx(t) = - x(s) tis,
t 0
( 6.13)
considered as an operator from LI to LI.
By (2.12) we have
I Hx(t)1 = .!.. (l x(s) ds
t )0
The last function is decreasing, and so
I l t
- x*(s) ds.
t 0
(HX)*(t)fPL,(t) = t(Hx)*(t) ( t x*(s) ds)* t 00 x*(s) ds,
132
II. SPACES OF MEASURABLE FUNCTIONS
i.e. H is an operator of weak type. Now let x(t) = X[O,h](t). Assuming that
t > h, we compute
(Hx )**(t)t = £ 1 [ £$ X[O,h]( r) dr]* tis
= £1 min{ 1, } tis = h + h In .
This implies that the inequality
t 1 00
(H(X[O.h»)**(t)t = h + h In h C 0 X[O,h)(S) tis = Ch
is impossible.
4. Interpolation theorems. From Theorem 6.1 and what was said following it
we obtain
THEOREM 6.1'. Let the operator A be of weakened types {Eo, Fo} and
{EI' F I }, and assume that the following conditions hold:
1) The function <PEo(S)[ <PE1(S)]-1 is decreasing.
2) The range of the function <P (s)[ <P F (s)] - I for 0 < s < 00 contains the
o 1
range of the function <PE (s)[ <PE (S)]-I (0 < s < 00).
o 1
If for a symmetric space E we have
I Ilal/TIlE dM'PE,(r) + I oo II aI/TilE dM'PEO(r) < 00, (6.14)
then A admits closure to a bounded operator from E to EI(,6' where (t) is a
measurable positive solution of the equation
<P F 0 ( ( t ) ) [ <P F 1 ( ( t» ] - I = <P Eo ( t) [ <PEl ( t) ] - \
( 6.15)
and
K(t) = <PFo( (t»[ <PEo(t)] -I = <PFl( (t»[ <PE1(t)] -I.
(6.16)
COROLLARY I. If y; and ; are the lower and upper dilation exponents of the
functions <PE' then under the hypotheses of Theorem 6.1' the operator A is
I
bounded, acting from any symmetric space E for which
II 0 t II E = o( t 6 0 ) as t 0
into the space EI(,6.
and
II0lilE = O«(Yl) as t 00 (6.17)
COROLLARY 2. If under the hypotheses of Theorem 6.1' we have Eo = Fo and
EI = F I , then the operator A is bounded, acting from a space E for which (6.14)
or (6.17) is satisfied into a space E; l' i.e. into a space with norm "xii E =
, 1,1
II x**" E.
6. WEAKENED AND WEAK TYPE OPERATORS
133
REMARK. If under the hypotheses of Theorem 6.1' the space Eo coincides
with Loo, then condition I) is satisfied automatically, and (6.14) takes the
form
I Ilal/TIlE dM'P,(r) < 00
Indeed, in this case we have fPE (t) = sgn t (sgn 0 = 0).
o
Now we consider an operator A which is of weak types {Eo, Fo} and
{EI' F 1 }, and assume that M m (so) < So for some So > 1. Then A is of
T'FI
weakened type {EI' F 1 }. The operator A satisfies the inequalities
(AXe)*( t)fPF( t) < C;fPE(mes e),
I I
which imply that for any (J E [0, 1]
(AXe)*(t)fP};0(t)fP:1(t) < Cd-oCffPl;O(mes e)fP:1(mes e). (6.19)
We write fPE,O(t) = fPl;o(t)cp:l(t) and fPF,O(t) = fP};0(t)fP:1(t). For the func-
tion fPF,O( t) for (J E (0, 1] we have
M'PF.. (so) ..; [ M'PFO (so) r -/J [ M'PF' (so) ] /J < sJ -/Jsg = so.
Repeating the calculation in (6.12), we obtain fr.om (6.19) that
(AXe)**(t)fPF,O(t) < CofPE,O(mes e) < CO<PE,O(mes e). (6.20)
As we noted on p. 130, this inequality implies that the operator A can be
extended by linearity and continuity to a bounded linear operator from A E.(J
into MF,o, i.e. to an operator of weakened type (A E.(J' McpF.,). Now we can
apply the main interpolation Theorem 6.1 to it.
As a preliminary, we show that every space E satisfying (6.14) satisfies a
similar condition with respect to the functions CPE,O(t) and fPI(t) for sufficiently
small (J. If we denote by Yo and o the lower and upper dilation exponents of
the function fPE,O(t), then by (1.29) we have o < 0(1 - (J) + I(J. By (6.14)
we have YI > {3 > a > o' and so for sufficiently small (J the inequality
Y 1 > {3 > a > o will be satisfied.
We may carry out similar arguments in the case where M m (so) < So for
T'FO
some So > 1.
We have arrived at the following assertion.
or
II (J t II = 0 ( t )' 1) as t 00. ( 6. 18)
THEOREM 6.2. If to the hypotheses of Theorem 6.1' the condition
McpF1(SO) < So or McpFo(so) < So (6.21)
for some So > 1 is added, then every operator of weak types {Eo, Fo} and
{ E I' F 1 } admits closure to an operator which acts boundedly from a space E
satisfying (6.14) into the space EI(.6.
134
II. SPACES OF MEASURABLE FUNCTIONS
REMARK. In the definitions of operators of weakened and weak types the
fundamental functions of the spaces are used. It would have been possible to
give similar definitions by replacing the fundamental functions by norms of
the dilation operators. For such operators, Theorems 6.1' and 6.2 remain
valid if in their formulation we replace the functions CPE.(t) and CPF(t) by II 0tll E.
, , ,
and II 0t II F' respectively. We note that in view of the submultiplicativity of the
,
nonn of the operator 0t the functions MrpE,(t) are replaced by the functions
11°7"IIE.
,
5. Optimal interpolation triples. Let cp(t) and 1/;(t) be two quasiconcave
functions. We introduce the operator
Sx(t) = 1/I t) tOO xes) dfj)(s).
By the concavity of ip(t) and the properties of rearrangements we have
(Sx )*( t) = 1/1 t) I tOO x(s) dfj)(s) I
1 roo 2 roo
" 1/1( t» )o x*( s) dfj)( s) " 1/1( t» )o x*( s) dq>( s).
Thus, an inequality of the fonn (6.11) is satisfied for the operator S. If
Mrp(so) < So for some So > 1, then according to Lemma 6.2 inequality (6.2)
will also be satisfied for S.
Now let two pairs CPo(t), 1/;o(t) and CPI(t), 1/;1(t) of quasiconcave functions be
given. Consider the operator
SOIX(t) = (00 x(s)d min {fj);(s)[ 1/I;(t) rl (6.22)
)0 I
By property 13° of rearrangements we have
I SOIX( t)I" (00 x*(s) d min { fj);(s) [ 1/I;(t) r l
)0 I
By 18°, on the right we have a monotone function. Therefore
(SOIX)*(t) " (00 x*(s) d min{ fj);(s) [ 1/I;(t)rl}
)0 I
" m}n {tOO x*(s) dfj);(s) [ 1/I;(t) r I}
2 n { ) (00 x * (s) dcp;( s ) } .
I 1/;; t ) 0
( 6.23)
Thus, the operator SOl satisfies an inequality of the type (6.11) both with
respect to the pairs CPo, 1/;0 and cP l' '¥ I.
6. WEAKENED AND WEAK TYPE OPERATORS
135
LEMMA 6.3. Let the functions rpo(t), rpl(t), 1/I0(t) and 1/I1(t) satisfy the hypothe-
ses of Theorem 6.1, and assume that the following conditions hold:
a) MC(>;(so) < So (i = 0, 1) for some So > 1.
b) Mcpl(sll) < 1 for some Sl > 1.
If for any operator A satisfying the hypotheses of Theorem 6.1 the function
K(t)(Ax)**( (t» belongs to E, then x E E.
PROOF. According to Theorem 3.4 there is an operator TEL such that
Tx = x*. By (6.23) and condition a) the operator SOl acts boundedly from Ai(;;
into Mo/ i , i = 0, 1. Therefore, so does the operator SOl T. By the hypothesis of
the lemma the function K(t)(SOI Tx)**( (t» then belongs to E, and, a fortiori,
K(t)(SOI Tx)*( (t» E E. As we mentioned above, the function SOl Tx = SOIX*
is decreasing, and consequently (SOl Tx)* = SOIX*. According to (6.22) we
have
K( t)( SOIX*)( 8( t» = K( t) oo x*(s) d min { cpj(s) [ 1/Il 8( t» r I}
K(t) oo x*(s) d min{ fPj(S) [ 1/Ij(8(t»] -I}.
From (6.8) it then follows that
K(t)(SOIX*)(8(t» oo x*(s) d min{ fPj(S) [ fP;(t)] -I}
ft x*(s) d min{ fPj(S) [ fPlt) r l }
t / $)
[1 - min{ fP;(t / SI)[ fPt(t) r I} ] x*( t)
> [1 - min { M CPi ( S 11) } ] x * ( t).
By condition 1) of Theorem 6.1 we have Mcpo(s} I) > Mcpt(SI-I) and conse-
quently
x*(t) ..;; [1 - M",,(si l ) rIK(t)(SOlX*)(8(t» E E.
LEMMA 6.4. Let the functions rpo(t), rpl(t), 1/I0(t) and 1/I1(t) satisfy the hypothe-
ses of Theorem 6.1 and condition a) of Lemma 6.3. Assume that the function (t)
determined from (6.7) is continuous and strictly increasing, and write ip;( - I( 'T)
= v;( 'T). If M" ( 'T - I) < 1 for 'T > 1, then for every function y E E" 6 there are an
I .
operator A acting boundedly from the couple {A o' A l} into the couple
{M.,. , M. , . }, and a function x E E such that Ax = y.
't'0. 't'1.
136
II. SPACES OF MEASURABLE FUNCfIONS
PROOF. We write x(t) = K(t)y**( (t)) and apply the operator SOl to this
function. Then we obtain
SOl x( t) = 00 lC(s)y **( 8(s» d min { ei';(s) [ 1/1;( t) ] -I}.
Using (6.8) for) = 0, 1, we obtain for TO > 1 that
SOl x( t) ;;. J.6 -I( '-0 1 ) 1/J.;( 8(s» [ Cf'i(s) ] -Iy**( 8( s» d min { ei';( s) [ 1/1;( t) ] -I}
«5 - 1 (t)
;;. 1f'l( t) [ cJJ( 8 - I( 'Tot)) r I y **( 'Tot) [ min { cp;( 8 -Ie-rot») [ 1/1;( t) r I}
- min { cp;( 8 - I ( t) ) [ 1/1;( t) r I } J.
By the continuity of the functions <p;( -I(t)) we may choose an index} and
a number 7'0 > 1 such that
min { <p; ( - I ( t) ) [ ;( t) ] - I }
are attained for this index). Then
SOlx( t) ;;. y**( TOt) { 1 - CPj( 8 -I(t» [ cJJ( 8 -I( Tot») r l
and
min{ <p;{ -I( 7' o t)) [ ;(t)] -I}
Since
Y**(TOt) = ('"0 1 y*(s) ds ;;. [I y*(s) ds = y**(t)
t t
and v j (t)[v j (7' o t)]-1 M')(7'O-I) < 1, we have
l' 0 [1 - M') ( 7'0) ] - I So I X ( t ) Y * * ( t ) Y * ( t) .
By Lemma 3.4 there exists an operator T E such that y =
7'0[1 - M p (7'O)]-ITS Ol x. By condition a) of Lemma 6.3 the operator
J
7'0[1 - M')(7'o)]-ITS oI acts boundedly from the couple {AcPo' A cP1 } into the
couple {M.,. , M. , . }.
'/"0. '/"1.
REMARK. If under the hypotheses of Lemma 6.4 the function (t) is strictly
decreasing, then the assertion of the lemma remains valid if Mp ( 1') < 1 for
,
7' > 1. The proof is analogous.
Lemmas 6.3 and 6.4 lead to the following assertion.
THEOREM 6.3. Let the functions fPo(t), fPI(t), o(t) and "-'I(t) satisfy the
hypotheses of Theorem 6.1 and the following conditions:
1) M "', ( so) < So (i = 0, 1) for some so> I.
2) Mcp,(s I) < 1 for some Sl > 1.
3) The function (t) determined from (6.7) is continuous and strictly increasing
(decreasing), and Mp(7'-I) < 1 (Mp(T) < 1) for i = 0, I and 7' > I, where
v;(t)=fP;( -I(t)). ' ,
6. WEAKENED AND WEAK TYPE OPERATORS
137
Then {A o' A I' E} is an optimal interpolation triple with respect to the triple
{ M,. , M,. , EI( 6} for any space E satisfying conditions (6.3)-(6.6).
't'0. 't'l. '
6. Operators of weakened types {Loo, Loo} and {Ll' L 1 }. The fundamental
function of the space L 1 is equal to t, and that of Loo is equal to sgn t.
Therefore every operator of weakened types (Loo, Loo) and {Ll' L 1 } is a
linear operator, defined for all simple functions, that satisfies the conditions
(Ax )**( t) Cox*(O) = Coil xii Loo'
t(Ax)**(t) ..;; Cl oo x*(t) dt = CdlxllL,.
From the first condition it follows that (Ax)*(t) Collxli L (weak type),
00
which in turn implies that IIAxl1 L = (Ax)*( +0) Collxll L , i.e. the operator
00 00
A is of strong type {Loo, Loo}. The second condition gives the inequality
J (Ax)*(t) dt C 1 11 x 11L 1 ' which implies that IIAxllLI C111x1lLI. Thus, the
operators of weakened type {Loo, Loo}, {Ll' L 1 } coincide with the operators
of strong type {Loo, Loo}, {Ll' L 1 }.
Corollary 2 and the remark to Theorem 6.1' lead to the following assertion.
THEOREM 6.4. If a linear operator A acts boundedly in the spaces Loo and L 1 ,
then it acts boundedly from any symmetric space E having the property that
"(Jtll E = o( t) as t 00, (6.24)
into any space E 1 ,t' i.e.
II(Ax)**IIE CllxllE.
( 6.25)
The conclusion of Theorem 6.4 is stronger than that of the interpolation
Theorem 4.3, and for this reason condition (6.24) is imposed on the space E.
It turns out that condition (6.24) is necessary for the validity of this conclu-
SIon.
THEOREM 6.5. Condition (6.24) is necessary for the boundedness of the identity
operator as an operator from a symmetric space E into the space El,t.
PROOF. If the identity operator acts and is bounded from E into E 1 , then
for any function x E E
IlxlIE";; 1. (s x*(t) dt ..;; qxli£. (6.26)
s)o E
We use (6.1'). Then
1I0'TXIlE";; 1. (s O'T x *( t) dt ..;; 'T 1. (S x**( t) dt
s )0 E 1 + In T s)o E
CT C 2 T
Il x** II E Il x il E (6.27)
1 + In T 1 + In T .
138
II. SPACES OF MEASURABLE FUNCTIONS
Hence II 0TII E = 0(7") as 7" 00.
REMARK. If condition (6.24) is satisfied for the space E, then it is satisfied
in every symmetric subspace of it. Therefore, every subspace of a symmetric
space E with condition (6.24) is an interpolation space between LI and Loo. In
5.7 we proved that in the spaces M", with the condition liml \[I(2t)/ \[I(t) =
1 there exist subspaces that are not interpolation spaces. Now if
inf \[I(2t)/\[I(t) > 1, then by (4.21) we have II0211M", < 2, which implies (6.24),
and for such spaces all subspaces are interpolation spaces.
7. Hardy-Littlewood and Hilbert operators. The operator
1 1 1
Hx(t) = - x(s) ds
t 0
is called the Hardy-Littlewood operator. It coincides with the operator SOl
constructed from the fundamental functions of the couples of spaces
(Loo, Loo) and (Ll' L 1 ), and is of strong type {Loo, Loo} and weak type
{Ll' L 1 }. The space Ll does not satisfy the hypothesis of Lemma 6.2, and, as
we saw on p. 132, the operator H is not of weakened type {LI' L 1 }. However,
if a symmetric space satisfies (6.24), then H acts in it. Indeed, by Lemma 4.7
we have
( 6.28)
IIHxllE = II I x(ts) tbt " I Ilollsll E tbllxllE,
( 6.29)
and the integral is finite under condition (6.24). This fact, together with
Theorem 6.5, leads us to the following important assertion.
THEOREM 6.6. For the Hardy-Littlewood operator (6.28) to act boundedly in a
symmetric space E it is necessary and sufficient that condition (6.24) be satisfied.
Inequality (6.29) implies that
IlxilE "llx**IIE = IIHx*IIE " £1 Iiolisil tbllxllEo
( 6.30)
Now we introduce the adjoint HI of H:
HI Y ( t) = f 00 Y ( 7") d7" .
I 7"
For nonnegative functions x(t) andy(t) we have by Fubini's theorem that
1 00 Hx( 7")Y( 7") d7" = 1 00 l T x(s) ds y( 7") d7"
o 07"0
= 1 00 x(s) f oo y( 7") d7" ds = 1 00 X(S)HlY(S) ds. (6.31)
o s 7" 0
6. WEAKENED AND WEAK TYPE OPERATORS
139
We now assume that the operator H acts from a space E into E II. Then for
x E E andy E E I we obtain
oo X(S)HIY(S) ds "IIHxIlE"llyIlE' "IIHIIE-+E"llxIIEilyIlE ' .
Hence IIHlyllEI IIH II E-.E II Ilyll E 1 . By the positivity of H and HI It IS
sufficient to verify their boundedness for nonnegative functions. Therefore, it
follows from the last inequality that HI acts in E I and IIHIIIEI-.E1
"H II E-.E II .
Conversely, if H I ac s in E I, then (6.31) implies that
00 Hx( t)y( t) dt "II H11IE.-+E.llxIIEllyIIE"
from which we obtain that IIHxllEl1 IIHIIIEI-.ElllxIlE' i.e., H acts from E
into Ell, and II H II E-.E II II HIli E1-.E 1 .
Thus, H acts from E into Ell if and only if HI acts in E I, and then
II H II E-.E II = 1/ HIli E1-.E 1 . (We note that in the proof of this statement we have
used no special properties of Hand HI; the assertion is therefore true for any
two positive integral operators which are adjoints of each other.)
Interchanging the roles of H and HI and taking account of the fact that
E 1 = E Ill, we conclude that HI acts in E I if and only if H acts in Ell. Thus,
H acts from E into Ell if and only if it acts in E 1I.
THEOREM 6.7. If the Hardy-Littlewood operator acts boundedly from a space
E into E II, then
II atll E-+E 11 = o( t) as t 00.
( 6.32)
PROOF. As was shown above, the operator H acts in Ell, and therefore,
from (6.1') we obtain
C 2 ".
IlaTxllEII 1 + In". IIxllE
in the same way as (6.27).
COROLLARY. If E is isometrically imbedded in Ell, then the condition that the
operator H act boundedly from E into E II implies condition (6.24).
From arguments based on (6.31) it follows that HI acts from E into E II if
and only if H acts in E I, and for this it is necessary and sufficient that
II at II E 1 = 0 ( t) as t 00. ( 6.33)
By (4.35) this condition is equivalent to
lIaTIIE-.EII = 0(1) as". O. (6.34)
140
II. SPACES OF MEASURABLE FUNCTIONS
We have proved the second part of the following assertion.
THEOREM 6.8. For the adjoint HI to act boundedly in a symmetric space E it
is necessary and sufficient that
1101"11 E = o( I) as 'T O. (6.35)
For it to act from E into E II it is necessary and sufficient that condition
(6.34) be satisfied, or, if E is isometrically imbedded in Ell, that (6.35) be
satisfied.
PROOF OF THE FIRST PART. By Lemma 4.7 we have
IIH1yliE = f oo y(s) ds = f oo Y(l'r) dr .;;; f oo II aI/TilE : IlyllE'
1" S E 1 'T E 1 '
If (6.35) is satisfied, the integral on the right is finite.
We show the necessity of (6.35). For 'T < I we have
f oo x*(s) ds > f '/T x*(s) ds > x* ( )f '/T ds = In .!llaTx*IIE'
1 s E 1 S E 'T 1 S E 'T
Hence
1101"XIIE = 1101"x*IIE (In(I/'T»-IIIHIx*IIE
(In( I / 'T» - III HIli E EII xII E,
and so (6.35) is satisfied. The theorem is proved.
The Hilbert operator
fx(t) = (00 x(s) ds
)0 t + s
is closely connected with the Hardy-Littlewood operator and its adjoint. The
Hilbert operator has positive kernel. Therefore it is sufficient to study it on
nonnegative functions. If x( t) 0, then
.! (' x(s) ds + f oo xes) ds 2 (1 xes) ds + 2 f OO xes) ds
t )0 1 S )0 t + SIt + s
= 2 (00 xes) ds .
)0 t + s
( 6.36)
On the other hand,
(00 xes) ds = (1 xes) ds + f OO x(s) ds .;;; .! (' x(s) ds + f oo xes) ds
)0 t + s )0 t + SIt + s t )0 1 S
Thus, for x(t) 0 we have
i(H + HI)x rx (H + Hl)X.
6. WEAKENED AND WEAK TYPE OPERATORS
141
This and Theorems 6.6-6.8 imply
THEOREM 6.9. For the Hilbert operator (6.36) to act boundedly in a symmetric
space E it is necessary and sufficient that
1I0lIIE = 0(1) as t O and 1I0lilE = o(t) as t 00. (6.37)
The notion of majorant function is connected with the Hardy-Littlewood
operator. We set
1 I t
0 1 ( t; x) = max x ( s) ds.
0<7"<1 t - 'T 7"
We establish the following important inequality:
0i(t; x) x**(t) (x E Ll + Loo).
Since /OI(t;x)1 OI(t;lxl), it is sufficient to prove the inequality for non-
negative functions.
Let O (to; x) = P > O (oo; x). We choose € <p - O (oo; x) and consider
the set e = {t: 0l(t; x) > p - €}. It has finite measure. According to (6.38)
the set e consists of those points t for which there exist points 'T E (0, t) such
that
( 6.38)
t x(s) tis - (p - e)t > T x(s) tis - (p - e)'T o
The function
g(t) = t x(s) tis - (p - e)t
is continuous. Therefore the set e is open, and consequently it consists of a
system of intervals. Let (a k , f3k) be an interval in e whose endpoints do not
belong to e. We show that on [ak' f3k] the function g(t) attains its minimum at
a k . Indeed, if the minimum were attained at t' E (a k , f3k) and the inequality
g(ak) > g(t') held, then there would not be a point 'T on (ak' t') where
g( 'T) < g(t'). There would not be such a point on (0, a k ], either, since the
condition ak fl e implies that g( 'T) g(ak) > g(t') for 'T E (0, a k ]. Thus, we
arrive at a contradiction to the fact that t' E e. In particular, from what has
been said it follows that g( f3k) g(a k ) or
(p - e)(l3k - a k ) ..;; f/3k x(s) tis.
CXk
Summing these inequalities over all intervals in e, we obtain
f (mes e
(p - €)mes e x(s) ds J{1 x*(s) ds.
e 0
142
II. SPACES OF MEASURABLE FUNCTIONS
By the definition of e we have mes e to. Therefore
p - € <: 1 l mese x*(s) ds..;; ! l t o x*(s) ds = x**(t o ).
mes e 0 to 0
Since € is arbitrary, we have
Or(t o ; x) <: x**(t o ). (6.39)
Now let Or(t o ; x) = p = Or( 00; x) > O. The set e defined as above for
€ E (0, p) has infinite measure and can contain an interval of the form
(a, 00). On this interval g( t) also attains its minimum at a; therefore for t > a
we have
it x(s) ds - (p - e)t l a x(s) ds - (p - €)a.
o 0
Dividing both sides by t and letting t tend to 00, we obtain
1 1 t 1 1 t
p - € <: lim - x(s) ds <: lim - x*(s) ds = x**( (0) <: x**(t o ).
t-+oo t 0 t-+oo t 0
Thus, in this case we also obtain (6.39).
We may introduce the function
1 f T
02(t; x) = sup x(s) ds,
7" >t 'T - t t
and obtain the inequality O!(t; x) <: x**(t) by similar arguments.
Finally, for the majorant function
O(t; x) = sup I I t x(s) ds
It - 7"\ > 0 t - 'T T
we have O(t; x) = max{ 0l(t; x), 02(t; x)}. Therefore
0*( t; x) <: max { Or ( t; x), O! ( t; x)} <: x * * ( t).
If a symmetric space E has the property that the Hardy-Littlewood
operator acts boundedly in it, then for'x E E we have
IIO(t; x)IIE = 119*(t; x)IIE <: Ilx**IIE <: CllxllE.
We note that for a decreasing nonnegative function x we have
1 I t
O(t; x) = - x(s) ds = Hx.
t 0
Therefore the inequality
IIO(t; x)IIE <: CllxllE (6.40)
implies that the Hardy-Littlewood operator acts In E and has norm not
exceeding c.
Using Theorem 6.6, we arrive at the following assertion.
@6. WEAKENED AND WEAK TYPE OPERATORS
143
THEOREM 6.10. Condition (6.24) is necessary and sufficient for the inequality
(6.40) to hold in a symmetric space E.
We still have to study the Hardy-Littlewood operator in symmetric spaces
with weight. Let E be a symmetric space and let p(t) be a positive function on
(0, 00). Consider the ideal lattice Ep(t)' i.e. the space of measurable functions
on (0, 00) for which
Ilxll Ep(/) = II xpll E < 00.
THEOREM 6.11. If p( t) is a submultiplicative function and
1 1 I
o IIOI/sIIEP(S-) ds < 00,
then the Hardy operator acts boundedly from Ep(t) into E;( ).
PROOF. We have
p(t) 1 t x('T) d'T I I Ix(ts)lp(t) ds
too
..;; I (al/slxlp)( t) p(s -) ds.
By the generalized Minkowski inequality (0.5) we have
IIpHxIIE" ..;; I II( al/slxlp)( t)11 EP(S -I) ds
1 1 I
..;; 0 IIOI/sIIEP(S-) dsllxplk
( 6.41 )
( 6.42)
( 6.43)
The theorem is proved.
REMARK 1. If the weight p(t) is decreasing, then the Hardy operator acts
boundedly in the space Ep(t) itself.
Indeed, from (6.42) it follows that
Ip( t)Hx( t)1 ..;; I (a l/SX*P)( t)p( s -I) ds,
and then by Lemma 6.7 we have
IlpHxliE ..;; I Ilal/Sx*pIIEP(s-l) ds ..;; I Ilal/sIIEP(s-l) dsllpxllEO
REMARK 2. Inequality (6.41) is equivalent to 8 E + I3 p < 1.
REMARK 3. If E = Lp and p(t) = (Y, then (6.41) assumes the form 'YP' < 1,
and (6.42) becomes the famous Hardy inequality
p'
I I tYHx l14 1 , ll tYx I14,
- 'YP
which is unimprovable (see [1 7]).
144
II. SPACES OF MEASURABLE FUNCTIONS
COROLLARY 1. If yp' < 1, then
1 00 tYPlx*(t)( dt .;;; 1 00 (YPlx**(t)IP dt .;;; ( p' , ) P 1 OO t¥Plx*(t)( dt.
o 0 1 - YP 0
COROLLARY 2. The Hardy-Littlewood operator of the form
1 l t
HIX(t) = tl+ 1 0 SIX(S) ds
is bounded in the spaces 4 with weight t Y , provided that (y - l)p' < 1.
Indeed,
,
II tYHIXll L = II t Y - I H(SIX)114 .;;; 1 ( P/)I II tYx114.
p - y- p
8. Operators of weakened and weak types {Lp;' Lq;}. We consider the case
where the spaces E; coincide with the spaces Lp;' and the F; with the spaces Lf/;
(i = 0, 1). We compute C{JE = t l / p ; and CfJF(t) = t1/f/;. From (6.15) and (6.16)
I I
we find that
8(t) = t<l/p,-l/Po)/(l/ql-l/qo) = t li and K(t) = t(1/Plqo-l/po'll)/(I/ql-l/qo) = t".
To satisfy condition 1) of Theorem 6.1' we have to assume that PI <Po.
Condition 2) is satisfied if qo =1= ql. Subject to the satisfaction of this inequal-
ity, at least one of the numbers q; is greater than 1. Therefore condition (6.21)
of Theorem 6.2 is satisfied. Because of (6.6), condition (6.14) imposed on the
intermediate space E can be written in the form
II (7tll E = o( 'T l/PI) as t 00 and II Otll E = o( t l/Po) as t o.
( 6.44)
Theorem 6.2 immediately implies the following theorem.
THEOREM 6.12. Let 1 PI < Po 00, 1 qo, ql 00, qo =1= ql and let A be
an operator of weak types {Lp;' Lq;}, i.e. let the inequalities
tl/ (Ax)*(t) .;;; c oo 'TI/Pi-IX*('T) d'T (i = 0,1) (6.45)
hold for the characteristic functiom of sets of finite measure if Po < 00, or for all
characteristic functiom if Po = 00. The operator A admits a bounded linear
extemion acting from every symmetric space satisfying conditions (6.44) into the
space Etr.t with norm
II x II Err ,tII = II t "X * * ( t Ii) II E.
6. WEAKENED AND WEAK TYPE OPERATORS
145
We remark that conditions (6.45) will automatically be satisfied if
tl/q;(Ax)*(t) Cllxll4,
(see the imbedding Theorem 5.5).
From Theorem 6.3 we obtain
THEOREM 6.13. If 1 PI <Po < 00, 1 < qo, ql 00, qo =1= ql' and condition
(6.44) holds, then {At'/po, Atl/ PI , E} is an optimal interpolation triple of spaces
with respect to the triple {Mt l - I/qo, Mt,-,/q" Et",t }.
Now we consider the case where the space E is an 4 with I/p =
(1 - A)/PO + A/PI (0 < A < 1). Since IIotl\4 = t llp , conditions (6.44) are
satisfied. In this case the space Et",t consists of all functions in LI + Loo for
which
oo (x**(tl')tvy dt < 00.
We carry out the change of variables t li = T. Then we obtain
oo (x**( r )Yr P / q - t dr < 00,
where I/q = (1 - A)/qo + A/ql. We have arrived at spaces which are usu-
ally denoted by Lr,p. In these spaces the norm is given by
{ 1 00 } lip
Il x l14. p = r- 2 (r - I) 0 [x**(t)Yt P / r - t dt .
Thus, under the hypotheses of Theorem 6.12 the operator A acts from Lp
into Lq,p (l/p = (1 - A)/PO + A/PI' 1/ q = (1 - A)/ qo + A/ ql' 0 < A < 1).
LEMMA 6.5. If 1 < r < 00 and P < PI' then Lr,p is imbedded in Lr,p,.
PROOF. Since x**(t) is decreasing, we have
(r - l)r-2 oo [x**(s)Ysp/r-t tis (r - l)r-2 ' [x**(s)Ysp/r-t ds
(r - I)(rp)-I[x**(t)JPt P / r .
Raising to the power (p I - p) / P > 0, we obtain
( r - 1 ) (PI-P)IP
Ilxllt::: rp [ x**(t) Y'-Pt(P,-p)/r,
and hence
( rp ) (Pl - p)lp -
[x**(t)JPltPllr-1 < r _ 1 [x**(t)JPtPlr-Illxll .pP.
146
II. SPACES OF MEASURABLE FUNCTIONS
Integrating with respect to t from 0 to 00, we finally obtain
( rp ) IIP-IIPI
IlxllL r , p1 r - 1 IlxIIL r . p .
The lemma is proved.
We note that for p = r > 1 the space Lr,p coincides with Lp. Indeed,
{ (CO } IIP
IlxllL,,= )0 [x*(t)Ydt
{ (00 } lip ( 2 ) lip
.;;; )0 [x**(t) Y dt = p 1 IlxllL",p'
On the other hand, since lIorllL" = t llp , condition (6.24) is satisfied, and the
operator ** is bounded in Lp' i.e. (see (6.30))
( 1 ) I IP
IlxllL",p = P Ilx**IIL p
( 1 ) IIP 1 I-rip
.;;; p 1 'T- Ilp d'TllxllL" = PI-II IlxllL p '
P 0 (p - 1) P
Thus, the norms in Lp,p and Lp are equivalent. Lemma 6.5 implies that for
p q (I < q < 00) the space Lq,p is imbedded in Lq,q = Lq.
Returning now to our interpolation Theorem 6.12, we see that for p < q the
operator A acts from Lp into Lq,p' which is smaller than Lq.
THEOREM 6.14. {Lpo' L p1 , Lp} is an optimal interpolation triple of spaces with
respect to the triple {LqO' L q1 , Lq,p}' if 1 < Pi qi < 00.
PROOF. Every operator acting boundedly from 4, into L'lJ' i = 0, 1, is of
weakened types {Lp" Lq,}. Therefore, as was shown above, such an operator
acts from Lp into Lq,p' and consequently the first triple of spaces is an
interpolation triple with respect to the second one.
Since 1 < Pi' qi < 00, there exist two numbers Ao < 0 and Al > 1 such that
1 1 - A. A.
0<--;= 7 +-2<1
Pj PoP 1
and
1
0<-=
q;
1 - A. A.
7+.2<1
qo ql
(j = 0, 1).
Every operator of weak types {L " Lq;} acts from the Lp, into the Lqi,Pi by
Theorem 6.12, and into the Lq, since Pi qi. The optimality of the triple
{Lo, L l' Lp} with respect to {LqO' L q1 , Lq,p} then follows from Theorem 6.13.
The theorem is proved.
6. WEAKENED AND WEAK TYPE OPERATORS
147
It is natural to call the operators acting from 4; into LilJ' with Pi qi'
"improving". Theorem 6.14 refines the classical Riesz- Thorin theorem (see
Chapter I, 4.2) for "improving" operators.
We now consider the case where the space E is an Lr,p with PI < r <Po. A
calculation shows that II (It 11 4 ,p = t Ilr. Therefore (6.44) is satisfied.
In this case the space Et",t consists of all functions in LI + Loo for which
oo [(x**(tl')tP)**Ytplr-I dt < 00.
Since (11r - IIp)p' < 1, we can apply Corollary 1 of Theorem 6.11 with
'Y = II r - lip. It implies that the quantity on the left is equivalent to
00 [( x ** ( t I' ) t P) * Y t pi r - 1 dt.
If we use properties 21 0 and 22 0 of rearrangements, we obtain that the last
quantity is equivalent to
(00 1 1 00
J [x**(tl-L)]PtPv+plr-1 dt = - [x**(t)]PtP«v+l/r)/I-L)-1 dt.
o 0
We have arrived at the following assertion.
THEOREM 6.15. If the hypotheses of Theorem 6.12 are satisfied, then the
operator A admits an extension to a linear operator acting boundedly from the
space Lr,p (PI < r < Po) into LSJJ' where
r lip I - 1 I Po
s = rv + 1 ' = 1 I q I - 1 I qo '
v=
1 I (PI qo) - 1 I (Poql)
l/ql-l/qo
9. Convolution operator. For functions defined on (0, 00) the convolution
operator is defined by
x*y(t) = t x(s)y(t - s) tis.
( 6.46)
It can be verified immediately that this bilinear operator acts boundedly
from LI X Loo into Loo and from LI X LI into LI. If we fix the first factor
x ELI' then the linear operator thus obtained acts boundedly in the spaces
LI and Loo, and so it acts in every interpolation space G between LI and Loo
(see Theorem 4.3 and what follows). Thus, the convolution operator acts
boundedly from LI X G into G.
Let G I be the space associated with G. For x E G I andy E G we have
Ix*y(t)1 =I t x(s)y(t - s) tis I =I oo x(s)Zt(s) tis I '1IxlkIIZtIIG'
148
II. SPACES OF MEASURABLE FUNCflONS
where
z/(s) = { (t - s)
fors t,
for s > t.
Clearly Zt E G and IIZtllG IIYIIG. Therefore
Ilx*yIIL IlxlIGlllyllG.
This means that the convolution operator acts boundedly from G 1 X G
into Loo.
We now fix the second factor y E G. Then we obtain a linear operator
acting boundedly from G 1 into Loo and from LI into G. We apply the
interpolation Theorem 6.1' to the operator, putting Eo = G I, EI = L 1 , Fo =
Loa and F 1 = G. In this case we have
<PEO(S) [ <PE.(S)]-1 = <PFO(S) [ <PF1(S)]-1 =[ <PG(S)]-I,
and so, conditions 1) and 2) of this theorem are satisfied. In view of (4.40),
condition (6.24) on the intermediate space E takes the form
I II (J 1/ TII E dr + 00 II (J I / TII E d[ r McpG( )] < 00.
Using the equivalence of (6.3) and (6.6), this condition can be written in the
form
1I0tIIE=0(tl-YcpG) ast O and 1I0lll£=0(t) ast oo. (6.47)
The functions 8 and K determined from (6.1) and (6.6) are thus 8(t) = t and
K( t) = t - I<pG( t). Consequently the norm in the space F = EI(,6 is defined by
IlzilF = z**(t) %(t)
t £
( 6.48)
From Theorem 6.1' it now follows that the convolution operator acts
boundedly from every space E X G (where G is an interpolation space between
Loa and L 1 , and the symmetric space E satisfies (6.47») into th£ space F with
norm (6.48).
We note that in (6.47) and the expression for the norm (6.48) only the
fundamental function of the space G is present. Therefore, for spaces with a
given fundamental function <p( t) the preceding assertion will be strongest
when G is largest. By the imbedding Theorem 5.7, G will be largest if it
coincides with the Marcinkiewicz space with the same fundamental function.
Then our assertion takes the following form.
@6. WEAKENED AND WEAK TYPE OPERATORS
149
THEOREM 6.16. Let cp( t) be an arbitrary quasiconcave function. The convolu-
tion operator acts boundedly from E X *' where E satisfies the conditions
II at II E = 0 ( t 1 - Yep) as t 0 and II at II E = o( t) as t 00, ( 6.49)
into the space F with norm
II zll F = II z**( t)t -Icp( t) II E.
( 6.50)
Moreover,
Ilx*yllF CllxIIEIIYIIMcp.'
(6.51 )
where cp * = t I cp( t).
From the assertion formulated earlier we obtain the inequality IIx*yllF
C(x)IIYllcp.' which implies (6.51) in view of the uniform boundedness princi-
ple.
We consider the special case where cp(t) = t l / q and E = 4. Condition
(6.49) is satisfied if p > 1 and 1 I p + 1 I q > 1. The norm (6.50) can be
calculated by the formula
Ilzll'i = oo [z**( t) Ytp/q-p dt = oo [z**(t) YtP(I/P+ I/q-I)-I dt,
and consequently F coincides with Lr,p, where Ilr = lip + l/q - I.
THEOREM 6.17. lfp > 1 and lip + l/q - 1 > 0, then
II x* Y II L"p CII xii IIYII M,I-I/q,
where 1 I r = 1 I p + 1 I q - 1.
( 6.52)
This assertion is a sharpening, in the case p > 1, of Young's famous
inequality. Indeed, the space M t l-l/q is larger than the space Lq (see Theorem
5.7) and the space Lr,p is imbedded in Lr (see Lemma 6.5), and so, (6.52)
implies the Young inequality
Ilx X yllL, Cllxl14llYllL q .
(6.53 )
A classical example of an operator of convolution type is the Riemann-
Liouville integral of fractional order
"x = rL ) t (t - S)"-IX(S) ds.
( 6.54)
If 0 < a < 1, then to:- I belongs to the Marcinkiewicz space Mta. Then
Theorem 6.5 implies the following theorem.
150
II. SPACES OF MEASURABLE FUNCflONS
THEOREM 6.18. The operator (6.54) of fractional integration acts boundedly
from any space E satisfying the conditions lIatll E = o(t Q ) as t O and lIatll E =
o(t) as t 00 into the space F with norm
IlxilF = Ilx**(t)t-aII E .
In particular, if p > 1 and 1/ p - a > 0, then the operator a acts from 4
into Lr,p, where l/r = l/p - a.
7. lbe singular Hilbert operator
In this section we shall consider functions x(t) defined on the whole axis
(- 00, 00). The singular Hilbert operator* is defined by
Sx(t) = lim f x(s) ds = p.v. f oo x(s) ds, (7.1)
8-+0 It - sl ;;> 8 t - s - 00 t - s
where p.v. means that the integral is taken in the sense of principal value.
If x( t) = X[a,b]( t), then
SX[a,b](t) = Inl(t - a)/ (t - b)l. (7.2)
From this it is clear that the operator acts in neither Loo nor LI. The
operator S is not defined for the function Xra,oo](t), since the right side of (7.1)
is identically equal to - 00. In connection with this there arises the problem
of determining a class of functions for which the integral in (7.1) exists for
almost all values of t. It turns out that L I ( - 00, 00) may serve as such a class.
For the proof of this deep result we establish three lemmas.
LEMMA 7.1. For the function x(t) = L7 ILj/(t - a j ), where a l < a 2 < . . . <
an and ILj > 0, the following equality holds:
2 n
n 1xl ( 7") = - L ILj (7" > 0).
7". 1
}=
PROOF. We consider the set e of those points where x(t) > T. The function
x(t) is decreasing on each of the intervals (- 00, a l ), (aI' a0, . . . , (an, 00) and
is negative on (- 00, at). Therefore it assumes the value T once on each of
the intervals (aI' a 2 ), . . . , (an' 00) at the points aI' . . . , an. Thus, mes e =
L(a; - a;) = La; - La j . The numbers a j are roots of the equation
n
L
j=l
IL
t - a.
J
=7"
· Editor's note. This operator is commonly called the Hilbert transform.
7. THE SINGULAR IDLBERT OPERATOR
151
or the equation
n n
'T II (t - a j ) - J-Lj II (t - a;) = O.
j=1 j=1 ;=Fj
By Viete's theorem we have
'T Lj_1 a; + LJ= I J.L.;
lX i =
;=1 'T
From this we obtain that mes e = 'T -I L 7 J-Lj. It can be shown similarly that
the measure of the set of points at which x(t) < - 'T is equal to the same
number. The lemma is proved.
We write
S6X(t) = 1 x(s) ds.
It-sl>8 t - s
It is obvious that the operator S6 is defined for every function x E
LI(-oo,oo).
LEMMA 7.2. Let
S6. X ( ti) > 'T > 0
,
(i = 1, 2, . . . , n),
(7.3)
where the intervals Ii = [t i - 0i' t; + 0;] are disjoint. Then
n 8
0; -llxIILI.
;=1 'T
PROOF. First we assume that x(t) O. In the integral
S6X(t.) = f x(s) ds
,I [ S _ I;] ;;> 8; t; - s
we replace the function l/(t; - s) by a step function. We choose points
s;(1), . . . , Si(N;) outside Ii so that for any system of points (s(1),..., S(N»
containing these points we have
N-I 1
(j) f sV+I) x(s) ds - Ss,x(t j ) < f.
j = 1 t; - S s()
(7.4 )
We combine all points s;0) (i = 1, . . . , n,j = 1, . . . , NJ in one system,
arranged in increasing order: (J = (Sl' . . . , SN)' and denote by (Jk the collec-
tion of all points in (J which are in the interval (t k - Ok' t k + 8 k ). We consider
the functions
Yi(t) =
s) fl a,
J-Lj
,
t - s.
J
J-L.
Zj(t) = 2 '
sEa. t Sj
') )
N
z(t) =
j=1
J1y
,
t - S.
J
152
II. SPACES OF MEASURABLE FUNCflONS
where J1y = f +1 x(s) ds. It is obvious that z(t) = y;(t) + z;(t). If we choose f
so that f < S6.(t;) - 'T (i = 1, . . . , n), then by (7.4) we have
,
y;( t;) > S6.X( t;) - f > 'T. (7.5)
,
The function y;( t) is decreasing on each of the intervals where it is defined,
and therefore (7.5) implies that y;( t) > 'T for 'T E [t; - 0;, tj]. Then for these 'T
we have either z(t) 'T /2 or z;(t) < -'T /2. We denote by eo the set of those t
for which z( t) > 'T /2, and by e j the set of those t for which z;( t) < -'T /2.
Then
n
n
U [t; - 0;, t; ] C U e j .
;=1 ;=0
From the proof of Lemma 7.1 we obtain
n n 2 N-I 2 n
0; mes e; = - J-Lj + - J1y
. 1 . 0 'T . 1 'T. 1 E
I = I = J = I = j OJ
4 N - 1 4 00
- J-Lj - f x(s) ds.
'T j=1 'T-oo
Similarly, for a nonpositive x(t) we can show that the inequality S6x(tJ <
-'T implies that L7=1 0; (4/'T)lIxIlLI.
For an arbitrary function x E Ll we have
S6. x (t) = S6. X +(t) - S6. X _(t).
, , ,
Inequality (7.3) implies that either S6.x +(t;) > 'T /2, or S6.x -(t;) < -'T /2. If
, ,
the index k denotes those indices for which the first inequality is satisfied,
and k' denotes those for which the second is satisfied, then
n 888
0; Ok + Ok' -llx+IIL I + -IIX-IIL I = -IIXIIL I .
k=1 'T 'T 'T
REMARK. If at the points t; we have
I S6,X( tj)1 > 'T > 0,
i = 1, 2, . . . , n,
then
n 16
0; -llxIILI.
;=1 'T
(7.6)
LEMMA 7.3. For x E L 1
mes { t: lim IS6X(t)1 > 'T } < 64'T-11IxIIL..
8--+0
(7.7)
7. THE SINGULAR IDLBERT OPERATOR
153
PROOF. The measure of the set e being considered is the supremum of the
measures of the compacta contained in it. Therefore it is sufficient to prove
the inequality mes K < 64'T- 1 1IxII L for any compactum K at whose points
- I
we have lim8 IS8x(t)1 > 'T. For every t E K the inequality IS8x(t)1 > 'T, is
satisfied for some 0, and so, the intervals (t - 8, t + 8) cover K. From this
cover we choose a finite cover (t l - ° 1 , t l + 0 1 ), . . . , (t N - ON' t N + 8 N ) so
that in this system only consecutive intervals intersect each other. Then the
intervals with even and with odd indices form two systems of disjoint
intervals. At least one of them (for example the one with even indices)
contains a part of K with measure not less than i mes K. Thus 4 mes K
L 20 2k . From (7.6) it follows that mes K 64'T- I ll x II LI .
From Lemma 7.3 it follows that
mes{t: 6 , O IS/)x(t) - S6,x(t)I>'T}';;; mes{t: I S/)x(t) I > ; }
128
7l1 x ll L I.
THEOREM 7.1. Iff ELI' then the limit in (7.1) exists almost everywhere. The
operator S is of weak type {LI' L I }, i.e.
t(Sx)*(t) CllxllLI. (7.8)
PROOF. We prove the first part of the theorem. We represent the function
x(t) as the sum of a step function xl(t) of compact support and a function
x 2 (t) with sufficiently small norm in LI' Then
lim I S8 x (t) - S8,x(t)1
8,8' -+0
lim
8,8' -+0
I S8 X I(t) - S8,X I (t)1 + lim 1 S8 X 2( t) - S8'x 2 ( t)l.
8,8'-+0
At the points of continuity of xl(t) the first term equals zero, since
S8XI(t) = S8,x l (t) for sufficiently small 8 and 8'. The measure of the set
where the second term is greater than 'T does not exceed the number
128'T-Illx21IL1. Thus, the left side does not exceed 'T anywhere outside a set of
measure 128'T- I llx 2 1IL . Since IIx 2 11 L can be made as small as we wish, and'lT
1 1
is arbitrary we have
lim IS8X(t) - S8,x(t)1 = 0
8,8' -+0
almost everywhere. The Cauchy test implies the first assertion of the theorem.
154
II. SPACES OF MEASURABLE FUNCflONS
Inequality (7.7) implies that
mes{t: ISx(t)1 > T} < 64T- I llxllL I
almost everywhere, and this is equivalent to (7.8).
Inequality (7.8) implies the following corollary.
COROLLARY 1. If a sequence of functions X n converges to a function x in Lh
then SX n converges to Sx in measure.
COROLLARY 2. If x E 4, 1 p < 00, then the limit in (7.1) exists almost
everywhere.
In fact, x(t) = x(t)X(-a,a)(t) + x(t)(1 - X(-a,a)(t». The first term belongs
to L I , and therefore SXX(-a,a)(t) is defined almost everywhere. For the second
term with t E ( - a, a) the corresponding integral is absolutely convergent.
Consequently, S8x(t) is defined for almost all t in (- a, a), and thus for
almost all t in ( - 00, 00).
LEMMA 7.4. If a sequence of functions xn(t) converges to a function x(t) in 4,
then Sxn(t) converges to Sx(t) in measure on every set of finite measure.
PROOF. We have xn(t) = x n (t)X(-2a,2a)(t) + xn(t)(1 - X(-2a,2a)(t». The se-
quence of the first terms converges to x(t)X( -2a,2a)(t) in L I , and consequently
it converges in measure according to Corollary 1. The second terms converge
uniformly on ( - a, a):
ISxn(I - X(-2a,2a»)(t) - Sx(I - X(-2a,2a»)(t)1
( xn(s) - x(s) ds
= J lsl >2a t - s
{ i IXn(s) - x(s)( ds } lip
Isl >2a
{ 1 ds } lip' .so;; { J'X> ds } lip' II X n - xll-4'
Isl>2a It - dsl P ' aSP'
LEMMA 7.5. Let e be a measurable set of finite measure. Then
. 2 mes e
(SXe)*( t) = arcslnh .
t
(7.9)
PROOF. By Corollary 1 of Theorem 7.1 and property 11 0 of rearrangements
it is sufficient to prove (7.9) for a set e consisting of a finite number of
7. THE SINGULAR IDLBERT OPERATOR
155
disjoint intervals [ai' bi] (a l < b l < a 2 < b 2 < . . . < an < b n ). Then by (7.2)
we have
n t - a.
SXe(t) = L In t _ ;
j=l }
We consider the equation
SXe(t) = 'T > o.
(7.10)
This equation has at least one root in every interval (a j , b j ) and (b j , a j + I) and
in (b n , 00). Tnus, it has at least 2n roots. For tEe our equation is equivalent
to
n
II
j=l
t - a.
J = _e'T
t - b. '
}
and for t fl e to
n t - a.
II J = e'T.
j= 1 t - b j
Each of these equations has n roots, and so our equation has exactly 2n
roots. A similar calculation applies to the case 'T < O. Hence it follows that
the function SXe(t) is monotone in every interval where it is continuous.
Denote by the root of (7.10) in the interval (a j , b j ) and by f3 j the root in
(b j , a j + I) (a n + I = 00). For'T > 0 we have
n n n
mes(t: SXe(t) > 'T) = (13.i - a j ) = L f3 j - L a j .
j=l j=l j=l
By Viete's theorem we have
n
j=l
a. + e'T b.
J }
a -
j - 1 + e'T
and
n 2a. - e'T b.
/3j = 7 1 _ e'T J
j=l
Hence
2( a. - b.)e'T
mes(t: SXe(t) > 'T) = J }
1 - e 2 'T
mes e
sinh T .
Similarly,
mes( t: SXe ( t) < -'T) = mes e j sinh 'T.
Thus,
nI SXeI ( 'T) = 2 mes e jsinh 'T,
which is equivalent to (7.9).
156
II. SPACES OF MEASURABLE FUNCflONS
THEOREM 7.2. For the singular Hilbert operator to act boundedly in a
symmetric space E it is necessary and sufficient that
lIalll E = 0(1) as -t 0 and 11 0 /11 E = o(t) as t 00. (7.11)
PROOF. For every p E (1, (0) inequality (7.9) implies that
I / . 2 mes e
t l P(SXe)*(t) = t l P arcslnh
t
.s;;; (mes e)I/P sup r l / p arcsinh 2 = C p (00 X:(t) dt I / p .
O<-r<oo 7")0
Thus, S is an operator of weak type {Lp, Lp} (1 < p < 00), and hence also
an operator of weakened type {4, 4} according to Lemma 6.4. From (7.11)
and Theorem 1.3 on submultiplicative functions it follows that there exist
numbers Po > Ijpo and PI > 1 - IjpI such that lIalll E = o(tJ'o) as t 0 and
lIalll E = o(tl-J'I) as t 00, where Po and PI are numbers from (1,00). Then
(6.5) is satisfied for the space E and the spaces Eo = Lpo and EI = 41.
Theorem 6.1 implies that the restriction of S to strictly simple functions
admits a bounded extension to an operator acting in E. By Lemma 7.4 this
extension will coincide with the operator S itself.
We now show that the hypotheses of the theorem are necessary for the
operator S to act in the symmetric space E. We consider an arbitrary
function x E E equal to zero on the negative semiaxis. Then the function
x( - t) belongs to E, as well:
f oo x( - s) 1 00 x(s) ds E E.
p.v. ds = p.v.
-00 t-s 0 t+s
If we restrict ourselves to t > 0, we see that
L oo x(s)
X ( O 00 ) ( t) ds E E.
, 0 t + s
If we introduce the symmetric space E(O, (0) of functions on the semiaxis,
consisting of the restrictions of the functions from E to (0, (0), then it is clear
from the preceding that the Hilbert operator r acts in this space. By Theorem
6.8 the space E(O, 00), and consequently the space E as well, has properties
(7.11 ).
8. Interpolation theorems for spaces with different measures
1. The operators II and R. Let 9.R be a space with a-finite measure p..
Similarly to 2, we may introduce the notion of distribution function for any
nonnegative measurable function u(m), m E WC, as follows:
nu ( 7") = p.( m: u ( m) > 7") (7" > 0).
8. SPACES WITH DIFFERENT MEASURES
157
We assume that a one-to-one continuous linear mapping II is defined from
S(9.R, J-L) into S(O, 00) having the following property:
n1ul( T) = n 1IIul ( T) (T > 0). (8.1)
In the same way as in 2, the rearrangement u*(t) of a function u(m) is
defined to be the function
u*(t) = inf{ T: n1ul( T) < t}.
This function is defined on (0, (0). If the operator II has property (8.1), then
(IIu)*(t) = u*(t) (0 < t < 00). (8.2)
(We note that on the right and left sides the operation * is applied in function
spaces of different nature.)
I t is easy to verify that
f lu(m)1 dp. = 1 0 00 u*(t) dt and ess sup lu(m)1 = ess sup u*(t). (8.3)
mEID1 O<t<oo
Let E(O, (0) be a symmetric space of functions on (0, 00). Denote by E(9.R)
the inverse image of £(0, 00) under IT. Obviously E( ) is a linear manifold in
S(9.R, J-L). We may introduce a norm in it by the formula
IluIIE( ) = IIITuIIE(O,oo) = II(IIu)*IIE(o,oo) = Ilu*IIE(O,oo). (8.4)
With this definition the operator IT will realize an isometric imbedding of
E(9.R) in £(0, 00). From (8.3) it follows immediately that the spaces Ll(9.R)
and Loo(9.R) coincide with the ordinary spaces Ll and Loo on measure spaces.
The space E(9.R) is Banach. Indeed, if a sequence Un is Cauchy in E(IDl),
then the sequence ITu n is Cauchy in E(O, 00), and consequently converges to
an element v. By Theorem 4.1 the sequence ITu n converges to v in L1(0, 00) +
Loo(O, 00), and so Un converges to some element u in L1(WC) + Loo(WC). The
continuity of II, which we assumed, implies that v = IIu, and consequently
v E E(9.R). Since IIu n ITu in E(O, (0), we have Un u in E(9.R).
I t is useful to note that to construct the space E(9.R) it is sufficient to know
only that the operator II with the indicated properties exists, without knowing
its concrete form. Indeed, £(9.R) may be defined as the collection of all
measurable functions u on for which u* E E(O, 00), with norm II ull E( ) =
lIu*IIE(o,oo)" In fact, if u* E £(0, 00), then by (8.2) we have (IIu)* E E(O, 00),
and, since the space £(0, 00) is symmetric, ITu E £(0, 00), i.e. u E
IT-1E(0, 00) = E(9.R). Conversely, if u E E( ), then by (8.4) we have u* E
E(O, 00) and the norm of u is defined by the above formula.
We now assume that the restriction of IT to L1(WC) + Loo(WC) has a left
inverse R that maps L1(0, 00) + Loo(O, 00) onto Ll(9.R) + Loo( )' acting with
norm 1 from Ll (0, 00) into Ll (9.R) and from Loo(O, 00) into L oo(WC).
158
II. SPACES OF MEASURABLE FUNCfIONS
We give the most important examples of measure spaces and the operators
II and R.
1) One-to-one image of the semiaxis. We assume that there exists a one-to-
one measure preserving mapping w(m) of the measure space we with measure
J-L onto the semiaxis with Lebesgue measure. Then II is defined as IIu(t) =
u(w -}(t». This operator has all the necessary properties and has the inverse
operator Rx(m) = x(w(m».
In particular, if we = (- 00, 00) with Lebesgue measure, then as w we may
take the mapping shifting the intervals [k, k + 1) (k 0) to the intervals
[2k, 2k + 1), and the intervals [- k, - k + 1) (k 1) to the intervals
[2k - 1, 2k).
If 9R is the semiaxis (0, 00) with the measure J-L (dJ-L = ds / s), invariant with
respect to dilations, then the function w(1n s), where w(t) is the same as above,
has the desired properties.
2) The interval (0, 1) with Lebesgue measure. Denote by II the operator
which maps a function u E S(O, 1) onto the function
IIu(t) = { u(t), 0 < t < 1,
0, t 1,
and by R the operator of restricting functions defined on the semiaxis to the
interval (0, 1). All properties are obviously satisfied.
3) Sequence spaces. Let 9R consist of countably many points with measure
equal to one. The functions on 9R are identified with sequences {u m } in a
natural way. To a sequence u = {urn} we assign the step function IIu(t) = U m
for m - 1 t < m. The operator IIu has the required properties. Its left
inverse is the averaging operator
Rx = {{m_l x(s) ds}.
It is obvious that R has norm equal to one as an operator from L}(O, 00)
(Loo(O, 00» to I} (/ 00 ), which coincides with L}(9R) (Loo(9R» in our case.
2. Interpolation theorem. Let 9R and 9R' be two measure spaces for which
there exist the operators II, R and II', R' with the properties indicated above.
If the operator V maps functions on 9R onto functions on 9R', then the
operator T = II'VR takes functions on (0,00) into functions on (0,00). If
T = II'VR acts boundedly from a symmetric space E(O, 00) into a symmetric
space F(O, 00), then V acts boundedly from E(9R) into F(9R'), and has smaller
norm. Indeed, the inequality IIII'VRxIIF(o,oo) CllxIlE(o,oo) implies that for
u E E(9R) and x = IIu we have
/I Vull F(9.n') = II II' Vul/ F(O,oo) = /I II' VR IIull F(O,oo)
ell IIull E(O, 00) = ell ull E(9.n).
8. SPACES WITH DIFFERENT MEASURES
159
The converse will be true if the operator R has norm one as an operator
from E(O, 00) to E( ). In fact, then we have
IIII'VRxIIF(o,oo) =11 VRxIIF(W1') ell RxIIE(9.n) cllxIIE(O,oo).
Denote by (9R, ') the collection of all linear operators acting bound-
edly, with norm not exceeding one, from L}(9R) to L}( ') and from Loo( )
to Loo( ').
From the interpolation Theorem 4.3 we obtain
THEOREM 8.1. For the spaces (L}(WC), Loo(9R), E) to form an interpolation
triple with respect to the triple (L}(9R'), Loo(WC'), F) with interpolation constant
equal to one it is necessary and sufficient that the inequality
tt v*(r) dr tt u*(r) dr (0 < t < (0) (8.5)
for u E E and v E L}(9R') + Loo(9R') imply that v E F and Ilvil F IlulI E .
PROOF. Necessity. (8.5) and (8.2) imply that
t (II'v)*(r) dr t (IIu)*(r) dr.
By Theorem 3.4 we then have II'v = TIIu, where T E . Then v =
R'TIIu. The operator R' TII obviously belongs to ( , 9R'), and therefore
from the interpolation properties of E and F it follows that v E F and
IIvll F = IIR'TIIuII F IlulI E .
Sufficiency. Let V E (9R, WC'). We have II'VR E , and by (3.16) for
x E L}(O, 00) + Loo(O, 00) we have
tt (II'VRx)*(r)dr tt x*(r) dr.
For u E E we set x = IIu. Then
t (II'Vu)*(r) dr tt (IIu)*(r) dr,
or, by (8.2),
tt (Vu)*( r) dr tt u*( r) dr.
From the properties of E and F we obtain that Vu E F and II Vull F
IlulI E .
3. Con-espondence between interpolation spaces. From the general discussion
in 4.6 of Chapter I it follows that if E(O, 00) is an interpolation space
between L}(O, 00) and Loo(O, (0) with interpolation constant equal to one,
then so is E(9R) such a space between L}( ) and Loo(WC). (Indeed, if
V E (9R, 9R), then II VR E L, and so it acts in E(O, 00); and then V acts in
E(9R) with not greater norm.)
160
II. SPACES OF MEASURABLE FUNCfIONS
In 4.6 of Chapter I a scheme is given according to which from every
interpolation space G between L}(W"l) and Loo( ) an interpolation space EG
(with the notation of subsection 6, a space E( G» can be constructed between
L}(O, 00) and Loo(O, 00). This space consists of all functions x for which
RTx E G for any operator T E . The norm in EG is introduced by the
formula
IlxllE G = sup IIRTxIIG.
TE
(8.6)
Under a certain assumption on the operator R this norm can be calculated
explicitly. We shall say that R satisfies condition A) if there exists a set
e R C (0, 00) having the following two properties:
1) For t E e R and x E L}(O, 00) + Loo(O, 00)
1 t (Rx*)*( r) dr = i t x*( 'T) d'T. (8.7)
o 0
2) If for u, v E L}(W"l) + Loo( ) the inequality
t U * ( r) dr " t V * ( r) dr
holds when t E e R , then it holds for all t E (0, 00) as well.
Assuming that condition A) is satisfied, we pass to the calculation of the
norm (8.6). By (3.16) for TEL we have
t (Tx)*(r) dr " t x*(r) dr. (8.8)
The projection operator IIR also belongs to , and therefore
t (IIRy)*( r) dr " t y*( r) dr. (8.9)
Settingy = Tx and combining (8.8) and (8.9), we obtain
it (IIRTx)*(r) dr " i t (Tx)*('T) d'T i t X*('T) d'T.
o 0 0
From this by (8.2) and (8.7) for t E e R we obtain
L t (R Tx) *( r) dr " i t (Rx *) * ( 'T) d'T.
o 0
By condition (2) this inequality is satisfied for t > O. Without loss of
generality we may assume that the interpolation constant of G is one. Then
Theorem 8.1 implies that
IIRTxllG IIRx*IIG.
8. SPACES WITH DIFFERENT MEASURES
161
On the other hand, by Theorem 3.4 there exists an operator To E such
that x* = TaX, and therefore RTaX = Rx*. From this and the preceding it
follows that
IlxllE G = IIRx*IIG. (8.10)
In passing we have proved that the space EG consists of all functions from
L}(O, 00) + Loo(O, 00), for which Rx* E G.
We note that (8.7) applied to x = IIu implies, by (8.2), that for t E e R
t (Ru*)*( r) dr = t u*( r) dr,
and so this equality is satisfied for all t. Then by Theorem 8.1
IIRu*IIG = IlulIG.
From this it follows that the space EG( ) coincides isometrically with G.
Indeed,
"ull EG(Wl) = II u* II EG = II Ru* II G = II ull G. (8.11)
We consider the following situation, as well: Let E(O, 00) be an interpola-
tion space between L}(O, 00) and Loo(O, 00) with interpolation constant one.
From it we construct the space E( ) = G, and then the space EG. We
calculate the norm in E G :
IlxllE G = IIRx*IIG = II Rx*IIE(Wl) = IIIIRx*IIE(o,oo).
Since IIR E , we have IIIIRx*IIE(o,oo) IlxIIE(o,oo)' and so in this case EG is
larger than the original space E(O, 00).
In examples 1)-3) condition A) on the operator R is satisfied. In example
1) we have e R = (0, 00) and (Rx*)*(t) = x*(t). In example 2) we have
e R = (0, 1) and (Rx*)*(t) = x*(t) for t E (0, 1). Property 2) is obviously
satisfied. In example 3) the set e R consists of all natural numbers, (Rx*)*(t)
= IIRx*(t), and
n n 00 m
i IIRx*(r)dr= i f x*(s)tisX[m-l.m)(r)dr
o 0 m= 1 m-l
= f fm x*(s) tis = in x*( r) dr.
m=l m-l 0
Property 2) is also satisfied.
4. Operators of weakened type. If E(W"l) and F( ') are symmetric spaces,
then a linear operator V, defined on all strictly simple functions on if
flJE(O, (0)( 00) = 00, or on all simple functions if flJE(O, (0)( 00) < 00, is called an
operator of weakened type {E(WC), F( ')}, if Vu E S(W"l') and
( Vu )**( t)cpF(o.oo)( t) " C 00 u*( t) dcpE(O.oo)( t). (8.12)
162
II. SPACES OF MEASURABLE FUNCfIONS
We consider the operator II'VR, assuming that the operator R takes
strictly simple (simple) functions on (0, 00) into strictly simple (simple)
functions of 9R. For a strictly simple (simple) function x(t) on (0, 00) we then
obtain
1 1 t
(II' VRx )**( t)cpF(O,oo)( t) = t 0 (II' VRx )*( r) drcpF(o,oo)( t)
1 1 t
= t 0 (VRx)*(r) drcpF(o,oo)(t)
, C oo (Rx)*( t) dCPE(O,oo)( t)
= C oo (IIRx)*(t) dcpE(O,OO)(t),
The operator IIR belongs to L. Therefore (3.16) is satisfied for IIR, and by
property 18° of rearrangements we then have
(II' VRx )**( t) CPF(O, 00)( t) , c 00 x*( t) dCPE(O, 00)( t).
Thus, II'VR is an operator of weakened type {E(O, 00), F(O, oo)}. Then
Theorem 6.1' allows us to obtain the following assertion.
THEOREM 8.2. Let Eo, Fo, Eland FI be symmetric spaces of functions on the
semiaxis for which the hypotheses of Theorem 6.1' are satisfied, and let V be an
operator of weakened types {Eo(WC), Fo(9R')} and {EI(WC), F I (9R')}. The opera-
tor V admits closure to a bounded operator acting from E(9R) to E",8(9R'), where
E(O, 00) and E",8(0, 00) are the spaces described in Theorem 6.1'.
PROOF. The hypotheses of Theorem 6.1' are satisfied for the operator
II'VR, and therefore it admits closure to a bounded operator from E(O, (0) to
E" 8(0, 00). Then, as described above, V admits closure to a bounded operator
,
from E(9R) to E" 8(9R'). The theorem is proved.
,
For the symmetric space E(WC) we can introduce the notion of a fundamen-
tal function by setting CfJE( )( J-L(e» = IlXeII E( ). By the definition of E(9R),
from this it follows that flJE( )( J-L(e» = IIX(O,lL<e»IIE(O,oo) = CfJE(O,oo)(J-L(e».
Thus, the functions CfJE( ) and CfJE(O,oo) coincide at all values assumed by the
measure J-L on 9R. Denote by ep' the closure on (0, 00) of the set of all values of
J-L, and assume that the function CfJE( ) is extended by continuity to ep.. It then
coincides with the restriction of flJE(O,oo) to ep.. The distribution function of any
measurable function on WC does not assume values in the intervals comple-
mentary to ep" and therefore the rearrangement of any measurable function is
constant on these intervals. This enables us to modify condition (8.12) a little.
8. SPACES WITH DIFFERENT MEASURES
163
Let (t}, t;) be an interval complementary to the set ep' and let the inequali-
ties
(Vu)**(tj)<pF(O,oo)(t j ) Q
(i = 1, 2)
or
L Ii (Vu )*( r) dr .s;;; t j ( ) Q .s;;; 1[;( tJ Q
o <PF( oo)
(8.13)
holp, where o/(t) is the smallest concave majorant of the quaslconcave
function t / f{JF(O,oo)(t). From the preceding it follows that the function
f (Vu)*d'r is linear on [t}, t 2 ], and so (8.13) implies that it is smaller than the
concave function Q\J;(t) on the whole interval. Consequently, by (1.7) we have
1 t t
( Vu )*( r) dr .s;;; 1[;( t) Q .s;;; 2 ( ) Q
o f{JF(O,oo) t
( t} t t 2).
Applying these arguments to (8.12), we conclude that it will be satisfied if
the inequality
( Vu )**( t ) CPF(IDl') ( t) .s;;; c I 00 u*( r) dCPE(IDl) ( r) (t E e,J (8.14)
is, where C} = C /2 and the function f{JE( ) in the integral is arbitrarily (for
example, linearly) extended to the intervals complementary to ep.. Thus, the
condition that an operator has weakened type {E(WC), F(WC')}, needs to be
verified only at the points t E ep" and can be expressed in terms of the
fundamental functions of the indicated spaces.
Theorem 8.2 is formulated in terms of spaces of functions on the semiaxis.
We obtain a somewhat different assertion formulated in terms of spaces of
functions on 9R and 9R'. In this connection, we relax the requirements on E
somewhat, but obtain a cruder inequality for operators of weakened type. Let
G be an interpolation space between L}(9R) and Loo(9R) with interpolation
constant 1, and let EG be the space constructed in subsection 3. From the
proof of Theorem 6.1 we can see that under its hypotheses any strictly simple
(simple) function satisfies (6.10), which takes the form
K( t)(II' VRx )**( 8( t» .s;;; c oo x*( tr) d min { M'I'E i (r) }
In our case.
For the function x = IIu this implies that
K(t)(II'Vu)**(8(t)) < C 1 00 (JI/TU*(t) d min{ Mrn(r)}.
o T
164
II. SPACES OF MEASURABLE FUNCTIONS
By Lemma 4.7 and (8.10) we then have
IIK(t)(II'Vu)**(8(t»II E G " C oo Iial/TU*IIE G d min{ M'I'E,(r)}
= C oo IIRal/Tu*IIGdmin{M'PE,(r)}.
We now make the following assumption: For u E LI(W"l) + Loo( )
ITRu* = u*. (8.15)
This property is equivalent to the condition that the function u* belongs to
the range of the mapping IT.
If (8.15) is satisfied, then by (8.11)
II Rol/,.u*( t)II G =" Ra 1/,.ITRu* II G
II Ral/,.ITIIG IIRu*IIG =IIRol/,.ITII G IlulIG.
Thus,
IIK(t)(II'Vu)**(8(t»II E G " C oo II Ral/TIIII G dmin{ M'PE,(r)}lluIIG, (8.16)
and if
I II Ral/TIIII G dM'PE, + oo II Ral/TIIII G dM'PEO < 00, (8.17)
then V acts boundedly from G to (E G )K,6(W"l').
We consider the operator Rol/,.IT in the examples 1)-3). In example 1) this
operator has the form Rol/,.IIu(m) = u(w-I('Tw(m»). The transformation
w -I( 'Tw(m» may have quite complicated structure. However, it has the
property that the measure of the image of any measurable set is 'T times
greater than the measure of the set itself. Since the space G is symmetric, any
one-to-one mapping ,.(m) of onto itself having the same property induces
an operator 0l/,.u(m) = u( ,.(m», having the same norm in G. Therefore the
operator Rol/,.IT in (8.17) can be replaced by the operator 0 1 /,.. For example,
for W"l = (-00,00) with Lebesgue measure it is natural to set 0l/,.u(m) =
u( 'Tm). In the case of the semiaxis with measure dp. = ds / s we may set
0l/"U(S) = u(s,.). Using this, we obtain
1) Condition (8.17) for W"l = ( - 00, 00) with Lebesgue measure is satisfied if
I Iiol/TIIG dM'PE,(r) + oo Iiol/TIIG dM'PEO(r) < 00, (8.18)
where ol/,.u(m) = u( 'Tm).
The operators at form a semigroup, and so the function II at II G is submulti-
plicative. Therefore (8.18) is equivalent to the following condition:
II0tliG = o(t13o) as t 0 and !l0tlIG = 0(t Q1 ) as t 00, (8.19)
8. SPACES WITH DIFFERENT MEASURES
165
where a l is the lower dilation exponent of C{JE and 13 0 is the upper dilation
1 ,
exponent of C{JEo.
In example 2) the operator 0l/T = ROl/TII is given by
0 - u(t) = { u( '1"t) if 0 '1"t 1, (0 1)
1/ t .
T 0 if '1"t > 1
The operators 0t do not always commute in this case. However, if t I < 1 2 ,
then 0 11/2 = 0 12 0 /1 , and so 110 1 II is again submultiplicative. Thus,
2) Condition (8.17) for 9R = (0, 1) with Lebesgue measure is satisfied if the
operator (8.20) satisfies (8.18) or (8.19).
In example 3) the operators 0l/T = ROl/TII are described in a natural way
for '1" = . . . , 1 /3, 1 /2, 1, 2, 3, . . . . I t is easy to verify that for u = {uk} we
have
(8.20)
On U = {U I' . . . , U I' U 2 ' . . . , U 2 ' . . . , Uk' . . . , Uk' . . . },
\.. V -I\.
(8.21 )
a 1/ n U = { U 1 + . . + Un
un + I + . . . + U 2n
,...,
n
U(k-l)n+l : . . . +u kn ,...}. (8.22)
The quantity II ROl/TIIII G is a decreasing function of '1", and so by relaxing
the condition (8.17) we arrive at the following conclusion:
3) Condition (8.17) for WC = {I, 2, . . . } with J.L( n) = 1 is satisfied zf
n l Ilan+dIG[ M EI( ) - M El( n 1 )]
00
+ L IloI/nIIG[ Mf/JEo(n + 1) - Mf/JEo(n)] < 00, (8.23)
n=l
where the operators On and 0l/n are defined by (8.21) and (8.22).
We pass to the description of the space (E G )",8(WC') into which the operator
V maps. According to the definitions, we have
II V II(E G )IC.6(gn,) = IIK(t)(II'v)**(o(t))IIEG
= II R[ K( t)(II' v )**( o( t)) ] * II G.
(8.24)
It is sometimes convenient to replace the inequality
II Vull (E G )IC.8(9n') C II u II G (8.25)
by a cruder one, using the following arguments: If we can replace the
function K(t)(II'v)**(o(t» by a smaller function of the form IIv, where
v E L I (9R) + Loo(WC), then by (8.11) and (8.2) we obtain
II vii G = II Rv* II G = II R (II v) * II G = II II 1311 EG II vii (E G )",8(9.n').
166
II. SPACES OF MEASURABLE FUNCfIONS
Thus, (8.25) implies the inequality
II Vull G ell ull G. (8.26)
We illustrate this by the case where 9R = {I, 2, . . .}. We write on =
sUPn-I I n o(t) and Kn = infn-I I n K(t). Then, using the fact that the func-
tion (II' v)* *( t) is monotonic, we can replace K( t)(II' v )**( o( t» by a smaller
function assuming the constant values Kn(II' v)* *( on) on the intervals
(n - 1, n]. Inequality (8.26) then takes the following form:
II {Kn(II' Vu )**( on)} II G ell {un} II G. (8.27)
If 9Jt' = {I, 2, . . . }, then II' v is constant on the intervals (n - 1, n], and
so is (II'v)*. Denote by v: the value of this function on (n - 1, n]. It is more
convenient to also replace the function (II'v)** by a smaller one assuming the
value
1 n
v** = - v:
n n £.J
1
on (n - 1, n].
If 9Jt = 9Jt' = {I, 2, . . . }, then (8.27) implies the inequality
II { Kn v :)} II G ell { Un } II G'
(8.28)
where {v n } = V { un} and v( n) is an integer no smaller than On.
We note that the expression (8.24) for the norm becomes essentially simpler
if Eo = Fo and EI = Fl. Then K(t) - 1 and o(t) - t. Therefore
II vii <EG)IC.sr.m') = II R(II' v )** II G. (8.29)
We shall not formulate the theorems which can be obtained from Theorem
8.2 in the various cases. The above arguments enable us to apply these
theorems in concrete situations.
5. Spaces of measurable functions on (0, 00), with a measure tlult is invariant
under dilation. We consider the measure on (0, 00) defined by the differential
form dt / t. This measure is invariant under dilation. We denote by Lp,_ the
spaces Lp«O, (0), dt / t), constructed from this measure (for p = 00 we obvi-
ously have 4.- = Lp, but we keep the *, for the sake of unifonnity). We shall
be interested in interpolation spaces E between L I ,_ and Loo,_. From what we
said in subsection 2 it follows, for instance, that Lp,_ are such spaces.
We consider the simplest operators acting in interpolation spaces E.
1) The dilation operator (J".fIJ( 7") = fIJ(t /7") acts in the spaces L I ,_ and Loo,-
and has norm 1 there. Therefore
II (J".flJll E kE11 flJll E,
where k E is the interpolation constant of E relative to the couple L I ,_, Loo,_.
8. SPACES WITH DIFFERENT MEASURES
167
2) The power-type transformation T'ACP(t) = cp(t'A) acts in Loo.- with norm 1
and in L I ._ with norm IAI- I , and therefore
IIT'AcpIIE =llcp(t'A)IIE k E max{l, IAI-I}llcpIIE. (8.30)
3) Multiplicative convolution transformation (cf. 6.9). If 0/ ELI. _, then for
cp E Loo.- we have
( 00 ( t ) ds
)0 I[; -; cp(s)-s L,.,.. .s;;; 1II[;IIL,..llcpIIL,.,..,
while if cp ELI _, then
,
00 1[;( ) cp( s) L....s;;; 111[;11 L...II cpll L,...
Therefore we have the inequality
(00 o/ ( ) cp(s) ds .s;;; kEIII[;IIL,..llcpIIE'
)0 SSE
which we shall call Young's inequality.
4) The operators of Hardy-Littlewood type
1 (t
Hlcp( t) = t 1 +1 )0 Slcp(S) ds
can be expressed in terms of convolution:
(8.31 )
(8.32)
Hlcp(t) = oo C f(l+I)X<I,OO) C )cp(s) . (8.33)
The function t -(/+ I)X(1,oo)(t) belongs to L I ,_ for I > -1, and its norm in L I ,_
is equal to 1/(1 + I). Therefore (8.31) implies Hardy's inequality
1 (I / k E
tl+I )O scp(s)ds E.s;;; l+/ llcpIIE
(8.34)
for I > -1.
We construct an averaging operator by the formula
00 f ell + I ds
Tavg = }: X(e",e"+ 1]( t) cp(s) - .
- 00 e" S
It is easy to see that this operator acts, and has norm 1, in the spaces L I ._ and
Loo - (similarly to the situation in 3.2). Therefore its norm in E does not
,
exceed k E .
In what follows we need spaces with power weights generated by interpola-
tion spaces between L I ,_ and Loo,_. We shall denote by Eto, or, for short, E a ,
the space with norm IlxllEo = Iltaxll E . We consider how the above operators
act in these spaces.
(8.35)
168
II. SPACES OF MEASURABLE FUNCTIONS
Dilation operators.
IIO,.<pII£a =IIta<p(,.-lt)ll£ = ,.all(,.-lt)a<p(,.-lt)II£' kETall<pIIE a . (8.36)
Power-type transformLltion. By (8.30) we have
II t aA<p( t A ) II £ ' k£ max { 1, 1i\1- I } II t a<p( t) II E,
and therefore
II T A <p11 £a.\ , k£ max { 1, 1i\1- I} II <p11 Ea.
Multiplicative convolution transformation. From (8.31) we obtain that
(8.37)
t a (00 ( ) cp(s) ds = (00 ( ) a\[l ( ) s«cp(s) ds
)0 s S £ )0 s s S £
II t C)[;( t) II Ll..11 <p11 Ea. (8.38)
Thus, the convolution operator acts in E a if IltC)[;IIL l .. < 00.
Hardy-Littlewood type operators. We write the expression for the operator
in a still more general form,
1 1 1
HI m<p(t) = I SI<p(S) ds.
, t +m 0
(8.39 )
We have
tm+a-IH m ( t ) = 1 i 1 sl-asam ( s ) ds.
I,m't' I + 1- a 't'
t 0
Assuming that I > a - I, from (8.34) we obtain
k E
1\ HI,m<P1I E m + a - I , 1 + I _ a II <PI! Ea.
We may consider similarly the adjoint operator
H/ m<p(t) = tl f oo cp(s) m '
1 S
(8.40)
We have
(00 ( t ) a+m ( t ) ds
tm-I+«H/mcp(t) =)0 -; X(O,I] -; s«cp(s)s'
The function ta+mX(o,llt) belongs to LI,. if a + m > 0, and its norm in LI,.
is equal to I/(a + m). Under this condition we have
IIH/mcpIIE m - ' + a , ex :E m IlcpllE a . (8.41)
A veraging operator. If en < t , en + I, then we have
f e n + t ta ds f e n + t cis
I taTavg<p(t) I asal<p(s)l- e 1al sal<p(s)I-.
en S S en S
8. SPACES WITH DIFFERENT MEASURES
169
Thus, for all t
\taTavg<p(t)1 < elaITavg(tal<p(t)l).
Then
" Tavg<PII EG < kEelall1 <p11 E G .
(8.42)
Without loss of generality (see Chapter I, 4.6) we may assume that the
space E is constructed from LI,. and Loo,. by means of some interpolation
functor. This enables us to construct, for any Banach couple (A, B), a space F
intermediate between A and B such that (L I ,., Loo,., E) is an interpolation
triple relative to the triple (A, B, F). We apply this reasoning to the trivial
case where A = B = R (or q. The integrals
l a dt f oo dt
tl3<p(t)- (a > 0), t-l3cp(t)- (b > 0)
o t b t
and
fb cp( t) dt
a
are finite for /3 > 0 for functions from LI . and for functions from Loo., and
, ,
therefore they realize bounded linear operators from these spaces into R (or
q. From this it follows that these operators act boundedly from E into R (or
C). In particular, all functions from E are locally summable. If <p E E a , then
l a dt
t -13<p(t) - < CII<pIIEG
o t
f 00 dt
t -13q;(t)- < CIIq;IIEG
b t
for /3 > lr,
(8.43)
for /3 > -lr.
(8.44)
6. Operators in spaces of measurable functions on the interval [0, 1]. Conju-
gation operator. We consider the Hardy-Littlewood operator on [0, 1]. It is
defined by the same formula as in 6.7:
I l t
Hx(t) = - x(s) ds.
t 0
As in 6.6 and 6.7, with the aid of (6.1') and (6.29) we can show that H
acts in a symmetric space E if and only if lIatll E = o(t) as t 00. The adjoint
operator has the form
f l ds
Hlx(t) = x(s)-
t S
(0 t < 1).
In the same way as on p. 139 we can verify that IIHIIE-+E II = IIHIIIE1-+E1,
and it can be shown that in the case where H acts from E into E II we have
II atll E-+E 11 = o(t) as t 00.
170
II. SPACES OF MEASURABLE FUNCTIONS
For HI to act boundedly in a symmetric space E it is necessary and sufficient
that IIatll E = o(t) as t O. We repeat the proof of the necessity. For 0 < t
'T I we have
f I ds f liT ds 1 ( t )
x*(s)- x*(s)- In -x* - .
1 S 1 S T T
The integral on the right is nonnegative for t > 'T, and therefore, using the
definition of the operators at (p. 164), we may write
f l ds 1
x*(s)- In - a'Tx*(t).
1 s 'T
Hence for T < 1
IlaTxli E = IlaTx*IIE I_I IIH1x*IIE I_I IIH11IEllxIIEO
In 'T In T
The assertion is proved.
We may prove similarly that the conditions
lIatll E = o( 1) as t 0 and l/atll E = o(t) as t 00 (8.45)
are necessary and sufficient for the operator
rx(t) = 1 1 x(s) ds (0 t I)
o t + s
to act in a symmetric space E.
Now we consider the singular Hilbert operator
Sx(t) = p.v. 1 1 x(s) ds
o t - s
(0 t 1).
(8.46)
From (7.9) it follows immediately that
. 2 mes e
(SXe)*(t) arcsinh
t
(0 < t 1, e C [0, 1]).
This inequality implies again that the operator S is of weakened type {Lp' 4}
for any p E (1, (0). If conditions (8.45) are satisfied, then repeating the
arguments in the proof of Theorem 7.2 (using the submultiplicativity of
II at 1/ E)' we may apply the interpolation theorem described in 8.2. Then from
(8.16) we obtain that IISxll E CllxllE.
Now we show that conditions (8.45) are necessary for the boundedness of
the Hilbert operator.
We can write (8.46) in the form
sx( 'I" ; 1 ) = p.v. f_\ x«a'l" /2) da
( - 1 T < 1). (8.47)
8. SPACES WITH DIFFERENT MEASURES
171
On [-1, 1] we introduce a symmetric space E( -1, 1) of functions with
norm IIYII£(-I, I) = Ily(2t - 1)11£. If the Hilbert operator acts in E, then by
(8.47) it acts in E( - 1, 1) as well. Reasoning in the same way as in the proof
-
of Theorem 7.2, from this we obtain that r acts in the space E(O, 1) consisting
of restrictions of the functions from E( - 1, 1) to (0, 1). This implies that
-
conditions (8.45) are satisfied for E(O, 1), and consequently for E( -1, 1) as
well. Under the isometric correspondence between the spaces E( - 1, 1) and
E, to a dilation operator in E( -1, 1) there corresponds the dilation operator
in E with respect to the center 1/2 with the same dilation coefficient. It is
easy to verify that the latter operator has the same norm as the dilation
operator with respect to the center O. Thus, condition (8.45) is satisfied for E.
Thus, condition (8.45) is necessary and sufficient for the singular Hilbert
operator to act boundedly in E.
In the theory of trigonometric series, an important role is played by the
conjugation operator, i.e. the operator that passes to the conjugate function
(see [2] and [50]). It can be written in the form
1 1 2TT x(s)
Qx(t) = p.v. 'TT 0 2tg«t _ 5)/2) ds.
Let a symmetric space of functions on [0, 2'1T] have property (8.45). Then
the Hilbert operator
Sx(t) = p.v. 1 2TT x(s) ds
'IT 'IT 0 t-s
acts boundedly in it.
Consider the subspace EI of E consisting of functions with support in the
interval ['IT /2, 3 'IT /2]. On such functions the operator Q takes the form
1 1 3TT/2 x(s)
Qx(t) = p.v. - «)/ ) ds.
'1T TT /2 2 tg t - s 2
The quantity (t - s)/2 runs over the interval [- 3 'IT /4, 3'1T /4]. On this
interval the function 1/(t - s) differs from 1/2 tg«t - s)/2) by a bounded
function. Therefore Q differs from the Hilbert operator S / '1T on the subspace
EI by an operator mapping EI into Loo C E. Hence it follows that Q acts
from EI into E.
We consider a subspace E 2 complementary to EI. It consists of functions
with support in [0, 'IT /2] U [3 'IT /2, 2'1T]. On E 2 we have
Q ( ) _ 1 l TT/2 x(s) ds 1 1 2TT x(s)
x t - p.v. - + p.v. -
'IT 0 2 t - S 'IT 3TT /2 2 t - s
tg 2 tg 2
ds.
172
II. SPACES OF MEASURABLE FUNCTIONS
If we assume that the function x(t) is extended periodically with period 2'11',
we have
x(z - '1T)
2 t-z+'1T
tg 2
x(z - '1T)
2 t-z+'1T
tg 2
where the function y(z) = x(z - '1T) has support in ['1T /2, 3'1T /2]. Then
IIQxllE =IIQyIIE IIQIIEI EIIYIIEI =IIQIIEI Ellx"E.
Thus, Q acts boundedly from E into E.
Conversely, if Q is bounded, then it acts from El into E and from E 2 into
E. From the preceding it follows that the operator S/'11' also acts from El into
E. We consider this operator on E 2 . Let the support of x(t) be contained in
[0, '1T /2]. Then
1 1 3TT /2
Qx(t) = p.v. -
'1T TT
d 1 I TT x(z + '11')
Z + p.v. -
'1T /2 t - z - '11'
TT 2 tg
2
dz
= p.v. -.!. (3TT/2
'1T J TT /2
dz = Qy(t + '1T),
l TT/2 x(s) I TT
Sx(t) = p.v. ds = p.v.
o t - S 'IT /2
I TT y(a)
= p.v. /2 do,
TT /2 t - a + '1T
x( 0 - '11'/2) do
t - 0 + '11' /2
where y( a) = x( a - '1T /2) for a E ['1T /2, '1T] and y( a) = 0 for a fl ['1T /2, '1T].
The functiony is equimeasurable with x. For 0 t 3'1T/2 we have Sx(t) =
Sy(t + '1T /2), and for 3'1T /2 t 2'1T we have
1 1 'IT
I Sx( t)1 - Iy( a) Ida,
'1T 'IT/2
theref ore
II Sxll E IIX[O,3'1T/2]Sxll E + IIX[3'1T/2,2TT]Sxll E
< IISYIIE + IIYIIL 1 < (IISIIEI-+E + )IIYIIE < qxlk
This inequality can be established similarly for functions from E with
support in [3'1T /2, 2'1T]. Thus the Hilbert operator acts in E, and hence
condition (8.45) is satisfied. We have arrived at the following assertion.
THEOREM 8.3. For the conjugation operator to act in a symmetric space E it is
necessary and sufficient that conditions (8.45) be satisfied.
9. APPLICATIONS TO ORTHOGONAL SERIES
173
9. Applications to the theory of orthogonal series
1. The generalized Paley theorem. We assume that an orthonormal system of
functions un(t) is given on the interval [0, 1]. We denote by {c n } the Fourier
coefficients of a function x(t) E L1(0, 1):
C n = £ I X ( t) Un ( t) dt. (9.1 )
By the Bessel inequality
00
L c; Ilxll
1
the Fourier operator Vx = {c n } is a bounded linear operator from L 2 (0, 1)
into 1 2 .
Let H(O, (0) be a symmetric space on the semiaxis and let H(O, I) be the
corresponding space on [0, 1]. We assume that, in addition, we know that the
system {un} is bounded in the space H(O, 1): Ilunll H C. Then we obtain the
estimate
ICnl = I £ I x(t)u n ( t) dtl " qxll H'(O,I)
for the Fourier coefficients. This implies that the Fourier operator V acts
boundedly from H1(0, 1) into '00. We denote by cp(t) the fundamental
function of H 1(0, (0) and assume that it has the following properties:
cp( 1) = 1, the function q;( t) / Vi is increasing and q;( t) / Vi 0 as t O. We
are thus able to invoke Theorem 8.2. In our case,
q;£ (t) = Vi, q;£ (t) = q;(t), \[IF (t) = Vi, \[I F (t) = sgn t.
o I 0 I
Therefore 8(t) = t / q;2(t) and K(t) = 1/ q;(t).
Repeating the reasoning in the proof of Theorem 8.2, we consider the
operator II' VR, where R is the operator of restriction of a function given on
the semiaxis to the interval [0, 1], and II' { c n } is the function equal to C n on
(n - 1, n]. The operator II' V R acts boundedl y from L 2 (0, 00) to L 2 (0, (0)
and from H 1(0, (0) to Loo(O, (0). Applying Theorem 6.1' to it, we obtain
(rr' VRx)** ( t )
q;(t) q;2(t)
CII xii £(0,00).
£(0,00 )
If x(t) = 0 for t > 1, then
( II' { c }) ** ( t )
q;(t) n q;2(t)
CII xii £(0,1).
£(0,00 )
174
II. SPACES OF MEASURABLE FUNCTIONS
If we denote by c; the rearrangement of the sequence of Icnl in decreasing
order, then (II' { c n } )*( t) = c:X(n -I,n]( t). The function (II' { c n } )**( t) can
be replaced by a smaller function, namely
00 1 n
c:*X(n-l,n](t), where c:* = - c:. (9.2)
1 n k= 1
If we now denote by an the solution of the equation t / cp2(t) = n (n =
I, 2, . . . ), then from the last inequality it follows that
1 00
- ( t ) L c:*X(an,an_d(t) Cllxll£(o,I).
<P 1 £(0,00)
This inequality may be made even more crude by replacing 1/ cp( t) on the
intervals (an, an-d by 1/<p(a n _ l ) and the norm in £(0, 00) by the norm in
£(0, I). Then
00 c**
L (n ) X(a".a._i t ) , Q x IIE(O.l). (9.3)
2 <P an - 1 £(0,1)
The functions X(an, _d(t) form an orthonormal system o n [0, 1]. It IS
sometimes convenient to normalize them, i.e. divide them by V an -I - an .
Since ani <p2( an) = n, we have
2
( ) an - 1
n - I an - 1 - na n =
cp2( an _ I)
in view of the quasiconcavity of cp(t). Hence
a 2
n 0
cp2( an)
a n - I
V an - 1 - an
<p( an - I)
(n - 1)(a n _ 1 - an)
= R
a n - I
1
-.
V2
n(a n _ 1 - an)
Therefore (9.3) implies the inequality
00
C * * ( a - a ) - 1/2 ( t )
n n-l n X(an,an_d
2
We have arrived at the following assertion.
THEOREM 9.1. Let {un} be an orthonormal sequence on [0, 1] that is bounded
in the norm of a symmetric space H(O, 00). Assume that the fundamental
function <PH(t) of H(O, 00) has the following properties:
I) <PH(I) = 1.
2) The function Vi I <PH(t) is increasing.
3) Vi I <PH(t) 0 as t O.
CII xii £(0,1).
£(0, I)
(9.4)
9. APPLICATIONS TO ORTHOGONAL SERIES
175
If E(O, (0) is a symmetric space for which
IIOtll£=o(Vi) ast O and lIorll£=0(tl-8H) ast 00,(9.5)
where 8 H is the upper dilation exponent of CfJH(t), then (9.4) holds, where the c:*
are calculated according to (9.2) and the an are the solutions of the equations
cp1(t)/ t = n.
In the formulation we have used relation (4.39): cp(t) = t / cpJlt).
We consider special cases of Theorem 9.1. Assume that E = Lp. Conditions
(9.5) will be satisfied if 1/2 < p -I < 1 - 8 H . Then we obtain
00 1/
** ( ) -1/2 { }
t en a n - 1 - an X(a.,a._d 4 = (e:*Y(a n - 1 - a n )I-P/2 p
< ellxll .
(9.6)
In the case of a Lorentz space Acp = E conditions (9.5) are satisfied if
1/2 < Y", < 8", < 1 - 8 H , and then
00
** ( ) -1/2
£.J C n a n - I - an X(an,an-d
2
I 00
;;;.1 e:*(a n - 1 - a n )-1/2 X (a.,a._i t ) dIf;(t)
o 2
00
= c:*(a n _ 1 - a n )-1/2[ 1/;(a n - l ) - 1/;(a n )J.
2
Therefore
00
c:*(a n _ 1 - an) -1/2[ 1/;(a n - l ) - 1/;(a n ) ] < ell xllA,y. (9.7)
2
Finally, for a Marcinkiewicz space we have
00
** ( ) -1/2
£.J C n an - I - an X(an,a n _ d
2
1 1 a k ** ( ) -1/2 ( ) d
, ( ) C n an - I - an X ( a a ] t t
1/; a k 0 2 n' n-I
k
1 ** ( ) 1/2
- 1/;(a k ) t C n a n - I - an .
Thus for 1/2 < 1 - 8", < 1 - Y", < 1 - 8 H we obtain
sup /) f e:*(a n - 1 - a n )I/2 .;;; qxIIM;,'
k 1/; a k 2
176
II. SPACES OF MEASURABLE FUNCTIONS
We consider still another special case, namely where the sequence un(t) is
uniformly bounded, i.e. bounded in Loo. Then CPH(t) = sgn t and cp1(t)/t =
1/ t. Hence an = 1/ n. Inequality (9.4) then takes the form
00
c:* [n(n + 1) ] 1/2 X (l/n,I/(n_I)]
2
, ell xii £(0,1)'
£(0, I)
or the equivalent form
00
**
LJ C n nX(l/n,I/(n+ I)]
2
< ell xii £(0, I).
£(0, I)
For the spaces Lp with 1 <p , 2 we obtain
{ 00 } I/p
(c:*Yn P - 2 qxll .
If we replace c:* by the smaller number c:, the inequality thus obtained is
the content of the classical Paley theorem.
2. Generalized Hardy-Littlewood theorem. For trigonometric series with
decreasing coefficients, the preceding assertions admit essential sharpening.
THEOREM 9.3. Let un(t) = cos '!Tnt, and let the space G(O, 1) have the
following properties:
IIOtllG = 0(1) as t 0 and IIOtllG = o(t) as t 00.
If the sequence c n is decreasing and converges to zero, then for the function
x(t) = C n COS '!Tnt to belong to G it is necessary and sufficient that
00
kCkX(I/(k + 1),1/ k) < 00, (9.8)
I G
or
00
CkX(O,I/k) < 00. (9.9)
1 G
PROOF. We use some inequalities for the trigonometric system (see [50], vol.
II, Chapter XII, 6). The first asserts that
(l/n
c n B )0 Ix(s)1 ds.
9. APPLICATIONS TO ORTHOGONAL SERIES
177
Hence, since x**(t) is monotone, we obtain
00 00 1 1/k
kc k X( I / (k + I), I / k) ( t) B k 0 I x ( S ) I ds X( 1/( k + I), I / k) ( t)
I I
00 1 1/k
" Bt k 0 x*(s) ds X(1/(k+I),I/k)(t)
B 1 t x*(s) ds.
t 0
The Hardy-Littlewood operator acts continuously in G (Theorem 6.4), and
therefore
00
kCkX(I/(k+I),I/k) CllxllG.
I G
To prove the converse we use the inequality
nIl
Ix(t)I"B Ck for n+I "t"-;;
(see [50], vol. II, Chapter XII, 6). Then
Ix(t)I" B ( C k )X(1/(n+I),I/n)(t).
(9.10)
We calculate the value of the adjoint HI of the Hardy-Littlewood operator
at the function y(t) = Lf kCkX(l/(k+I),I/k/t). For I/(n + 1) < t < Iin we
have
( ) f l Lf kCkX(l/(k+I),I/k)(S) ds
Hly t =
t S
f l Lr' kCkX(I/(k+I),I/k)(S) ds
I/n S
n-I n-I
- kC k In (1 + Ilk) In 2 c k '
I I
I.e.
Hly(t) In 2 I ( n 1 1 )
" LJ £.J c k X(I/(n+I),I/n)(t)
= In 2 ( c k )X(1/(n+2),I/(n+ 1»( t).
178
II. SPACES OF MEASURABLE FUNCTIONS
Then
HIY( ) In 2 (* c k )X(l/(n+2).I/(n+ 1»( )
In 2 (* c k )X(l/(n+ I), I/n)( t).
(9.11 )
Thus,
IlxliG ..; B f ( i Ck ) X(l/(n+ 1),l/n)
I I G
B ( 1 ) B
In 2 Hly 2 In 2 1IHIIIGllo21IGIIYIIG.
We have established the inequalities
Ilxll G ..; B f ( i Ck ) X<I/(n+ 1),l/n)
1 1 G
00
C I L kC k X(1/(k+I),l/k) C 2 1I X IIG.
1 G
Since
f ( i C k ) X(1 / (n + 1 ),1/ n) ( t) = f C nX(O,1 / n) ( t ),
1 1 1
the theorem is proved.
3. Bases. We recall that a basis in a separable space E is, by definition, a
system {uk} r:' of elements of E such that every x E E can be represented
uniquely in the form of a convergent series:
00
x = L CkU k .
1
If {Uk} is a complete minimal system (see [25]) and {fk} is the system of
linear functionals biorthogonal to it, then the bounded operators
n
Sn x = L fk(X)U k
1
are defined. For the system {Uk} to form a basis it is necessary and sufficient
that the operators Sn be uniformly bounded.
9. APPLICATIONS TO ORTHOGONAL SERIES
179
A basis is said to be unconditional if it remains a basis under any permu-
tation of its elements. For a basis to be unconditional it is necessary and
sufficient that operators of the form
n
L fk;(X)Uk,
j= I
(n = I, 2, . . . )
be uniformly bounded.
Thus, determining whether a complete minimal system is a basis or an
unconditional basis reduces to estimating norms of certain operators. From
this it follows that if Eo => E I , E is an interpolation space between Eo and E 1 ,
EI is complete in E and the elements {uk} of EI form a basis (unconditional
basis) in Eo and in £1' then they form a basis (unconditional basis) in E.
On [0, I] consider the Haar orthonormal system defined by
hg ( t) = I; hb I) ( t) = { I, 0 t < I /2,
-I, 1/2<t 1
and
h k)( t) =
2 n / 2
,
_ 2 n / 2
,
(2k - 2)/2 n + 1 t < (2k - 1)/2n+l,
(2k - 1)/2 n + 1 t 2k/2n+1
(k = I, 2, 3, . . . , 2 n ),
o otherwise.
The functions hnk(t) are arranged in a simple sequence um(t), m =
0, I, . . ., in order of increasing n and for fixed n in order of increasing k. It
is easy to see that the kernels QN(t, s) of the operators SN corresponding to
the partial sums of the Fourier series in the functions h k) are nonnegative.
Therefore
I I N I
in IQN(t, s)1 ds = in L um(t)um(s) ds = in uo(t)uo(s) ds = 1.
o 0 m=O 0
Similarly,
fa I I QN ( t, s) I dt = 1.
These equalities say that the operators SN are uniformly bounded in LI and
Loo. By the corollary to the interpolation Theorem 4.6 these operators are
bounded by I in any separable symmetric space E. It is well known that the
Haar system is a basis in LI. This in particular implies that the system is
complete in E. Indeed, otherwise there would be a nonzero function in
E' = E I C LI orthogonal to all functions um(t), which contradicts the basis
property of the system {u m } in LI.
180
II. SPACES OF MEASURABLE FUNCTIONS
We have arrived at the following assertion.
THEOREM 9.4. The Haar system is a basis in any separable symmetric space.
We now consider the trigonometric system on [ - '1T, '1T].
THEOREM 9.5. For the trigonometric system to be a basis in a separable
symmetric space E it is necessary and sufficient that
Ii0TIIE = o( 'T) as 'T 00 and Ii0TIIE = 0(1) as 'T O. (9.12)
PROOF. Denote by Snx the nth partial sum of the Fourier series of x(t). For
this sum (see [2])
1 f '1T sin n(t - s) 1 f '1T
Snx(t) = - x(s) ds + _ 2 x(s)cos n(t - s) ds
'TT - 'IT 2 tg t ; S 'TT - 'IT
= -cos nt . ! f'IT x(s)sin n.s ds + sin nt . ! f'IT x(s)cos n.s ds
'TT - 'IT 2 tg t ; S 'TT - 'IT 2 tg t ; s
1 f '1T
+ _ 2 x(s)cos n(t - s) ds.
7T - '1T
If conditions (9.12) are satisfied, then the operator of passage to the
conjugate function is bounded in E (see Theorem 8.3), and therefore
IISnxllE C(llx(s)sin nsllE + Ilx(s)cos nsilE + IlxIIL.)
C111x11E.
The sufficiency of conditions (9.12) is proved.
Now we assume that the trigonometric system is a basis in E. Then
IISnxllE C(E)llxIIE for any function x E E. First we assume that x(t) is a
trigonometric polynomial. Then for the conjugate function x(t) we have the
identity
S2n x (t) = -2 cos nt . Sn( - x sin nt) + 2 sin nt . Sn(x cos nt),
which can be verified in an elementary way for the functions sin kt and
cos kt. Now it follows that
IIS2nxilE 4C(E)ll x IIE.
But for large n we have S2nx = X, and therefore
IlxilE 4C(E)llxIIE. (9.13)
Now let x(t) be an arbitrary function in E. By assumption, SNX x in E
as N 00. Then (9.13) implies that the sequence SNx = SNX is Cauchy in E,
and consequently converges to a function y E E. The relation x E E C Ll
9. APPLICATIONS TO ORTHOGONAL SERIES
181
implies, as we know (see [2], Chapter VIII, 21), that SNX x in 4 with
P < 1, and therefore y = x. From this we conclude by passing to the limit
that (9.13) holds for all x E E. Thus, the conjugation operator acts bound-
edly in E, and, by Theorem 8.3, conditions (9.12) are satisfied.
4. Unconditional bases.
THEOREM 9.6. For the Haar system to be an unconditional basis in a
separable symmetric space E it is necessary and sufficient that conditions (9.12)
be satisfied.
PROOF. As we noted above ( 6.6), conditions (9.12) imply that E is an
interpolation space between some spaces 40 and Lpi (1 <Po <PI < (0). By
what was said in subsection 3, the Haar system is an unconditional basis in E.
For the proof of the necessity we consider the subsystem of the Haar
system consisting of the functions uo(t) - 1 and un(t) = h _I(t) (n =
1, 2, . . . ).
The kernel of the integral operator corresponding to the sum SNx in the
system {un} is the function
N
KN(t, s) = un(t)un(s).
n=O
We fix t in the interval (2-2k-l, 22k). Then qJn(t) = 2(2n-I)/2 for n , k and
qJn(t) = 0 for n > k, and
k
1 +
22n - 1
,
o < s < 2- 2k ,
n=1
KN(t, s) =
j-l
1 +
n=1
2 2n - 1 _ 2 2 )-1
,
2- 2 ) < s < 2- 2 )+1
(j = k, k - 1, . . . , 1),
j
1 +
n=O
22n - 1
,
2 - 2) - 1 < s < 2 - 2) (j = k - 1, k - 2, . . . , 1),
0, 2 - 1 < s < 1.
As s changes, the function KN(t, s) assumes positive and negative values
alternately on the intervals (0, 2-2k), (2- 2k , 2-2k+I), . . . , (2- 2 ,2- 1 ) and is
equal to zero on (2 -I, 1). Its indefinite integral
loT KN(t, s) ds
182
II. SPACES OF MEASURABLE FUNCTIONS
with respect to s is a continuous function having local minima at the points
2 -2j+ 1 (j = k, . . . , 1). The first of them is equal to
( 1 + f 2 2n - 1 + 1 + kil 2 2n - 1 _ 2 2k - 1 ) 2- 2k = 2 (1 + 22k-l)2-2k.
n=l n=l 3
Moreover, the integral of KN(t, s) over (2- 2j - 1 , 2- 2j + 1 ) is equal to
( 1 + f 22n-l ) 2-2j-l + ( 1 + j'i 1 2 2n - 1 - 22j-I ) 2-2j > 0
n=l n=l
(j = k - 1, . . . , 1),
and so all local minima are greater than the first one.
We consider the function KN(t, s) = (1 + 22k-l)X(O,2-2k+I)(s). By construc-
tion,
T KN(t, s) ds T KN(t, s) ds (0 < 7" ..;; 1).
By property 18° of rearrangements, for x = x* we then have
SNX(t) = t KN(t, s)x(s) ds l KN(t, s)x(s) ds
1 (2-2k+1 1 1 (t
= 3(1 + 22k-l) )0 xes) ds ;> 12 t)o xes) ds
(_2- 2k + 1 < t < 2- 2k ).
Hence
1 1 (t ( N )
SN X ( t) 12 t)o x(s) ds k l X(2-2k+',2- 2k )( t) ·
By the assumption that the Haar system is an unconditional basis, the
partial sums corresponding to a subsystem of it are uniformly bounded:
IISNXIlE Cllxll£. Then
N
Hx X(2- 2k - 1 ,2- 2k ) 12Cllxll£,
k=l £
where H is the Hardy-Littlewood operator.
Reasoning similarly to the proof of the theorem, we may pass to the limit
as N 00 and obtain the inequality
00
Hx X(2- 2k - 1 ,2- 2k ) 12CII xii E.
k=l £11
It is obvious that
00 00
02 X(2- 2Jc - 1 ,2- 2k ) = X(2- 2Jc ,2- 2Jc + I ).
k=l k=l
9. APPLICATIONS TO ORTHOGONAL SERIES
183
Next, Hx(t) is decreasing, and therefore
Hx( t) f XW 2k . 2 - 2k + ')( t) .;;; C1 2 [ Hx X(2-2k-,.2-2k) ] (t).
k=l k=l
Hence II HXX(o,I/2)1I Ell 36C Ilxll £.
Similarly,
Hx 02[ HXX(O,I/2) J.
Therefore
IIHxllE11 72Cll x ll£.
Since IHxl Hx*, this inequality holds for any x E E. By Theorem 6.7
and 8.6, the condition II 0T II £ = o( 'T) is satisfied as 'T 00. The kernels
KN(t, s) are symmetric, and therefore IISNII£ = IISNII£,. This implies that
II 0TII £1 = o( 'T) as 'T 00.
Since E is separable, the imbedding of E in E II is isometric. Then by (4.36)
we have
'TIIOI/TII£ = o( 'T) as 'T 00, i.e. lIoTII£ = 0(1) as T o.
The theorem is proved.
In [251] it was announced that if there exists an unconditional basis in a
symmetric space then the Haar system forms an unconditional basis in it (the
proof is to be found in [33a]). Thus, conditions (9.12) are necessary and
sufficient for the existence of an unconditional basis in a symmetric space.
5. Series in the Rademacher system. The Rademacher system on [0, 1] is the
orthonormal system of the functions rk(t) = sgn sin 2 k - l 7Tt. For every se-
quence u = {un} E /2 we construct a function
00
vet) = Vu = Ukrk(t).
k=l
(9.14)
(We recall that for {un} E /2 the series (9.14) converges almost everywhere;
otherwise it is divergent almost everywhere-see [50]).
Thus, we obtain an operator from sequence spaces into function spaces.
THEOREM 9.7. Let G be a symmetric sequence space satisfying the conditions
IIOnllG = o(n), IloI/nlIG = o(ljYn) as n 00.
ffu = {un} E G,then {v**(e-n)jn} E G,and
{ v**(e-n) }
n G CllullG.
(9.15)
(9.16)
184
II. SPACES OF MEASURABLE FUNCTIONS
PROOF. Since Irk(t)1 1, we have
00
II VuIIL lukl = Ilull/ i .
k=l
If u E /2' then according to Khintchine's inequality the function Vu be-
longs to all spaces 4, 1 p < 00, and, moreover (see [50], Vol. I, Chapter V,
8), for II ull /2 1 we have
(I e(VU(t)/A)2 dt , k ( 2 ) 2k
J o k=O k. A
for any A > O. Since e k kkj k!, for Ao = vse we have
(I e( Vu(t)/ )2 dt , ( 4e ) k = 2.
J o k=O A5
This inequality means that in the Luxemburg norm in the Orlicz space LiI
constructed from the N-function M(u) = e ril - 1 (see [22]) we have
II VullLZt vse .
Thus, the operator V acts boundedly from /1 into Loo and from /2 into LZt.
We apply Theorem 8.2. In our case M m (t) = Vi and M m (t) = t. Therefore
TEa TEl
from (9.15) it follows that the space G satisfies condition (8.17). According to
(4.27) we have
CfJ Llt ( t) = [ In( 1 + 1 j t) ] - 1/2,
and CfJL = sgn t. Consequently, equations (6.7) and (6.8) for detennining ,,(t)
and 8(t) take the form 1 = K(t)t (i.e. K(t) = 1 j t) and 8(t) = 1 j(e t - 1). We
calculate Kn = infn-I t n K(t) = n -1 and
8n = sup (e t - 1) -1 = (e(n -1) - 1) -1.
n-l<.t<.n
From (8.27) we obtain that
{ v**«(en - 1)-1) } CII{un}IIG.
n + 1 G
Since V**(at) V**(t)j a for a 1, from (9.17) we obtain (9.16).
Theorem 9.7 is sharp in the following sense.
THEOREM 9.8. Under the assumptions (9.15), for every function of the form
(9.14)
( 9. 1 7)
{ v**(e-n) }
II { un}ll G ' n G'
9. APPLICATIONS TO ORTHOGONAL SERIES
185
PROOF. We use the fact (without recalling its proof) that the functions
m m
Vm(t) = L ukrk(t) and Ym(t) = L u:rk(t)
k=1 k=1
are equimeasurable, and consequently v:(t) = y:(t). Using property 11 0 of
rearrangements, we obtain that v*(t) =y*(t), wherey(t) = L u:rk(t). This
in particular implies that any function v(t) of the form (9.14) is equimeasura-
ble with the function - v(t), and consequently mes{ t: v(t) O} 1/2.
We representy(t) in the form
i
yet) = L u:rk(t) +
k=1
00
L
k=i+ 1
u: r k ( t).
The first term is equal to L u: on (0, 1/2;-1). The second term is periodic
with period 1/2;-1, and so it is nonnegative on a set of measure not less than
1/2; on (0, 1/2; - I). Thus, y( t) L; u: on a set of measure not less than 1/2;.
Then
00
v*(t) = y*(t) L U:X(O,2- k ](t).
k=l
We calculate
n
v*(2- n ) L u: nu:.
k=1
Therefore
{ v** -n) } I G { v** -n) } G { v*( -n) } G
I
II {u:} IIG = IlulIG.
CHAPTER III
SCALES OF BANACH SPACES
1. Scales of Banach spaces. Related spaces
1. Scales of Banach spaces.
DEFINITION 1.1. A family Ea, ao a (30, of Banach spaces is called a
scale of Banach spaces if the following conditions are met:
1) For (3 > a the space Ep is densely imbedded in Ea' and consequently
IIxll E C(a, (3)llxllE:
a p
2) There exists a function C( a, (3, y), finite at all points of the domain
a o a < {3 < y {3(p such that
IIxilEp C(a, (3, Y)llxll a-P)/(y-a)IIxll -a)/(y-a) (1.1)
for any x E Ey.
A scale of spaces is said to be compact if the imbedding of EfJ in Ea is
compact for (3 > a.
If the family of Ea has property 1) and Ep is an interpolation space
between Ea and Ey of type JL = ({3 - a)/(y - a) for a < {3 < y (see Chapter
I, 4.3), then these spaces form a scale. Indeed, let fo be a functional from E o.
Then (see Chapter I, 2.I)fo E E for all a E [ao, {30]. Let x E Ey. Consider
the operator Txu = fo(u)x acting in Ey and all spaces E9 for () < y. We
determine the norm of Tx in the spaces E9 for () < y:
II fo( u)x II EfJ
" Tx" Eo = : 9 II u II Eo =" x II Eo II fo II EO.
( 1.2)
Since Ep is an interpolation space between Ea and Ep of type JL, we have
li T II c ( a {3 "" )11 T II (y-P)/(y-a) 11 T 11 (I3- a )/(y-a)
x Ep-.Ep 1" I X Ea-.Ea x Ey-.g y .
Substituting expression (1.2) for the norm of the operator Tx into this
inequality, we obtain (1.1).
187
188
III. SCALES OF BANACH SPACES
In what follows, for brevity we write
JL = (y - /3)/ (y - a) and p = (/3 - a)/ (y - a), (1.3)
where JL + p = 1 and JliX + pY = /3.
Inequality (1.1) can then be written in the form
Ilxll Ep C(a, /3, Y)lIxll Clllxll .
2. Properties of scales of Banach spaces. 1 0. If in a scale Ea' ao <;; a /3fP
we replace the index a by an index a' according to the formula a = ka' + 9,
where k > 0, then the spaces Fa' = Eka'+8 will form a scale on the interval
[(a o - 9)/ k, (/3 0 - 9)/ k] relative to the index a'.
In connection with this, without loss of generality we may assume that
a o = 0 and /3 0 = 1.
2 0. If equivalent norms are introduced in the spaces Ea of a scale, then they
will again form a scale.
3°. On the interval [ao, /3d let a family of Banach spaces Ea be given which
form a scale on the intervals [ao, /30] and [/30' /3d (a o < /3 0 < /3.). For the
spaces Ea to form a scale on the whole interval [afP /3d it is necessary and
sufficient that for all a < /3 0 and y > /30
II x II Epo C( a, /3ep y) II x II CI II x II .
4 0. If a family of normed spaces is given which satisfies conditions 1) and
2) of Definition 1.1 and at least one of the spaces is not complete, then we
shall say that an incomplete scale of spaces is given. Consider the completions
Ea of the spaces Ea. Assume that the following condition is satisfied:
'7T) If {x n } is a Cauchy sequence in Ep and Ilxnll E o as n 00, then also
cro
II X n II E o.
'fJ
Then the spaces {Ea} form a scale under the natural imbeddings EfJ C Ea
(a < /3).
We often encounter the special case of incomplete scales where all spaces
Ea coincide as sets: Ea = M for a o < a < /3, but differ in norm. Such an
incomplete scale is called an incomplete scale with base M. Every scale {Ea}'
ao a /3, can be obtained by completing the scale with base EfJo and the
norms of the spaces Ea.
3. N017lUl1 scales.
DEFINITION 1.2. A scale Ea' ao a /30' of Banach spaces is said to be
normal if we can set C(a, /3) = C(a, /3, y) = 1, i.e. if the following conditions
are satisfied:
1) Ilxll Ea Ilxll Ep (a < /3, x E E p ),
2) IlxilE Ilxll Ilxll .
p a"y
The last inequality says that the function f[Jx( /3) = II x II Ep (x E Ey) IS
logarithmically convex on [a o , y].
1. RELATED SPACES
189
Let {Ea} be an incomplete normal scale with base M, i.e. a linear space M
on which a family IIxllE (ao a /30) of norms is given satisfying the
CI
preceding inequalities 1) and 2). The logarithmically convex function cpx(a) =
II x II E (x E M) is continuous on [a o , /30] except perhaps at the endpoint /30. If
CI
the norm with index /30 is replaced by a new norm by setting
II x II = lim II x II E , ( 1.4)
Po 13 /3o Il
then a normal scale is obtained again.
Inequalities 1) and 2) imply that condition '7T) is satisfied for all /3 < /30. The
spaces obtained by completing the set M in the norms IIxllE form a normal
CI
scale on every interval [ao, /3d with /3 1 < /30' and if the norm with index /30 is
defined by (1.4), the construction is valid for the whole interval [a o , /30].
DEFINITION 1.3. A normal scale {Ea} (ao a /30)' for which the function
cpx(a) = IIxIlE CI ' x E Epo' is continuous on the interval [a o , /30] is called a
continuous normal scale. In this case we say that the scale {Ea} connects the
spaces Eao and Epo.
4. Related spaces.
DEFINITION 1.4. Two Banach spaces Eo and E 1 are said to be related if E 1 is
normally imbedded in Eo and there exists a continuous normal scale on the
in terval [0, 1] connecting Eo and E I.
LEMMA 1.1. If {Ea}' 0 a 1, is a normal scale, then
lim II x 1/ E 1/ x II E ,
a l CI 01
for x EEl' where EOl is the completion of EI relative to Eo.
( 1.5)
PROOF. By the definition of the norm in E 01 , there exists a sequence of
elements x n E EI such that IIxnll EI = IIxllEOI and X n x in Eo. Then
I l x - xnllE I l x - xnllk-allx - xnll ( 11x1l E + IIxllE ) a llx - xnllk- a O
CI 0 1 1 01 0
as n 00. In particular, II X n II ECI II x II ECI . We choose a so that II x II Eo
lima-+lllxl/E - f, and then n so large that
CI
IIxll E = Ilxnll E IIxnll E IIxll E - f lim IIxllE - 2f.
01 1 CI CI a 1 CI
Since f is arbitrary, inequality (1.5) is proved.
COROLLARY. For /3 < 1 the space EfJ is isometrically imbedded in its comple-
tion relative to Eo.
Indeed, for x E £1 the function IlxilE is continuous for a < 1, and there-
fore
II x II E lim II x II E = II x II E .
Oil a 13 - 0 CI Il
190
III. SCALES OF BANACH SPACES
The reverse inequality always holds, and so IIxllE = IIxII E . Since EI is
OfJ 'fJ
dense in Eo, this equality extends to all elements of Ep.
THEOREM 1.1. Let the space EI be normally imbedded in the space Eo. For Eo
to be related with EI it is necessary and sufficient that EI be isometrically
imbedded in its completion relative to Eo.
PROOF. Necessity. If Eo is related with E 1 , then there exists a continuous
normal scale {Ea} connecting these spaces. By Lemma 1.1 we then have
IIxll EI = lima-+tllxll Ea Ilxll E 01 ' and since the reverse inequality always holds,
the norms of the spaces EI and EOl coincide on EI.
Sufficiency. In the linear space EI we introduce a family of norms by the
formula
II xII = sup 11(x)1
Eo IE E6 II fll k-:- a II fll a E , .
o 1
( 1.6)
The quantity Ilxil E obviously has all properties of a norm. Moreover, for
a
every x EEl and f E E the function
If(x)1 If(x)1 ( IIfllEo ) a
IIll1k alllI1E; = 1I111Eo lilliE,
( 1.7)
is continuous and logarithmically convex in a on [0, 1]. Since IIfll E' IIfll E'
1 0
(f E E ), the function (1.7) is increasing in a. Then the supremum of all
functions (1.7) over f E E , i.e. II x II E , is a continuous increasing logarithmi-
o
cally convex function of a. 1'he set EI with the norms (1.6) is a continuous
incomplete normal scale of spaces with base EI. Moreover, for a = 0 we have
If(x)1
IIxIl E ._ o = sup 11111 = IlxllE o '
IE E6 Eo
and, by the completeness of EI in Eo in the norm of Eo, the completion of EI
in the norm IlxllE can be identified with the space Eo.
o
From the hypotheses of Theorem 1.1 and Theorem 2.1 of Chapter I it
follows that the set E is normative in the space E 1 . Therefore
If(x)1
II xII E._. = sup 11111 = II xII E."
IE E6 EI
Thus, the completion of EI in the norms II xII E gives a continuous normal
o
scale Ea connecting Eo with EI.
The theorem is proved.
By Theorem 2.1 in Chapter I, the hypothesis of Theorem 1.1 admits various
equivalent formulations. Combining them, we obtain the following assertion.
1. RELATED SPACES
191
THEOREM 1.2. For a space EI normally imbedded in Eo to be related with EfP
it is necessary and sufficient that one of the following equivalent conditions be
satisfied:
1 o. The hypothesis of Theorem 1.1 holds.
2 0 . T"lze space E is normative for E I.
3 0 . The ball of EI is closed in the topology induced by the topology of Eo.
Taking Lemma 1.3 of Chapter I into account, we obtain the following
corollaries.
COROLLARY 1. If the space EI is normally imbedded in Eo, then the comple-
tion EOl of E I relative to E I is related with Eo.
CoROLLARY 2. If the space EI is normally imbedded in Eo and is complete
relative to it, then Eo and EI are related. In particular, a reflexive space is
related with any space in which it is imbedded.
Lemma 2.4 of Chapter I leads to the next assertion.
COROLLARY 3. If the space E I is normally imbedded in Eo and the dual space
E is densely imbedded in the dual space E{, then Eo and EI are related.
Theorem 2.2 of Chapter I implies the following corollaries.
COROLLARY 4. Under the hypotheses of Corollary 3, the spaces E and E{ are
related.
COROLLARY 5. Let E 2 be normally imbedded in EI and EI normally imbedded
in Eo. If Eo and E 2 are related, so are EI and E 2 .
COROLLARY 6. Let E 2 be related with EI and imbedded in EI compactly, and
EI normally imbedded in Eo. Then Eo and E 2 are related.
5. Condensation of a nonnal scale by means of relative completion. We
consider a normal scale of Banach spaces Ea' 0 a 1. We construct a
family of spaces EOa which are completions of Ea relative to the space Eo
containing them, and establish a number of properties of this family.
1 o. For a < y the space EOy is imbedded in EOa with imbedding constant 1.
Indeed, if x E Eoy, then there exists a sequence X n x in Eo such that
" X n II J;' = II x II E . Then II X n II E II X n II J;' = II x II E , which implies that x E
a
EOa and Ilxll Eaa IlxIIE .
2 0 . The completion Ear of Ey relative to Ea (a < y) coincides with EOy.
Let x E Eay. Then there exists a sequence X n x in Ea such that IlxnllE., =
II x II E . By the imbedding Ea C Eo the sequence x n converges to x in EfP and
a:y
therefore x E En.u and IIxllE IlxilE . Conversely, if x E En.u, then one can
vr Oy a:y vr
construct a sequence X n x in Eo so that IIxnll J;' = IIxllE . Now we use the
......, Oy
192
III. SCALES OF BANACH SPACES
fact that the spaces Ea form a normal scale. We have
IIxn - xmll E IIx n - xmllk-a/Yllx n - xmlli Y
a 0 y
< IIXn - xmllk ah(2I1xIIE(,Jah.
Thus, the sequence X n is Cauchy in Ea' and by virtue of the imbedding
Ea C Eo it converges to the same element x in Ea. This means that x E Eay
and II x II E II x II E . The reverse inequality proved earlier implies the equal-
ay Oy
ity Ilxll E = IIxll E .
ay Oy
Property 2 0 and the properties of relative completion immediately imply
3 0 . The space EOy is imbeddea in every space Ea for a < y and does not
coincide with it (if the spaces Ea and Ey did not coincide).
From this we obtain
4 0 . If the space Ea is not complete relative to Eo, i.e. Ea =1= E Oa , then the
spaces EOy are imbedded not densely in EOa for a < y.
Indeed, Ea is related with Eo, and therefore, if it is not complete, relative to
Eo, then by Theorem 1.2 it is a proper subspace of EOa. On the other hand,
EOY C Ea for a < y, and consequently EOy is not dense in EOa.
50. For a < /3 < y the following inequality holds:
IlxilE Ilxll IIx!!'E (x E E ).
Op Ocr Oy v r
Let x E Eoy, X n x in Eo and II X n II E., = II x II Eoy. Then
IIxnllE p IIxnll Jlxnll = IIxnll Jlxll£Oy. (1.8)
In the proof of property 2 0 we showed that X n x in Ea. Theorem 1.1
implies that Ea is imbedded isometrically in E Oa , and therefore X n x in EOa.
Then for any f > 0 and sufficiently large n we have IlxnllEa IIxllEoa + E. It
follows from (1.8) that
II x II Eop (II x II EOcr + E)IL II X II£oy'
and, since f is arbitrary
II x II Eop II x II Ocr II x II £Oy .
Thus, the family of spaces EOy has all properties of a normal scale except
that of dense imbedding of one space in another.
2. Maximal and minim al normal scales
1. Maximal normal scales of Banach spaces. Let a space FI be normally
imbedded in a space Fo. On the interval [0, 1] we consider all incomplete
continuous normal scales Ea with base FI such that
II xII Eo IIxilFo' X E F I ,
IIxll EI Ilx11F1, X E Fl.
(2.1 )
(2.2)
2. MAXIMAL AND MINIMAL NORMAL SCALES
193
We note that there always exists one such scale, the trivial scale for which
Ilxll Ea = IIxliFo (0 a 1).
In FI we now introduce a family of norms IIxlla, 0 a 1, by the formula
IIxli a = sup IIxllE
a
(x E F I ),
(2.3)
where the supremum is taken over all scales with the above property. For a
fixed scale Ea and fixed x E FI the function IIxllE is increasing, logarithmi-
a
cally convex, and depends continuously on a in [0, 1]. The collection of such
functions is uniformly bounded for fixed x:
II xii Ea II x II EI Ilxll F 1 '
and therefore supllxllE for this collection of functions is an increasing,
a
logarithmically convex and continuous function on [0, 1]. The completions of
the space FI in the norms II x II E form a continuous normal scale of spaces
a
Earnax on [0, 1]. From (2.1), (2.3) and the remark on the trivial scale II x liE =
CI
IIxil Fo it follows that Ilxlio = IlxllFo' and consequently the spaces Er:ax and Fo
coincide. The scale thus obtained is called the maximal normal scale con-
structed from the spaces Fo and FI on the interval [0, 1].
Thus, the maximal normal scale constructed from the normally im-
bedded spaces FI and Fo has the following properties.
1) Etr ax = Fo.
2) FI is normally imbedded in E;nax.
3) If for some continuous scale Ea on [0, 1] the space FI IS normally
imbedded in Eland (2.1) is satisfied, then
II x II E II x II E maJt
a CI
(x E F I , 0 a 1).
(2.4)
From the last property it is obvious that the maximal normal scale is
defined in a unique way.
If Fo and FI are related, then the maximal normal scale constructed from
Fo and FI connects these spaces. The following assertion is stronger.
LEMMA 2.1. If a space FI is normally imbedded in a space Fep then the space
E l rnax coincides with the closure of FI in its completion FOl relative to Fo.
PROOF. By Corollary 1 of Theorem 1.2 the space FOl is related with Fo. If
FOa is a continuous normal scale connecting FOl with Fo (Foo = Fo), then,
since IlxliFol IlxllFI (x E F 1 ), properties (2.1) and (2.2) are satisfied for it.
Therefore Ilxll Fol IlxIIEF.
Now let Ea be an arbitrary continuous normal scale having property (2.2)
and such that Eo = Fo. For any element x E FI there exists a sequence
X n x in Fo such that Ilxnll FI = Ilxll FOI.
194
III. SCALES OF BANACH SPACES
Then IIxnll EI IIxnll FI = IIxIiFOI. This means that IIxll Eol II X n II FOI. The
space EI is related with Eo, and so by Theorem 1.2 EI is imbedded isometri-
cally in EOl. From this it follows that IlxliEI IlxliFol (x E F I ). Now taking
the scale EaffiaX as Ea and taking account of the inequality obtained earlier, we
obtain the equality IlxllEmaJt = IIxll (x E F I ), which proves the lemma.
a 01
LEMMA 2.2. Assume that the space FI is normally imbedded in the space Fo
and a functional (x, a), 0 a 1, is defined on F I , which is a seminorm for
fixed a and is a logarithmically convex function of a for fixed x for which
<I>(x, a) o. If (x, 0) Ilxll Fo and (x, 1) Ilxll F I ' then
(x, a) IIxliEIDAX (x E F I , 0 a 1). (2.5)
a
PROOF. We define the functional i'(x, a) = sUPal<a (x, a l ), i'(x, 0) =
(x, 0). This functional is also a seminorm for fixed a and is a continuous
increasing logarithmically convex function of a for fixed x. Moreover,
i'(x, 0) Ilxll and i'(x, 1) IIxIi F . We construct an incomplete continu-
o I
ous normal scale with norm max{i'(x, a), II xii EmaJT.}. This scale has properties
a
(2.1) and (2.2), and consequently inequality (2.4) holds for it, from which (2.5)
follows.
THEOREM 2.1. The maximal normal scale EafDa.X constructed from the spaces
Fo and FI on the interval [0, 1] is the maximal scale constructed from the spaces
Ea rnIiX and EF on every interior interval [aI' I3d.
1
PROOF. Let Ha be the maximal scale constructed from the spaces Ea7 and
E;:X on [aI' I3d. Then
IIxllEamax Ilxll Ha for x E HfJ and a E (aI' 131)' (2.6)
and IIxliEIDAX = IIxliH for a equal to a l or 131. We construct a family of spaces
a a
Ea on [0, 1] such that Ea = EaTnaX for a fl [aI' I3d and Ea = Ha for a E
[aI' I3d. By what was said in 1.2, the spaces Ea form a normal scale on [0, 1].
Besides, this scale is continuous and has properties (2.1) and (2.2), and so
(2.4) holds.
In our case, for a E [aI' I3d inequality (2.4) becomes
Ilxli H IIxllEmaJt (x E F I ). (2.7)
a a
Since FI C E l ffiaX C EfJ ' from (2.6) and (2.7) it follows that IIxli Ha =
IIXIlEmax for x E FI and a E [aI' I3d. However, the space FI is densely
a
imbedded in both Eamax and Ha, and therefore Ha = Eamax..
The theorem is proved.
2. MAXIMAL AND MINIMAL NORMAL SCALES
195
We consider two families Ea and Fa' of Banach spaces depending on the
parameters a E [ao, 130] and a' E [a , 13 0 ], respectively, and having the prop-
erty that for a < 13 the space Ep is imbedded densely in Ea (for a' < /3' the
space Fp' is imbedded densely in Fa').
DEFINITION 2.1. We shall say that the family Ea' ao a /3ep has
1) the interpolation property relative to the family Fa' of Banach spaces Fa' if
the triple (Eao' Epo' Ea) (a o a /30) of Banach spaces is an interpolating
triple relative to the triple (Faa' Fpo' Fa')' where a' satisfies the equality
a - a o
13 0 - ao
a' - a'
o .
a' , ,
PO - a o
(2.8)
2) the normalized interpolation property relative to the family Fa' of Banach
spaces if the triple (Eo. , ED , Ea) of Banach spaces is a normalized interpola-
o Po
tion triple of type (J = (a - ao) / (130 - ao) relative to the triple (Fa'o' Fpo' Fa');
3) the strong interpolation property relative to the family Fa' (a a' /30)'
if each part of it consisting of spaces whose index runs through any interval
[ai' 13d c [ao, 13 0 ] has the normalized interpolation property relative to the
corresponding part of the family of the spaces Fa'. The correspondence is
given by (2.8).
THEOREM 2.2. The maximal normal scale has the strong interpolation property
relative to any normal scale.
PROOF. Let E a ffiaX , 0 a 1, be the maximal scale, let Fa' 0 a 1, be an
arbitrary normal scale, and let A be a linear operator mapping E;nax into FI
such that
IIAxll FI CtllxllEF (x E E;nax), IIAxli Fo Collxll EcF (x E E;nax).
The function C;-ICI-<XIIAxIiF = <I>(x, a) has the properties indicated in
a
Lemma 2.2. From (2.5) it follows that <p(x, a) < IlxllEmax, or, in other words,
a
IIAxliF c -aCfllxIlEmax.
a a
Thus we have proved that the scale Eama:x has the normalized interpolation
property relative to Fa on [0, 1]. Since Ea mu is maximal on any interval
[ai' 13d c [0, 1], this implies that it has the strong interpolation property.
2. Regular scales. Let Eo., 0 a 1, be a family of Banach spaces such
that Ep is densely imbedded in Ea for a < /3. In the space Eo dual to Eo we
introduce a family of norms 11111 E'. We denote the completion of Eo in the
norm III" by Ea. We call the family {Ea} dual to the family Ea. We
196
III. SCALES OF BANACH SPACES
emphasize that the spaces Ea of the dual family may not coincide with the
-
spaces E dual to Ea if Eo is not dense in E . The space Ea is a subspace of
E . __
We note that Ea C Ep for a < {3. This follows immediately from the
construction of Ea and the fact that E is imbedded in E/3 for a < {3.
DEFINITION 2.2. A continuous normal scale Ea' 0 a 1, is said to be
regular if the norm in the dual spaces E is a logarithmically convex function
of a, i.e.
IIfilE p IIfll lIfIlE;' (2.9)
for a < {3 < "I andf E E , where
J.L = ("I - {3) ("I - a) -I and p = ({3 - a) ("I - a) -I .
The family of spaces 4(0, 1) (I Po P PI 00) is a regular scale.
Indeed, the family 4(0, I) (Po P PI) is a normal scale Ea (a o a a l ),
if we set a = I - 1 I p. In view of the Holder inequality, the norm in the dual
spaces E = 4' (lip + lip' = 1) is a logarithmically convex function of a.
If the spaces Ea form a regular scale, than the spaces Ea constitute a
normal scale with respect to the index 0 = 1 - a (0 0 I). This scale is
continuous. Indeed, the function
f(x)
qJ( a) = sup II xII = IIfll E,_.
x EEl E I - a
is left continuous at 0 = I, as a supremum of continuous increasing functions
-
of o. We shall call the scale E I -_ o (0 = 1 - a) the dual scale.
It is obvious that the scale E I - o connects the spaces E; and E , provided
that Eo is densely imbedded in E;.
THEOREM 2.3. The maximal normal scale is regular.
PROOF. Assume that a < {3 < "I and 1 E (Eamax)', where EarDa.X is the maxi-
mal scale connecting Eo and EI. Since (EarDaX)' C (Eymax)', for every element
x E Eymax the functional
<p(x (j) = If(x)1
, II 111 fEClDWt)'11 fll(E y max )'
where J.L = ("I - {3)(y - a)-I and p = ({3 - a)(y - a)-I. This functional is a
seminorm for fixed {3 and an increasing, logarithmically convex function of {3
for every fixed x for which (x, {3) O. Besides,
(x, a) IIxliEIDAX and (x, "I) IIxilEmax.
Q y
(a {3 "I),
2. MAXIMAL AND MINIMAL NORMAL SCALES
197
Since the scale E p ff13X is the maximal scale connecting Eamax with Eyrna.x, on
the basis of Lemma 2.2 we have
( x {3) = 11(x)1 II II
, II 1 II t E: U )' II 1 II ( Eymax)' X EprDax
(x E Eyma:x.) .
The space EyrfW{ is dense in E p ff1aX , and so the last inequality implies that
II 111( E;u)' 11111 tEam&X)' II 111( Eym&X)'.
3. Minimal scale of spaces. In the proof of Theorem 1.1, as the criterion for
the two spaces Eo and E I to be related, we constructed a continuous normal
scale of spaces by completing the space EI relative to the system of norms
Il xl l = su p 11(x)1
Eo fEEo IIll1k allllla E ' .
o J
(2.10)
If the spaces Eo and E I are related, then the scale connects them. We call
the scale minimal and denote its spaces by Ea min .
THEOREM 2.4. Let E 1 c Eo and FI C Fo be two couples 01 related spaces. The
minimal scales Ea min and Fa min have the normalized interpolation property
relative to each other.
PROOF. If the linear operator A satisfies the condition
IIAx II Fo Coil x II Eo'
IIAxll FI Clllxll EI
(x EEl)'
(2.11 )
then the adjoint operator A * maps F into E and F{ into E{, and
IIA *111 Eo Coil 111 Fa' II A *111 EI C 111111 Fl.
Let x EEl. Then
IIAxll n = sup 1{ x)1 a - sup 1 (x)1 a
cz f E Fo II 1 II Fa II 1 II Elf E F6 II 1 II Fa II 1 II FI
, c1-ac a sup IA*f(x)1 , c1-ac a sup I g(x)1
o I f E F6 II A * 1 II k a II A * 1 II I 0 I g E E6 II g II k"b a II g II I
= cd-aC IIXIlEmm.
a
(2.12)
DEFINITION 2.3. Let Ea and Ea' a o a ai' be two normal scales of
Banach spaces co nn ecting the space Eao with the space Ea J . We say that the
scale Ea majorizes Ea if
Ilxil E Ilxil E
a a
(x E EaJ' ao a a 1 )
for every x E EaJ.
198
III. SCALES OF BANACH SPACES
THEOREM 2.5. Let Eo and E 1 be two related spaces. The minimal scale EarrUn
constructed for Eo and E 1 is majorized by any regular scale Ea connecting Eo
and E 1 .
PROOF. Let x EEl. The spaces Ea and Eo are related, and therefore
If(x)1
Ilxil E = sup Ilfll
CI f EE' E'
o CI
(see Theorem 1.2). In view of the regularity of the scale Ea we have
Ilfll E' Ilfllk a Ilfll ,
CI 0 I
(f E Eo).
Then
II II > su p If(x)\ = Il x ll DUD.
X Ea fEE;' Ilfllk allfllE; Ea
(2.13)
REMARK. In the proof of Theorem 2.5 the regularity property (2.9) was used
only for a = 0 and y = 1.
Theorems 2.4 and 2.5 imply that every regular scale Ea has the nonnalized
interpolation property relative to any minimal scale. Indeed, by means of
(2.12) and (2.13) we obtain from (2.11) that
IIAxllFnun Cri-aCfllxllE (x EEl)' (2.14)
CI CI
which implies our assertion. The following more general theorem holds (see
[ 186]).
THEOREM 2.6. For a continuous normal scale to have the strong interpolation
property relative to any minimal scale, it is necessary and sufficient that it be
regular.
The following assertion establishes a connection between the notions of
minimal and maximal scales.
THEOREM 2.7. Let Ea min be the minimal scale connecting the spaces Eo and E 1
of a reflexive couple. If its dual family E (0 (J 1) forms a normal scale,
- -
then this scale is the maximal normal scale connecting Fo = E 1 and F 1 = Eo =
Eo.
PROOF. The spaces E 1 and Eo are related. Let Fa be a onnal scale
connecting Fo and Fl. From the dense imbedding Fo C Fo C E 1 we obtain
the imbedding (E 1 )' C F . We always have the natural imbedding E 1 C (£1)"
2. MAXIMAL AND MINIMAL NORMAL SCALES
199
and consequently EI C F . Using the density of FI = E in Fo for x EEl' we
obtain
IIxll£' = sup If(x)1 sup If(x)1
a jEE6 IlfllFa jEE6 Ilfll ollfIIFI
- sup If(x)1 - Ilxil E nuD (x EE l ).
- jEE6 Ilfllk;ollfll o - I-a
(2.15)
Next, the spaces Fo and Fo are related, and therefore the set Fo = (£1)' is
-
nonnative for Fo. By the hypothesis of the theorem (E I )' = £1' and so
IlfilF = sup If(x)1 = sup If(x)1 sup If(x)1 .
a X E (E I)' II x II F: X EEl II X II F: X EEl II X II E a
Now if f E Eo, then the quantity on the right is equal to IlfilE _ . Since E
_ I a_
is dense in E I - o , the preceding inequality implies the imbedding E I - o C Fo
and the inequality
Ilfll Ilflli\_a
(f E EI-o).
Now if Fo is the maximal scale connecting Eo with £1' then the reverse
inequality is satisfied on E , and therefore Fo and £1-0 coincide.
COROLLARY. If the minimal scale connecting the spaces of a reflexive couple
Eo and EI is regular, then the scale dual to it is the maximal normal scale
-
connecting Eland Eo.
The following assertion is dual to Theorem 2.7 in a certain sense.
-
THEOREM 2.8. Let Ea, 0 a 1, be a continuous normal scale, and let Ea be
the family dual to it. If
II f II - " su I f( x ) I
Ea '" x E I IIxil k- a II x liE '
o I
(2.16)
for every f E E , then Ea is the maximal normal scale connecting Eo and EI.
PROOF. If Earnax is the maximal normal scale constructed for Eo and E I ,
then Ilxl!EQ IlxllEamax. This implies that Ilfll(E a max )' Ilfllia for every f E Eo.
On the other hand, by (2.16) we have
-
Ilfll - sup If(x)1 sup If(x)1 Ilfll
(Eam&X)' - II x II max II II I-a II Ii a Ea.
xEE 1 Ea xEE 1 X Eo X EI
200
III. SCALES OF BANACH SPACES
The inequalities obtained above imply that Illll(Ea IDAX )' = Ilfll Ea (f E E ). Since
the spaces Eo and Ea max are related, we have
Il(x)1 If(x)1
IIXIIE.max = sup 11111 = sup 11111 = IIxIl E . (x E E) .
f E E6 (£::0.)' f E E6 £
3. lbe scale of Holder spaces
The main notions of the theory of scales of spaces can be well illustrated by
classical examples of function spaces, from which we only choose the Holder
spaces. For simplicity we consider functions on the interval [0, 1].
1. Holder spaces. The Holder spaces Co, a (0, 1), 0 a 1, are intermediate
between the space C(O, 1) of functions continuous on [0, 1] and the space
C. (0, 1) of functions continuously differentiable on [0, 1]. They consist of all
continuous functions satisfying the Holder condition
Ix(t) - x(T)1 elt - Tla
with index a for every t, T E [0, 1]. The norm in CO,a(O, 1) is defined by
IIxli co . = max Ix(t)1 + sup Ix(t) - x r)1 (3.1)
. 0<1<1 0<1,7"<1 It - TI
An equivalent norm can be introduced by the formula
II xII co. = Ix(O)1 + sup Ix(t) - x r)1 (3.2)
. 0<1,7"< 1 It - TI
If in C 1 we introduce the norm
IIxll c = max Ix(t)1 + max Ix'(t)l,
I 0<1<1 0<1<1
then the space CO,a with norm (3.1) will be imbedded in C and in it
will be imbedded the space C. with imbedding constants equal to 1, i.e.
. 1
C. C CO,a C C.
The presence of two terms in the norms (3.1) and (3.2) causes some
inconvenience. Therefore, we shall consider the quotient space of CO,a modulo
the one-dimensional subspace consisting of the constant functions. We shall
denote this quotient space by Ra. If we introduce the norm (3.2) in CO,a' then
this quotient space will be isometric to the subspace of all functions in CO,a
vanishing at zero. Consequently Ha consists of classes of functions differing
by a constant and satisfying a Holder condition of order a, the norm being
IIxll H = sup Ix(t) - x(r)1 (0 ..;; a ..;; 1). (3.3)
a 0<1,7"<. It - Tla
We note that Ho is isometric to the quotient space of C modulo the
subspace of constant functions, if in C we introduce the norm Ilxll c = Ix(O)1
+ maXo t.T llx( t) - x( T)I.
3. THE SCALE OF HOLDER SPACES
201
It is obvious that Hp is imbedded in Ha if {3 > 0:. This imbedding is not
dense for 0: > o. Indeed, xo(t) = t a E Ha' but Ilx o - xIIH > I for all x E
a
Hp. Of course, all spaces Ha are densely imbedded in Ho.
According to (3.3) the norm of a given function x E HI is a continuous
logarithmically convex function of 0:, being a supremum of continuous
logarithmically convex functions. However, the family Ha does not form a
scale of spaces in our sense, since the spaces Ha are not imbedded densely
into each other. We may consider the incomplete normal scale with base HI
and norms (3.3) and complete it.
2. The Holder scale. We denote by H the completion of HI in the norm
(3.3) of Ha. In view of what has been said in 1.3, the spaces H form a
continuous normal scale connecting Ho and HI. This scale is called the
Holder scale.
THEOREM 3.3. The space H a o , 0 0: < 1, consists of all functions (classes of
functions) in Ha such that
lim Ix(t) - x 'T)1 = o.
It - 1"1 O I t - T I
(3.4)
PROOF. All functions in HI obviously have property (3.4). Now let X n E HI
and Ilx - xnll H O. We have
a
Ix(t) - X(T)I Ix(t) - Xn(t) -[X(T) - xn(T)]1 IXn(t) - Xn(T)1
+
It - Tla It - Tla It - Tla
Ilx - xnll H + Ilxnll H It - TII-a.
a I
First choosing n to make the first summand sufficiently small, and then
choosing It - TI to make the second summand sufficiently small, we obtain
that the left side is arbitrarily small if It - TI is sufficiently small. Conse-
quently, every function x E Ha has property (3.4).
Now let x be an arbitrary function in Ha having property (3.4). We show
that it belongs to H . We consider the functions
f t+ l/n 1 1 / n
xn(t) = n X(T) dT - n X(T) dT
t 0
setting x(t) = x(l) if t > I. The functions xn(t) are continuously differentia-
ble, and consequently belong to HI. Performing the change of variables
T = (J + t in the first integral, we obtain
l l/n
xn(t) = n 0 [x(t + 0) - x(O)] dO.
(n = I, 2),
Hence
l l/n
xn(t) - x(t) = n 0 [ x(t + 0) - x(O) - x(t)] dO.
202
III. SCALES OF BANACH SPACES
We write [xn(t + h) - x(t + h)] - [xn(t) - x(t)] = i'(xn - x, h). With
this notation we have
IIX n - xIIH = sup
a O<t+h<l
Ii'(xn - x, h)1
h a
In view of condition (3.4), for every E > 0 we may choose ho > 0 such that
h-alx(t + h) - x(t)1 E/2 for all t and h hoe Then
h-al'l'(xn - x, h)1
= h--<Xnl I/"[ x(t + 0 + h) - x(t + h) - x(t + 0) + x(t)] dOl
r l/n [ Ix(t + h + (J) - x(t + (J)I Ix(t + h) - x(t)1 ] -1IJ
nJo h a + ha uu
n' ( + )=f
for h ho and n > 1. Now let ho h 1. Then
h-al'l'(xn - x, h)1
r 1/ n !!:... [ I x (t + h + (J) - x (t + h) I I x (t + (J) - x( t) I ] dO
n J o ha (Ja + (Ja
l l/n (Ja 211 x IIH
2nllxIIH - h a dO (1 h: a '
a 0 + a) on
and we again have h-ali'(x n - x, h)1 E for sufficiently large n. Thus,
Ilx - xnll H < E for sufficiently large n, i.e. x E H a o.
a
LEMMA 3.1. The space H3 is imbedded compactly in H for {3 > a.
PROOF. Let X n be a bounded sequence in H3. We assume that xn(O) = O.
Then the sequence xn(t) is uniformly bounded and equicontinuous, and
therefore, compact in C. Let xn,(t) be a uniformly convergent subsequence.
We show that it is a Cauchy sequence in H . Indeed,
IXn,(t) - xm,(t) - xn'( 'T) + xm'( 'T)I
It - 'Tla
= It _ 71 P - a x",( t) - x"' ( 7) _ X m '( t) - X m '( 7)
It - 'TIP It - 'TIP
It - 'T1 13 -a(llx n , IIH 3 + IIx m , IIH 3) < E (3.5)
for
( ) l/(p-a>
E
t-'T < =h
I I 2 max II x" II HI' ·
n 0
3. THE SCALE OF HOLDER SPACES
203
For It - TI > h the expression on the left side of (3.5) can be made smaller
than f for sufficiently large m' and n', in view of the uniform convergence of
xn,(t). For such m' and n' we have Ilxn' - xm,IIHo < f.
a
REMARK. The same proof shows that the assertion of the lemma also holds
for the spaces Ha.
We have obtained the Holder scale H by completing an incomplete scale
with norms of the spaces Ha. The family of the spaces Ha can be recovered
from the scale Ha o by completing the spaces H relative to the space Ho
described in 1.5.
First we establish an auxiliary assertion. Let 0 = to < t I < . . . < t N -I <
t N = 1 be an arbitrary partition of [0, 1] and let XN(t) be a continuous
function which is linear on every interval [t k - I , t k ]. This function obviously
belongs to HI.
LEMMA 3.2. For a continuous piecewise linear function,
Ix N ( ti) - X N ( )I
IIXNIIH = max I l a .
a 0 <:.i J " <:.N t. - t"
I J
i=t=j
(3.6)
PROOF. Let us denote the right side of (3.6) by O(x). It is obvious that
IXN(t) - xN(r)1 IXN(tJ - xN( )1 _ O( )
IlxNllH = sup I la ". max I l a - x.
a 0<:'1,7"<:.1 t - T o <:.i,j<:.N t i - 1.i
If the points t and T belong to [t k - I , t k ], where the function XN(t) is linear,
then
IXN(t) - XN(T)I
It - Tla
_ XN(t k ) - XN(t k - 1 ) It _ rll-a
t k - t k - I
Ix N ( t k ) - x N ( t k -1)1 O( )
I l a x.
t k - t k - I
- -
Assume that for the points t, t k - I , and t k , where t fl [t k - I , t k ], we have
proved that
IXN(i - xN(tk-1)1 "O(x) and
It - tk_Ila
Ix N ( t) - x N ( tk)1 O( )
_ x.
It-tkl a
(3.7)
We show that
IxN(i) - xN(t)l/li - tl a <: O(x), (3.8)
where t = At k _ I + (1 - A)t k (0 A 1) is an arbitrary point of [t k -I' t k ].
204
III. SCALES OF BANACH SPACES
Indeed, by virtue of the linearity of XN(t) on [t k - I , t k ] and (3.7) we have
IXN(t-) - xN(t)1 IA[ xN(t-) - XN(t k - I )] + (I - A)[ xN(t-) - XN(t k )] I
-
It - tl a IA( t- - t k - I ) + (I - A)( t- - tk)la
Alt- - tk_Ila + (I - A)lt- - tkl a ( ) ( )
a (}x (}x.
IA(t--t k - l ) + (1 - A)(t--tk)1
The last inequality follows from the concavity of the power function with
exponent lX, 0 < ex 1.
Now let'T E [t i - 1 , ti] and t E [t k - I , t k ]. Inequalities (3.7) are obviously true
- -
for t = t i - 1 and t = tie Then it follows from the preceding that (3.8) holds for
- -
these values of t. The t inequality
IxN('T) - xN(t)I/I'T - tl a (}(x)
follows from this by the same arguments.
Thus, IlxN11 H (}(x), and so (3.6) is satisfied.
a
THEOREM 3.4. The completion of H relative to Ho coincides with Ha.
PROOF. Let x E Ha. We construct a sequence of piecewise linear functions
XN(t) uniformly converging to a function x(t) on [0, I] as N 00 so that the
graphs of these functions are inscribed in the graph of x(t). Then by Lemma
3.2 we have
IxN(t i ) - xN( )1 Ix(t i ) - x( )1
IIxNllH o = max I a = max I la ";; Ilxli H .
a 1 <:.i,j <:.N t i - I 1 <:.iJ<:.N t i - l.i a
i=t=j i=t=j
The sequence XN belongs to HI C H , converges to x in H o , and is
bounded in Hao. Therefore x belongs to the completion of H relative to H o .
Conversely, if X n x in Ho and IlxnllHo M, then
a
Ixn(t) - xn('T)I/lt - 'Tla < M,
for every t, 'T E [0, I], and, by the uniform convergence of xn(t) to x(t),
Ix(t) - x('T)I/lt - 'Tla M,
i.e. x E Ha. The theorem is proved.
Thus, the spaces H are not complete relative to H 0' and in accordance
with what has been said in 1.5 the spaces Ha are not imbedded densely into
each other.
In order to elucidate the interpolation properties of the Holder scale, we
have to study some properties of the dual spaces of the Ha. First we exhibit
these properties in the case of finite-dimensional analogues of Holder spaces.
3. THE SCALE OF HOLDER SPACES
205
For a given partition 0 = to < t l < . . . < t N - I < t N = 1 of the interval
[0, 1] the collection of continuous piecewise linear functions that appears in
Lemma 3.2 forms an N-dimensional subspace of Ha, which we denote by H:.
The norm in H: is given by the right side of (3.6). We denote by G: the
space dual to H:. The functionals
gt.t. = x( t;) - x( )
'1
belong to this space. As follows from (3.6), we have
II gt; II G,:' II gt; II H It; - Ia.
Let t; < . We consider the piecewise linear function
o if 0 t t;,
xo(t) = (t - t;)lt; - Ia-I if t; t ,
It; - I a if t < 1.
For it we have Ilxoil Ha = Ilxoll H,:' = 1 and I gt; (xo)1 = It; - Ia. Thus,
II gt; II G,:' = It; - Ia. (3.9)
We write hj(x) = It; - I-agt, .(x). Then Ilh)11 G,:' = 1, and formula (3.6) for
the norm of the space H: can be written in the form
IlxllHN = max I l lj oo(x)l. (3.10)
a l i,j<:'N
i=l=j
This implies that the unit ball of G: coincides with the convex hull of the
functionals fij. Indeed, if f does not belong to this convex hull, then there
exists an element x E H: such that f(X) > Ifij(X)I. Then by (3.10) we have
f(x) > IlxIIHN, i.e. IlfllGN > 1. Thus, all extreme points of the unit ball of G:
a a
are among the functionals Jij.
Let S be a face of the unit sphere of G:. This means that there exists an
element Xs with IlxsIIHN = 1, such that J(xs) = 1 for all J E S. If the
a
functional fij belongs to S, then no functional of the form fld belongs to S.
Indeed, if Jk; E S, then
.h .(x ) = XS(t k ) - xs< ) = XS(t k ) - xs(t;) + xs(t;) - xs( )
k.J S Itk - Ia Itk - Ia Itk _ Ia
_ Itk - t;la It; - Ia
- fk;(x S ) Itk - Ia + fy(xs) Itk _ Ia
Itk - t;la + I - t;la
I l a > 1,
t k -
which contradicts the equality II fig II G,:' = 1.
206
III. SCALES OF BANACH SPACES
Thus, for the system of functionals fij lying on the face S, the set of first
indices does not intersect with the set of second indices.
Now let 1 be any functional with IIfllGN = 1. It belongs to some face S of
the unit sphere in G:, and consequently a
1 = aijfij
iJ
(fij E S),
where aij > 0 and ijaij = 1. We calculate the norm off in Gt. We have
IIIIIGt' lrijllJ;jIIGt' = aijlt j - I-a.
iJ iJ
Now we consider the continuous piecewise linear function x E H: equal
to 1 for all t j for which aij > 0, and equal to zero for the remaining tj. In view
of the fact that the sets of first and second indices for which aij > 0 do not
intersect, the definition of x(t) is not contradictory. It is obvious that
IlxIIHt' = 1 and
_ x(t j ) - x( )
f(x) = (Xij It. _ t.la = aylt; - I-a.
I,) I J I,}
We have shown that
11111 Gt' = aijlt j - I-a.
iJ
(3.11 )
Now let 0 f3 a. We calculate
IIIII G ; aijllfijllG; = aijlt j - 1,8-a
iJ iJ
= ( :: l/a lt . _ t . I) -<a- P ) / /a
£.J a y I J a y .
iJ
Applying the Holder inequality with exponents a/(a - {3) and al {3 and
taking account of (3.11), we obtain
( ) <a-p)/a ( ) p/a
IIfll G; t (Xijlt; - I--a (Xij = II fll <;;iP)/a.
We recall that 11111 GN = 1. Then
a
IIIII G : IIIII iP)/allfll a (3.12)
for any 1 E G:.
We have prepared everything for the proof of the following assertion.
LEMMA 3.3. For any functional 1 E H
Ilfll< Hi)' Ilfll o- 13)/ allfllfk '.
(3.13)
3. THE SCALE OF HOLDER SPACES
207
PROOF. Letf E Ho. We denote by x O a function in H3 such that IIxoll H3 =
1, and f(x > Ilfll(Hj)' - f. This function can be approximated with any
accuracy in the norm of Ho by a continuous piecewise linear function x (t)
whose graph is inscribed in the graph of xO(t). Since f is continuous in the
space Ho, we may choose xZ so thatf(xZ) > Ilfll(H ' - f; then Ilx IIHo < 1,
p - p
as we saw in the proof of Theorem 3.4. If we denote by f the restriction of f to
the subspace Ht, then it follows from the preceding that IllIIG; > Ilfll(Hj)'
- f.
It follows from (3.13) that
IIfll(H3)' - f <: IlllIG; Illll fP)/alljll a Ilfll / )/allfllfhc?)'.
Since f > 0 is arbitrary, the lemma is proved.
THEOREM 3.5. The Holder scale H is the minimal scale connecting the spaces
Ho and HI.
PROOF. It can be proved by means of Theorem 2.5 that inequality (3.13) for
0: = 1 implies that the Holder scale majorizes the minimal Scale constructed
for Ho and HI' i.e. IlxllHmm IlxllHo (x E HI' 0 0: 1). We prove the
a a
reverse inequality. We recall that
IIxll nun = sup If(x)1
Ha f E H 0 II f II k a II f II '1. .
As above, we may show that for the functional gT"T' = x( 7") - x( 7"') we have
II grr,1I H = 17" - 7"'la. Therefore
a
sup
o < T. T' < 1
I grr'(x) I
II gT"T,1I k b a II grr,II'1;
IX(7") - x(7"')1 = II II
I ' I a X HO .
7" - 7" a
IlxllHrDm
a
sup
o < T. T' < 1
From Theorems 2.4 and 2.6 we obtain the following assertion.
THEOREM 3.6. Every minimal scale has the normalized interpolation property
relative to the Holder scale. Every regular scale has the strong interpolation
property relative to the Holder scale.
By Lemma 4.4 of Chapter I and Theorem 3.4, the fact that the Holder scale
is a normalized interpolation family implies that the family of spaces Ha is
also a normalized interpolation family.
CHAPTER IV
INTERPOLATION METHODS
Introduction
1. Spaces of strongly measurable fimctions with values in a Banach space. We
recall (see [18]) that a function x(t) defined on a space WC with measure J.L and
taking values in a Banach space A is said to be strongly measurable if there
exists a sequence of strictly simple functions on WC converging to x( t) almost
everywhere. The collection S(WC, J.L, A) of all strongly measurable functions
(classes of functions) is obviously a linear space. This space is metrizable by
means of the metric
( Ilx(t) - y(t)IIA
p(x,y) = J<.rn 1 + IIx(t) _ y(t)IIA v(t) dp.(t),
The space S(WC, J.L, A) is complete.
For functions belonging to S(WC, JL, A) the analogue of Egorov's theorem
holds. From it we can derive the following assertion.
I'(t) > 0, I' E L.(WC, JL).
LEMMA 1. Every strongly measurable function is the limit of a uniformly
convergent sequence of measurable elementary functions.
For a strictly simple function x(t) assuming the nonzero values Xi on the
sets e i the integral is defined by
f x(s) dp. = L x;p.(eJ
A function x(t) is said to be (Bochner) integrable if there exists a sequence
of strictly simple functions xn(t) converging to x(t) almost everywhere and
such that
lim f Ilx(s) - xn(s)lIdJ.L = O.
n oo
209
210
IV. INTERPOLATION METHODS
The integral of the function xes) is, by definition, the limit
lim f Xn(s) dJL = f x(s) dJL.
n oo
For a function x(t) to be integrable it is necessary and sufficient that it be
strongly measurable and
IIxll = fllx(s)1I dlL < 00.
(1)
The space of integrable functions with the norm (1) is a Banach space and
is denoted by L 1 (A).
Let E be an ideal lattice of functions of WC. We denote by E(A) the
collection of strongly measurable functions x(t) with values in A for which
Ilx(t)IIA E E. It is easy to verify that the space E(A) is linear.
LEMMA 2. E(A) is a Banach space relative to the norm
II x II E( A) = 1111 x ( t) II A II E.
PROOF. Let X n E E(A) and L lIxnIIE(A) < 00. Since the space E is com-
plete, the series L llxn(t)IIA is convergent in E, and so its sum is finite almost
everywhere. It follows from the completeness of A that L xn(t) converges to
a (strongly measurable) function x(t) almost everywhere, and
00
}: Ilx n ( t)11 A Ilx(t)11 A.
n=l
(2)
Since E is an ideal lattice, this inequality implies that
00
IlxIIE(A) IlxnIIE(A) < 00.
n=l
Thus, x E E(A). Applying (2) to the function x( t) - = I x n ( t) =
L _N+ 1 xn(t), we obtain
N
X - }: X n
n=l
00
}: Ilxnll E(A) 0 as N 00.
E(A) n=N+I
The lemma is proved.
By arguments similar to the proof of Theorem 1 of Chapter II we can
establish that E(A) is imbedded in S(WC, J-L, A).
Lemma 1 admits the following sharpening.
LEMMA 3. If x E S(WC, J-L, A) and p(t) is a measurable function, positive on
the support of x, then there exists a measurable elementary function x(t) such
that II x( t) - i( t) II A p( t) (t E WC).
INTRODUCTION
211
PROOF. We set WCn = {t E : 2 n < p(t) 2n+l}. By Lemma 1 there exists
an elementary function xn(t) with support in n such that
Ilx(t) - xn(t)IIA < 2 n < p(t) for t E WCn. Then the function x(t) defined by
x(t) = xn(t) for t E WCn and zero outside the support of x has the required
properties.
CoROLLARY. The set of elementary functions in E(A) is dense in E(A).
Indeed, if x E E(A), then, choosing a function p(t) positive on the support
E such that Ilpll E f, according to Lemma 3 we construct an elementary
function x(t) such that Ilx(t) - x(t)IIA p(t), from which it follows that
IIx - xii E(A) Ilpll E f and
x = x + (x -x) E E(A).
LEMMA 4. If the lattice E is regular, then the set of strictly simple junctions
from E(A) is dense in E(A).
PROOF. It is sufficient to prove that any elementary function
00
x(t) = Xk lc(t)
k=l
(x k E A, e; n e j = 0, i =1= j)
in E(A) can be approximated by a sequence of strictly simple functions. We
set xn(t) = Lk-l Xk 1c (t). The intersection of the decreasing sets en =
u :: 1 e k is empty, and therefore the absolute continuity of the norm in E
implies that
Ilx - xnIIE(A) = Ilx:xe,.IIE(A) = IIXen(t)llx(t)IIAIIE O
as n 00.
We need a modification of Lemma 4.
LEMMA 5. If WC = U n' where the measurable sets WCn are disjoint and
have finite measures, then the collection of those strictly simple functions whose
supports can be covered by a finite number of sets WCn is dense in E(A).
PROOF. Let x E E(A). The function XN(t) = x(t)XN(t), where XN(t) is the
characteristic function of U '( WCn' converges to x in E(A) because of the
absolute continuity of the norm in E. According to the above, the function x N
can be approximated by strictly simple functions with support in U'( 9Rn.
2. The dual spaces of the spaces E(A). It is natural to assume that the dual
space (E(A»' has to coincide with E'(A') by the duality
f (x( t), x'( t) )d/L,
212
IV. INTERPOLATION METHODS
where ( , > is the value of a functional in A' at an element of A. However,
this is not true in the general case. For example, there is no such coincidence
in the case where A = II and E = L 2 (0, 1) (see [I5Ia]). We indicate a class of
ideal lattices for which the indicated coincidence takes place.
We assume that we = N + = {I, 2, . . . } and JL({n}) = J.Ln > O.
THEOREM 1. If E is a regular lattice on N + (with support = N +), then the
space (E(A))' is isometrically isomorphic to E'(A ').
PROOF. Let f E [E(A)]'. We consider a numerical sequence cP =
(CPI' CP2' . . . ) E E and an element Xo EA. Then cpxo = {CPlx(p CP2 X O' . . . } E
E(A) and
If( cpX o ) I II fll(E(A»'11 cpxoll E(A) = Ilfll(E(A»'11 cpll Ell xoll A . (3)
I t is obvious from this that f( cpx o ) is a bounded linear functional on E, and
so by the regularity of E there exists a numerical sequence cp'(x o ) = {cp (xo),
CP2(x O )' . . . } E E', such that f(cpx o ) = L =I CPnCP (xO)J.Ln. The uniqueness of
the sequence cp'(x o ) implies that it depends linearly on Xo EA. On A we
define the linear functionals (xo, fn> = cp (xo) (xo E A). Setting cP = 1/;n =
(0, . . . , 0, 1,0, . . . ) (I in the nth place) in (3), we obtain
l(xo,fn>1 = Icp (xo)1 = JL;llf(1/;n x o)1 IL;lllfIlE(A)'II1/;nIIEllx o II A ,
which implies that cP is bounded on A. From the definition we obtain
00
f( cpx o ) = CPn (xo, fn > J.Ln.
n=l
Any finitary sequence x = (XI' x 2 , . . . ) E E(A) (only finitely many terms
are different from zero) can be represented in the form L kXk. Therefore
N N
f(x) = f( kXk) = (xk,fk>JLk.
k=l k=l
(4)
Now we take cP = (CPI' CP2' . . . ) 0 from E and in place of x k we substitute
the elements CPkYk in (4), where IIYkllA 1. We have X =
{CPI YI' . . . , CPNYN' 0, . . . } E E(A), and the norm of this sequence in E(A)
does not exceed II cpll F. Consequently If(x)1 Ilfll(E(A»'llcpli E. Making Yk run
independently through the unit ball of A, we can make the right side of (4)
arbitrarily close to L CPk II f k II A' JLk. Hence
N
CPk II fk II A' ILk II fll(E(A»'11 cpll E.
k=l
Letting N tend to 00, we obtain
00
CPkllfkllA'lLk IIfll(E(A»'llcpIIE.
k=l
INTRODUCTION
213
From this it follows that the sequence Tf = {fl' f2' . . . } belongs to E'(A')
and
IITfl1 E'(A') IIfll(E(A»'.
On E(A) we consider the functional
(5)
00
j(x) = (xk,fk)ILk.
k=l
(6)
We have
00
Ij(x)1 = L (xk,fk)JLk
k=l
00
L Ilxkll A IlfkllA' ILk IIxll E(A)II'Tfll E'(A').
k=l
Th us,
Ilfll(E(A»' II'Tfll E'(A') Ilfll(E(A»'. (7)
By (4) and (6), the bounded linear functionals f and f coincide on finitary
sequences from E(A), which are dense in E(A) by Lemma 4. Consequently
f= f.
Thus, to every functional f E [E(A)]' there corresponds the functional
'Tf = (f l , f2' . . . ) E E'(A '), and
00
f(x) = (xk,fk)ILk.
k=l
(8)
From (5) and (7) it follows that Ilfll(E(A»' = II'TfIIE'(A'). It is clear from the
construction that the operator 'T is linear.
We note that equality (8) determines the sequence {f k } uniquely, as can be
verified by replacing x by sequences having only one nonzero element.
If {fk} is an arbitrary sequence from E'(A'), then, as was shown earlier, (8)
determines a linear functional on E(A). From what we said above, it follows
that 'Tf = {fk}. Thus, the mapping f 'Tf is suIjective, i.e. the operator T
effects an isometric isomorphism between the spaces (E(A»' and E'(A').
The theorem is proved.
In the case of lattices on spaces with continuous measures the situation
becomes more complicated. We mention one theorem, without proof, which
will be used in the sequel (see [12], p. 807).
THEOREM 2. Let E = Lion the axis (-00, (0) with Lebesgue measure. For
any linear functional F on the space E(A) = LI(A) there exists a function 1/;(t)
with values in the space A', bounded in the norm of A' and such that the function
(x, 1/;(t) is measurable for every x E E and
F(x) = Joo (x(t), \¥(t» dt
-00
for x(t) E LI(A). Moreover, II FII(L1(A»' = ess supll1/;(t)IIA'.
214
IV. INTERPOLATION METHODS
1. lbe complex method of interpolation
1. Scalar hannonic and analytic .functions in a strip. We recall that a
complex-valued function u( ) (I I 1) continuous and harmonic in the unit
disk of the complex plane is representable in terms of its boundary values by
means of a Poisson integral:
1 f 'IT . e i8 +
u( ) = _ 2 u(e'8)Re i8 dO.
'IT -'IT e -
If cp(O) is an arbitrary continuous 2'lT-periodic function, then the formula
1 'IT e i8 +
u( ) = _ 2 f <p(O)Re i8 dO (1.2)
'IT -'IT e -
(1.1)
defines a function, harmonic inside the disk and continuous in the closed
disk, that coincides with <p(0) on the boundary of the disk (at the points ei8).
Formula (1.2) retains its sense for any bounded measurable function <p(0).
In this case the function u( ) is also hannonic inside the disk and bounded in
it. Moreover, at almost every point of the circle the boundary value of u( )
exists and is equal to cp(O) on any path which is not tangent to the circle. If
we denote this limit value by u(e i8 ) (assigning it arbitrarily at all points where
there is no limit value), then the Poisson formula (1.1) remains true for every
bounded harmonic function (see, for example, [19]).
A special case of (1.2) is the assertion of the mean value theorem:
1 f 'IT
u(O) = 2'1T -'IT cp( 0) dO,
( 1.3)
There arises the question of when (1.2) generates a function analytic inside
the disk. For this it is necessary and sufficient that all Fourier coefficients of
<pC 0) with negative indices be equal to zero, i.e. that
f'IT cp( O)e in8 dO = 0 (n = 1, 2, . . . ). (1.4)
-'IT
These conditions can be formulated in a more concise form: For the
function (1.2) to be analytic in the disk it is necessary and sufficient that, for
any function analytic inside the disk, continuous in the closed disk and equal
to zero at = 0, the equality
f'IT cp(O)t[;(O) dO = 0 (1.5)
-'IT
hold.
The necessity of this condition follows from the fact that the product
u( ) ( ) is an analytic function, and consequently by (1.3) the integral is
equal to 2'lTU(0) (0) = O. This condition is sufficient, since it implies (1.4) (for
n( ) = n).
1. THE COMPLEX METHOD
215
Under a conformal mapping of the unit disk onto some domain, harmonic
functions turn into harmonic functions and analytic functions into analytic
functions. Therefore every such domain has its Poisson formula with proper-
ties similar to those described above. The Poisson formula for the half-plane
is well known:
1 J oo 1
u(O = '1T -00 u(s)Im s - r ds.
( 1.6)
We shall be interested in functions harmonic and analytic inside the strip
II: 0 Re z 1. The mapping = e-rriz maps this strip conformally onto
the half-plane. By making the appropriate substitutions in the integral (1.6),
we arrive at a Poisson formula of the form
1 J 00 sin 'lTS
u z = - U iT dT
() 2 -00 ( ) cosh '1T( 'T - t) - COS'1TS
1 J 00 sin 'lTS
+ - u 1 + iT dT,
2 -00 ( ) cosh '1T( 'T - t) + cos '1TS
where z = S + it. The kernels
1 sin 'lTS
J.LO z, 'T) = 2 cosh '1T( 'T - t) - COS '1TS
and
1 sin ?TS
J.LI (z, 'T) = 2 cosh '1T( 'T - t) + cos '1TS
are nonnegative, and
J 00 J.Lo( a, 'T) d'T = 1 - a, J 00 J.LI( a, 'T) d'T = a. (1.7)
-00 -00
A criterion of analyticity of a function representable in the form
u( z) = i: go( 'T) JLo( Z, 'T) d'T + i: g I ('T )J.LI( Z, 'T) dr, (1.8)
where the functions gO(T) and g.(T) are continuous and bounded on (-00, (0),
can be obtained from the analyticity criterion for the disk formulated above:
For a function u(z) of the form (1.8) to be analytic inside the strip it is
necessary and sufficient that, for any function h(z) continuous in the closed
strip, analytic inside the strip, having limits as 1m z + 00 and vanishing at
z = lX,
J 00 h(i'T )go( 'T )/Lo( a, 'T) d'T + foo h(l + i'T )gl( 'T )J.LI( a, 'T) dr = 0
-00 -00
hold.
216
IV. INTERPOLATION METHODS
We denote by the space of functions analytic inside II, continuous and
bounded in its closure, and equipped with the norm
II ulh)' = max { sup I u(i'T)I, sup I u(l + i'T)I}.
'T 'T
The completeness of this space follows easily from the maximum principle.
It is obvious that the product of two functions from again belongs to .
Therefore is a Banach algebra.
2. Abstract analytic .functions in a strip. Let A be a complex Banach space.
We consider the collection (A) of all functions with values in A, analytic
inside II, and continuous and bounded in the closed strip. If j E ffiA), then
we have the representation
f( z) = f 00 f(i'T) /La ( z, 'T) d'T + f 00 j( 1 + i'T) ILl ( Z, 'T) d'T. ( 1.9)
-00 -00
To prove this it is sufficient to apply (1.8) to scalar functions cp(j(z»,
where cp E A', and use the fact that the integrals in (1.9) are convergent in A
by virtue of the boundedness of j(z) and the integrability of the functions
JLo(z, 7') and JLI(Z, 7').
We introduce the norm
IIfll5<A) = max{ sup II f(i'T) II A' sup IIf(1 + i'T) II A }
'T 'T
in (A), which makes it a Banach space. (We recall that the maximum
principle remains valid for abstract analytic functions (see [18], 3.I3).) The
space 3(A) is a module over the Banach algebra 3.
3. The space (Ao, A 1). Now let Ao and A 1 be a Banach couple of complex
spaces. By (Ao' A 1) we denote the linear space consisting of all functions
j(z) defined in the strip II: 0 Re z 1, with values in the space Ao + Al
and having the following properties:
1) j(z) is continuous and bounded in the norm of Ao + Al in the closed
strip II.
2) j( z) is analytic relative to the norm of A 0 + A 1 inside the strip.
3) j(i7') assumes values in the space Ao and is continuous and bounded in
the norm of this space, while j( 1 + i7') assumes values in A 1 and is continuous
and bounded in the norm of A 1.
We equip the linear space (Ao' A 1) with the norm
II f I15<A o.A ,) = max { sup II f(i'T) II Ao' sup II f(1 + i'T) II A.}.
'T 'T
(1.10)
1. THE COMPLEX METHOD
217
We note that, by the maximum principle, for an analytic function we have
IIf(z)IIAO+A, " max{ sup IIf(i'T)IIAo+A" sup IIf(1 + i'T)IIAo+A,}
7 7
" max{ sup II f(i'T) II Ao' sup IIf(1 + i'T)IIA,} = IlfliffiAo.A,) (1.11)
7 7
for f E (Ao' A I). From this it follows in particular that f(z) = 0, provided
IlfllffiA I) = o.
We prove the completeness of 3(Ao, A I). Let fn be a Cauchy sequence
in 3(Ao, AI). From (1.11) we obtain that Ilfn(z) - fm(z)IIAo+A 1
IIfn - fmllffiA I)' and consequently the sequencefn(z) is uniformly convergent
in the norm of Ao + Al in II. Its limitf(z) is a function having properties 1)
and 2). The functionsfnU + iT) (j = 0, 1) converge uniformly in the norm of
Aj' and by virtue of the imbedding Aj C Ao + A I their limit is the function
fU + iT), which is continuous and bounded in the norm of Aj. Thus, f E
3(A(p A I) and IIf - fnllffiA I) 0 as n 00.
We denote by 30(Ao,AI) the subspace of ffiA(pAI) consisting of all
functions f(z) for which fU + iT) converges to zero in the norm of Aj
(j = 0, 1) as ITI 00.
If f E (Ao' A I)' then the function
fo(z) = e 6z y(z) (1.12)
obviously belongs to o(Ao, A I) for every > o. It is easy to verify that the
functionsf6(z) converge tof(z) in the norm of (A(p AI) as O.
For the sequel we need several properties of the spaces ffiAo, A I) and
30(A(p A I).
LEMMA 1.1. If f E 3(Ao, AI)' and if the sequence of functions gn E
30(Ao, A I) is bounded in (A 0' A I) and has the property that the junctions
gnU + iT) converge uniformly to fU + iT) in the norm of Aj U = 0, 1) on every
bounded subdomain of T values, then there exists a sequence of positive numbers
2
sn such that the functions e SnZ gn(z) converge to fez) in 3(Ao, AI).
PROOF. As we said above, the functions eSnzy(z) converge to f(z) in
30(Ao, A I) as sn O. Therefore it is sufficient to choose a sequence of sn so
that sn O and e SnZ2 (gn(z) - f(z) O in (Ao' AI). Let us consider the
functions
CPk(t) = ax { SUP Ilf(j + iT) - gn(j + iT)IIA i } .
J - 0, I 171 <;; t
n>k
218
IV. INTERPOLATION METHODS
The functions CPk(t) are increasing in t, decreasing in k, and converge to
zero as k 00 for every t 0 by the hypothesis of the lemma. Finally, the
sequence CPk(t) is uniformly bounded: sUPo<;t<oo CPk(t) c.
We denote by [0, an] the largest interval on which tCPn(t) 1. Obviously
an a n + I . Moreover, an 00, because otherwise an < a < 00 and CPn(a)
1/ a, which contradicts the fact that CPn(a) 0 as n 00. Now we choose
sn = a;I. Then sn 0 and
IleJ;,Z>( gn - f)II3<Ao>Al) = j { s p e-J;,r'+jJ;,IIfU + iT) - gnU + iT )IIA j }
e S lcpn(a n ) + eSle ax { SUP ItJ(J + i'T) - gn(J + i'T)IIA j }
J - 0,1 I 'T I > an
< eS1a;I + eSle (II.flI A AI) + Ilgnllrn:A I») 0 as n 00.
LEMMA 1.2. For every function f E tlo(A(V A 1) there exists a sequence of
functions gn E tl(Ao, A 1) satisfying the hypotheses of Lemma 1.1 and consisting
of functions periodic in 'T.
PROOF. It is sufficient to prove the lemma for functions of the form (1.12),
where f E tlo(Ao, A 1). We define the functions
00
gT(Z) = f6(z + ikT)
k=-oo
(T 1).
(1.13)
To study the convergence of this series we consider the function series
00
e-8('T+ kT) .
k=-oo
The sum of this series depends on 'T periodically with period T. Therefore it
is sufficient to study the convergence of the series on the interval 0 'T T.
For such 'T we have
00 00 00 00
e-8(T'+kT)2 + e-8(T'-kT)2 < e-8k?-T 2 + e-8(k-I)2r 2
o 1 0 1
00
< 2 e-8k?- = C(8) < 00.
o
The inequality
I I f ( z + ikT )11 e6d2-6(T'+kT)2 Ilfll
1-'6 3(A()7A 1 ),
which is true when we have on the left side the norm of Ao + A 1 for Z inside
II and the norm of A.i for z = J + i'T U = 0, 1), implies the convergence in
(Ao, AI) of (1.13) to the function gT E ffiAo, AI). Moreover, II gTIIrn:A I)
e6c(8)llfllrn:A 1). It is obvious that the functions gr<z) are periodic in 'T with
period T.
z = (J + i'T,
1. THE COMPLEX METHOD
219
We show that the functions grU + iT) converge tof6U + iT) in the norm of
Aj uniformly on every bounded subdomain of T values as T 00. Let
ITI M. We choose T 2M. Then
2
IlgT(iT) - k(iT)IIA o e (T+kT) Ilf(iT + ikT)IIA o
00 2
2 e (kT- T12) 1 '11 1
£.J IJ I Ao.A I)
1
00
2e--8T2/4 e k(k-I)T21 Irn:A I) 0 as T 00.
1
LEMMA 1.3. If a function h E ffiAo, AI) is periodic in T with period T, then it
can be represented as a limit in i)-(Ao, A I) of junctions of the form
N
h N ( z) = x k e 2k 'ITZ I T,
k=-N
where x k E Ao n AI.
PROOF. For every 0 we expand h(o + iT) in Fourier series in T:
00
h(o + iT) = b k (0)e i2k 'ITTIT,
k=-oo
where
1 f T /2
b k ( 0) = - h( 0 + iT )e- i2k 'ITT IT dT.
T -T/2
Writing a k ( 0) = b k ( o)e- 2k 'TTO I T, we obtain
00
h( z) = a k ( 0 ) e2k'ITzl T,
k=-oo
(1.14)
where
1 f T /2 .
a k ( 0) = - h (0 + iT) e -2k'1T{ a + 'T) IT dT.
T -T/2
( 1.15)
Due to the periodicity of the integrand, the last formula can also be written
as
1 f mT /2 . .
a k ( 0) = - h (0 + IT) e -2k'IT( a + IT) IT dT.
mT -mT/2
( 1.16)
We show that the coefficients ak(o) are in fact independent of 0. Let
o 00 0 1 1. Denote by r m the contour formed by the intervals {oo
Re z 01' 1m z = + mT /2} and {Re z = CJ.i U = 0, 1), 11m zl < mT /2}.
220
IV. INTERPOLATION METHODS
Since h(z)e 2k7TZ / T is analytic and continuous within II in the norm of Ao +
A l' its integral over this contour is equal to zero. Then
i f mT / 2 .
- - h( 0 0 + iT )e- 2k 'IT(oO+IT)/T dT
mT -mT/2
i f mT / 2 -' . )/
+ - h( 0 1 + iT ) e- 2 k-1I \OJ + IT T dT
mT -mT/2
+ f a l h ( o - i mT ) e- 2k7T (O-imT/2)/T do
mT 00 2
- f Olh ( o + i mT ) e- 2 k-rr(o+imT/2)/T do = O.
mT 00 2
The function h(z)e 2k 'ITz/T is bounded in the strip 0 Re z 1 in the norm
of Ao + AI. Therefore the two last tenns converge to zero as m 00. The
first two integrals do not depend on m according to (1.15) and (1.16). Passing
to the limit as m 00, we obtain that a k ( (0) = a k ( 0 1) = a k . In particular,
1 f T /2 . 1 f T /2 .
a k = - h(iT)e- 2Ik 'ITT/T dT = - h(I + iT)e- 2k 'IT(I+IT)/T dT.
T -T/2 T -T/2
Since h(iT) is continuous in the topology of Ao, the first integral belongs to
Ao. Similarly, the second integral belongs to A l' and therefore a k E Ao n AI.
Now we form the Cesaro sums of the series (1.14):
hN(z) = f ( 1 - Ikl ) a k e 2k 'tTZ/T = f x k e 2kvrz / T .
k=-N N + 1 k=-N
Since the functions h(iT) and h(I + iT) are continuous in the topologies of
Ao and AI' respectively, by Fejer's theorem the functions hN(iT) and
hN(I + iT) converge to the functions h(iT) and h(I + iT) in the norms of Ao
and A 1 uniformly in T as N 00. Thus,
II h N - hllffiAo.AJ) O.
By combining the lemmas proved above, we obtain the following important
assertion.
THEOREM 1.1. The set of functions of the form
N
g(z) = e 6z2 xneA"z,
n=l
( 1.17)
where X n E Ao n AI' the A" are real, and 8 > 0, is dense in i)JAo, AI).
REMARK. It is obvious that the elements X n may also be chosen from an
arbitrary set M dense in Ao n AI.
1. THE COMPLEX METHOD
221
4. The interpolation functor [Ao, A da. We denote by [A(p A I]a (0 a 1)
the collection of all elements x E Ao + A I representable in the form x = f( a)
for some functions f E i)(Ao, A I). This collection is linear. In this collection
we introduce the norm
IIXII[A Ada = Ilxll a = inf Il.fllffiA I).
f( a) = x
( 1.18)
We denote by N a the linear manifold of all functions from ffiA (p A I)
vanishing at z = a. It follows from (1.11) that N a is a closed subspace of
i)(Ao, A I). The definition of the norm in [A(p A da shows that this space is
isometric to the quotient space of i)(Ao, A I) modulo N a , and consequently
[A(p A I]a is a Banach space.
From (1.11) we obtain
IlxIIAO+AI = Ilt(a)IIAo+AI 111l1i){A AI)'
for a functionf(z) withf(a) = x. Therefore for x E [Ao, AI]a we have
IlxIIAO+AI inf Il.fllffiA AI) = l!xll[Ao,Ada.
f( a) = x
Hence it follows that [Ao, A da is imbedded in Ao + A I with imbedding
constant 1.
The space Ao n A I is imbedded in [Ao, A I]a. To verify this, we consider an
arbitrary element x E Ao n Al and functionfx(z) = x. Thenf x E ffiA(p AI)
and
Ilxll a IltxllffiAo.AI) = max{ IlxllAo, Ilx11A 1 } = IlxllAonAI.
1 1
Thus, Ao n Al C [Ao, Ada C Ao + AI' and the space [Ao, AI]a turns out to
be intermediate between Ao and AI.
We note that the mapping j(z) f(1 - z) is an isometry of the space
i)(Ao, A I) onto tl(A I' Ao), and therefore [A I' AO]a = [A(p A dl-a.
THEOREM 1.2 (interpolation theorem). Let AO' Al and B(p BI be two Banach
couples. The triple (Ao, A I' [Ao, A I]a) of Banach spaces is a normalized interpo-
lation triple of type a relative to the triple (Bo, B I , [B(p Bda).
PROOF. Let T E L(AoA I' BoBI). We consider an element x E [A(p Ada.
For any E > 0 there exists anf E tl(Ao, AI) such thatf(a) = x and IIfliffiA I)
Ilxlla + E.
222
IV. INTERPOLATION METHODS
The function g(z) = II T" -:Boll TIIAZI BIT(f(z» belongs to i5(Bep B 1 ). Next,
II gll3< B",B I) = max { sup II g(iT) II Bo' sup II g( 1 + iT) II B, }
'T 'T
.s;;; max { s p 11111 A -> Bo II 111 "o->Bo IIJ(iT ) II "0'
sup II 11IA ->B,lll1IA,->B,I (l + iT)IIA 1 }
'T
= 1I.flI3(A I) Ilxll a + E.
This implies that
II xII a + E II gll3{ B o.BI) II g( a) II[B Bda
= IIII TII -:Boll TII;: BIT(f(a»"[B Bda
= II TII Boll TII;:--+B111 TXII[Bo.Bda.
Since E is arbitrary, we have
II Txll[B(pBda II TII :Boll T" I Blllx"[A I]a.
REMARK 1. In calculating the norm IIxll[A I]a by (1.18), the infimum may
be taken over functions from i)(Ao, A I) only.
Indeed, if f E ffiAo, AI) and f(a) = x, then fl(z) = e6(Z2_ (z) E
i)o(Ao, AI) andfl(a) = x for any 8 > O. Now let IIfll3(A I) Ilxll a + E. For
sufficiently small 8 we have Ilflll3(A I) Ilfll3(A I) + E. Therefore
Ilxli a = inf Ilcpll3{A I) Ilflll3(A I) IIxll a + 2E,
Effi:Ao,A 1)
(a)=x
which proves our assertion.
REMARK 2. If x E Ao n Al and 0 < a < 1, then in the calculation of the
norm according to (1.18) the infimum may be taken over functions of the
form f ak(z)x k only, where x k E Ao n A I.
Indeed, according to the preceding remark we choose the function f E
i)o(Ao, AI) so thatf(a) = x and Ilfll3{A I) Ilxlla + E. Let r(z) be a function
mapping the strip 0 < Re z < 1 conformally onto the unit disk so that
r(a) = O. Then Ir(iT)1 = Ir(1 + iT)1 = 1, and therefore
2
f(z) - e 6 (z-a) x
r(z) E o(Ao. AI).
By Theorem 1.1 there exists a function g(z) of the form (1.17) such
that lI[f(z) - e 6 (z-a)2 x ]r- l (z) - g(z)II3(A I) < E. We set fl(z) = e8<z-a)2x +
r(z)g(z). This function has the required form,fl(a) = x, and
Ilflll3(A I) Ilfll3(A I) + Ilf - flll3(A I) Ilxll a + 2E.
1. THE COMPLEX METHOD
223
THEOREM 1.3. The space Ao n Al is densely imbedded in any space [A (V Ada
for 0 a 1.
PROOF. Let x E [Ao, Ada and f E i)o(Ao, AI) with f(a) = x. By Theorem
1.1, for any E > 0 we can construct a function g(z) of the form (1.17) so that
IIf - gllffiA I) < E. Then
Ilx - g(a)II[Ao,Ada = Ilf(a) - g(a)II[A()tAda Ilf - glli){A I) < E
and the the0rem is proved, since g(z) takes values in Ao n A I.
Remark 2 and Theorem 1.3 show that the spaces [Ao, Ada could be defined
in the following way: We consider the space i)(Ao n A I)' for the elements
x E Ao n A I we introduce the norm
inf II glli){A I)'
g( a) = x
gE3(A o nA .)
and we complete A 0 n A I in this norm.
We investigate what the extreme spaces [Ao, A do and [Ao, A Ih are. If
x E [Ao, A do, then f(O) = x for some f E ffiA(p A 1). On the other hand,
f(iT) E Ao for all T, and thereforef(O) = x E Ao. Moreover,
Ilxlla =
111113( A o.A,) = max{ sup 111U r )IIA o ' sup 111(1 + i'T)IIA,}
'T 'T
Ilf(O)IIA o = IIxiI Ao .
Hence it follows that
Ilxll[A Ado Ilxli Ao . (1.19)
On the other hand, on the basis of Theorem 1.3 for E > 0 there exists an
element Xl E Ao n A I such that IIx - xilIrA IJo < E. We construct the func-
tionsfn(z) = e z2 - nz xl E i)(Ao, AI). We havefn(O) = Xl and
II fn I13(A I) = max {lIxlll Ao' e I - n II xIII AI}.
Consequently
II X III [A do max { II X III A 0' e I - n II x III AI} .
Letting n tend to 00, we obtain that IIxIII[A IJo IlxII1Ao. By (1.19) we have
IIx - xIIiA o Ilx - xIII[A do < E, and therefore
Ilxll[A()tAd o < IlxIII[A()tAdo + E IlxIIIA o + E IlxliAo + 2E.
Since E is arbitrary, this inequality and (1.19) imply
II x II [A ()tA do = II x II A 0 (x E [ A 0' AI] 0).
Thus, [Ao, A do is isometric to some subspace Ao of Ao.
224
IV. INTERPOLATION METHODS
It can be proved similarly that [A(p Adl is a subspace Al of AI. It follows
from Theorem 1.3 that [Ao, A d j = Aj U = 0, 1) coincides with the closure of
Ao n A I in the norm of Aj. In particular, if Ao n A I is dense in Aj' then
[Ao, A d j = Aj.
From the equivalent definition of the spaces [Ao, A da given above it
- -
follows that [Ao, Ada = [Ao, A da' and so without loss of generality we may
always assume that the intersection of the initial spaces is dense in each.
We establish some additional important properties of [Ao, AI]a.
LEMMA 1.4. Let f E (Ao' A I) and a E (0, 1). Then
In II J( IX) II [A",A da .;;; f 00 [In II J(i'T) II AJ Po( IX, 'T) d'T
-00
+ foo [InIlJ( I + i'T) IIA,] ILl (IX, 'T) d'T,
-00
[ 1 00 ] I-a
II J( IX )11[A",A,la';;; I _ IX i} J(i'T )11 Ao Po( IX, 'T) d'T
[ 1 00 ] a
X ioo II J( I + iT) II A, ILl (IX, 'T) d'T ,
II J( IX) II [A",A.la .;;; f 00 II JU'T) II A 0 Po( a, 'T) d'T
-00
+ J 00 II J( I + iT) II A I ILl (IX, 'T) d'T.
-00
( 1.20)
( 1.21 )
( 1.22)
PROOF. We choose bounded functions f(Jo( T) and f(JI( T), infinitely differentia-
ble on (-00, (0), such that f(JO(T) lnllf(iT)IIAo and f(JI(T) ;> Inllf(1 + iT)IIA..
We construct a function <I>(z) analytic inside II whose real part is bounded
and continuous in the closed strip, and with Re <I>(iT) = flJO(T) and
Re <1>(1 + iT) = f(JI( T). For Re <I>(z) we have
Re <1>( z) = f 00 q:>o( 'T) Po( z, 'T) d'T + f 00 q:>1( 'T )ILI(Z, 'T) d'T.
-00 -00
It follows from the differentiability of the functions f(Jo( T) and f(JI( T) that
<I>(z) is continuous in the closed strip II. Therefore e-4(z>j(z) E ffiAo, A I)' and
SInce
Ile (i1")f{iT)IIAO = e-cpo(1")llf{iT)IIA o 1,
lIe (l +;1")f{1 + iT)IIA. = e-cp.(1")llf{1 + iT)IIA. 1,
we have
II e- (z)f{z) II3< A o.A 1) 1,
1. THE COMPLEX METHOD
225
and consequently
II e- (a)f( a) II[Ao,A d a 1.
This implies that
Inllf(a)II[AQtAda Re <I>(a)
= f 00 CPo( 'T )1Lo( a, 'T) d'T + f 00 cp\( 'T) IL\( a, 'T) d'T. (1.23)
-00 -00
Now choosing decreasing sequences of functions f(Jo( 'T) and f(JI( 'T) converg-
ing to Inllf(i'T)II A and Inllf(1 + i'T)IIA , respectively, and passing to the limit,
° 1
we obtain (1.20) from (1.23).
The convexity of the function e t and (1.7) imply that
exp [ 1 a i: [In II J(i'T) II AJ 111J( a, 'T) d'T ]
1 f oo
1 - a II f( i'T) II Ao 1Lo( a, 'T) d'T,
-00
ex p [ i:[lnIlJ(l + iT) II A.] 1L\(a, 'T) d'T]
1 f oo
- Ilf(1 + i'T)IIA1ILI(a, 'T) d'T.
a -00
( 1.24)
( 1.25)
Multiplying and dividing the first and second terms on the right side of
(1.20) by 1 - a and a, respectively, taking powers, and using (1.24) and
(1.25), we obtain (1.21). Finally, if in the inequality
exp(a + b) = exp[ (1 - a)a(1 - a)-I + aba-I]
(1 - a)eQ(I-a)-l + ae ba - 1 (1.26)
we replace a and b by the first and second terms on the right side of (1.20)
and again use (1.24) and (1.25), we obtain (1.22).
LEMMA 1.5. Let the sequence 1n(z) be bounded in tl(Ao, A I) and convergent
for every i'T from some set e of positive measure on the imaginary semiaxis in the
norm of Ao. Then the sequence fn(a) converges in [Ao, A1]a (0 < a < 1).
226
IV. INTERPOLATION METHODS
PROOF. Applying (1.20), we obtain
Inllfn(a) - fm(a)II[A da
![Inllfn(i'T) - fm(i'T) II A.] lLo(a, 'T) dr
e
+ 1. [lnllfn(iT) - fm(iT)IIA ] 1Lo( a, T) dT
R\e 0
+ f 00 [In II fn(1 + i'T) - fm(1 + i'T) II A.J ILl (a, 'T) d'T.
-00
( 1.27)
By hypothesis, the first integral tends to -00 as n, m 00 and the last two
are bounded above since the sequence fn is bounded in i)(A(V A 1) and the
functions JLo(a, T) and J.LI(a, T) are integrable. Therefore the left side of (1.27)
tends to -00, and consequently II fn( a) - fm( a)II[A I]a 0 as n, m 00.
5. Conditions for the reflexivity of [Ao, Ada.
THEOREM 1.4. If the space Ao is reflexive, then the spaces [A(p Ada' 0 < a <
1, are a/so reflexive.
PROOF. By Alaoglu's theorem it is sufficient to prove that the unit ball of
[Ao, A da is compact in the weak topology, and by V. L. Smul'jan's theorem
the ball will have this property if every decreasing sequence of closed convex
subsets Qn of the ball has a nonempty intersection (see [11], Chapter V, 6.2,
Theorem 2). In every Qn we choose an arbitrary element X n . Since IIxnIIrA I]a
1, there exist functionsh E ffiAo, AI) such thatfn(a) = X n and IIfnllrn:A I)
2. We consider the restrictions of the functions fn(z) to some interval I of
the imaginary axis as elements of L 2 (A o ). By a theorem of Phillips (see [18])
this space is reflexive, and therefore every ball in it is compact in the weak
topology. Since the indicated sequence of elements of (Ao) is bounded, it
contains a weakly convergent subsequence with indices n'. Then there exists a
sequence of convex combinations of these elements converging in the norm of
L 2 (Ao). Thus, from the sequence fn'(z) we can construct another sequence
gk ( z) = akn'fn'( z )
n'>k
(a n') 0, n' a n') = 1, and only a finite number of coefficients ak n ') =F 0 for a
given k) having the property that the restrictions of the functions gk(Z) to I
converge in L 2 (A o ), and consequently converge in measure on I. Then there
exists a subsequence g (z) which converges in the norm of Ao almost
everywhere on I. By construction, II g IIrn:A I) 2, and so by Lemma 1.5 the
sequence
gk, ( a) = at n '%,( a) = a n')xn'
n' >k; n' >k;
1. THE COMPLEX METHOD
227
converges in [Ao, Ada to an elementy. Since the sets Qn are convex, we have
gl4(a) E QI4' and since they are closed, the elementy belongs to all Qn.
6. Completion of the spaces [Ao, AI]a. The spaces [A(V AI]a are imbedded in
Ao + A I' and so we may consider their completions [A(p AI]a relative to
Ao + A I. From the interpolation Theorem 1.2 and Lemma 4.5 of Chapter I
we immediately obtain the following theorem.
......-........
THEOREM 1.5. The triple (Ao, AI' [Ao, Ada) of Banach spaces is a normalized
interpolation triple of type a relative to the triple (B(V B I , [B(V BI]a).
An example shows that the space [A(V A da need not be complete relative
to Ao + AI. We consider the space Ao = Loo(O, 1) with norm Ilxllo and the
space Al consisting of all functions with finite norm Ilxil i = ess sup t-1Ix(t)l.
In subsection 9 we shall show that in this case [A(V Ada consists of all
functions for which Ilxll a = ess sup t-alx(t)1 and t-alx(t)1 0 as t o. The
completion of [Ao, Ada relative to Ao (here Ao :J AI) consists, however, of all
functions with finite norm Ilxllo. Indeed, if for such a function we set
xn(t) = X(l/n,I](t)X(t), then X n E [Ao, A da and
Ilxnll[Ao.Ada Ilxll a ,
Ilx n - xll Ao = ess sup Ix(t)1 <:: (l/n)all x lla O as n 00.
O<t<I/n
7. Duality. We assume that Ao n Al is dense in Ao and AI. Then the dual
spaces A and A form a Banach couple imbedded in (Ao n AI)" and on this
couple the functor [Ao, A ]a is defined. By Theorem 1.3 the space Ao n Al is
dense in [Ao, Ada' and so we have the imbedding ([A(V A1]a)' C (Ao n AI)'.
Thus, both spaces [Ao, A ]a and ([Ao, Ada)' are imbedded in (Ao n AI)" and
we may pose the question of their mutual position in (Ao n AI)'.
LEMMA 1.6.
I
[ A 0' A; ] a C ([ A 0' A I ] a)' .
( 1.28)
PROOF. Let u E [Ao, A ]a. This means that there exists a function \/; E
tJ(A , A ) such that \/;(a) = u. The function (z) is analytic inside II and
continuous in the norm of Ao + A in the closed strip. By Theorem 3.1 of
Chapter II the space Ao + A is isometric to (Ao n AI)" and so \/;( z) is
analytic in II and continuous as a function with values in (Ao n A 1)' in the
closed strip. If now f E tl(Ao n A I)' then the scalar-valued function
<f(z), \/;(z) is analytic in II, and by the maximum principle we have
I<f(z), (z»1 max{supl<f(iT), \/;(iT»I, sup I <f(1 + iT), \/;(1 + iT»I}.
228
IV. INTERPOLATION METHODS
Let X E Ao n AI. For the function f l E ffiAo n AI) with fI(a) = x this
implies that
1 <x, u)1 = 1 <fI( a), (a) >1 IlfIIIB(A AI)II IIB(AO,AD.
Taking the infimum over f l and , we obtain (by virtue of Remark 2) that
I<x, u)1 Ilxll[A AdJI ull[AO,Ai]a (x E Ao n AI).
This inequality says that the functional u as an element of (Ao n AI)'
belongs to the set ([Ao, A da)' and
Ilull[A Ad Ilull[AO,Ai] .
COROLLARY. By Theorem 2.2 of Chapter I the space ([Ao, A da)' is complete
relative to (Ao n A 1)'. Therefore it follows from the imbedding (1.28) that
1
[ A 0' A ] a C ([ A 0' AI] a)' ,
where - is the symbol for completion relative to (AO n AI)'.
Our next goal is to prove that in the above inclusion we have the equality
SIgn.
( 1.29)
LEMMA 1.7. Assume that the functions (z) (0 < Re z < 1), O(T) and (T)
(-00 < T < (0) with values in Ao + A have the following properties:
1) (z) is analytic and bounded inside II.
2) For any x E Ao n Al the function <x, (z» has boundary values for
z = iT and z = 1 + iT coinciding with the values of the functions <x, o( T) > and
< x, I ( T) >, respectively.
3) j(T) E A; and II j(T)IIA;' c (j = 0,1). (1.30)
Then
.........-.....
(a)E[Ao,A Ja and 11 (a)II[ a'c. (1.31)
PROOF. We consider the function
<p(z) = i Z 1/JW d
2
(0 < Re z < 1).
Since (z) is bounded, it follows that cp(z) can be extended continuously in
the norm of Ao + A (or, similarly, the norm of (Ao n AI)') to the closed
strip. Next, for x E Ao n A 1 we have
(x, <p(z + ih» - (x, <p(z» = fZ+ih (x, 1/J( » d
z
= i h (x, 1/J(z + iT» dT (0 < Re z < 1).
1. THE COMPLEX METHOD
229
Setting z = s + it and letting s tend to zero, we obtain
(x, qJ(it + ih) - (x, qJ(it) = i £h (x, 1/;oU + T) dT.
By (1.30) the expression on the right gives a continuous linear functional on
Ao whose norm does not exceed ch. Consequently
cp(it + ih) - cp(it) E A and "* [ cp(it + ih) - cp(it)] /lA o < c.
It can be shown similarly that
,,*[ cp(I + it + ih) - cp(I + it)JIIAi 'c.
From the last inequalities it follows that the function
4>h(Z) = * [cp(z + ih) - cp(z)]
belongs to 3(A , A ), and II<I>h IlffiA(pA'I) c. This implies that
<I>h( a) E [ A , A ] a and II<I>h( a) II[AO,AiJ a ' c.
On the other hand, <I>h(a) \[;(a) as h 0 in (Ao n AI)" and so (1.31) is
sa tisfied.
THEOREM 1.6. If Ao n A 1 is dense In Ao and A l' then the dual space
([Ao, A da)' is isometric to [A , A ]a.
PROOF. Let u E ([Ao, Ada)'. Since [Ao, Ada has been defined as the quo-
tient space of 3(Ao, A 1) modulo the subspace N a , the functional u induces a
linear functional ii on 3(Ao, A 1) with the same norm and equal to zero on N a .
Now we consider the mapping 0 of 3(Ao, A 1) into LI(AJ X LI(A 1)' de-
fined by
O(f) = {f(iT)JLo(a, T),f(I + iT)J.LI(a, T)}.
This mapping is linear and one-to-one. On its range we define a linear
functional A by the formula
< O(f), A) = <f, ii).
By (1.21) we have
I<O(f), A)I = I<f, u)1 = I<f(a), u)1 ' Ilf(a)II[A()tAdJlull([A()tAda)'
II U II ([Ao> A .]aY[ i: 111(;T) IIA o /Lo( a, T) dT
+ i: 1I1 (1 + iT)IIA, p-.(a, T) dT]
= II u 11([ A()t A dayll O(f) II LI(Ao) x L1(A I). (1.32)
230
IV. INTERPOLATION METHODS
Thus, on the range of () the functional A is bounded in the norm of
LI(Ao) X LI(A I). We extend this functional with preservation of the norm to
the whole space LI(Ao) X LI(A I). By the theorem on the general form of a
linear functional on LI(A) (see the Introduction to this chapter, Theorem 2)
there exist two bounded functions \[Io(t) and \[I1(t) with values in Ao and AI'
respectively, such that
(1, ii) = (O(J), A) = j'>O (JUT) /lo( a, T), 1/;o( T» dT
-00
+ fOO (J(1 + iT)/l\(a o , T), 1/;\(T» dT. (1.33)
-00
Moreover, by (1.32) we have
m.r { ess supil 'h( T) II AJ ' II u II ([ AD> A ,JaY' (1.34)
Let X E Ao n A I. We consider the function \[I(x, z) defined by the follow-
ing Poisson integral:
1/;( x, z) = f 00 (x, 1/;o( T» J1.o( z, T) dr + J 00 (x, 1/;\( T »/l\ (z, T) dT. (1.35)
-00 -00
This function depends linearly on x for given z, and by (1.34) we have
/\[I(x, z)/ < m {ess supl<x, 'he T»I} < IIxIiAonAlllull([A da)'. (1.36)
J
Thus, there exists a functional \[I(z) E (Ao n A I)' such that \[I(x, z) =
<x, \[I(x». From (1.36) it follows that \[I(z) is bounded in II. We show that it
is analytic inside II. For this we take a scalar-valued function h(z) analytic
inside II and continuous in the closed strip, having limits at + ioo, and such
that h(a) = O. In (1.33) we putf(z) = h(z)x. Then
o = <h(a)x, u) = <f(a), u) = <f, u)
= f 00 h(iT )(x, 1/;o( T» J1.o( a, T) dT
-00
+ foo h(1 + iT)(X, 1/;\( T»/l\(a, T) dT.
-00
By virtue of what we said in subsection 1 it follows that <x, \[I(z» is
analytic for any x E Ao n A I' and so \[I(z) is analytic in the norm of
(Ao n A I)'.
The functions \[I(z), \[Io(z), and \[II(Z) satisfy the hypotheses of Lemma 1.7,
therefore \[I(a) E [Ao, A;]a and
II\[I( a) 11[ ]a < II ull([A Ot Ada)'.
Settingf(z) = X E AO n Al in (1.33), by (1.35) we obtain
<x, u) = \[I(x, a) = <x, \[I(a»,
I. THE COMPLEX METHOD
231
i.e. (a) = u as an element of (Ao n AI)" and consequently u E [A , A;]a
and
II ull(:;:A;]a II ull([A A da)'.
The theorem now follows by virtue of the imbedding (1.28).
REMARK. If Ao is reflexive, then ([ Ao, A da)' and [A , A ;]a are isometric.
Indeed, this follows immediately from Theorems 1.6 and 1.4, and Theorem
2.3 of Chapter I.
8. A reiteration theorem. From a Banach couple Ao, A 1 we construct the
spaces Aa = [Ao, A da and AIJ = [Ao, A 1]13 (0 (; a < /3 (; 1). These two spaces
again form a Banach couple imbedded, for example, in Ao + AI. Therefore
we may construct the space [Aa' AIJ]o (0 a (; 1).
LEMMA 1.8. The space [Ao, A ds, where s = (I - a)a + 0/3, is imbedded in
[Aa' AIJ]o with imbedding constant not exceeding one.
PROOF. Letf E 3(Ao, AI) andf(s) = x E [Ao, A I]s. We introduce the func-
tion g(z) = f«1 - z)a + z/3). Then g( a) = f(s) = x,
g(iT) =f(a + i(/3 - a)T) and g(1 + iT) =f(/3 + i(/3 - a)T).
The function1r(z) = f(z + i(/3 - a)T) belongs to 3(Ao, AI) for every fixed T,
and therefore g(iT) E Aa and g(1 + iT) E AIJ. Next,
II g(iT)IIA a 111r11 (A AI) = IIfI13(A AI).
Similarly, II g(l + iT)IIA p (; Ilfll3(A I). This implies that x E [Aa' AIJ]o and
IIxll[Aa.AP]a max{ sup II g(iT)IIA a ' supll g(1 + iT)IIA p } (; IlfI13(A AI).
Taking the infimum over f, we obtain
1
[ A 0' AI] s C [ A a' A 13 ] o.
( 1.37)
We shall be interested in conditions under which the inclusion reverse to
(1.37) holds. For this we study the problem of imbedding the dual spaces of
the spaces considered in the lemma. However, if the space [Ao, A ds is not
dense in [Aa' AIJ]o, then the dual spaces are not imbedded. In this connection,
we make the additional assumption that Ao n A 1 is dense in the spaces
Ao, AI' and Aa n AIJ. By Theorem 1.3 the space Aa n A/3 is densely im-
bedded in [Aa' AIJ]o, and therefore Ao n A l' and so also [Ao, A ds, is densely
imbedded in [Aa' AIJ]o.
Due to the assumption made above, the dual spaces of all indicated spaces
can be considered as sets in the space (Ao n AI)' = A + A;.
LEMMA 1.9. Under the above assumption, the spaces ([Ao, A I]s)' and
([Aa' AIJ]o)' coincide.
232
IV. INTERPOLATION METHODS
PROOF. The imbedding (1.37) immediately implies the imbedding
1
([Aa' A,8]o)' c ([Ao, Ads)'.
By Theorem 1.6 the space ([Aa' A,8]o)' coincides isometrically with
[A , A ]:,8, where a/3 means completion relative to A + A . This last
space is imbedded in Ao + A;, and ([Aa' A,8]o)' is complete relative to
(Ao n AI)' = Ao + A; by Theorem 2.2 of Chapter I. We find ourselves in the
situation considered in Corollary 2 to Lemma 1.6 of Chapter I, according to
which ([Aa' A,8]o)' = [A , A ]a' where denotes completion relative to Ao +
A; (a notation used without reference in the sequel).
1
According to Lemma 1.8, [Ao, A;]s C [[Ao, A;]a' [Ao, A;],8]o. Further, from
Theorem 1.6 we obtain
1 1
[ A 0' A; ] a C [ A 0' A; ] a = A and [A 0: Ai] ,8 C [ A 0' Ai],8 = A,8'
1
and therefore [Ao, A ;]s C [A , A ]a.
I
Passing to completions relative to Ao + A i, we obtain [Ao, Ai]s C [A , A,8]O'
1
or, similarly, ([Ao, Ads)' C ([Aa' A,8]o)'.
We have obtained two mutually reverse imbeddings. The lemma is proved.
Lemmas 1.8 and 1.9 immediately imply the following theorem.
THEOREM 1.7. If Ao n A 1 is densely imbedded in Ao, A l' and Aa n A,8' then
the spaces [Ao, A I]s' where s = (1 - a)a + a/3, and [Aa' A,8]o (0 (; a (; 1)
coincide isometrically.
9. Connection with the theory of scales. We consider the family of spaces
[Eo, EI]a' where E I is normally imbedded in Eo. In this case Eo n EI
coincides isometrically with E I and Eo + £1 with Eo. By the maximum
principle for a function f E 3( Eo, E 1) we obtain
II f( z) II Eo II fll E EI). (1.38)
We show that for a < /3 the space E,8 = [Eep Ed,8 is normally imbedded in
Ea = [Eo, Eda. Let the function f E 3(Eep E I ) be such that IlfIl3(E EI)
Ilxll E + E andf( /3) = x. We consider the function
fJ
(z) = = z + = ; ).
I t is analytic inside the strip II and continuous in the closed strip. Moreover,
by (1.38) we have
s pll (ir)IIEo = s p = ; + i = T) Eo ' 111113(Eo.E,)
1. THE COMPLEX METHOD
233
and
. J. 1 - /3 )
s pll<ll(1 + 17)IIE, = s p J\1 + 1 1 _ a 7 E." II 11 I 3<Eo>E,)'
Consequently <1> E ffiEo, E I ) and 11<1>IIB<E EI) (; IIfllB<E EI). Finally, <1>(a) =
f( {3) = x. This means that x E Ea and
Ilxll Ea II <1>11B{Eot E 1 ) (; Ilfll (E()tEI) Ilxll E,g + e.
Since e is arbitrary, we have IlxilE IlxilE (x E E Q ), and since EI is dense
a 'fJ 1J
in all spaces Ea (Theorem 1.3), the space E/3 is normally imbedded in Ea.
The spaces [Eo, Elt U = 0, 1) coincide with the spaces Ej.
We prove that the function Ilxil E (x EEl) is logarithmically convex in a.
a
We consider the function
cp(z) = (1IxIIEo/llxIIE)z-ax.
Obviously cp E 3(Eo, E I ) and
sup IIcp(iT)IIE = IlxII E I-allxlla E '
'T 0 0 I
sup Ilcp(1 + iT)IIE I = IIxllk allxll l.
Next, cp(a) = x, and therefore
II x II Ea II cp 113( E E I) = II x II k a II X II I.
( 1.39)
Now let 0 a < /3 < y 1. By the reiteration Theorem 1.7 the space E/3
coincides isometrically with [Ea' E1']o, where (J = (/3 - a)/(y - a). Applying
(1.39) to Ea and Ey, for x E E1' we obtain
II xii E,g (; II xii k o II xll = II xll a- /3)/(1' -a) II xll -a)/(1'-a).
Thus, the spaces {Ea} form a normal scale.
The spaces [Eo, E{]a also fonn a normal scale (we recall that [E , Enl is the
closure of Eo in the norm of E;). By Theorem 1.6 the spaces E coincide with
the completions of the spaces [Eo, E{]a relative to E;, or, what is the same,
relative to the subspace [Eo, E{]I of it. Thus, the family E coincides with the
condensation of the nonnal scale [Eo, E{]a (see Chapter III, 1.5). From
properties 1 0 and 50 of the condensation of a normal scale it follows that
IIfll E' is a decreasing logarithmically convex function of a. In particular, it is
a
continuous at a = 1. This circumstance permits us to obtain the properties of
the scale {Ea} as a 1. Indeed, let x EEl. Then by (2.1) of Chapter I we
have
f(x)
II x llE Ol = sup 11111 .
f E E6 EI
234
IV. INTERPOLATION METHODS
We choosef o E Eo so that
IIxll EOI Ifo(x)l/ilfoII Ei + E.
Using the continuity of IIfoll E' at a = 1, we choose a so close to one that
a
Ifo(x)1
II X II EOI < 1\ fo II E' + 2e < II X II E. + 2e.
a
This implies that IlxllEol lima-+lllxIIEa. By Lemma 1.1 of Chapter III the
reverse inequality holds, and therefore
lim l llxll[EOtEda = Ilxli EOI (x EEl).
a
The preceding considerations lead to the following assertion.
THEOREM 1.8. If the spaces Eo and E 1 are related, then the family [E(p Eda
fortm a regular normal scale.
COROLLARY. If the spaces Eo and E 1 are related, then the scale [E(p Eda has
the strong interpolation property relative to any minimal scale.
We now introduce the notion of analytic scale of spaces. Let M be a
normed linear space in which a family of linear operators T(z) acts satisfying
the following conditions:
1 0. For every x E M the function T( z)x is an entire function of the
complex variable z.
2°. The function II T(z)xII M is bouned on every straight line parallel to the
.. .
Imagmary axIS.
3°. T(O)x = x.
4°. sUPlLtpl1 T(a + iJ-L)T(/3 + iv)IIM (; sup,. II T(a + /3 + iT)xIl M .
5°. T(ip.) T(z + Llzl - T(z)x T(ip.)(T(z)x)'
uniformly in J-L as z 0 (property 5° follows from property 1 ° if the
operators T(iJ-L) are uniformly bounded in norm).
We introduce the family of norms
Ilxll a = sup II T(a + iT)xllM
-00<7<00
in the space M and complete M to a Banach space Ea in each of these norms.
The family Ea' - 00 < a < 00, of Banach spaces will be called an analytic
scale of spaces.
By the three lines theorem for the logarithmically subharmonic function
II T(z)xII M (see [39]), the function Ilxlla, is a logarithmically convex function
of a. If Ilxll a (x E M) is in addition an increasing function of a, then the
analytic scale will be a continuous nonnal scale on any interval [a o , /30].
1. THE COMPLEX METHOD
235
Property 4 ° can be written in the following fonn:
II T(,B + iv )xll Ea IIxll a +,8.
Finally, from property 3° it follows that
Ilxil M = II T(O)xII M IIxiIEo.
( 1.40)
(1.41 )
We give an example of an analytic scale. We consider the set of all
continuous functions on [0, 1], equal to zero in a neighborhood of zero (the
neighborhood may vary with the function). On this set we define the family
of operators T(z)x(t) = t-Zx(t). The operators T(z) will be considered as
linear operators in the space M equipped with the norm of Lp(O, 1) (1 (; p (;
00 ). We denote by a (- 00 < a < (0) the scale of spaces constructed from
these operators. The space a consists of measurable functions for which
{ (I } l/P
II x 11 4 " = }o It-ax(t)IP dt < 00,
ifp < 00 and
IIXllLa = ess suplt-ax(t)1 < 00,
CIO
ess sup It x(t)1 0 (T 0)
0<;1<'T
if p = 00.
THEOREM 1.9. Let Ea' 0 (; a 1, be an analytic scale oj spaces, and let EI
be normally imbedded in Eo. The space Ea coincides isometrically with [Eo, El]a.
PROOF. Let x belong to the set M from which the scale Ea is constructed.
By property 5° the functionj(z) = T(a - z)x is analytic in the norm of Eo.
Next, by 4° we have
Ilj(z)IIE o = IIT(a - z)xll Eo = supIlT(iT)T(a - z)xll M
'T
(; supllT(a - Rez + iJL)xlI M (; max{llxII Ea , IlxIIEa_I},
p.
for 0 (; Re z 1, and so Ilj(z)IIE o is bounded in the strip II.
According to (1.40), on the boundary of II we have
II j( iT) II Eo = II T( a - iT) x II Eo (; II x II Ea'
II j( 1 + iT) II Eo = II T( a-I - iT) x II E I (; II x II Ea .
Finally, by property 3° we have j(a) = T(O)x = x. Consequently, x E
[Eo, Eda and Ilxll[E Eda (; Ilxll Ea.
We prove the reverse inequality. By virtue of the remark to Theorem 1.1
and the proof of Remark 2 of subsection 4 for x E M we may construct a
functionj(z) = Lf ak(z)x k such that a k E 3(c), x k E M, j(a) = x and
IljllB{E EI) IIxll[E Eda + E. (1.42)
236
IV. INTERPOLATION METHODS
We consider the function 'I'(z) = T(z + iJL)f(z), where JL is a fixed real
number. This function is analytic in Eo inside II and continuous and bounded
in the closed strip. Further,
11'¥(iT)IIE o sup II T(it)f(iT) II Eo (; sup Ilf(iT)IIE o IlfIIB(E EI)'
t,'T 'T
11'1'(1 + iT)IIE o sup II T(I + it)f(I + iT)IIE o (; sup Ilf(I + iT)IIE I (; IIfI15(E EI).
t,'T 'T
By the maximum principle, taking account of (1.42), we obtain
1I'¥(a)IIE o = II T(a + iJL)f(a)IIEo = II T(a + iJL)xll Eo (; Ilxll[EOtEda + E.
Finally, from (1.41) it follows that
Ilxll Ea = supllT(a + iJL)xlI M (; supllT(a + iJL)xll Eo (; Ilxll[E Eda + E.
Thus, IlxllEa = IlxlhEOtEda on M. On the other hand, since Mis dense in all
spaces Ea and also in [Eo, EI]a by Theorem 1.3, the theorem is proved.
From the interpolation Theorem 1.2 and the reiteration Theorem 1.7 we
obtain the following theorem.
THEOREM 1.10. Two analytic scales connecting two pairs of normally im-
bedded spaces have the strong interpolation property relative to each other.
REMARK. It follows from the theorem that the spaces L;(O, 1) considered
above are obtained by the complex method from the spaces 4(0, 1) and
L (O, 1).
10. Hilbert scales. In applications an important role is played by Hilbert
scales, which constitute a special class of analytic scales.
Let Ho be a complex Hilbert space with inner product (., .). Let an
unbounded positive definite selfadjoint operator j be given in Ho with domain
D(j), such that
Ilxll HO Iljxll Ho
(x E D(j».
( 1.43)
We denote by E A the spectral resolution of the identity corresponding to j,
and consider the set M of elements representable in the form x = If A dE"}..x
for some N < 00. The set M is dense in Ho. On this set the operators
T(z) = jZ are defined:
fx = NAZ dE-Ax.
It is easy to verify that these operators have properties 1 0 _5 0 necessary for
the construction of an analytic scale from them. On M we introduce the
norms
Ilxll a = sup Ill .a+i'T xll Ho = { f. N\, (d fr;' X x) } 1/2 III xii
I {\2a LA , =.a Ho.
-00<'T<00
I. THE COMPLEX METHOD
237
The completions Ha of M in these norms are Hilbert spaces; they form an
analytic scale which is called a Hilbert scale. By virtue of (1.43) the norm
Ilxll a is an increasing function of a, and therefore the scale {Ha}' - 00 < a
< 00, is a continuous normal scale on any interval.
For a > 0, Ha C Ho and Ha constitutes the domain DU a ) ofja. For a < 0
the space Ha already contains generalized (ideal) elements relative to the
initial space. Let -a = 13 > o. We clarify the structure of the space H dual
to Ha. If cp E H , then for elements of M we have
Icp(x)1 ' IlcpllH,lIxll a = IlcpIIH,llj-,8xll o .
a a
We perform the change of variables j-,8x = y. Then IcpU'Y)1 ' IIcpIIH,lIyllo'
a
i.e. the functional cp(j'Y) is bounded in the norm of Ho on the setj-13M. By
the density of this set in Ho, this functional admits a unique representation of
the form cpul3y) = (y, z), where z E Ho. Performing the inverse change of
variables, we obtain
cp(x) = (j- 13 x, z) = (x,j-fJ z ) = (x, u),
where u = j-,8z E D(j/3) = H,8. Conversely, for u E H,8 every functional
(x, u) is a linear functional on M, bounded in the norm of Ha:
I(x, u)1 = 1 (j- 13 x, j,8u) 1 II ull H II xii H ,
fJ a
and therefore it can be extended by continuity to a functional belonging to
H . From the last inequality it follows that II cpll H' , II ull H . We show that we
a fJ
have equality here. Let Un be a sequence of elements of M such that Un u in
H13. We set x n = j 213 u n . We have
(x n , u) = (j2,8U n , u) = (j,8u n ,j,8u) Ilull .
For any e > 0 and sufficiently large n we then have
cp(x n ) = (x n , u) (1 - e)llull
(1 - 2e)lIunll,8llull,8 = (1 - 2e)llx n ll a llull,8.
From this and the above it follows that IlcplIH = Ilull,8. We have proved that
H is isometric to H,8 = H -0.; an isometry is given by the equality cp(x) =
(x, u) (cp E H , u E H -0.' X EM). In what follows all values of the func-
tional cp(x) will be denoted by (x, u) (x E Ha' u E H ).
Since Hilbert spaces are reflexive, Ha is isometric to H'-a,.
Theorem 1.10 shows that two Hilbert scales have the strong interpolation
property relative to each other. We consider a special case of this assertion.
238
.. IV. INTERPOLATION METHODS
THEOREM 1.11. In a Hilbert space Ho let two positive definite selfadjoint
operators) and) I having property (1.43) be given. If DU I) :) DU) and
II) I xII Ho lI)x II Ho' (1.44)
then for 0 a 1 we have DUr) :) Dua) and
lI)ixll Ho lI)axll Ho (0 a < 1, x E D()a)).
PROOF. We construct the Hilbert scales Ha and H generated by the
operators ) and) I' respectively. Inequality (1.44) says that the identity
operator I (the imbedding operator) is a bounded operator from HI into Hl
with nonn not exceeding 1. Obviously I is bounded as an operator from Ho
into Ho. By the interpolation theorem the identity operator acts from Ha into
Hal and has nonn not exceeding 1. This gives the assertion of the theorem.
THEOREM 1.12. If a Hilbert space HI is imbedded normally in Hep then there
exists a unique Hilbert scale connecting these spaces.
PROOF. By virtue of the inequality
\(X'Y)Ho\ IIYIIHollxllH o IIYII Hollxll H,
for a fixed Y E Ho the functional (y, x) is bounded on HI' and so there is a
uniquely determined element z E HI such that (x, Y)H = (x, Z)H (x E HI).
o ,
We write z = Vy. Then
(X'Y)Ho = (x, VY)H, (x E HI). (1.45)
Moreover, IIzllH = II VylIH IlyII H . The operator V is linear, defined on
, , 0
the whole of Ho, and bounded: II VYIlH o < II VYIIH, IIYIIH o . The operator V
is selfadjoint and nonnegative. In deed, by ( 1.45) for X,Y E Ho we have
(VX,Y)H o = (Vx, VY)H, = (Vy, Vx) H, = (Vy, x) Ho = (x, VY)H o
and (Vy, Y)H = (Vy, VY)H O. Next, if Vy = 0, then (x, Y) n = 0 for every
0' 0
x E HI' and by the density of HI we have Y = (). Thus, V is positive. We
write V-I = U. Then D( U) = R(V) C HI' and (1.45) can be written in the
fonn
(Uz, X)Ho = (z, X)H,. (1.46)
We introduce the operator) = U I / 2 . The domain of U I / 2 can be obtained by
closing D( U) in the nonn
IIU I / 2 zII HO = V (Uz, Z)H o = V (z, Z)H, = IIzIlI.
( 1.47)
Thus, D()) is contained in HI and is closed in it. We show that DU) coincides
with HI. Otherwise there would exist Xo E HI such that (z, X H, = 0 for all
z E DU) and, in particular, for all z E D( U). Then it follows from (1.46) that
1. THE COMPLEX METHOD
239
(Uz, xo) = 0 (z E D( U), and consequently Xo = 0, sInce R( U) = Ho. Fi-
nally, it follows from (1.47) that lI}zllH o = IIzllH1 (z E HI).
The Hilbert scale generated by the selfadjoint operator} connects Ho and
HI.
We show the uniqueness of the Hilbert scale connecting H 0 and H I. If
there were two such scales Ha and H , then for their generating operators}
and}1 the relations DU) = HI = DUI) and lI}xll Ho = II}, XIiHo would be
satisfied. By Theorem 1.11 we would then have DU 1/2 ) = DU: 12 ) and
11}1/2x1IHo = 1I}:12xIlHo. In particular, for x E HI we would have Ux, X)H o =
Ulx, X)H o . From the equality of the quadratic forms we obtain the equality of
the bilinear forms: (}X'Y)H o = UIX'Y)Ho (x,y E HI). By the density of HI in
Ho this implies that}x = }IX for x E HI' and consequently the scales Ha and
H: coincide. The theorem is proved.
Now, by means of interpolation we establish an inequality important in
operator theory.
THEOREM 1.13. In a Hilbert space Ho let there be given two positive definite
selfadjoint operators} and}1 normalized by condition (1.43) and a linear
operator T with D( T) ::> DU). If for any x E D(}) and Y E DUI)
(Tx,y) II}xIlHoIIYIIH o and (Tx,y) II xII Holl}1 YIIH o ' (1.48)
then
(Tx,y) II}axIIHoll}:-CYIIH o (0 < a 1). (1.49)
PROOF. We denote by Ha and H: the Hilbert scales generated by the
operators} and}l. From the first inequality in (1.48) it follows that II TxllHo <
lI}xIl Ho ' and so T acts boundedly from HI into Ho with norm not exceeding 1.
The second inequality in (1.48) may be written in the form UI ITx , }IY)
IIxIlHoll}IYIIHo' which implies that II}lITxIlHo IlxliHo. In other words, we
may say that T is defined on the set DU) dense in Ho, and acts boundedly
from Ho into H I with norm not exceeding 1. The operator T can be extended
by continuity to an operator f defined on the hole space Ho with the same
norm. Applying the interpolation theorem to T, we obtain that it acts from
any space Ha into H:_ I (0 a 1) with norm not exceeding 1, i.e. IlfxIIH _1
IIxIi H .
a
If now x E D(}) andy E DUI)' then
(Tx,y) = (} -ITx,}:-cy)
II}f-ITxIIIHolI}:-CYIIHo II}QxIIHolI}:-CYIiHo.
REMARK. Condition (1.43) has been imposed on the operators} and}1 only
to simplify the exposition. A multiplication of these operators by normalizing
240
IV. INTERPOLATION METHODS
factors and a repetition of the arguments leads again to the conclusion that
(1.48) implies (1.49). Moreover, by passing to the limit, this conclusion can be
extended to the case where) and)1 are only positive.
11. The complex method of interpolation in ideal lattices. Let Eo, E I be a pair
of ideal lattices of functions on a space with a a-finite measure JL. In the
Introduction to this chapter we described how the Banach spaces Eo(A) and
EI(A) are constructed, where A is an arbitrary Banach space. We consider the
simple case where A coincides with the field C of complex numbers, and write
Eo(C) = Xo and EI(C) = XI. We shall call these spaces ideal lattices of
complex-valued functions.
We consider the space [X o , Xda. It turns out that this space is also an ideal
lattice. Indeed, let x E [X o , Xda and \y(t)1 Ix(t)l. We consider the function
a(t) = y(t)/ x(t), assuming that a(t) = 0 if x(t) = o. Clearly la(t)1 < 1, and
so the operator of multiplication by a(t) acts in the ideal lattices and has a
norm not exceeding one there. By the interpolation Theorem 1.2 this operator
also acts in the space [X o , Xda with norm not exceeding 1. Then y E
[X o ' Xda and
IIYII[xo.xda = lIaxll[xOtxda II x II [XOtXd a .
We establish a series of properties of the ideal lattice [X Xda.
If x E Xo n XI' then by virtue of Remark 2 of subsection 4 there exists a
function
N
g(t, z) = L ak(z)xk(t)
1
such that X k E Xo n XI' a k E (C), and
g(t, a) = x(t), II gll (XOtXl) (1 + E)llxll[x Xl]a. (1.50)
For fixed t we apply (1.21) to the scalar-valued function g(t, z). We obtain
[ 1 00 ] I-a
Ix(t)J = I g(t, a)I" 1 _ a Lool g(t, iT)1 (a, 'T) dr
[ 1 00 ] a
X -;;. L 00 I g( t, 1 + iT ) IlL. (a, 'T) d'T ·
( 1.51 )
We consider the functions
1 f oo
Y o(t) = 1 _ a _ 00 I g( t, iT) I (a, 'T) dT
and
I f 00
Yl(t) = -;;. _ooJg(t,l + iT)JIL.(a, 'T) d'T.
1. THE COMPLEX METHOD
241
The right sides of these equalities may be understood as Riemann integrals
of functions with values in Xo and Xl' respectively. Therefore
II Yo II x 0 ' 1 a f_ 0000 II g( t, iT) II x 0 JI.o( a, T) dT
sup II g(t, i'T)llx o (1 + e)lIxll[xOtx.]a (1.52)
'T
(the last inequality holds in virtue of (1.50) and
IIYdlx,' foo Ilg(t, 1 + iT)lIx,lLl(a,T)dT
-00
supll g(t, 1 + i'T)lIx l (1 + e)llxll[x x.]a. (1.53)
Inequality (1.51) takes the form
Ix(t)1 [Yo(t)]I-a[YI(t)]a.
If we now write xo(t) = [IlYollxo]-yo(t) and xl(t) = [IIYlllx.J- I(t), by (1.52)
and (1.53) we obtain
Ix(t)1 (1 + e)lIxll[xOtx.]a[xO(t)]I-a[xl(t)]a, (1.54)
where Ilxollxo = 1 and IIxlllx 1 = 1.
It is interesting to consider the problem of when the functions of the fonn
of the right side of (1.54) belong to [X o , Xda.
LEMMA 1.10. If Xo E Xo and Xl E Xl and the values of Ixo(t)1 and Ixl(t)1
have finite positive lower and upper bounds, then Ixo(t)ll-alxl(t)la E [X o ' Xda
and
IIlxoll-alxllall[x x.]a IIxolI allxllI l. (1.55)
PROOF. First let Ilxollxo = IIxlllx 1 = 1. We write xa(t) = Ixo(t)ll-alxl(t)la.
Let e be the support of xa(t). The characteristic function Xe of the set e
belongs to Xo n Xl. Indeed, the functions Ixo(t)1 and Ixl(t)1 have positive
lower bounds mo and m l on e, and therefore Xe(t) Ixo(t)11 mo E Xo and
Xe(t) Ixl(t)11 m l E Xl.
We consider the function
f( t, z) = I xo( t)II-Z Ix I (t)lz.
Since Ixo(t)1 and Ixl(t)1 are bounded by numbers Mo and M I , for every z
we have
If(t, z)1 MJ-RezMfezXe(t) E Xo n Xl.
For fixed tEe the functionf(t, z) is entire, and
z [ f( t, z + z) - f( t, z)] f: ( t, z)
242
IV. INTERPOLATION METHODS
uniformly in t. Since in the last relation all terms are equal to zero for t fl e,
we may write the inequality
I [ f( t, z + z) - f( t, z)] - f: ( t, z) I 11 ( z )Xe ( t),
where 1J( z) 0 as z O. From this it follows that f(t, z) is entire as a
function with values in Xo n XI. By the above, it is bounded in the strip
o Rez 1. Next,
Ilf(t, iT)lIx o = Ilxollxo 1 and IIf(t, 1 + iT) II XI = IIxlllx l 1.
Finally,f(t, a) = xa(t). This implies that
xa E [Xo, XI Ja and ilxall[xOtx.]a 1.
If now Xo and XI are any functions from Xo and XI' then
Illxo(t)ll-alxl(t)lall[Xo.x.]a
1 - a a I Xo( t)
= II Xo II X 0 II X 111 X I I II X Ii I X 0
I-a
a
X 2 ( t)
Ilxlllx l
IIxolI allxllI l.
[XOtX.]a
COROLLARY. If [X o , X.]a has the Fatou property (see the Introduction), then
(1.55) holds for every Xo E Xo and XI E XI.
Indeed, denote by em the set of all t for which 11m Ixo(t)1 m and
1 I m IXl(t)1 m. The union of all sets em' m = 1, 2 . . . , forms the support
of xa(t), and therefore Xe (t)xa(t) xa(t) almost everywhere. By virtue of the
m
lemma we have
IIXe m Ixoll-a Ix Iia II[xo,x.]a = II [Xe m Ixol ] 1- a[ Xe m Ix II] all [Xo.X.]a
IlXe Xoll -a IIXe X III II Xoll -a II X III .
mOm I 0 I
Due to the Fatou property, this inequality implies (1.55).
Inequality (1.55) is a further generalization of the Holder inequality.
The above properties suggest the introduction of the following space: For
fixed a E (0, 1) denote by X the class of complex-valued functions x(t) on we
such that
IX(t)1 Alxo(t)ll-alxl(t)la (1.56)
for some A > 0, Xo E Xo and XI E XI with Ilxollxo 1 and II xIII XI 1. The
space X is linear. Indeed, if (1.56) is satisfied for X and ly(t)1
,uIYo(t)1 1 - a l y1 (t)la with IIYolix o 1 and IIYllix l 1, then
Ix(t) + y(t)1 Alxo(t)ll-alxl(t)la + ,uIYo(t)ll-alyl(t)la,
1. THE COMPLEX METHOD
243
and by Holder's inequality with weights A and p. we have
Ix(t) + y(t)1 (Alxo(t)1 + p.IYo(t)l)l-a(Alxl(t)1 + p.IYt(t)l)a
( A ) 1- a
= (A + IL) A + IL I Xo(t) I + A IL IYo(t)1
X ( A : IL IXJ(t)1 + A IL IYJ(t)!f. (1.57)
The functions in parentheses belong to the unit balls of Xo and XI'
respectively, and therefore x + y E X. It is obvious that ex E X if x E X. In
X we introduce the norm
Ilxllx = inf A, (1.58)
where the infimum is taken over all A satisfying (1.56) for some Xo and XI.
From (1.57) it follows that the quantity (1.58) satisfies the triangle inequality.
It is obvious that Ilexlix = lelllxli x . Now let Ilxllx = O. This means that
there exist sequences of numbers +0 and functions X On E Xo and
X ln E XI with IIxonllx o ' IIxlnllx. 1 such that
I x ( t) I I x On ( t) II - a I X In ( t) I a .
The sequences /2(I-a)lxonl and /2aIXlnl converge to zero in Xo and XI'
respectively. Since Xo and XI are imbedded in S(WC, p.), these sequences
converge to zero in measure on every set of finite measure. The sequences of
functions IXonll-a(t) and Ixlnla(t) have the same property. Since
Ix(t)1 does not exceed the product of these functions, we have Ix(t)1 = 0
almost everywhere. Thus, (1.58) has all properties of a norm.
It is obvious that if lu(t)1 Ix(t)1 and x E X, then u E X and Ilullx
II x II x.
We prove the completeness of X. Let X n E X and lIxnllx < 00. Given any
e > 0, for each function X n we choose a number and functions XO n E Xo
and X ln E XI' with IIxonllx o ' IIxlnllx. 1, such that
..
IIxnllx + e/2 n and Ixn(t)1 Anlxon(t)ll-alxln(t)la.
Then < 00, and the same arguments as in the proof of (1.57) show that
00 [ 00 ] 1- a [ 00 ] a
t IXn(t)1 " L t L IXon(t)1 t L Ix1n(t)1
( 1.59)
for almost all t.
By virtue of Corollary 2 to Theorem 1 in the Introduction to Chapter II the
functions in square brackets are defined almost everywhere and belong to the
unit balls of Xo and XI. This implies that lxn(t)1 belongs to X. We write
244
IV. INTERPOLATION METHODS
x(t) = L xn(t). Then Ix(t)1 L lxn(t)l, and therefore x E X. Inequality
(1.59) implies that Ilxllx IILlxnlllx L , and by the arbitrariness of E we
have
00
Ilxlix }: IIxnllx.
1
Applying this inequality to the function x(t) - L xn(t) = L +l xn(t), we
obtain
N
X - }: x n
1 X
00
}: Ilxnllx O as N 00.
N+l
The completeness of X is proved. Thus, X is an ideal lattice. We denote this
lattice by XJ -ax .
The space XJ -ax is intermediate between Xo and Xl. Indeed, if x E Xo
n Xl' then
Ix(t)1 = Ix(t)ll-alx(t)la = Ilxlll-allxlla ( Ix(t)1 ) l-a ( Ix(t)1 ) a
Xo x. Ilxlix o Ilxllx.'
which implies that x E xJ-aXf and
I/xl/xJ- a Xj IIxll allxll . max{ II xII Xo' IIxllx.}.
Next, if x E xd-ax , then (1.56) implies that
Ix(t)1 A«1 - a)lxo(t)1 + alxl(t)1)
and so x E Xo + XI and
I/x(t) llx o+x. A[ (1 - a)lIxolixo + allxlll x .] A.
Thus, Ilxll x +x I IXll x .-ava.
o. 0 A.
From inequality (1.54), true for any function x E Xo n Xl' Theorem 1.3,
and the definition of the norm in XJ -aXf it follows that [X Xda is
imbedded in xd-ax with imbedding constant not exceeding one.
The reverse imbedding holds under certain conditions.
THEOREM 1.14. If the space X = XJ -ax is regular (has absolutely continu-
ous norm), then it coincides isometrically with [X o ' Xda.
PROOF. We denote by U E the collection of all functions x in X for which the
inequality
Ix(t)1 (1 + E)llxllxxci-a(t)x (t) with IIxollxo 1 and Ilxtllx. < 1
is satisfied and the nonzero values of the functions xo(t) and XI(t) have finite
positive lower and upper bounds. By Lemma 1.10 and the fact that [X o ' Xda
1. THE COMPLEX METHOD
245
is an ideal lattice, we have
IIxll[xo.xda (1 + e)lIxlix (x E U e ). (1.60)
We show that the set ll,; is dense in X. Lety E X, let e be the support of y,
and let
ly(t)1 (1 + e/2)lIyllxIYo(t)l l - a l yI (t)la,
where IIYollx o ' IIY.llx. 1. We denote by em the set of all t at which
ly(t)1 > 0, l/m IYo(t)1 m and l/m IYI(t)1 m. Then the functions
zm( t) = Xe m (t)y( t) converge to y( t) almost everywhere, and
IIzm - yllx = IlXe\e yllx 0
m
by virtue of the absolute continuity of the norm in X.
In particular, IIzmllx lIyllx, and so
IZm(t)1 (1 + e/2)IIYllxIXe (t)Yo(t)ll-aIXem(t)YI(t)la
m
(1 + e)lIzmllxlXe".(t)Yo(t)lIl-aIXem(t)YI(t)la
for sufficiently large m. We note that IIXmYolix o 1 and IIXmYlllx. 1; thus
zm E U e .
Now we may apply Lemma 3.4 of Chapter I, from which it follows by
virtue of (1.60) that the space X is imbedded in [X o , Xda with imbedding
constant not exceeding 1 + e.
Since e is arbitrary, the theorem is proved.
REMARK. The hypothesis of the theorem is satisfied for 0 < a < 1 if at least
one of the spaces Xo or Xl has absolutely continuous norm.
Indeed, let Xo have this property. If en is a decreasing sequence of sets with
empty intersection and x E X, then from (1.56) we obtain
IXe (t)x(t)1 AIXe (t)xo(t)ll-aIXe (t)xl(t)ll-a
n n n
Xe (t)xo(t) I-a
'AIlXe.xoll a liXe.xllx o Ix\(t)j«.
This implies that IIXenxllx AIlXenxolI a O as n 00.
The family of spaces XJ -aXf is called the Calderon family connecting Xo
and X I .*
We note that in the general case the space XJ -aXf may not coincide with
[X o , Xda. Moreover, XJ -aXf may not be an interpolation space between Xo
· Editor's note. In the Russian edition Calderon families were called Calderon scales. However,
a Calderon family may not be a scale in the sense of Chapter III, since the spaces Xo and Xl
need not be related by an imbedding.
246
IV. INTERPOLATION METHODS
and Xl (see Notes on the Literature on p. 352). Nevertheless, if the linear
operator T is positive and acts boundedly in Xo and Xl' then it acts in
Xci-aXf as well. Indeed, if x(t) > 0 and x E xJ-aXf, then by (1.56)
Tx(t) AT(lxoll-alxlla)(t).
Similarly to the way we did in subsection 1 of the Introduction to Chapter II
(property 6°), from this we obtain that
Tx(t) A(Tlxol)l-a(t)(Tlxll)a(t),
and, since Tixol E Xo and Tlxll E Xl' we get that Tx E xJ-ax . For any
real functions the assertion follows from the equality Tx = Tx+ - Tx _, and
for complex-valued functions from the equality Tx = T(Re x) + iT(lm x).
If Xo = 40 and Xl = 41' 1 Po 00, then XJ-aXi coincides isometri-
cally with 4, where l/p = (1 - a)/po + a/pl. Indeed, (1.56) and Holder's
inequality imply that for x E xJ-aXf we have
flx(t)IP dlL ;\.P[ flxo(t)IPO dlL r-a>p/PO[ !lx\(t)IPI dlL r>P/PI ;\P,
from which it follows that x E 4 and IIxllz" IIxIl40-a4 . Conversely, for
any function in 4 the representation
( Ix(t)IP/Po ) l-a ( Ix(t)IP/PI ) a
Ix( t)1 = II xii z,. II IxlP/poll II Ixlp/p'll
4 0 41
holds (we have used the fact that l/p = (1 - a)/po + a/PI)' and therefore
x E L;o-aLp and IIxllz,:o-az;; IIx1l4.
The norm is absolutely continuous in at least one of the spaces (if Po =F PI),
and thus by Theorem 1.4 we have the isometric equality
T.I-aLa - [ T L ] - T. ( ! = 1 - a + ) . (1.61)
o PI - -Po' PI a - P Po PI
The interpolation Theorem 1.2 leads us in particular to the proof of the
Riesz-Thorin theorem formulated in Chapter I, 4.2.
2. The methods of constants and means
(:K- and -methods)
1. The method of constants (:K-method). We recall that if A is a Banach
space and E is an ideal lattice on the space with measure JL, then by E(A)
we denote the Banach space of functions (classes) u(t) with values in A,
strongly measurable in A, and such that lIu(t)IIA E E, with the norm lIuIlE(A)
= IIlIu(t)IIA liE. This space is imbedded in the metric linear space S( , JL, A)
of all strongly measurable functions with values in A.
2. METHODS OF CONSTANTS AND MEANS
247
Now let Ao and A I be a Banach couple, and Eo and EI be two ideal lattices
on the same measure space. The space Ej(A j ) is imbedded in Ej(Ao + A I)'
and, as we said above, these spaces are imbedded in S(9R., JL, Ao + A I).
Therefore Eo(Ao) and EI(A I) form a Banach couple. We denote by
(Ao, A 1);OtE I the part of Eo(Ao) + EI(A I)' consisting of all functions which are
constant almost everywhere. The imbedding
Eo(Ao) + EI(A I ) C S(WC, JL, Ao + AI)
implies easily that (Ao, A 1);o.E 1 is a closed subspace of the sum EJ..A I) +
EI(A I ). It can be identified with the Banach space of all elements x E Ao +
A I for which there exist functions U j = u;(t) E Ej(A j ) (i = 0, 1), such that
x = u o ( t) + u I (t) (2.1)
with norm
Ilxll(A-.d )9C = inf (1luoll E (A \ + IIUIIIE (A »). (2.2)
()?"" 1 E()oE 1 X = Uo(t) + u. (t) 0 01 1 1
The space (A o , A I); E may contain nonzero elements only in the case
Ot 1
when e E Eo + E I , where e = e(t) = 1. Indeed, if x E (A AI) E. and
x =1= 0, then there exist u;(t) E Ej(A j ) such that uo(t) + ul(t) = x. By the
definition of the norm in Ao + Al we have 0 < IIxIlAo+AI lIuo(t)IIA o +
II ul(t)11 A I. This implies that
e.(t) = lIu;(t)IIA, , lIuj(t)IIA, E Ej"
I lIuo(t)IIA o + lIul(t)IIA I IIxIlAo+A.
Then e(t) = eo(t) + el(t) E Eo + EI.
In what follows we always assume that e E Eo + EI.
LEMMA 2.1. The spaces (Ao, AI) OtEI are intermediate between Ao and AI.
PROOF. Let x E Ao n A I. If e(t) = eo(t) + el(t), where ej(t) E Ej, we set
uj(t) = ej(t)x. Then x = uo(t) + ul(t), and Ilu;(t)IIA. = le;(t)lllxIlA. E Ej. Con-
..
sequently, x E (Ao, A I)Eo.E 1 and
IIxll(A I) ()oEI II eo II EollxllAo + lIelllElllxllAI
(II eoll Eo + II elll E) IlxliAonA I.
Taking into account that i nf e ==e + e (II e o II E + II elll E ) = II ell E +E' we ob-
o 1 0 1 0 1
tain
Ilxll(Ao.AI)}O-EI IleIIEo+E11IxIIAonAI'
i.e. Ao n A I C (Ao, A 1) I" .
Now let x E (Ao, A I)Eo.E.. The equahty x = uo(t) + ul(t), u;(t) E E;(A;),
implies the inequality Ilxii A +A lI u o (t)IIA + lIul(t)IIA . By calculating the
o 1 0 1
248
IV. INTERPOLATION METHODS
norm in Eo + EI of the left and right sides, we have
II x II A 0 + A I II e II Eo + E I II II u o ( t) II A 0 + II u I ( t) II A III Eo + E I
II uoll Eo(Ao) + II ulll EI(A I).
This implies that
1
IIxIlAo+A, ..;; lIell Ilxll(A.,.AI) "EI'
Eo + E I
i.e. (Aij) A 1) o.EI C Ao + A I.
THEOREM 2.1 (interpolation theorem). Let Ao, A I and Bij) B I be two Banach
couples. The triple Ao, A I' (Ao, A I) E of spaces is an interpolation triple
9(0.1
relative to the triple Bo, B I' (Bo, BI)Eo.EI.
PROOF. If x E (Ao, AI) o.EI and T E w(Aij) AI; Bo, B I ), then x = uo(t) +
ul(t), where U; E E;(A;), and so Tx = vo(t) + VI(t), where v;(t) = Tu;(t).
Moreover, IIv;(t)IIB. II TIIAo-+B.llu;(t)IIA.. Consequently v; E E;(B;) and
I I I I
II Txll(Bo.BI) ()O£1 IIvoll Eo(Bo) + Ilvlll EI(B I )
II TIIAo-+BolluoIlEo(Ao) + II TIIAI-+BllluIIIEI(AI)
max { II TII A;-+B;} (II uoll Eo(Ao) + II ulll EI(A I»).
Taking the infimum over all representations (2.1), we obtain
II TII(A I) ()O£I Bo.BI) ()'£1 II TII'1T(Ao.AI;Bo.BI).
Theorem 2.1 shows that we have defined an interpolation functor ( , ) E .
I
REMARK. If e E Eo n E I , then (Ao, A 1) o.EI is isomorphic to Ao + A I.
Indeed, if x E Ao + AI' then x = Xo + XI' x; E A;. We set u;(t) = e(t)x;.
Then II u;II Ei(A;) = II ell Eillx;IIA;. Thus, x E (Aij) AI) o.EI and
IIxll(A I) ()O£1 II ell EollxollAo + IlellElllxlllAI.
Taking the infimum on the right, we obtain
II x II ( A - A ) x II ell EnE II x II A + A ·
a I E()o£ I 0 I 0 I
2. The method of means ( -method). In the notation of the preceding
subsection we assume that e E EJ + El, where E;I is the ideal lattice
associated with E;.
LEMMA 2.2. For any function U E Eo(Ao) n EI(A I ) the integral J u(t) dp.
exists in Ao + A I. The mapping : U J u( t) dp. is a continuous linear operator
from Eo(Ao) n EI(A I ) into Ao + AI.
2. METHODS OF CONSTANTS AND MEANS
249
PROOF. Let e(t) = eo(t) + el(t), where e; E E/. Without loss of generality
we may assume that the functions e;(t) are nonnegative. We have
Jllu(t)IIAo+A. dp. J eo(t)lIu(t)IIAo dp. + J e1(t)llu(t)IIA. dp.
lIeoIIEJlluIlEo(Ao) + lIeIIlEllluIIEI(AI)
(lleollEJ + IIelllEI)lluIlEo(Ao)nEI(AI).
Taking the infimum over all representations e(t) = eo(t) + el(t), we obtain
J u(t) dp. lIellEI+ElllullE ( A ) nE ( A ) .
Ao+AI 0 I 0 0 I I
The lemma is proved.
COROLLARY. The null space N( ) of the operator is a closed subspace of
Eo(A n EI(A I ).
We denote by (Ao, AI)t EI the quotient space Eo(Ao) n EI(A I )/ N( ),
which can be identified with the Banach space of all elements x E Ao + A I
representable in the form
x = f u(t) dp., (2.3)
where u E Eo(Ao) n EI(A I) with norm
II x II(A."A ,)t()oE, = x= j t) d,.11 u II Eo(Ao>n E.(A ,>. (2.4)
The space (Ao, A l)t EI may contain nonzero elements only in the case
where Eo n EI =1= {O}. Indeed, if x =1= 0, x = J u(t) dp., lIu(t)IIA. E E;, then
I
u =1= 0 and the function min(llu(t)IIA o ' Ilu(t)IIA) belongs to Eo n EI and is
different from zero, because the functions II u(t)II A and II U(t)IIA vanish at the
o I
same time.
In what follows we assume that Eo n EI =1= {O}.
LEMMA 2.3. The spaces (Ao, AI)t EI are intermediate between Ao and AI.
PROOF. The continuity of the imbedding (A AI) EI C Ao + Al follows
from the preceding lemma and the definition of the norm in (Aij) A l)t EI.
Now let x E Ao n AI. We take a function = (t) such that E Eo n EI
and J 1/;(t) dp. = 1. We set u(t) = 1/;(t)x. Then u = u(t) E Eo(Ao) n EI(A I )
and x = J u(t) dp.. Thus, x E (Ao, AI)t EI and Ilxll EI 111/;IIEonElllxllAonAI'
which means we have the imbedding Ao n A I c (Aij) A l)t EI.
THEOREM 2.2 (interpolation theorem). Let Ao, Al and Bep BI be two Banach
couples. The triple of spaces Ao, A I' (Ao, A I)t E is an interpolation triple
I
relative to the triple Bo, B I , (Bo, BI)E EI.
250
IV. INTERPOLATION METHODS
PROOF. If x E (Ao, AI)t EI' then x = J u(t) dp., u E Eo(A n EI(A I ). The
operator T E w(Ao, AI; Bo, B I ) acts from Ao + Al to Bo + BI. Therefore
Tx = J Tu(t) dp. = J vet) dp.. Next, IlvIIE;(B;) II rl/A;-+B;I/ullE;(A;). From this
we obtain
I/vllEo(BI)nEI(BI) max{11 TIIA;-+B;} I/ul/Eo(Ao)nEI(AI).
By the definition (2.4) we have
II Txll(B BI)iC)oEI IIvIIEo(Bo)nEI(BI) II TI/?T(A AI;B BI)llul/Eo(Ao)nEI(AI).
Taking the infimum over all representations (2.3), we obtain
II TII(B BI)iC)oEI AepAI)iC)oEI II TI/?T(A AI;B BI).
Thus, the interpolation functor ( , )t EI is defined.
Sometimes in constructing the spaces (Ao, A I) EI and (A A l)t EI It IS
convenient to effect the discretization of the lattices Eo and E I beforehand
(see Chapter II, the Introduction, subsection 2). Let 9R = U IDl n be a
partition of 9R into disjoint sets of finite positive measure JL" and such that
X9n E Eo n E} (such a partition exists if the supports of Eo and EI coincide
n
with 9R). We assume that the averaging operator T constructed for this
partition acts boundedly in the spaces E;. Then it maps E; onto the subspace
E;c of functions constant on the sets 9Rn. In order to obtain a quantity
equivalent to the norm of (Ao, A }) EI' we need consider only those represen-
tations x = uo(t) + u}(t) in (2.2) in which the functions u;(t) are constant on
the sets 9Rn. Indeed, if x = vo(t) + VI(t), where V; E E;(A;), we set u;(t) =
Tv;(t). Then x = Tx = uo(t) + ul(t), and
lIu;IIEi(A;) = II II Tv;IIA;IIE; I/r(I/v;IIA)IIE; IITII Ei -+E;I/ v IIE;(A;).
Thus, the infimum in (2.2) calculated only over functions constant on the
sets 9Rn does not exceed the norm of (2.2) by more than a factor of
max{11 TIIEi-+E;}. A similar assertion holds for the spaces (Ao, AI) EI.
In the terms introduced in subsection 2 of the Introduction of Chapter II,
the assertions obtained so far can be formulated in the following way.
REMARK 2.1. If the averaging operator corresponding to the partition
9R = U 9JC n is bounded in the spaces E;, then
(Ao, AI) I = (Ao, AI) ! EI
up to isomorphism.
3. The methods of constants and means for spaces with power weights. In this
subsection the semiaxis (0, 00) with the measure defined by the differential
form dt / t will serve as the space [n. The spaces 4 constructed for this
measure will be denoted by 4,.. Let ao and a l be real parameters, and
2. METHODS OF CONSTANTS AND MEANS
251
assume that ao < 0 and a l > O. Let Eo and E I be ideal lattices which are
intermediate spaces between L I ,. and Loo,.. We consider the spaces with
weights E; ta; = E;Q; (i = 0, 1). We show that the function e(t) = 1 belongs to
,
Eoo + Efl. Indeed, e(t) = X(1,OO)(t) + X(O,I)(t). Next, t (I,oo)(t) ELI,. n
Loo,. C Eo and tQ1X(0,1)(t) ELI,. n Loo,. eEl' as required. This implies that
(Ao, A 1) o-o,Erl is an intermediate space between Ao and A I.
We shall assume that both Eo and EI contain functions that are positive
almost everywhere. Then Eo n EI =1= {O}. The associated spaces EJ and El
are ideal lattices on (0, 00). It is easy to see that they are also intermediate
spaces between L I ,. and Loo,.. Next, the spaces (E;lXt)1 = E/(-a;) are associated
with the E;lXt. The same reasoning as above shows that e E (E;O)1 + (E I)I.
Therefore we may define the intermediate space (A o , A I)t Eal.
0' 1
Now we assume that the lattices Eo and EI are interpolation spaces
between L I ,. and Loo,.. We recall that in 8.5 of Chapter II we considered
certain operators acting in the spaces E; and obtained estimates for their
norms.
For the dilation operator we have
II (J l/,r<P11 E; = II cp( 'Tt) II Ei < k; II <p1I Ei'
where the k; are the interpolation constants of the spaces E;.
For a power transformation we have
II T A <P II Ei = II <P ( f' ) II E; k; max { 1, IAI- I } II <P II E;.
For the convolution we have the following Young inequality:
(2.5)
(2.6)
1 00 ( t ) ds
o 0/ S <1>(S)-; E..so;; kdlo/IIL,)<1>IIE"
I
(2.7)
Finally, we considered the special averaging operator
00 f en+1 ds
TI<p(t) = L X(en en+I](t) <p(s)-,
- 00 en S
(2.8)
for the norm of which we have obtained the estimates
II TICPll Ea; k;ellXtlll<p11 E..
I I
From Remark 2.1 and the last assertion we obtain
REMARK 2.2. To obtain a quantity equivalent to the norm of (A o , A I) E al'
1 , 1
in (2.2) we need consider only representations of the form x = uo(t) + ul(t)
in which the functions u;(t) are constant on the intervals (en, e n + l ] (n =
0, + 1, + 2, . . . ).
A similar remark holds for the space (Ao, A l)t!lO E at"
1 , 1
252
IV. INTERPOLATION METHODS
THEOREM 2.3. If the ideal lattices Eo and EI are interpolation spaces between
L I ,. and Loo,., then the spaces (A A 1) ao,Eil and (Ao, A l)t 8 o,Ei 1 coincide.
PROOF. Let x E (Ao, A l)loo,Ei 1 . This means that there exists a represen-
ta tion
1 00 dt
x = 0 u(t)t'
where u E Eoo(Ao) n Efl(A I). We consider the functions
vo(t) = 1 1 u(s) ds and vt(t) = f oo u(s) ds .
o S t S
Then x = vo(t) + VI(t). The function vo(t) assumes its values in Ao. Indeed,
1 1 II U(S)IIA ds < (I s-ao(saoll u(s)IIAJ ds
o 0 s J o s
< II s-aCX(O,t]( s) II EJ II s (X°II u( s) II Aoll Eo. (2.11)
Since ao < 0, we have s-aCX(O,t](s) ELI,. n Loo,. C EJ. Therefore (2.11)
implies that the first integral in (2.10) is convergent in the norm of Ao, and
consequently vo(t) E Ao.
It can be verified in a similar way that vl(t) E A I.
We show that V; E E;CX;(A;). Using the Young inequality (2.7), we obtain
(2.9)
(2.10)
I ltaollvo(t)IIA o IIE o < taO r111u(s)IIA ds
J o 0 S Eo
- oo( : rOX(I,OO)( : )saollu(s)IIAo Eo
< kO oo taoX(I,oo)(t) IIsaollu(s)IIAoIIE o
kO
= - -lluIlE A ).
aO 0 0
(2.12)
A similar check shows that
k l
II t(Xlllv l (t)IIA II E < -II Ull EGl ( A 1 ) .
1 1 a 1
I
(2.13)
From the representation x = VO(t) + VI(t) and (2.12) and (2.13) it follows
that x E (Ao, A 1) OO,Eil' and so we have the imbedding
(Ao, A 1) oO,Eil C (A A 1) 8<>,Eil.
For the proof of the reverse imbedding we consider x E (Ao, A 1);oO,Ef 1 .
Then x = vo(t) + VI(t), where tQ,v; E E;(A;) (i = 0, 1). We show that in this
2. METHODS OF CONSTANTS AND MEANS
253
representation of x the functions vo(t) and vIet) may be assumed to be
smooth. For this we consider an infinitely differentiable nonnegative function
(t) on (0, 00) with compact support such that J (t)t-I dt = 1. We set
v;(t) = £00 \fI( )V;(s) .
The functions viet) assume values in A;, as the functions viet) do, and are
infinitely differentiable, and x = vo(t) + vIet). Next,
ta;lIv;(t)IIA. l oo ( ) a; ( i ) sa;lIv;(s)IIA ds
I 0 S S i S
and by the Young inequality (2.7) we have
II V; II.&ai(A,) ..;; (£ 00 t ( t) t ) II V; II.&a'(A,)o
Now we set
u(t) = tv (t) = - tv;(t).
Then we have
taou( t) = t ao + Ivo( t) = 00 ( ) a., + 1 \fI'( )saOVo(s) '
and by the Young inequality (2.7)
II u II EOO<A 0) ..;; ( OO t ao + 1\fI' ( t) t ) II V o II EOO<Ao)'
Similarly,
II u II Er'{A I) ..;; (£ 00 t al + 1\fI' (t) t ) II vIii Efl(A I)'
Thus, u E EOO(Ao) n E I(A 1). Finally,
l tu(s) ds = l tv (s)ds = vo(t) - vo(+O) = vo(t),
o s 0
f 00 u(s) ds = - f 00 V'I(S) ds = - [ v l ( (0) - vl(t)] = v l ( t),
t S t
(2.14)
(2.15)
and thus x = J U(S)S-I ds. In the calculation of (2.14) and (2.15) we have
used the fact that
"v;( t)" A. 1 00 1/1 ( ) II v;(s) II A ds ..;; s--a;\fI ( ) II v; II J;".G; ( A. ) .
I . 0 S I SSE: '"-1 I
The function (s / t)-<X; (t / s) can be obtained from s o/(s) by means of the
power transformation T_I and the dilation (Jr. From (2.5) and (2.6) it follows
254
IV. INTERPOLATION METHODS
that these operators act in the spaces E;' and have norm k; there. Therefore
II v;{ t) II Ai k; t-<l; II s a;1/J( s) II ' II v; II E/.i(A i ) -+ 0
for i = 0 as t -+ 0 and for i = 1 as t -+ 00.
In what follows, under the hypotheses of Theorem 2.3 we shall omit the
letters % and in the notation of our spaces, and, defining them up to
isomorphism, we shall write (A 0' A I) EGO,Ef..
It turns out that the number of real parameters describing these spaces may
be reduced to one.
LEMMA 2.4. Under the hypotheses of Theorem 2.3, for A =1= 0
(Ao, A I) E ,E I = (A A I) E&o,Eil
up to isomorphism.
PROOF. If x E (Ao, AI)E&o,Ei l , then x = vo(t) + VI(t), where t«V(t) E E;(A;).
Then x = vo(ti\) + vl(ti\), and by (2.6), ta;>V;(ti\) E E;(A). This means that
x E (Ao, AI)E ,E I. Moreover,
IlxII(Ao,AI) AO:O AO:I Iltaohllvo(ti\)IIAoIlEo + IItali\llvl(ti\)IIAIIiEI
EO lEI
..; max { 1, I ' I } (II taoll VO(t) IIAoll Eo + II ta'il VI( t) II A.II E.).
From this it follows that
Ilxll(Ao,Al)E aO.E al ..; max{ 1, I ' I } Ilxll(Ao,Al)E O.E I'
The reverse inequality with A replaced by I/A can be established in a
similar way.
COROLLARY. If we choose A = 1/(a l - ao) and write () = -ao/(a l - ao),
then for 0 < () < 1
(Ao, A I)EoO,Eil = (Ao, A I)E ,E"-' = (Ao, A 1)9,E E.
up to isomorphism.
An important special case is where Eo and EI coincide. It turns out that the
general case may always be reduced to this special case (see [118]). For the
spaces E; = L Pi this will be proved in subsection 7.
We mention yet another property of the spaces (Ao, A 1) 9 E E .
, I
LEMMA 2.5. The inequality
Ilxll (Ao A ) kJ- 9 kf inf lluIIEL71 ) lIull (lI ( A )
, I EO:O EO:I 0"\ 0 I I
o I I
holds, where (j = -ao/(a l - ao) and the infimum is taken over all u E E;o(AeJ
n Efl(A 1) yielding the representation (2.9).
2. METHODS OF CONSTANTS AND MEANS
255
PROOF. Let u E EOo(AO) n E I(AI) and x = J u(t)t- I dt. If 'T > 0, then
the function uT(t) = u('Tt) belongs to E;,Ao) n E I(AI) and x =
J UT(t)t- 1 dt.
By virtue of (2.5) we have
II x II OO,Eil max( II t Qou T II Eo(Ao)' II t QIU T II E I(A I»)
max( ko'T-a°11 t Qoull Eo(Ao)' kl'T-a11i tQluli EI(A I»).
From this inequality, setting
'T = ( kllluIlEil(AI) ) 1/(QI-CXo)
kollull Ego(A I ) ,
we obtain
Ilxll 8<>,Eil < kJ-9kfllullk&:(Ao)llull il(Alt
CoROLLARY. There exists C = C(O, Eo, E I ) such that
II X II (Ao.A 1)9,£0,£1 C IIxll 91Ixll I
for all x E Ao n A I.
Indeed, in the representation (2.9) of the element x we may take u(t) =
(t)x. Then
II x II(Ao.A 1)9,£0,£1 C III t Q'\f;( t)x II kJ o)1I t Q ( t)XIl I(A I)
c I max(11 t 1I k 9, II tQ;JIII J IIxll 81Ixll I.
Lemma 2.5 implies the following sharpening of the interpolation theorem in
our case.
THEOREM 2.4 (interpolation theorem). Let Ao, A I and Bfj) B I be two Banach
couples, and let the lattices Eo and EI satisfy the hypotheses of Theorem 2.3.
Then the triple Ao, A I' (Ao, A 1)9 E E is an interpolation triple of type 0 relative
- , 0. I
to the triple Bep B I , (Bep B I )9,Eo.E I .
PROOF. Let x E (Ao, A I )9,Eo.E I and T E '7T(Ao, AI; Bo, B I ). From (2.9) we
obtain
1 00 dt
Tx = Tu(t)-.
o t
By Lemma 2.5 we have
II Txll(Bo.BI)9'£O'£1 C II Tull k Bo>lI Tull l-'(BI)
C II TII !:Boll T II I--+BIII ull ki (Ao>lI u lI l-'(A I)
C II TII !:Boli T II I--+BIII ull Eo-'(Ao)n El-'(A I).
256
IV. INTERPOLATION METHODS
Taking the infimum over u giving representation (2.9), we obtain
II TII(Ao.A1)o.Eo.E(-+(Bo.B1)o.Eo.EI CII TII Boll TII I-BI.
REMARK 2.3. As can be seen from the proofs of the assertions in this
subsection, the interpolation property of the lattices Eo and E 1 has been used
only with respect to four concrete operators ((2.5)-(2.8)). The fact that these
operators act in Eo and E 1 can often be verified immediately, without
establishing the general interpolation properties of Eo and E 1 . In this case all
assertions of this subsection remain valid.
4. Imbedding theorems and the reiteration theorem. Let four ideal lattices Eo,
E 1 , Fo and F 1 be given on the semiaxis (0,00) with measure dtlt which are
interpolation spaces between L 1 ,. and Loo,.. We assume that there exist two
ideal lattices Go and G 1 containing all functions of compact support and such
that for the convolution operator we have the inequality
£ 00 0/ ( ) qJ ( S) F,' kd I 0/ II G, II qJ II E;
(see Chapter II, 6.9).
THEOREM 2.5 (imbedding theorem). If condition (2.16) is satisfied, then
(A A 1)9,Eo.E 1 is imbedded in (A A 1)9,Fo.F 1 .
PROOF. Let x E (Ao, A 1) 9 E E . By means of a nonnegative smooth function
, 0. 1
1/;(t) of compact support with J 1/;(t) dt = 1 we pass from (2.9) to the
representation
(2.16)
1 00 dt 1 00 ( t ) ds
x = v(t)-, where v(t) = - u(s)-.
o t 0 s s
(2. i 7)
Then
II t vll Fo<Ao) , £ OO( ) 0/( )S II u(s)IIA o Fo
II t 11 GolIs II u(s)11 Aoll Eo.
Similarly,
IItl-9vIlFI(AI) IItl-9 IIGIllsl-91Iu(s)IIAIIIEI.
From these inequalities and (2.17) it follows that
IIxll(Ao.A1)o.Fo.F 1 max{ II t 11 Go' II tl-9 11 G 1 } II ull E:(Ao)nEl-'(A 1 ).
Taking the infimum over all u(t) providing representation (2.9), we obtain
the required imbedding:
(A A 1)9,Eo.£1 C (A A 1)9,Fo.F 1 .
2. METHODS OF CONSTANTS AND MEANS
257
COROLLARY.
(Ao, A 1)8,L I ..,L I .. c (Ao, A 1)8'£1>'£1 C (Ao, A 1)8,L oo ..,L oo ...
(2.18)
Indeed, the first imbedding follows from the theorem and (2.7), and the
second from the fact that inequalities (2.16) hold for F; = L ,. and G i = E;.
DEFINITION 2.1. The space B intermediate between Ao and A I has type
(j, 0 < (j < 1, if
(Ao, A 1)8,L I ..,L I .. C B c (Ao, A 1)8,L oo ..,L oo ...
From the preceding it follows that the space (Ao, A 1)8,£1>'£ I (0 < () < 1) has
type ().
LEMMA 2.6. 1£t B be an intermediate space between Ao and AI. For
(Ao, A 1)8 L L to be imbedded in B it is necessary and sufficient that for any
, I..' I..
xEAonAI
Ilxil B < Cllxll 81Ixll I'
where C does not depend on x.
PROOF. Let B :J (Ao, A 1)9 L L . Then there exists a constant C I such that
, I..' I..
IlxilB < ClllxII(Ao,Al)e,Ll,.,Ll,. for all x E (A(p A 1)8,L I ..,L I .. By the corol-
lary of Lemma 2.4 we have
(2.19)
II X II(AI>'A I)O.LI...LI.. < C21Ixll 911xll l.
Combining the inequalities, we obtain (2.19).
Conversely, let (2.19) be satisfied. If x E (Ao, A 1)8 L L , then represen-
, I..' I..
tation (2.9) holds, where u E L * (Ao) n L:'-.9 (AI). From (2.19) we obtain
that II u(t)11 B < C II u(t)II 811 u(t)II I. From this and Holder's inequality it
follows that
1 00 dt 1 00 I 9 9 dt
Ilxll B < Ilu(t)IIB- < C (t llu(t)IIA) - (t l - 9 11 u (t)IIA) -
- 0 toO It
'" C lIull lJ (Ao) II ull tt:. 8 (AI)'
Consequently, x E Band IIxll B < Cll x ll9L L .
, I..' I..
LEMMA 2.7. For a space B intermediate between Ao and AI to be imbedded in
(Ao, A 1)8 L L , it is necessary and sufficient that there exist a constant C such
. 00..' 00..
that, for every x E Band s, x; = x;(s) E A; may be chosen so that x = Xo +
XI and
Ilxoll Ao < Cs811x11B'
II x III A I < Cs 9 - III x II B.
(2.20)
258
IV. INTERPOLATION METHODS
PROOF. Let B c (Ao, A 1)9 L L ; consequently there exists a constant C I
, 00..' 00..
such that Ilxll L L Clllxli B (x E B). As is shown in the proof of
, 00..' 00..
Theorem 2.3, there exist smooth functions vo(s) and VI(S) such that x = vo(s)
+ VI(S) and
Ils- 8 voIILoo..(Ao) + Ilsl-8 vIIIL oo ..(Al) < C21Ixll L",...L",.:
Setting Xo = vo(s) and XI = vl(s), from the last two inequalities we obtain
that
IlxoII AO C I C 2 S 9 11 x llB and IIxlll AI CIC2S9-lllxIIB.
Conversely, let conditions (2.20) be satisfied. Let X E B and So > o. There
exist Xo E Ao and x I E A I such that
x = Xo + XI' Ilxoll Ao Cstllxll B , Ilxlll AI Csg-IllxII B .
In some interval I containing So the inequalities
Ilxoll Ao 2Cs 9 11 x 11B' Ilxlll AI 2CS 9 - l ll x ll B
will be satisfied. It is now clear that (0, 00) can be partitioned into intervals
II' 1 2 , . .. so that for every n there exist x E Ao and xf E A I such that
X = x + xf and
Ilx IIAo < 2Cs 9 11 x 11B' IlxfliA I < 2CS 9 - l ll x li B
for s E In. To complete the proof it remains to set v;(s) = xt for s E In.
From this it will follow that
x E (Ao, A I )9,L oo ..,L oo .. and IIxll(AQtAI)S.Loo...Loo.. 2Cllxll B .
REMARK. It is easy to see that in Lemma 2.6 we need only require that
B :J Ao n AI' and in Lemma 2.7 that B C Ao + AI.
THEOREM 2.6. Let 0 < 0 0 < ° < 0 1 < 1. If the spaces (Ao, A 1)0. L L are
I' I..' I..
imbedded in the spaces B;, i = 0, 1, intermediate between Ao and AI' then
(Aij) A I) 9,E()tE I c (Bij) B 1)9',FQtF I ,
where Of = (0 - ( 0 )/(° 1 - ( 0 ) and the F; = EJ-9;Ef; are spaces of the
Calderon family connecting the lattices Eo and E I (see 1.10).
PROOF. If x E (Ao, A 1)9,EQtE I ' then there exists a representation (2.9) for x
with u E Eo-fJ(A o ) n El-9(AI). By Lemma 2.6 we have
II u( t) II B; < C; II u( t) II 9;11 u( t) II I.
Hence it follows that
r(8- 8 0 )11 u( t) II Bo < C o ( r-911 u( t) IIAY - 8 0 ( t l - 8 11 u( t) IIAl o
2. METHODS OF CONSTANTS AND MEANS
259
for i = O. This inequality implies that the function on the left belongs to the
space Fo = Ed- 9o Efo and
II t-(9-9 0 )u( t) II Fo(Bo) Coil ull k o>lI ull l-'(A I). (2.21)
The inequality
II t 91 -9u( t)1I F1(B 1 ) < CIII u II k o>ll ull :-'(A I) (2.22)
may be obtained similarly.
From (2.21) and (2.22) it follows that u E Fo-<9-9O>(Bo) n Ffl-9(B I ), and
since a l - 9 ;b 9 ; < max{a, b}, we have
II ull Fo(9- 90)(B o )n FfI-'(B 1 ) < max { C()) C I} II ull EQ'(Ao)n El-'(A I).
Thus, x E (Bo, BI)FO<'-'O>,F(I-'. On the other hand, by Lemma 2.4 the last
space coincides with (Bo, B I )9',Fo.F 1 up to equivalent norms.
THEOREM 2.7. Let 0 < 0 0 < 0 < 0 1 < 1. If some spaces B; intermediate
between Ao and Al are imbedded in the spaces (Ao, A 1)0. L L (i = 0, 1), then
" CIC..' CIC..
( A ()) A I) 9, Eo. E 1 => (B 0' B 1 ) 9', F (ft FI '
where Of = (0 - ( 0 )/(° 1 - ( 0 ) and F; = Ed -9;Ef; are spaces of the Calderon
family connecting the lattices Eo and E I.
PROOF. Let x E (Bo, B I )9',Fo.F 1 . We again use the fact that by Lemma 2.3 we
have x E (Bo, BI)F, 9-'O>,FfI-'. Then there exists a representation x = vo(t) +
v l ( t), where t-</I- 3 0>1I vo(t) II Bo E Fo and t/l,-/lil v l ( t)1I B, E FI" According to
Remark 2.2 the functions v;(t) can be assumed to be step functions. Since
vo(t) E Bo, by Lemma 2.7 for every s > 0 we have the decomposition
vo(t) = voo(t, s) + VOI(t, s), where
IIvoo(t, s)IIA o CoS 9 °llv o (t)IIB o
and
Ilvol(t, s)IIAI CoS 9 o- l lI v o(t)IIB o .
As we can see from the proof of Lemma 2.7, voo(t, s) and VOI(t, s) can be
assumed to be step functions in s for every t and, obviously, not depending
on t on every interval of constancy of vo(t). The following estimates hold:
t-8 II voo(t, s)IIA o Cot-8os90(t-(9-90)llvo(t)IIBo)' (2.23)
t l - 9 11 VOl (t, s) IIA 1 < Cot 1-9os90-1( t-(9-9 0 )11 vo( t) II Bo). (2.24)
The function in parentheses belongs to Fo = Ed -9°Efo, and so there exist
functions CPoo E Eo and CPOI E EI with II CPo; II E, < 1 such that
t-(9- ( 0 ) II v o ( t) II Bo < 211 voll F o (9-90)(B o )l<1>oo( t)II-9 0 \ CPo I (t)1 9 o.
260
IV. INTERPOLATION METHODS
Combining this inequality with (2.23) and (2.24), we obtain
t-IJ II v oo ( t, s) II Ao 2 Coil voll Fo(fJ-fJO)(Bo)t OS901 ((Joci t)II- 9 0 1 <1>01 (t )1 90 ,
t I -911 VOl (t, S) II A I 2 Coli voll Fo(fJ- fJO)(Bo)t I-9 0s 9 0 - 1 1 ((JOO( t)II- 9 0 1 ((JOI (t)1 9 o.
Now we choose s = so(t) from the equality
so(t) = tl((Joo(t)l/l((Jol(t)l.
Then
t-IJ II voo(t, so(t»)IIA o 2CollvoIIFo(fJ-fJo)(Bo)\((Joo(t)l,
t 1- 9 11 VOl (t, so( t») II A I 2 Coli voll Fo(fJ-fJO)(Bo)1 ((JOI (t)\.
Similarly, vl(t) can be decomposed into two parts:
V I ( t) = V I o( t, s I ( t») + V 11 ( t, s I ( t) ),
so that
t-IJ II vIO(t, sl(t»)IIA o 2C l llv I IIFfI-'(B 1 )\((Jlo(t)\,
t I - 911 V II ( t, s I ( t) ) II A I 2 C III V III Ff I -'( B I) 1 ((J II ( t) \ ,
where ((JIO E Eo, ((J11 EEl and II <1>li II E. 1.
I
We set
vo(t) = voo(t, so(t)) + vIO(t, Sl(t»), vl(t) = VOI(t, so(t») + v l1 (t, Sl(t»).
The functions vo(t) and vl(t) are strongly measurable in Ao and A I'
respectively, due to the fact that the vij(t, s) are step functions. By construc-
tion, x = vo(t) + vl(t). Finally, by the preceding inequalities we have
t-IJllvo(t)IIA o 2 max{ Co, CI}(lIvoIIFo(fJ-fJo)(Bo) + Ilv l ll/1I-'(B 1 »)
X max(\((Joo(t)\, \((JIO(t)l),
t 1- 9 11 VI (t) II A I 2 max { C C I} (II voll Fo(fJ-fJO)(B o ) + II vIII FfI-'(B 1 »)
X max(1 ((JOI (t )1, 1 ((J11 (t)l).
The right sides of these inequalities belong to Eo and E I , respectively, and
therefore x E (Ao, A 1)9.E EI and
Il x ll(Ao,A1)8 I E O I E1 8max{Co, Cl}ll x II(Bo,B1) -(8-80) 81- 8 .
FO IF1
Combining Theorems 2.6 and 2.7, we arrive at the following very important
assertion.
2. METHODS OF CONSTANTS AND MEANS
261
THEOREM 2.8 (reiteration theorem). If the spaces Bo and B I , intermediate
between Ao and A I' have types 0 0 and ° 1 , respectively (0 < 0 0 < ° 1 < 1), then
(Ao, A 1)0,£0,£. is isomorphic to (Bo, BI)o',Fo,F. for ° 0 < ° < ° 1 , where Of =
(0 - ( 0 )/(° 1 - ( 0 ) and F; = Ed-o;Ef; are the spaces of the Calderon family
connecting Eo and E I.
REMARK 1. The operators (2.5), (2.6), and (2.8) are pOSItive, and the
convolution operator (2.7) in the proof of Theorem 2.3 was used in the case
where its kernel was positive. Eo and EI are interpolation spaces relative to
L I ,. and Loo,., and therefore the operators (2.5)-(2.8) act in them and also in
the spaces Fo and F I , in view of their positivity, according to what we said on
p. 246 and in 1.10. By Remark 2.3, Theorem 2.3 holds for these spaces, i.e.
the spaces (Bo, BI);,FrpF I and (Bo, B l)i',FrpF I coincide, which fact we used in
the proof of Theorems 2.6-2.8.
REMARK 2. The proof of the reiteration theorem also works in the case
where Bo = Ao (B I = AI) if we set ° 0 = 0 (° 1 = 1).
The reiteration theorem allows us to describe intermediate interpolation
spaces between the spaces Bo and B I "of complex nature" by means of spaces
constructed from the "simpler" spaces Ao and A I. Moreover, it is interesting
that for this description we only need to know the type of the spaces Bo and
BI.
5. Construction of spaces by means ofthefunctionals %(t, x) and (t, x). In
some cases it is more convenient to calculate the norm in intermediate spaces
constructed by the %-method or -method in terms of the functional %(t, x)
defined for x E Ao + A I by
%(t, x) = inf (lIxoll A + tllxlll A ), (2.25)
X=Xo+ XI 0 ·
or (t, x) defined for x E Ao n Al by
(t, x) = max{ IIxiIAo' tllxIl A .}. (2.26)
It is obvious that for given t the functionals %(t, x) and (t, x) are
equivalent to the norms of Ao + A I and Ao n A I' respectively.
For fixed x, %(t, x), as the infimum of functions linear in t, will be an
increasing continuous concave function of t.
Now let x E (Ao, A I)'ff E , where p(t) is a continuous almost everywhere
, P(I)
positive weight. This means by definition that there exists a representation
x = uo(t) + ul(t), where U o E E(Ao) and U I E EP(I)(AI). Then
%(p(t), x) , inf (lIxoli A + p(t)llxIII A )
X==XO+XI 0 I
lIuo(t)IIA o + p(t)llul(t)IIA..
262
IV. INTERPOLATION METHODS
Both terms on the right belong to E, and therefore %(P(t), x) E E and
11%(p(t), X)IIE < IIlIuo(t)IIAoIlE + IIp(t)lIu1(t)IIA1IIE.
Taking the infimum over all representations (2.1) on the right side, we
obtain
II %(p( t), x)1I E Ilxll ( A A ) 9C .
0, 1 E,Ep
Conversely, if %(p(t), x) E E, then for a given to there are elements
Xo E A 0 and Xl E A 1 such that x = Xo + x 1 and
Ilxoll Ao + p(tO)lIx1Il AI < 2%(p(t o ), x).
Since the functions on the right and left are continuous in t, the inequality
IlxoliAo + p(t)lIx1Il AI < 2%(p(t), x)
holds in a neighborhood I of the point to. Hence it is clear that (0, 00) can be
partitioned into intervals In such that for t E In there exists a decomposition
x = Xo + x such that
IlxollAo + p(t)lIxfIIAI < 2%(p(t), x).
We set uo(t) = Xo and u1(t) = x for t E In. Then x = uo(t) + u1(t), and
Iluo(t)IIAo + p(t)lIu1(t)IIA I < 2%(p(t), x).
From this we obtain IluoIIE(A o ) < 211%(P(t), x)IIE and Ilu1IlEp(A 1 ) <
211 %(p(t), x)11 E' and therefore
Ilxll(A AJE.Ep 411%(p(t), x)IIE.
We have proved the assertion.
LEMMA 2.8. The space (Ao, A l) E consists of all elements x for which
, p
%(p(t), x) E E, and the norm of the space is equivalent to 1I%(P(t), x)IIE.
We may establish the following lemma similarly.
LEMMA 2.9. The space (Ao, A l)t,E p consists of all elements x representable in
the form (2.9) for which (p(t), u(t)) E E, and the norm of the space is
equivalent to infll (p(t), u(t))11 E' where the infimum is taken over all represen-
tations (2.9).
We leave the verification of this simple fact to the reader.
We note that the space (Ao, A1)'iE p may contain nonzero elements only
under the condition that e(t) = 1 E E + Ep. This condition is satisfied if
cp(t) = min{ 1, p(t)} E E. Indeed, e(t) = Xl(t) + X2(t), where Xl(t) is the
characteristic function of the set where p(t) < 1. Then P(t)Xl(t) < cp(t) E E,
and X2(t) < cp(t) E E, and therefore e E E + Ep. Conversely, if cp fl E, then
2. METHODS OF CONSTANTS AND MEANS
263
one of the functions P(t)XI(t) or X2(t) does not belong to E, since cp(t) =
P(t)XI(t) + X2(t). We claim that then one of the functions XI(t) or X2(t) does
not belong to E + Ep. Indeed, if X2(t) = JLI(t) + J.L2(t), JLI E E and J.L2 E Ep'
then we may assume that ILI(t) 0 and J.L2(t) 0 and the functions ILl and J.L2
have the same support as X2. Therefore PIL2 E E implies that J.L2 E E. Then
X2 = ILl + IL2 E E. If XI(t) = PI(t) + P 2 (t), PI E E, P2 E Ep' then, under the
same assumptions, PP I E E, and so PXI = PP I + PP 2 E E. Thus, either XI fl
E + Ep or X2 fl E + Ep' and therefore e = XI + X2 fl E + Ep.
Thus, for the space (Ao, A I) £ to be nontrivial it is necessary that
, p
min{ 1, p(t)} E E.
In the construction of the spaces (Ao, A l)to,£1 we have used the fact that
e E EOI + El. Therefore (Ao, A I)t E is defined if e EEl + (E )1 = E I +
, p p
E :-1, which is equivalent, by virtue of the preceding, to the relation
min(l, P -I( t)) EEl.
Under the assumption that the %- and -methods coincide, we obtain the
following theorem.
THEOREM 2.9. If E is an ideal lattice which is an interpolation space between
L I ,. and Loo,., then (Ao, A I)O,£,E consists of all elements x for which %(t, x) E
E-IJ and also all elements x representable in the form (2.9) with (t, u(t)) E
E-IJ. The norm of (Ao, A 1)0,£ is equivalent to
11%(t, x)IIE-8 and infll (t, u(t))II£-9,
where the infimum is taken over all representations (2.9).
We apply this theorem to obtain another version of the reiteration theorem.
THEOREM 2.10. Let Fo, F I , Go and G I be ideal lattices which are interpolation
spaces between L I ,. and Loo,., and let Bo = (Ao, A I)O,Go,G o and BI =
(Ao, A 1)0,G1,G 1 . Then
(Bep B 1)0',Fo,F 1 = (Ao, A I)O,H,H'
up to isomorphism, where
H = (Go, GI)0',F()JF 1
(0 < 0, Of < 1).
PROOF. Let x E (Bo, BI)0',Fo,F 1 . Then
( 00 dt
x = J o u(t)t' r 9 'lIu(t)IIB o E Fo, t l - 8 'lIu(t)IIB, E Fl' (2.27)
To calculate the norms in Bo and B I , we apply Theorem 2.9. We obtain
t-IJ'lIs %(s, u(t))IIG o E Fo and tl-O'lls-IJ%(s, u(t))IIG I E Fl. (2.28)
264
IV. INTERPOLATION METHODS
From (2.9) it follows that
1 00 dt
s %(s, x) < s %(s, u(t))-.
o t
By virtue of (2.28), the function on the right belongs to H = (Go, GI)0',Fr;F 1 '
and therefore so does s %(s, x). This implies by Theorem 2.9 that x E
(Ao, A I)O.H.H.
Conversely, if x E (Ao, A I)O.H,H' then by Theorem 2.9 we have the repre-
sentation
1 00 ds
x = u(s)-,
o s
Then by the definition of H we may construct the decomposition
s (s, u(s)) = s ((Jo(s, t) + S ((Jl(S, t),
' ( I 0' ( )
t lis ((Jo s, t)1I Go E Fo, t - lis ((JI s, t II G 1 E Fl.
s (S, u(s)) E H = (Go, GI)0',Fr;F 1 .
(2.29)
(2.30)
We set
1 00 ((Jo( s, t) d.s
vo(t) = 0 (s, u(s» u(s)-,;-,
1 00 ((JI(S, t) cis
v l ( t) = 0 Hs, u(s» u(s) -,;- .
(2.31 )
Using Theorem 2.9 again, we calculate the norm of vo(t) and vl(t) in Bo
and Bl. We have
( ((Jo(s, t) )
Ilvo(t)IIB o ..;; Co s Hs, u(s» u(s) Go
( <1>1(S, t) )
II v 1 ( t)11 B, ..;; C 1 s Hs, u(s» u(s) G,
Then (2.30) implies that t 'lIvo(t)IIBo E Fo, tl-O'llvl(t)IIB I E Fl. Finally, by
(2.29)-(2.31) we have x = vo( t) + VI (t), i.e. x E (Bep B 1)0',Fo.F 1 .
6. Intennediate spaces between LI and Loo. As in the main part of Chapter
II, we shall consider the spaces LI(O, 00) and Loo(O, 00) with Lebesgue
measure. We construct the space (LI' Loo)o,Lp..,Lp... According to Theorem 2.9
we may calculate an equivalent norm in this space using the functional
%(t, x). For this functional in Chapter II, 3.1 we saw that
:JC( t, x) = t x*( 7") d7" = tx**( t).
Then for the norm we obtain
= Coli S ((Jo( s, t) II Go'
= CIlls ((JI(s, t)IIG I .
{ 1 00 dt } 1/ p
Ilxll(Ll,Loo)e,Lp,.,L p ,. = 0 (t l - 9 X**(t)Y t ·
2. METHODS OF CONSTANTS AND MEANS
265
Up to a constant we obtain the norm of the spaces considered in Chapter
II, 6.8, with 1/ r = 1 - O. Thus, up to isomorphism we have
(L I , L oo )l/r l ,L p ..,4.. = Lr,p (1 < r < 00, 1 P 00). (2.32)
In particular cases we obtain
(L I , L oo )l/pl,4..,4.. = Lp,p = Lp,
(L I , Lrx:)I/r',Loo..,L oo .. = Mtl/r (Marcinkiewicz spaces),
(L I , Loo)l/rl,LI..,L I .. = Atl/r (Lorentz spaces).
From (2.32) it follows that Lr,p has type 1/ r' with respect to LI and Loo.
We consider two numbers ro and r l , 1 < ro < r l < 00, and apply the reitera-
tion Theorem 2.8 to L rOtPO and Lrl,PI. Then for 1 / r < 0 < 1/ r; the space
(LI' L oo )(J,4..,4.. is isomorphic to (L rOtPo ' L rl ,p)9 1 ,4..,4..' where 0' =
(0 - I/r )/(I/r - I/r ). Taking account of (2.32), up to isomorphism we
obtain
1 1 - 0' 0'
( Lr_n,Lr ,p)9 1, , = L r p , where-= +-.
\,PrO I I '-p..''"?.' r r 0 r I
(2.33 )
Using the interpolation Theorem 2.4, by means of the relations obtained so
far we may derive a number of concrete interpolation theorems. For example,
setting ro = Po = I/a o , r l = PI = I/a l and r = p = I/a, where a =
(1 - O')ao + 0' aI' and then ro = Po = 1/ /30' r l = PI = 1/ /31 and r = 1/ /3,
where /3 = (1 - 0')/30 + 0'/31' we obtain the triples (L I / ao ' L I / al , L I / a ) and
(L I //3o' L I //3I' L I //3,I/a) mentioned in Theorem 6.14 of Chapter II, which are
interpolation triples relative to each other.
Setting ro = Po = l/a o , r l = PI = l/a l and r = P = l/a again, where
a = (1 - O')ao + 0' aI' and then ro = 1/ /30' Po = 00, r l = 1/ /31' PI = 00 and
r = 1/ /3, where /3 = (1 - 0')/30 + 0'/31' we obtain that (L I / ao ' L I / al , L I / a ) is
an interpolation triple of type 0' relative to the triple
(Mtl-I IIJo , Mtl-I /PI , L I //3,I/a). If we now assume that /3; a;, then 1/ /3 > 1/ a,
and L I //3,I/a is imbedded in L I //3,I//3 = L I //3. We have arrived at the follow-
ing assertion.
MARCINKIEWICZ'S THEOREM. If 0 < /3; < a; < 1, then the triple
(L I / ao ' L I / al , L I / a ) of spaces is an interpolation triple of type 0 relative to the
triple (Mtl-I/Po, Mtl-I/P!) L I //3)' where
a = (1 - 0) a o + Oa 1 , /3 = (1 - 0) /30 + 0/3 1 .
As we have seen above, a sharper assertion holds in which LI//3 must be
replaced by L I //3,I/a.
266
IV. INTERPOLATION METHODS
7. Intermediate spaces between abstract LP spaces with weight. In this
subsection we consider a more general situation than in the preceding one.
Let Ao and Al be a Banach couple. We introduce the spaces 4:o(A and
Lp I(A 1) of functions defined on the semiaxis (0, 00) with values in Ao and A I'
where wl(t) and w 2 (t) are weights-in other words, positive measurable
functions on (0, (0).
THEOREM 2.11. If at least one of the indices Po or P I is finite, then
( LWo ( A ) T.wl ( A )) - T.w -'wf ( A A ) )
Po 0' -PI I 9.40... 4 1'. - 0' I 9. L p o-..4 1 ..
up to isomorphism, where l/p = (1 - O)/Po + O/PI.
PROOF. Let x(s) belong to (4:o(A o ), 4 I(A 1))9.40...41'" Then the representa-
tion
1 00 dt
x = u(t)-
o t
holds, where u E 4-:.(L p :O(A o )) n L;I:9(Lp I(A I))' u(t) is constant in each of
the intervals (en, e n + I), n = 0, + 1, + 2, . . ., and the integral is taken in
4: O (A o ) + Lp I(A 1). For every t > 0 the abstract function u(t) is a function
from Lp:O(A o ) n Lp I(AI); we denote it by u(s, t) and assume that u(s, t) =
u(s, t') for en < t, t' < e n + I (n = 0, + 1, . . . ). For a given s, u(s, t) is a step
function in t (and consequently, strongly measurable in Ao n A I) and, in
addition, belongs to 4-:'.(A o ) n 1:9(AI) (as we shall see below when apply-
ing Fubini's theorem). Hence in Ao + A I the integral J u(s, t)t- I dt exists
and is an element of (Ao, A 1)9.40...41'" Next, since J: n u(t)t- I dt and
J::n u(s, t)t- I dt obviously coincide as functions of s, the sequence of the
functions
f en dt
xn(s) = u(s, t)-
-n t
e
converges to x(s) in 4:o(A o ) + 4 I(A I). This implies the equality
1 00 dt
x(s) = u(s, t)-.
o t
Now from Lemma 2.5 we have
(n = 1, 2, . . . )
{ (00 dt } (1-9)/po
Ilx( s )1!cAo,Al)e,Lpo,' ,Lpl " C J o (t--811 u(s, t)IIAJo t
{ (00 dt } 9/pl
X J o (t l - 9 I1 u (s, t)IIA,Y't ·
2. METHODS OF CONSTANTS AND MEANS
267
We note that x(s) is strongly measurable in (Ao, A 1)9,40-.,41'.. This is a
consequence of the fact that the sequence of the functions xn(s), strongly
measurable in Ao n A I' converges to x(s) almost everywhere in
(Ao, A 1)9,40-.,41'. in view of the inequality
Ilx(s) - xn(s)II(Ao,AI)9,Lpo,.,LPb.
CII(1 - X(e-n,en»)u(s, . )IIL o-.(Ao)II(1 - X(e-n,en»)u(s, . )1141-.: (AI)
and the fact that one of the indices Po or PI is finite. Next,
(W5- 8 (s )w (s )llx( s)1 i(Ao,AI)e,L po .* ,LPb*)P
{ 1 00 dt } (I - 9)p / Po
..;; C P 0 (r..9 wo (s)lIu(s, t)IIA.)°t
{ 1 00 dt } 9p/PI
X 0 (t l - 9 W 1 (S)llu(s, t)IIA,tt .
Integrating both sides of this inequality with respect to s and applying
Holder's inequality and Fubini's theorem on the right side (the function
II u(s, t)IIA. is obviously jointly measurable in all variables), we obtain
,
Ilxll w l - 9 w 9
Lp ° 1((Ao,AI)9,L PO ,.,L PI ,.)
{ rOO f dt } (I - 9) / Po
..;; C J o (t--8W o (s) II u(s, t)IIA,J° ds t
X {{'" !(t l - 9 W 1 (S)lIu(s, t)IIA,t ds t riP,
= C II uII4: (4 O(Ao»1I uI1 4 \:' (4 I(A I»"
Con versel y, let
1 - 'w ' ( ) )
x E Lpwo I (A A I 9,40-.,4 1 ,. .
Let the elementary functions wo (s) and w l(s) be such that w; (s) w;(s)
2 w; (s). We shall assume that x(s) is strictly simple and has support contained
in the union of a finite number of sets on which both functions wo es) and
w l(s) are constant.
For every s we have the representation
1 00 dt
x(s) = u(s, t)-,
o t
268
IV. INTERPOLATION METHODS
where u(s, .) E L p -: .(Ao) n L;I : (A I) and u(s, .) is constant for en < t <
e n + 1, n = 0, + 1, . . . . We may assume, first, that if x(s) = x(s'), then u(s, t)
= u(s', t), and second, that
max( II t-B II u(s, t) II Aollz,,()o.' II t 1- 9 11 u(s, t)11 A ,II L.J 211x( s )1i(Ao,Al)o,L pQ ,..L P1 .. .
We consider the function
v(s, t) = u(s, w (S) W f(S)lIx(S)II(A.,A,)..",()o..""..t),
where the exponents A, 11., and " will be chosen later (v(s, t) = 0 when
x(s) = 0). For each fixed t the function v(s, t) is a strictly simple function of s
with values in Ao n A I. It is also easy to see that v as an abstract function of t
with range in the space of strongly measurable functions of s assuming values
in Ao n A 1 is a step fu. ction. Indeed, the support of x(s) can be written in
the form U'( M n , where the measurable sets M n are disjoint and the func-
tions wo (s) and w l(s) are constant on every M n . Setting
r n = w (s ) w f(s) IIx(s) II(A.,A I).."'...."',.. and un(t) = u(s, rnt),
for s E M n we have the equality
N
v( s, t) = L Un ( t )XM II ( s).
n=l
Now the required assertion follows from the fact that un(t), n = 1, . . . , N,
. . th . 1 m -I < /' m+ 1 -I - 0 + 1 N t
IS constantIn e In terva s ern t ern , m - , _ , . . .. ex,
Ilvll ..(z,,7I'(Ao») = oo J (t-Bwo(s) II v(s, t)IIAJo ds
= J ( wo(s) w (s) w j'1( s) II x( s) II .,A ,)"""".,,,,,..r
1 00 dt
X ( t-811 u ( s, t) II A yo - ds
o 0 t
2<IAI+II1I)/Jpo+po J W&1 +A/I)Po(s)w o(s)llx(s)II 1 r:.L,.o-..l.p,.. ds.
Similarly,
II II PL 2 (IAI + I ".\)(1- 9)PI +PI J w-A{l-9)PI ( S )
V 41-'..(4 I(Ao») 0
X IIx(s)II-. (lA9)PI +PI ds.
( 1)'.1-.....,1-
-P..,.. -PI'.
2. METHODS OF CONSTANTS AND MEANS
269
I t is natural to try to choose A, 11., and " so that
(I + "AO )Po = (I - O)p, pDpo = fJp,
(p(J + I)po = P
and
-"A(I - O)PI = (I - O)p, -11.(1 - O)PI + PI = fJp,
-,,(1 - O)PI + PI = p.
It turns out that this can be done: "A = -PPI I , 11. = pPOI and v = p(pOI - PII).
Thus, in 4: 0 (A o ) + 4 I(A I) the integral of v exists and is an element x of
the space (Lp:O(A o ), Lp I(A 1))9,400.,41'.. On the other hand, since
J v(s, t)t- I dt = x(s) in Ao + A I' we have x = x. Besides,
Ilxll ( LWO (Ao) LWl ( Al )) 8 L L < C'llvll l po .(L'tl p O(Ao))llvllt p l- l (LW pl l(Al))
po 'Pl ' PO ,.' Pl,. '0 '
(!-8)p/po
< C'l\xll w l - 8 w 8
- Lp 0 l((Ao,Al)8,L PO ,.,L pl ,.)
X Ilxll 1£1 1 -'1£1'
L" 0 1 ((Ao,Al)8,L Po ,.,L Pl ,.)
= C"llxll w l - 8 w 8 .
Lp 0 l((Ao,Al)8,L po ,.,L pl ,.)
Finally, it can be seen easily that the strictly simple functions x of the form
considered are dense in the space
1 -lIw ' ( )
Lpwo 1 (A(p A 1)9,40..,41'. .
This follows from the regularity of 4 wb -B.wf (p < (0) and Lemma 5 in the
In trod uction.
Consequently, we have the continuous imbedding
Lpw -"wr( (Ao, A \)/1,4...,4,..) C (L oO(Ao), L ll (Al))e,Lpo,.,Lpl'.'
COROLLARY. Up to isomorphism,
(Lpo(Ao), Lp,(A \) )/1,40..,4,.. = LA (Ao, A 1)/1,40..,4,..)'
where lip = (I - O)lpo + 0lpl.
This corollary implies, for example, the following interpolation theorem.
THEOREM 2.12. For 1 I P = (I - 0)1 Po + 01 PI the triple
(Lpo(Ao), Lp,(A 1)' LA (A 0> A 1)/1,40..,4,..))
of spaces is an interpolation triple of type 0 relative to the triple
(4o(Bo), 41(B I ), Lp«Bo, B 1 )9,4o..,4,..))' where Ao, A 1 and Bo, B 1 are arbitrary
Banach couples.
270
IV. INTERPOLATION METHODS
Theorem 2.11 permits us to simplify the construction of the spaces
(A ep A 1)9.L,,(}..41'.
THEOREM 2.13. Up to isomorphism,
(A(p A 1)9.4(}..41'. = (Ao, A 1)9.4...L"..'
where lip = (I - O)lpo + 0lpl.
PROOF. We choose 0 0 and 0 1 so that 0 < 0 0 < 0 < 0 1 < 1. We write
(I - OJ)1 Po + Ojl PI = II Pj and introduce the spaces (Aep A 1)8. L. L. hav-
, P I.' P I.
ing types OJ according to the corollary to Theorem 2.5.
By the reiteration Theorem 2.8 we have
(Ao> A \)S,4",.,4,.. = ((A o , A \)S",L"",..L"",., (Ao, A \)S"Lp,..,Lp,J S',F",F,' (2.37)
where 0' = (0 - ( 0 )/(0 1 - ( 0 ), F; = (4OJ.)1-9;(L p ;..)9; = LA...
Now let x E (Ao, A 1 )9.4(}..L"I'.. Then by (2.37) and Theorem 2.9 we have
x = 1 00 u(t) dt , t-B's-Bo%(s, u( t» E L p .( L p .)
o t OJ
t l - 9 'S-8I%(S, u(t)) E L pl ..( L pl ..). (2.38)
Hence
1 00 dt
Xes, x) %(s, u(t))-.
o t
By virtue of (2.38), the function on the right, and so also the function
%(s, x), belongs to (L p -8o . , L p -8 l . )9' L L , which is isomorphic to L p -8" by
OJ I. · p().' p().
Theorem 2.11, where 0" = (I - 0')0 0 + 0'0 1 = 0 and
I _ I - 0' 0' _ ( I 0 ' ) ( I - 0 0 0 0 ) 0 ' ( I - 0 1 0 1 )
-- +-- - +- + +-
P Po PI Po PI Po PI
1-0 0 I
+ - =-.
PI P
Po
Thus, %(s, x) E L p -8, and hence x E (Ao, A 1)9.L"...4..
Conversely, from the last membership relation it follows that
x = oo u(s) , where s-BHs, u(s» E 4,.,
Taking into account that T_ = (L p -8 , L p -8 )9' L L , we obtain the
....". OJ I' · Poo.' PI..
existence of a decomposition
s-8 (s, u( s)) = s-8cpo(s, t) + S-8CPl(S, t),
2. METHODS OF CONSTANTS AND MEANS 271
' I - 9'
where t S °cpo(s, t) E LpOJ*(LpOJ.) and t S tcpl(S, t) E Lpl,.(L pl ,.). The
functions <Po(s, t) and <PI(S, t) may be assumed nonnegative. We set
(00 <pj(S, t) ds
Viet) =}o Hs, u(s» u(s)s'
Then
1 00 dJ.
vo(t) + VI(t) = u(s)- = x.
o S
(2.39)
Next,
t ' S o(J, (S' <Po(S, t) U(S)) - t ' s (s t) E L (L )
d" (s, u( s)) - '#"0' POJ. POJ.'
1-9' I(J, ( <PI(S, t) ( )) _ 1-9' ( ) ( )
t s tY S, Hs, u(s» u s - t S<j?) s, t E Lp,.. Lp,.. '
and so by Theorem 2.9 we have
Ilvo(t)II(Ao,Al)80ILPOI.,LPOI. E L;:'.
and
Ilv! (t)II(Ao,Al)81ILP11.,LP11. E L I :'.
By (2.39) and (2.37) this means that x E (Ao, A 1)9,L"o..,L"I'..
8. The interpolation functor (Ao, A 1)9.0. In applications we most often
encounter the interpolation functor (A(p A 1)9,L"..,L".. (0 < () < I), I < p , 00,
which is denoted briefly by (Ao, A 1)9.0. For the convenience of the reader we
give the main properties of this functor proved in more general situations
earlier.
1 0 . Up to isomorphism,
(Ao, A 1) 4....4,.. = (Ao, A l)t. 4 .... 4 ,.. = (Aij> A 1)8.p,
where 1 / P = (1 - ()) / Po + () / Po (Theorems 2.3 and 2.13).
2 0 ., The space Ao n A I is densely imbedded in (A(p A I)S,q for 1 'P < 00
(see Lemma 2.14, below).
3 0 . For P < q the imbedding (Ao, A 1)9,p C (Ao, A 1)9,q holds (Theorem 2.5
and Young's inequality).
4 0 . If the spaces Bj have types ()j «()o < ()I)' then for ()o < () < ()I
(Bo, B 1)9',p = (Ao, A 1)9,p,
up to isomorphism, where ()' = «() - ()o) / «() I - ()o) (Theorem 2.8).
Moreover, B has type () if
(A(p A 1)9,1 C B c (A(p A 1)9,00.
272
IV. INTERPOLATION METHODS
The space (Ao, A 1)9.0 has type () by virtue of 3°.
5°. The equivalent norms in the space (Ao, A 1)9.0 may be calculated as
/It-8:JC(t, x)lIlp,. or infll (t, u(t))IIz",.' where the infimum is taken over all
representations (2.9) (Theorem 2.9).
6 0. Up to isomorphism,
«A(p A I)9,po' (A(p A I)9,p.)9',p = (A(p A I)9,p, (2.40)
where lip = (I - ()')Ipo + ()' Ipl.
Indeed, by I ° we have
«A(p A I)9,po' (Ao, A I)9,PI)9',p = «A(p A I)9,po' (Ao, A I)9,p.)9',lpo..,lpl'..
By Theorem 2.10 these spaces coincide with (Ao, A 1)9 H H' where H =
(1 .,1 .)9'1 1 . By Theorem 2.11, taking into acco t'that ,. . = 1.- 1 ,
CP I' '-P(}.'-PI'. ,
we obtain that H = 4,., where lip = (I - ()')Ipo + ()' /PI. This proves
equality (2.40).
9. The study of extreme spaces. We consider the space (Ao, AI)tL L . Since
, 00' 00
e(t) = I E Loo, this space is intermediate between Ao and AI. Moreover, Ao is
imbedded in this space, since for x E Ao the equality x = uo(t) + u 1 (t),
where uo(t) = x and uI(t) = 0, shows that x E (A(p Al) Loo,Loo and
IIxll( ,Al) Loo,Loo < suplluo(t)IIA o + sup tllu1(t)IIA I = IIxiIAo.
Now let x be an arbitrary element of (Ao, A1)tL L . Then x = uo(t) +
, 00' 00
u1(t), where
suplluo(t)lI: o + sup tllu1(t)IIA I < (I + f)lIx/l I>o.Loo.Loo.
This implies that lIu1(t)IIA I 0 as t 00, and therefore uo(t) x as t 00 in
the norm of AI' and consequently in the norm of Ao + Al as well. Recalling
the definition of completion of a space relative to another one con tainin g it
(Chapter I, 1.4), we conclude that x belongs to the completion 10 of Ao with
respect to Ao + A l' and
II x II A 0 ..;; sup II u o ( t) II A 0 ..;; (1 + e) II x II .,.A ,>o.Loo. L",:
Conversely, let x E 10. For any f > 0 there exists a sequence X n E Ao such
that IIxnll Ao = IIxliAo and II x - XnIlAo+AI < fl n. The latter means that x - X n
= Y n + Zn' where IIYnllA o + IIznllA I < fin. We set vo(t) = X n + Yn for n - I
< t n, and vI(t) = x - vo(t). Then
II v o ( t) II A II X n II A + II Y n II A II x II A + fin II x II A + f
o 0 0 0 0
for n - I < t n. Next, tllvI(t)/lA I tllzn/l AI tfl n f. From the inequali-
ties obtained so far it follows that
x E (Ao, A I)O,Loo,L oo and IIxll(Ao.A1)o.Loo.L oo IIxliAo + 2f.
2. METHODS OF CONSTANTS AND MEANS
273
We have proved the following assertion for i = O.
LEMMA 2.10. The space (Ao, A 1) Loo,Loo (i = 0, 1) coincides isometrically with
the completion of Ai relative to Ao + A I.
We note that the spaces (Ao, A 1);cL L (i = 0, 1) consists of only zero
, I..' I..
elements, since the function min{ 1, t} belongs to neither L I ,. nor L I.. For
the same reason we do not consider the spaces (Ao, A 1)1. L oo.Loo.
We consider the space (Ao, A 1)6 L L . It is intermediate between Ao and
, I..' I..
A I. Moreover, it is contained in Ao. Indeed, if
r 00 dt
x =)0 u(t)t'
where lIu(t)IIA o ELI,. and tllu(t)IIA I ELI,., then the integral is convergent in
Ao, and so x E Ao and
IIxll Ao ';;; lIuIlL1..(A o ) .;;; IIX Il (Ao,A 1 )6.Ll..,Ll..'
If now x E Ao n A I' we consider the function
u(t) =[In(1 + l/f)]-I X (E,I+E)(t)x.
Obviously x = J u(t)t- I dt. Next,
lIu(t)IIL1..<A o ) = IIxli Ao and IItu(t)IIL1..<A 1 ) = [In(1 + l/f)]-llIxIIAI.
We choose f so small that [In(1 + 1/ f)]-lllxII A Ilxli A . Then
1 0
IIxll(Ao,Al)6.Ll. OO ,Ll. OO .;;; II xII AD -
The two inequalities just obtained show that the norms of the spaces Ao and
(Ao, A I )6L L coincide on Ao n AI. Below we prove Lemma 2.14 from
, I..' I..
which it follows that Ao n A I is dense in (Ao, A 1)6 L L . We arrive at the
, I..' I..
following assertion.
LEMMA 2.11. The space (Ao, AI)lL1..,L 1 .. (i = 0, 1) coincides isometrically with
the closure of Ao n A I in Ai.
10. The duality of the :K- and -methods. Assuming the hypotheses of
Remark 2.2 to be satisfied, we suppose that Eo and EI are two ideal lattices of
functions (sequences) on N + = {I, 2, . . . } with measure /L( {n}) = J.Ln > O.
LEMMA 2.12. If the lattices Eo and EI are regular, and for a Banach couple
Ao, A I the intersection Ao n A I is dense in both Ao and A I' then the set of
Ao n A I-valued finitary functions is dense in E;(A i ), i = 0, 1.
PROOF. The norm in Ei(A;) is absolutely continuous, and so in this space
the finitary sequences with values in Ai are dense. Every such sequence can be
274
IV. INTERPOLATION METHODS
approximated arbitrarily well by an Ao n AI-valued finitary sequence in
Ei{A).
COROLLARY. The space Eo(Ao) n EI{AI) is dense in Eo{Ao) and EI{AI)' and
consequently the spaces [Eo{Ao)]' and [EI{AI)]' form a Banach couple.
By virtue of Theorem 1 we have the isometric isomorphisms [E;{A;)]' =
E;'(A;) for the regular lattices Eo and EI. These isomorphisms T;I; =
{If, f , . . . } (f; E (E;{A;))', f E A;) are such that relation (8) is satisfied:
00
fi(X) = L <xk,f >JLk
k=l
(x = {XI' x 2 , . . . } E E;(A;»).
We consider (Eo{ A 0))' n (E I (A I))'. Let f belong to this intersection, and let
X E Eo{Ao) n EI(A I ). We have
00 00
f(x) = L <Xk,.ff>/lk = L <xk,fkl>JLk.
k=l k=l
This equality implies that the functionals .ff and f k l coincide on Ao n A I'
which is dense in A; by assumption. This implies in turn that the functionals
fk O and fk 1 as elements of (Ao n A I)' = Ao + Ao are the same: fk O = fk l = fk.
Since {II ff" A', "f II A" . . . } E E/, we have {II fIll A ' II f211 A ' . . . } E E/, and
, , , ,
consequently {fl,f2' . . . } E E {Ao) n E{{A ). We have obtained a mapping
Tf = {fk} of (Eo(Ao))' n (EI{AI))' into E {Ao) n E{{A ). This mapping is
isometric:
II Tfll Eo(Ao)n Ei(AJ) = maxll Tfll E,'(A;)
= maxIlIII(E;(A,»' = II fll(Eo(Ao»'n (E;(AI»'.
The suIjectivity of T can be proved in the same way as in the proof of
Theorem 1.
We may prove similarly that (Eo{Ao))' + (EI{AI))' is isometrically isomor-
phic to E (A ) + E{(A ).
Using Theorem 3.1 of Chapter I, we obtain the following asssertion.
LEMMA 2.13. If the lattices Eo and E I are regular and Ao n A I is dense in the
spaces of the Banach couple Ao, A I' then the isometric isomorphisms
[Eo(Ao) + EI(A I) J' = E (Ao) n E{(A )
and
[Eo(Ao) n EI(A I) J' = Eo(Ao) + E{(A;)
hold.
DEFINITION. We say that the lattices Eo and EI have property D) relative to
the couple Ao, A I if Ao n A I is dense in (Ao, A I) EI.
2. METHODS OF CONSTANTS AND MEANS
275
THEOREM 2.14. If Ao n Al is dense in the spaces of the Banach couple Acv Al
and the regular lattices Eo and EI on N+ have property D) relative to Acv A I'
and Eo + EI :3 e,* then the space [(Ao, AI) E)' is isometrically isomorphic to
(A A ;) Q,Ei.
PROOF. We note that by virtue of the inclusion (E )I + (E{)I :J Eo + EI :3
e the space (A , A;)lO,Ei is defined.
The space (Ao, A I) EI is a subspace of Eo(Ao) + EI(A I) consisting of
sequences of the form {xo, xo, . . . }, where Xo E Ao + A I. By Lemma 2.13,
E (A ) n E{(A;) is the dual of Eo(Ao) + EI(A I ). By the Hahn-Banach theo-
rem, every functional f E [(Ao, A I) E]' is the restriction sj of a functional f
I _
belonging to (Eo(Ao) + EI(A I))' having the form f = {fl' f2' . . . } (fk E A
n A;), and
IIfll[(A AI) o-EI]' = i IIfIlEo(Ao)nEi(A;).
Sf=f
By virtue of the definition of this space, every functional cp belonging to
(A , A ;) , E' can be obtained from a functional f = {fl' f 2 , . . . } E E (A ) n
I _
E{(Ai) by means of the operator f = Lr' fkJ-Lk = cp, and
IlcplI (A ' A , ) = inf II fll E'(Ao)n E'(AJ}.
0' 1 E' E' - 0
0' 1 fj,f=cp
If we now introduce the correspondence sf = f f = cp, to prove that it
is well defined we have to verify that the kernels of the mappings S and
coincide. Let Sf= 0 and x = {xo, xcv... }, where Xo E Ao n AI. Then by
(8) we have
j(x) = fk(XO)J-Lk = 0 (xo E Ao n AI). (2.40')
Since j = L fk J-Lk is convergent in A + A; = (A I n A I)' by Lemma 2.2,
this implies that j = o.
Conversely, if j = 0, then (2.40') holds, which means that sf is equal to
zero on the set Ao n A I dense in (Ao, A I) EI. From this it follows that
Sf = O. The isometricity of the correspondence f cp follows from the
formulas for the calculation of norms.
COROLLARY. If the hypotheses of the theorem are satisfied for two lattices Eo
and EI defined on an arbitrary measure space, and such that a common
averaging operator acts in them, then the space [(Ao, A I) E)' is isomorphic to
(A A ;) , E'.
v' I
· Editor's note. Concerning e see subsection 1 of this section.
276
IV. INTERPOLATION METHODS
The corollary follows immediately from the theorem and Remark 2.1.
We may establish the next results by similar arguments.
THEOREM 2.15. If Ao n Al is dense in the spaces of the Banach couple A(p Al
and the regular lattices Eo and E I on N + are such that E and E; have property
D) and E + E{ :3 e, then the space [(Ao, AI)t EI]' is isometrically isomorphic
to (A A ;) O,EI.
COROLLARY. If the hypotheses of the theorem are satisfied for lattices Eo and
EI defined on an arbitrary measure space and such that an averaging operator
acts in them, then [(Ao, A I) E.J' is isomorphic to (A , A;) O,EI.
The verification of condition D) becomes simpler in cases where the spaces
constructed by the :JC- and -methods are isomorphic. Indeed, the following
lemma holds.
LEMMA 2.14. If Eo and EI are regular lattices consisting of functions summa-
ble on every set of finite measure, then Ao n A I is dense in (A(p A l)t EI.
PROOF. We consider an increasing sequence of measurable sets 9Rn such
that JL(WC n ) < 00 and 9Rn jWC. Let x E (Ao, A l)l EI. Then x = J IDl u(t) dJL,
where u E Eo(Ao) n EI(A I ). We write X n = JIDl u(t) dJL. It can be shown that
II
u(t) is strongly measurable as a function with values in Ao n Al (we leave this
to the reader). Besides,
r lIu(t)IIA nA dJL = ( m ax { lIu(t)IIA, lIu(t)IIA } dJL < 00,
J 0 I)Wl 0 I
n II
by the hypothesis of the theorem. Therefore u(t) as a function with values in
Ao n A I is integrable on 9JC n , and consequently X n E Ao n A I. Next,
x - x n = J u(t)[ 1 - XID/. (t)] dJL,
and therefore
Ilx - Xnll( ,Al) o,El maxll u(1 - XIDl II ) II E;(A,) 0
as n 00, by virtue of the regularity of the Ej.
We recall that by Theorem 2.3 the spaces (Ao, A 1) OO,Eil and (Ao, A l)t ,Eil
coincide if Eo and EI are ideal lattices that are interpolation spaces between
Ll,* and Loo,.. Besides, if Eo and EI are regular, so are E;o and Er l . Then the
dual spaces of Eoo and E ' are (E )-ao and (E{)-QI.
Combining the above facts with the preceding results, we arrive at the
following assertion.
2. METHODS OF CONSTANTS AND MEANS
277
THEOREM 2.16. If Eo and E I are regular lattices that are interpolation spaces
between L I ,. and Loo,. and the intersection Ao n A I of the spaces of the Banach
couple Ao, Al is dense in each of these spaces, then the dual space of
(Ao, A I)EQO,E1QI is isomorphic to (A , A ;)(EO}-ao,(EI)-aU or, in other words,
[(A o , A 1)9,E Ot E I J' = (A , A;)9,EO,Ei (0 < () < 1),
up to isomorphism.
REMARK. As has been shown recently in [116], property D) is always
satisfied if the lattices Eo and EI are regular and do not contain the function
identically equal to one.
11. Almost interpolation properties of scales of spaces. The theory of spaces
constructed by the %- and -methods is connected with the theory of scales
of spaces which are not normal scales.
In this subsection we always assume that the space A I is imbedded in Ao.
LEMMA 2.15. If 0 < a < /3 < 1, then every space BI of type /3 intermediate
between Ao and A I is imbedded in every intermediate space B 2 of type a.
PROOF. By assumption, (Ao, A I)a L L C B 2 . By the remark to the reitera-
, I..' I..
tion of Theorem 2.8, the space (Ao, A I)a L L is isomorphic to
, I..' I..
(Ao, B l)a/p,L1..,L 1 .. and consists of the same elements. Therefore
BI c (A(p BI)a/p,L1..,L 1 .. C B 2 .
LEMMA 2.16. If 0 < a < /3 < y < 1 and the spaces Ba' Bp and By inter-
mediate between Ao and A I have types a, /3, and y, respectively, then
II xii BfJ Ca,p,y Ilxll - P)/(y- a) Ilx II - a)/(y-a).
(2.41)
PROOF. By assumption, (Ao, A I)P,L1..,L 1 .. C Bp. By the reiteration Theorem
2.8 we have (Ao, A l)p,L1..,L 1 .. = (Ba' B y )9,L 1 ..,L 1 ..' where () = (/3 - a)/(y - a).
Then (Ba' B y )9,L 1 ..,L 1 .. cAp, and by Lemma 2.6 we have (2.41).
In accordance with Definition 1.2 of Chapter III, 1, Lemmas 2.15 and 2.16
lead to the following assertion.
THEOREM 2.17. Let {Aa} be a family of Banach spaces having the property
that the spaces Aa' 0 < a < 1, are intermediate of type a between Ao and AI.
Then the family {Aa) consisting of the closures Aa of A I in the norms of Aa
fortm a scale of spaces.
278
IV. INTERPOLATION METHODS
The scales of spaces occurring in the theorem will be called scales of means.
The family of spaces (Ao, A I)a L L provides an example of a scale of
, I..' I..
means. According to Lemma 2.11 we have
( A A ) = ( 4 A ) = A
0' I I,LI..,L I .. - 0' I,LI..,L I .. I
and the space (Ao, AI)OL L = (Ao, A I )6L L = Ao coincides with the
, I..' I.. ' I..' I..
closure of A I in Ao. By Lemma 2.14 the space A I is dense in the spaces
(Ao, A I)a L L for 0 < a: < 1. Due to the property expressed in the next
, I..' I..
lemma it is natural to call the scale obtained in this way the maximal scale of
means .
LEMMA 2.17. The maximal scale of means is imbedded in every scale of spaces
A
{Ba}' 0 a: 1, where BI = Al and Bo = Ao.
PROOF. By the properties of the scale for x E Al we have IIxil Ba
CllxIl A I-allxll , from which by virtue of Lemma 2.6 we obtain the imbedding
o 1
(Ao, A I)a L LeBa.
, I..' I..
The following lemma gives an important property of scales of means.
LEMMA 2.18. Any scale of means is imbedded in the minimal scale constructed
from the spaces Ao and AI.
A A
PROOF. For any scale of means {Aa} we have the imbeddings Aa C Aa C
(Ao, A I)a L L . For x E A I we consider the representation
, 00' 00..
1 00 dt
x = 0 u(t)t'
where u E L:'.(A o ) n L : (A I). Next, for f E Ao we have
I f( u ( t ) ) I II f II A (/ a II t ( t) II A 0 < t a II fll A <> II u II L;:.. (Ao) n L : (A I)
and
If(u(t))1 IlfIIA,/a-Illtl- (t)IIAI ta-IllfIIA'llluIIL;:..(Ao>nL:.o : (AI).
Then
1 00 dt
If(x)1 If(u(t))I-
o t
( 1 Nt a II 1 II A' dt + f OOt a - III 1 II A' dt ) II u II L - (A ) n L I - a ( A )
o 0 t Nit 00.. 0 00.. I
.;;; ( N a II 111 Ai> + 1 a: N a -III 111 A'I) II u II L;',.(A o ) n L:"-:-: (A ,)'
2. METHODS OF CONSTANTS AND MEANS
279
By minimizing the right side with respect to N, we obtain
If( x)1 .;;; a( 1 a) II fll a II fll':t; II u II L;:..(Ao)n L:';:; (A ,).
Hence
I f( X) 1 .;;; a( 1 a) II fll a II fll':t; II x II (Ao-A ,)(a.L",...L"".r
Then
IIxllAaun = sup If(x)1
a Ilfll allfll '
o 1
1 C
.;;; a(1 - a) II x II (Ao-A ,)( a.L",...L",..) .;;; a( 1 _ a) II x II A. ;
A A .
by virtue of the density of A I in Aa this implies the imbedding Aa C A:m.
In Lemmas 2.17 and 2.18 we have used the first part of the following
defini tion.
DEFINITION 2.2. If {Aa} and {Ba}, 0 a 1, are two families of Banach
spaces such that AJ3 C Aa and BJ3 C Ba for a < {3, then we say that the first
family is imbedded in the second if Aa C Ba for every a, and almost imbedded
in the second if AJ3 C Ba for every a and {3 > a.
We recall that in 2.2 of Chapter III for a family {Aa}' 0 < a < 1, of
spaces for w ich AJ3 is densely imbedded in Aa for a < {3 we introduced the
dual family Aa which consists of the closures of Aij in the spaces A . Now let
{Aa} be an arbitrary scale. Since A I is dense in Aa' the space (Ao, A I)a L L
, I..' I..
is densely imbedded in Aa' according to Lemma 2.17. Then by Theorem 2.16
we have A C (Ao, A )a L L . If, in addition,
, 00..' 00..
IlflIA Cllfll allfll '1 (f E Ao), (2.42)
then (Ao, A ;)a L L C A , by Lemma 2.6. Consequently A has type a
, I..' I..
relative to the couple Ao, A; and type 1 - a relative to the couple A , Ao.
Theorem 2.17 leads to the following assertion.
THEOREM 2.18. Let {Aa}' 0 a 1, be an arbitrary scale of Banach spaces.
For the dual family {AI-a}' 0 (J 1, to form a scale it is necessary and
sufficient that condition (2.42) be satisfied.
REMARK. Under condition (2.42) the scale {A I -a} will be a scale of means
relative to A and Ao.
CoROLLARY. The dual family of a minimal scale forms a scale.
280
IV. INTERPOLATION METHODS
Indeed, for a minimal scale inequality (2.42) is established in Lemma 2.3 of
Chapter III.
We note that condition (2.42) implies that the scale {Aa} is imbedded in
the minimal scale. Indeed, for x E A I we have
sup
fEAo
If(x)1
IlflIA'
a
If(x)1 1 If(x)1 1
IIfIlA c /: o IIfll a IIfll ; = c IIxIlA:u"
II xii A = sup
a
fEA
which implies the imbedding Aa C A:m. Conversely, if the scale {Aa} IS
imbedded in {A;nn}, then (A:m)' c A , and therefore
IlflIA Cllfll(A: iD )' Clllfll allfll '1
according to Lemma 2.3 of Chapter III.
Thus, the following theorem holds.
THEOREM 2.19. Condition (2.42) is necessary and sufficient for the scale {Aa}
to be imbedded in the minimal scale constructed from Ao and AI.
LEMMA 2.19. The minimal scale {A;un} is almost imbedded in every scale
{ Aa} imbedded in it.
PROOF. By the above, the spaces A and (A:m)' have type 1 - a: relative to
A; and Ao. Then by Lemma 2.15 for a: < {3 we have (Aa)' C (A;m)'. For
x E A I we have
1
- sup
C fE(Ern)'
where C is the imbedding constant of A in (A;m)'. The inequality thus
obtained implies the imbedding A;nn C Aa.
IIxliA = sup If(x)l .;;; sup If(x)1
a fEA IlflIA fE(Epm)' IlflIA
If(x)1 1
IIfll(Ai")' = C IIxIlAi'"
THEOREM 2.20. The minimal scale {A:m} is almost imbedded in every scale
{ Aa}.
PROOF. It is sufficient to consider the scale {Aa} consisting of the closures
of £1 in A:m n Aa. It is imbedded in A;un. Therefore Apm C Aa C Aa for
a: < {3 according to Lemma 2.19.
DEFINITION 2.3. A family {Aa}' 0 a: 1, of Banach spaces has the almost
interpolation property relative to a family {Ba} of Banach spaces if the
condition that a linear operator acts boundedly from Ao into Bo and from A I
into Bl implies that it acts from every space AI3 into the space Ba for a: < {3.
2. METHODS OF CONSTANTS AND MEANS
281
THEOREM 2.21. Let {Ba} be an arbitrary scale and let {Aa} be a scale
satisfying condition (2.42). Then {Aa} has the almost interpolation property
relative to {B,e}.
PROOF. Let the operator Tact boundedly from Ao into Bo and from A I into
BI. By Theorem 2.3 of Chapter III the operator T acts from Apm into Bpn (in
the proof of Theorem 2.3 the relativity of spaces was not used). By Theorem
2.19 it acts from A,e into Brn, and by Theorem 2.20 it acts from A,e into Ba
for a < {3.
COROLLARY 1. A minimal scale has the almost interpolation property relative
to any scale.
COROLLARY 2. Any scale of means has the almost interpolation property
relative to any scale.
Indeed, by Theorem 2.18 any scale of means is imbedded in a minimal one,
and so it satisfies (2.42) by virtue of Theorem 2.19.
COROLLARY 3. Any regular scale has the almost interpolation property relative
to any scale.
12. Connection with the complex method. Let Ao and A I be a Banach couple
of complex spaces. We consider the spaces [Ao, Ada constructed by the
complex method of interpolation.
LEMMA 2.20. The space [Ao, A da has type a relative to Ao and A I.
PROOF. The fact that the space (A(p A I)a L L is imbedded in [A(p Ada
, I..' I..
follows from (1.39) and Lemma 2.6. We show that
.
[ Ao, A I] C (Ao, A l)a,Loo..,L oo ...
Let x E [Ao, A da and estimate the functional K(t, x) (see subsection 5). Let
the function f(z) from (A(p A I) be such that f(a) = x. The function fl(z) =
ta-'i(z) also belongs to (Ao' AI)' andfl(a) = x. We use the representation
(1.9) for the functionfl(z):
x = 11(0:) = j'X> 11 (i-r)/Lo(O:, -r) d-r + foo 11(1 + i'T)/ls(O:, -r) d-r = Xo + XI'
-00 -00
where Xo E Ao and Xl E AI. Then
K(t, x) IIxoll Ao + tllxlii AI
(1 - a) sup Ilfl(i'T)IIA o + at supllfl(I + i'T)IIA I
T
= (1 - a)t a sup IIf(i'T)IIA o + at a supllf(I + i'T)IIA I tallfll3<A I).
T
282
IV. INTERPOLATION METHODS
On the right side we take the infimum over all f E ffiAo, A I) for which
f(o:) = x. This gives us
t-<r.K(t, x) IIxll[A Ada.
By Theorem 2.9 we have x E (Ao, A I)a L L . The lemma is proved.
, I..' I..
Now let Al be imbedded in Ao. Since Al is dense in [A(p AI]a' the spaces
[Ao, .A da form a scale of means. Therefore the following theorem holds.
THEOREM 2.22. The family [Ao, A da of spaces has the almost interpolation
property relative to any scale of spaces.
Lemma 2.20 and reiteration Theorem 2.8 immediately imply one further
corollary, important in applications.
THEOREM 2.23. The space ([Ao, Ada' [Ao, A d,8))',£,£, where 0 < 0: < {3 < 1
and 0 < y < 1, is isomorphic to (Ao, A 1))",£,£, where y' = (1 - y)o: + y{3.
For completeness of the exposition we note that for A I C Ao the scale
[Ao, A da is not only imbedded in the minimal scale but also majorizes the
minimal scale (see Definition 2.3 in Chapter III). Indeed, let f E ffiAo, A I)
and let cp be a functional from A . The function <f( z), cp) is analytic inside
the strip 0 < Re z < 1, continuous in the closed strip, and bounded. By the
three lines theorem we have
I<f(o:), cp)1 [supl<f(i7.), cp)IJ I - a [supl<f(1 + iT), cp)IJa
" Ilcpll allcpIl1,[ supllf(i'T)IIAJI-a[ supllf(l + i'T)IIA.]a
II cp II a II cp II III fll3<A AI).
Settingf(o:) = x E [Ao, Ada' we obtain
Il x ll miD = su p I<x, cp)1 Ilfll
Aa , II II I - a II II a 3< A A I).
q:> EAo cp Ao cp A'I
This implies that x E A:m and IlxlIA:m Ilxll[A da.
CHAPTER V
INTERPOLATION IN SPACES OF SMOOTH FUNcnONS
...,
BY S. G. KREIN
1. Interpolation spaces constructed from an unbounded operator
and a smoo thing approximation process
1. Banach couples associated with unbounded operators. In a Banach space E
we consider a closed unbounded linear operator A with domain 6D(A) dense
in E. If in 6D(A) we introduce the graph norm Ilxll EA = IIxllE + IIAxIIE' then
6D(A) becomes a Banach space, which will be denoted by EA. Since IlxilE
Ilxll E A ' the space E A is normally imbedded in E.
We denote by N A the kernel of A, i.e. the collection of all elements x E E
for which Ax = B. The set N A is a subspace" of E. Indeed, if X n x in E and
AX n = 0, then Ax = 0 since A is closed, and consequently x E N A . On N A the
norms of E and E A coincide.
In applications A is often a differential operator; therefore the membership
relation x E 6D (A) expresses a certain smoothness. The problem of construct-
ing intermediate spaces between E and E A is connected with finding spaces of
functions having intermediate smoothness. This construction is often facili-
tated by the presence of a smoothing approximation process in E. By such a
process we understand a commutative family of bounded linear operators Qt
(0 < t < (0) having the following properties:
1) The operators Qt are strongly continuous in t, bounded uniformly in t on
(0, (0), i.e.
II QtXllE Mllxll E ,
(1.1)
and strongly converges to I as t o.
2) For every t the operator Qt maps E into 6D(A) (smooths) and commutes
with A on 6D (A).
283
284 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
We consider the operators St = AQt. From 1) and 2) it follows that
IIStXllE = IIQtAxllE MllAxll E MllxllEA (1.2)
for x E 6D (A).
Besides, St is bounded for every t > o. Indeed, it is closed as the product of
a closed operator and a bounded operator, and is defined on the whole space.
We note that if A is not bounded, then the function IIStIIE-+E cannot be
bounded near zero. Indeed, the inequality IIStIIE-+E K for 0 < t a would
imply IIAQtxllE KllxlIE. Next, AQtx = QtAx Ax on 6D(A), and there-
fore the operators AQt would be uniformly bounded and would converge to A
on the dense set 6D (A). By the Banach-Steinhaus theorem A would then be
bounded.
We denote by p(t) a positive increasing continuous function having the
property that
IIp(t)StIIE-+E 1
(0 < t < (0).
( 1.3)
From what has been said above, it follows that p(t) 0 as t O. Moreover,
p(t)StX 0 for every x E E. Indeed, for x E 6D(A) this follows from the
preceding: p(t)StX = p(t)QtAx 0, and since 6D (A) is dense in E, and the
operators p(t)St are uniformly bounded, this is also true for any x E E.
By (1.2), for the elements of 6D (A) we have
supll StXl1 E < 00. (1.4)
We pose the problem of describing all elements x for which (1.4) holds.
LEMMA 1.1. The set of elements x E E having property (1.4) coincides with
the completion E OA of E A relative to E.
PROOF. If x E E OA , then there exists a sequence x n E E A such that X n x
in E and IlxnllE A = IlxiIEoA. Then (1.2) leads to the inequality IIStXnllE <;
Mllxnli E = Mllxll E . Using the boundedness of the operator S t in E, we
OA
obtain that IIStxllE MllxllEoA. This implies that supllStxllE MllxllEoA.
Conversely, if (1.4) is satisfied, then QtX x in E and
II QtxllEA = II QtXllE + IIAQtXllE Mllxll E + supllStxllE.
Hence x E E OA and Ilxil EoA MllxliE + supllStXIIE.
The lemma is proved.
Since we always have IlxllE IlxlloA (x E E OA )' from the proof of the
lemma we obtain the inequality
C1(llxll E + supllStxllE) Ilxil EoA C 2 (II x il E + supIIStxIIE). (1.5)
LEMMA 1.2. The set E A is closed in E OA .
1. A SMOOTlDNG APPROXIMATION PROCESS
285
PROOF. Let X n E E A and X n x in E OA . By (1.5) we have X n x in E and
StXn StX in E unifonnly in t. For any € > 0 and sufficiently large n
and m we have II StXn - StXm II E < € for all t > O. From this we obtain
II Qt(Ax n - Axm)11 E < €. Passing to the limit as t 0, from property 1) of the
operators Qt we obtain that II AX n - AX m II E €. Since A is closed, from the
last inequality and the fact that X n x it follows that x E 6D(A) = EA. The
lemma is proved.
COROLLARY 1. The norms IlxllE + IIAxllE and IlxIIE + supllStXIIE are
equivalent on 6]) (A ).
COROLLARY 2. The space E A lS almost related with E (see Chapter III,
Theorem 3.1).
2. Construction of intermediate spaces. A smoothing approximation process
assigns, to every element x of a Banach space E, a function StX on (0, (0)
with values in E. Lemma 1.1 says that the space E OA consists of all elements x
for which IIStXllE E LoclO, (0), or, in other words, StX E Loo(E; (0, (0)). The
consideration of other function spaces on the semiaxis leads to the possibility
of constructing spaces intermediate between E A and E. Let F(O, (0) be an
ideal lattice on the semiaxis (0, (0) with respect to the measure dt / t. By
definition, we set x E EA,F if II StXl1 E E F(O, (0). The set EA,F is linear. In it
we introduce the norm
IlxIIE = IlxIIE + IIIIStxIIEIIF.
'A,F
( 1.6)
Then E OA is isomorphic to E A L in view of (1.5).
, 00
LEMMA 1.3. The space EA,F is complete.
PROOF. We consider a sequence of elements x n E EA,F for which
r'llxnIIE < 00. We show that x n converges to some element of E A F.
'A,F ,
From (1.6) and the completeness of E it follows that x n converges in E to
an element x. Then IIStXIIE = liSt xnll E IIStxnIIE. Since F is complete
and IIIIStXnIIEIIF is convergent, the function IIStxnIlE belongs to F and
since F is an ideal lattice, we have II StXl1 E E F. Consequently x E EA,F.
Moreover,
IIxIl EA . F =11 xnIIE +IIIIS/ XnIIEIIF
Ilxnll E + 1IIIStXnllEliF = IlxnIlEA,F. (1.7)
286 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
It remains to prove that f x n x in E A F as N 00. This follows from
(1.7) applied to the element x - f x n = 1 x n . We have
N 00 00
X - xn - X n Ilxnll E o asN oo.
'A,F
I E A . F N+ I E A . F N+ I
The lemma is proved.
It is obvious that IlxIIA,F IlxlIE, and so EA,F is imbedded in E.
We assume that the ideal lattice F has the following property:
a) The function min{I, p-l(t)} belongs to F.
By the obvious inequality min{M, p-l(t)} max{I, M}min{I, p-l(t)} we
have min {M, p -1( t)} E F for every M > O. Next, for every a > 0 we have
X[O,a] p(a)min{p-l(a), p-l(t)} E F
and
p-l(t)X(a,oo) min(p-l(a), p-l(t)) E F.
LEMMA 1.4. Under condition a) the space E A is imbedded in EA,F' and so EA,F
is intermediate between E A and E.
PROOF. If x E E A , then from (1.2) and (1.3) it follows that IIStxllE <
min{M, p-l(t)}llxIIEA. Hence
1IIIStXllEIIF Ilmin{M, p-l(t)}IIFll x II EA .
Condition a) implies that the first factor on the right is finite; that is, E A is
imbedded in E A F. The lemma is proved.
,
The operators Qs map E into E A , and therefore they act in E A F. Moreover,
,
they are uniformly bounded in EA,F. Indeed,
II QsXIl EA . F = IIQsxll E + 1IIIStQsxllEIIF = II Qsxll E + II II QsStxllEIiF
M ( II x II E + II II St x II E II F) M II x II E · ( 1.8)
A.F
LEMMA 1.5. If an ideal lattice F has the Fatou property, then EA,F is complete
relative to E and coincides with the completion of E A in the norm (1.6) relative
to E.
PROOF. If an element x belongs to the completion of EA,F relative to E,
then there exists a sequence X n x in E such that Ilxnll E C. Since the
A,F
operators St are bounded, StXn StX in E, and hence IIStXnllE IIStxllE for
all t. Then from the Fatou property it follows that
IIllStxllEllF limllllStxnllEllF limllxnllE C,
'A,F
and so x E EA,F. The completeness of EA,F relative to E is proved.
1. A SMOOTHING APPROXIMATION PROCESS
287
Now let x be any element in EA,F. Then Qsx E E A , Qsx x in E, and by
(1.1) we have II Qsxli E MllxllE . This means that E A F is contained in the
A.F 'A.F ,
completion of E A in the norm (1.6) relative to E, and since EA,F is complete in
E, it coincides with this completion.
The lemma is proved.
LEMMA 1.6. If the ideal lattice F has property a) and has absolutely continuous
norm, the set E A is dense in EA,F.
PROOF. Let x be an arbitrary element in EA,F. Using (1.3) and (1.7), we
have
IIx - QsxII EA . F Ilx[o,8](t)II S t(x - Qsx)IIEIIF
+ Ilx[8,oo)II S t(x - Qsx)IIEIlF + Ilx - QsxliE
(M + 1 )llx[O,8](t)1I Stxll Ell F
+ (1Ix[8,oo)(t)p-I(t)IIF + 1)llx - QsxlI E .
Due to the absolute continuity of the norm in F, the first term on the right
may be made arbitrarily small if we choose sufficiently small, and then so
can the second term, if we choose s sufficiently small. Thus, Qsx E E A and
Qsx x in EA,F.
The lemma is proved.
3. Interpolation intermediate spaces. There arises the problem, important in
applications, of the relations between the spaces EA,F constructed by means
of a smoothing approximation process and the spaces constructed from the
Banach couple E, E A by means of certain interpolation constructions.
THEOREM 1.1. The space (E, EA)'ff _I F constructed from the Banach couple
p (I)'
E, E A by the ex-method by means of the ideal lattices Fp-I(t>, F is imbedded in
E A F.
,
PROOF. If x E (E, E A )'{: _I F' then there exists a representation x = uo(t) +
p ,
uI(t), where U o E Fp-I(E) and U 1 E F(E A ). Then S,x = StUo(t) + StUI(t). By
(1.2) and (1.3) we have
IIStUo(t)IIE p-I(t)lluo(t)IIE E F
and
II StuI(t)11 E M I! uI(t)1I E A E F.
Hence II Stxll E II Stuo(t)11 E + II StuI(t)11 E E F. From the inequalities ob-
tained so far it also follows that
IlxII EA . F Mllxll(E,EA) p_,.F.
The theorem is proved.
288 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
From the point of view of approximation theory, Theorem 1.1 belongs to
assertions of the type of the classical Jackson theorem: a space is described
on which the process St behaves in a certain way (II St X II E E F) and the
construction of the space is not connected with the process itself. The
converse assertions in which from a certain behavior of the process St we
conclude that the element x belongs to some space are analogues of the
classical Bernstein theorem. To obtain such theorems we need a more delicate
use of the connection between the operator A and the process Qt' or, what is
the same, between the processes St and Qt.
In the examples given below the operators St and Qt satisfy a relation of
the following form:
I = fo1p(A1t)L(A, t)SNda(A) + I QNN(A, t) df3(A), (1.9)
where L(A, t) and N(A, t) are operator-valued functions unifonnly bounded in
A and t and commuting with S.,. and Q.,. for all 'T' > 0, and a(A) and f3(A) are
bounded increasing scalar-valued functions on (0, 1] and [8, 1], respectively.
THEOREM 1.2. Assume that representation (1.9) is valid, and that condition a)
and the inequality
fol Mp(A) II (JI/AII F-+Fda(A) + 11I(JI/AIIF-+Fdf3(A) < 00,
(1.10)
hold, where 0 1 />" is a dilation operator and
Mp(A) = sup p(At)p -I( t).
t>O
Then E A F is isomorphic to (E, EA)F _I F.
, p ,
PROOF. By virtue of Theorem 1.1 it is sufficient to obtain the imbedding
E A F C (E, E A ); _I F. According to Lemma 2.8 of Chapter IV, for this it is
, p ,
sufficient to show that for any x E EA,F we have p-l(t)%(P(t), x) E F, where
the function %(t, x) is constructed for the spaces E and EA. Using (1.9), we
have
X(p(t), x) ..;; II foIp(At)L(A, t)SAt X da(A)t
+p(t)IIf. 1 QAtN(A, t)x d f3 (A)tA
..;; C(folp(At)IISNXIlE da(A) + p(t) IIISNXIIE df3(A) + p(t)IIXIIE).
1. A SMOOTIDNG APPROXIMATION PROCESS
289
From this we obtain
p-l(t)X(p(t), x) ..;; CC( Mp(A)I!S]uxII E da(A) + IIIS]uXIlE d,B(A) + IIXIIE).
By definition, %(t, x) IlxilE for any x E E, and therefore
p-l(t)X(t, x) ..;; c 1"0 1 Mp(A) II S]uxll E da(A)
+ C IIIS]uXIlE d,B(A) + min{ C, p-l(t)}lIxIIE.
The last term on the right belongs to F by condition a). For the second
term we have
11 IIIS]uXlld,B(A)t ..;; IIIIIS]uXIIEIIF d,B(A)
..;; 11I(JIP\IIF--+Fd,B(A)III1StXIIEIIF
..;; III (J 1/" II F--+Fd,B(A) II xII E A . F '
It can be shown similarly that under condition (1.10) the first term also
belongs to F.
The theorem is proved.
To calculate the norm in EA,F the quantity
w(t, x) = sup IIp( 'T)S,.xIIE
O<,.<t
is sometimes introduced. By (1.2) and (1.3) we have
w( t, x) Ilxll E and w( t, x) sup p( 'T)MIIAxll E
0<,. < t
Mp(t)IIAxIl E
(x E 6D(A)). (1.11)
LEMMA 1.7. Under condition (1.10) the norm in EA,F is equivalent to
II w( t, x) II F _I.
p
PROOF. By means of (1.11) we obtain from representation (1.9) that
w(t, x) ..;; 1"o I w(t, p(M)L(A, t)S]ux) da(A) + Iw(t, Q]uN(A, t)x) d,B(A)
..;; 1"o 1 1IP(At)L(A, t) S]ux II E da(A)
+ M lp(t)IIAQ]uN(A, t)xll E d,B(A).
Hence
p-l(t)W(t, x) ..;; c 1"0 1 Mp(A) II S"txll Eda(A) + C IIIS]uXIlE d,B(A).
290 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
In the proof of Theorem 1.2 we proved that the function on the right
belongs to F and its norm does not exceed C II x II E .
A,F
By definition, p-l(t)w(t, x) IIStxIlE' and so the reverse inequality IlxllE
A,F
IIp-Iwll F is obvious.
The lemma is proved.
As a consequence, Theorem 1.2 implies the following theorem.
THEOREM 1.3 (interpolation theorem). Under conditions a), (1.9), and (1.10)
E A . F is an interpolation space between E and EA.
In what follows the function p(t) will be proportional to t r . Let G be an
interpolation space between LI.. and Loo... We set F = Gr-s, where 0 < s <
r. Condition a) turns into the condition min{ 1, t- r } E G r - s or, what is the
same, min{ t r - s , t- S } E G. The last function belongs to LI.. n Loo.., and
therefore also to G, i.e. condition a) is satisfied.
It is easy to verify that for the dilation operator we have
II a l/i\ II G'-S-+G'-S = A -(r- s)1I a l/i\ II G-+G.
Since G is an interpolation space between LI.. and Loo.., II al/i\1I G-+G is
bounded by a constant (see (2.5) in Chapter IV). The second term in (1.10) is
then finite. Since in our case the function Mp(A) is equal to A r , the first term
in (1.10) takes the form f6 A sda.(A), and so it is also finite. Thus, (1.10) is
satisfied.
The space E A . F = (E, EA) -I.F turns into the space (E, EA) -S.G'-s. By
Theorem 2.3 and Lemma 2.4 of Chapter IV this space is isomorphic to
(E, EA)fJ.G.G' where () = s / r.
4. Subordinate operators.
DEFINITION 1.1. A closed linear operator B is subordinate to the operator A
with order () (0 () 1) if 6D (B) :) 6D (A) and
6D(B):) 6D(A) and IIBxll E Cllxllk-fJllxll A (x E 6D(A». (1.12)
An operator B has order () with respect to the operator A and the smoothing
approximation process Qt if (1.12) is satisfied and
IIp( t)Stxll E CpfJ( t) II xII Es
(1.13)
for x E 6D (B).
From (1.12) it follows that
II xII Es max{ 1, C} II xII k- fJ II xll A
and by Lemma 2.6 of Chapter IV we have
(E, EA)fJ L L C E B .
. I,.. I,.
1. A SMOOTIllNG APPROXIMATION PROCESS
291
If p(t) is proportional to t r , then (1.13) implies that
t r - Or II St X II C II X II Es'
and consequently E B is imbedded in E A Lr-9r, which IS isomorphic to
, 00..
(E, EA)o L L under conditions (1.9) and (1.2).
, 00..' 00..
We have arrived at the following assertion.
LEMMA 1.8. If condition (1.9) is satisfied and the operator B has order (J with
respect to the operator A and the smoothing approximation process Qt' then E B
is an intermediate space of type (J between E and EA.
The reiteration Theorem 2.8 of Chapter IV and the remarks to it lead us to
the following important assertion.
THEOREM 1.4. If condition (1.9) is satisfied and the operators Bj (i = 0, 1)
have order (Jj with respect to the operator A and the smoothing approximation
process St, then for 0 < (Jo < (J < (JI < 1
(EBo' EB)Ol,G,G = (E, EA)o,G,G = EA,Gr-s, (1.14)
up to isomorphism, where (J' = «(J - (JO)/«(JI - (Jo) and s = (Jr. Moreover, for
o < (J < (JI
(E, EB)O",G,G = (E, EA)o,G,G'
up to isomorphism, where (J" = (J / (JI' and for (Jo < (J < 1
(EBo' EA)o"',G,G = (E, EA)o,G,G'
where (J'II = «(J - (Jo) / (1 - (Jo).
5. Smoothing by means of the resolvent. In a Banach space E let an
unbounded closed linear operator B be given with domain 6D (B) dens in E,
having the property that all points of the negative semiaxis (-00,0) are
regular for it and
sup IIA(B + A)-IIIE < 00.
A>O
From this property it follows immediately that
sup IIB(B + A)-IIIE = sup III - A(B + A)-IIlE < 00.
A>O A>O
( 1.15)
We write
a = sup IIA(B + A)-IIIE < 00 and b = sup IIB(B + A)-IIIE < 00. (1.16)
A>O A>O
If X E 6D(B), then A(B + A)-IX = x - (B + A)-IBx, and A(B + A)-IX x
as A 00 by (1.15). Since, again by (1.15), the operators A(B + A)-l are
292 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
uniformly bounded, we have
"A(B + "A)-1 x x as "A 00 (x E E).
We write t = 1/"A. Then (1.16) and the last relation can be written in the
following form:
a = supll(tB + 1)-I IIE < 00, b = suplltB(tB + 1)-I IIE < 00,
t>O 1>0
( 1.17)
(tB+/)-IX X ast O (xEE). (1.18)
We introduce the operator A = B r . This operator also has dense domain
6D(A) = 6D(B r ) in E (see [18], 2.16). The space E A consists of all elements
x E 6D(B r ) with norm
Ilxll Esr = IlxilE + IIBrxIl E .
It is sometimes convenient to introduce another equivalent norm in EBr.
This possibility is based on the following well-known assertion.
LEMMA 1.9. If A is a closed linear operator with domain 6D(A) and D is a
linear operator that admits closure, whose domain 6D(D) contains 6D(A), then
IIDxll E C(llxllE + IIAxII E ), (1.19)
where C does not depend on x E 6D(A) (the operator D is subordinate to A with
order () = 0).
PROOF . We consider the restriction of D to 6D (A) as an operator from E A to
E. It is closed. Indeed, if X n x in E A and DX n y in E, then X n x in E
- -
and AX n x in E. The operator D admits closure D in E; therefore y = Dx,
and then y = Dx, since x E 6D (A). The operator D is defined on the whole
space E A' and is therefore bounded. This assertion is equivalent to (1.19).
The lemma is proved.
By the definition of powers of operators, we have 6D(B k ) ::> 6D(B r ) for
k < r, and therefore Lemma 1.9 implies the inequality
II B kX II E C kr ( II x II E + II B r x II E).
( 1.20)
From this it follows that in EBr an equivalent norm can be introduced by
the formula
r
Ilxll Esr = IIBkxII E .
k=O
( 1.21 )
Now for the operator B r we construct a smoothing approximation process
in E by the formula
Qt X = (tB + I)-r x .
( 1.22)
1. A SMOOTHING APPROXIMATION PROCESS
293
The properties of the resolvent and relations (1.17) and (1.18) enable us to
verify easily that this process has the required properties 1) and "2). We may
set M = a r.
Next,
St = AQt = Br(tB + I)-r = [ B(tB + 1)-1 Jr.
From (1.17) we obtain that
( 1.23)
II St x II E b r / t r,
and so we may assume that
p( t) = b -r t r.
In the corresponding space EBr,Gr-s the norm is defined by
Ilxll Esr.Gr-s = II xii E + II tr-sll [ B( tB + 1)-1 J r xii Ell G.
Now we pass to the derivation of a representation of the type (1.9). For this
we observe that the polynomials (1 - u)r and u r in u are relatively prime, and
thus there exist polynomials p(u) and q(u) such that (1 - u)p(u) + urq(u) =
I. In this identity we substitute the bounded operator (tB + 1)-1 for u. Then
we obtain
( 1.24)
I =[tB(tB + I)-IJrp(tB + I)-I) + (tB + I)-rq(tB + I)-I). (1.25)
Taking account of (1.22)-(1.24), we see that (1.25) is precisely a representa-
tion of type (1.9) in which the functions a(A) and {3(A) are functions of unit
jump at the point
A = 1 (da(A) = d{3(A) = 8(1 - A)dA),
L(A, t) = bp(AtB + I)-I) and N(A, t) = q(AtB + I)-I).
We shall not reformulate all assertions proved earlier for the special form
of the operator and the smoothing approximation process considered here,
and restrict ourselves to the formulation of an assertion which follows from
Theorem 1.2.
THEOREM 1.5. Let the space EBr,Gr-s be constructed from the operator B r , the
smoothing approximation process Qt = (tB + I)-r and the space G that is an
interpolation space between LI,. and Loo,.. If the operator B satisfies condition
(1.15), then the space EBr,Gr-s is isomorphic to the space (E, EBr)fJ,G,G' where
O=s/r.
Now we pass to subordinate operators.
LEMMA 1.10. If condition (1.15) is satisfied, then for m < r the operator B m is
subordinate to the operator B r with order m/ r.
294 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
PROOF. We consider the polynomials (1 - u)r-m and u m in u. Since
they are relatively prime, there are polynomials Pm(u) and qm(u) such that
(I - u)r-mPm(u) + umqm(u) = 1. Hence (1 - u)m = (1 - u)Pm(u) +
um(l - u)mqm(u). In this identity we substitute the bounded operator (tB +
1)-1 for u. Then we obtain
[tB(tB + 1)-IJm = [tB(tB + 1)-IJrPm((tB + I)-I)
+ (tB + 1)-m(tB(tB + 1)-I)mqm((tB + I)-I).
If now x E Gj) (B r), then applying the operators to this element on both
sides of the above identity, we can write the result in the form
(tB + 1)-mBmx = tr-m(tB + 1)-rpm((tB + 1)-I)Brx
+ t-m(tB + 1)-m(tB(tB + 1)-I)mqm((tB + I)-I)x.
Since the operator (tB + I)-m is invertible, we have
Bmx = tr-m(tB + 1)-r+mpm((tB + I)-I)Brx
+ t- m ( tB( tB + I)-I)m qm( (tB + I)-I)x.
Taking account of (1.17), we obtain
II B m x II E C I ( a, b) [ t r - m II B r x II E + t -m II x II E] .
Minimizing the right side over t, we finally get
IIBmxll E C(a, b)lIxllk-m/rIlBrxll,;/r. (1.26)
The lemma is proved.
Now we show that (1.13) is satisfied. We have
IIp( t)S(xll E = b-rll [ tB(tB + I)-I]r xil E b-mll [tB(tB + I)-I] mxl\E
b-mtmamllBmxllE = ampm/r(t)IIBmxIIE.
Thus, the operator B m has order m I r with respect to the operator Brand
the process (tB + I)-r.
Then Theorem 1.4 implies the following assertion.
THEOREM 1.6. If the operator B satisfies condition (1.15), then, up to isomor-
phism,for 0 mol r < () < mIl r 1 we have
(EBmo, EBml)(J'.G.G = (E, EBr)(J.G.G = EBr.Gr-s, (1.27)
where ()f = (s - m o )/(m 1 - mo) and s = Or. (We assume that EIfJ = E.)
Hence we obtain the following important result.
COROLLARY I. Tht space EBr.Gr-s is independent of r > s up to isomorphism.
1. A SMOOTHING APPROXIMATION PROCESS
295
Indeed, from Theorem 1.6 it follows that for s < r l < r (r and rl are
integers) we have
EBrl,Grl- s = (E, EBrl)s/rl,G,G = (E, EBr)s/r.G.G = EBr.Gr-s.
COROLLARY 2. If the integer m is smaller than s, then EBr.Gr-s C 6fJ(B m ).
Indeed, in this case we have
EBr.Gr-s = (EBm, EBr)(s-m)/(I-m).G.G C EBm = 6fJ(B m ).
We attempt to describe the space on the left of (1.27) more effectively. On
the space E B'"O = 6fJ (B m o ) we may consider the operator B, which will satisfy
condition (1.15) in the norm of this space too. The domain of Bml-mo on
6fJ(B m o) is exactly the space 6fJ(B m l). Finally, the operators (tB + I)-(ml-mo>
form a smoothing approximation process for B ml - mo. By Theorem 1.5, up to
isomorphism, we then have
(EB'"O, E B m l)8'.G.G = (6fJ(B m o))B m l-'"O,G(I-9')(ml-mo)
= (6D(B m O))B m l-'"O.G m l-s.
This means that our space consists of all elements of 6fJ (Bmo) for which
t m I - s II [ B ( t B + I) - I ] m I - mo X 116D (B '"0)
= tml-sll [ B(tB + I)-I] ml-mo Bmo xll E
+tml-sll[ B(tB + 1)-IJml-moxII E E G.
We show that if the first term belongs to G, then x belongs to EBr.Gr-s. By
virtue of Corollary 1 it is sufficient to show that x E EB2ml,G2ml-s. We consider
the expression
t2ml-s[ B(tB + 1)-1]2m l = t2ml-sBmIBml(tB + 1)-2m l
and to estimate it we apply Lemma 1.10 with exponents r = m l + mo and
m = mI. Then
t 2m '-sll[ B(tB + I)-1]2mlxIIE
t 2ml - s {II Bml(tB + 1)-2m1xll E} mo/(mo+m l )
X {IIB 2 m l +m o (tB + 1)-2mlxIIE} ml/(mo+m l )
a mo b ml tml-sIIBml-mo(tB + I)-(ml-mo) BmoxilE E G.
Thus we have proved the following theorem.
296
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
THEOREM 1.7. The space E Br,Gr-s consists of all elements x E OJ) (B m o ) with
mo < s for which
Bmox E EBP,GP-(s-mo)
(p > s - m).
We have proved the theorem for p = m 1 - mo; nevertheless, by Corollary
I it also holds in the above formu1ation.
From general considerations it follows that
II xii E + II tP-(s- mo>11 [ B( tB + 1)-I]P Bmoxll Ell G
(p>s-m o )
is equivalent to the norm of E Br,Gr-s.
We mention two special cases of the last formula: if s is not an integer,
then by setting mo = [s] and p = 1, we obtain that
IlxilE + Ilt 1 -(S-[sDII[ B(tB + 1)-I]B[S]xIlEIiG (1.28)
is equivalent to the norm of EBr,Gr-s. If, on the other hand, s is an integer,
then, taking mo = s - I and p = 2, we obtain an equivalent norm of the form
Ilxil E +lltll[ B(tB + I)-IYBs-1xllEIIG"
( 1.29)
6. Smoothing by means of a semigroup.(l) A strongly continuous semi group of
operators is, by definition, a family of bounded linear operators U(t) in a
Banach space E which depends on the parameter t E (0, 00) continuously in
the strong operator topology and has the property that U( t + 'T') = U( t) U( 'T')
(0 < t, T < 00). If, moreover, U(t) strongly converges to the identity operator
as t 0, we say that the semigroup satisfies the Co-condition. For every
semigroup with the Co-condition we have
II U(t)11 E-.E Me wt .
We shall consider bounded semigroups, i.e. serrugroups with the C o -
condition and the estimate
II U(t)11 E-.E M.
( 1.30)
For a semigroup with the Co-condition the function U(t)x is continuous at
zero, and U(O)x = x. We consider the collection of all elements x for which
U(t)x is differentiable at zero (from the right). For these elements the linear
operator
U'(O)x = lim U(h)x - x = Dx
h +0 h
e) The basic information on semigroups of linear operators can be found in the books [18],
[11], [48], and [24].
1. A SMOOTInNG APPROXIMATION PROCESS
297
is defined, and therefore we denote the indicated collection by 6D (D). It turns
out that the operator D is closed and has dense domain if,(D) in E. The
operator D is called the infinitesimal generator of the semigroup U(t).
The infinitesimal generator of a bounded semigroup has a resolvent R("A) =
(D - "A)-l for "A > 0, and this resolvent is related to the semigroup by the
Laplace transformation
R(A) = - oo e-NU(t) dt. (1.31)
For a closed operator D with dense domain 6D(A) in E to be the infinitesi-
mal generator of a bounded semigroup it is necessary and sufficient that it
have a resolvent R("A) for "A > 0 satisfying the conditions
II"AkRk("A)IIE E M (k = 1,2,.. .). (1.32)
From this it is clear (for k = 1) that the operator B = -D belongs to the
class of operators satisfying condition (1.15) considered in subsection 5. We
would like to mention that this class of operators is larger than the class of
infinitesimal generators of bounded semigroups.
On if, (D) the semigroup U(t) commutes with D, and
dU(t)x
dt = DU(t)x = U(t)Dx
(x E 6D(A)).
( 1.33)
From this we obtain
B / U('T)xdr = -D / U('T)xd'T = (I - U(t»x. (1.34)
We consider the operator B r = (_D)r, and for it we construct a smoothing
interpolation process by the formula
Q/x = [ t U( 'T) d'T r x = t 1r t . . . 1"0 t U( 'T 1 + . . . + 'Tr)xd'T 1 . . . d'Tr.
( 1.35)
The operators Qt form a commuting family of bounded operators which is
strongly continuous in t and strongly converges to I as t 0 (by virtue of the
strong continuity of the semigroup at zero, the Co-condition). Next,
1 i t i t
II Qt x II r ... II U ( 'T 1 + . . . + 'T r) X II d'T 1 . . . d'T r M II x II.
too
Finally, by (1.33) and (1.34) the operator Qt maps E into if,(B r ) and
commutes with B r on 6D(B r ). Thus, all conditions concerning a smoothing
approximation process are satisfied.
298 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
From (1.35) we obtain
[ I i t ] r I
St x = IE 0 U('T)dr x=?[I- U(t)rx.
From the estimate
( 1.36)
IIStXllE < (M If Ilxil E
t
it follows that as p( t) we may take the; function
pet) = (M + I)-rt r . (1.37)
Now we attempt to obtain a relation of the type (1.9) for our operators St
and Qt. We have
x = ( ; r t/r . . . t/r[ I - U('T( + · . . + 'Tr) r x d'T l .., d'Tr
+ ( ; r t/r . . . t/r[ I - (I - U('T1 + . · · +'Tr>rJx d'T l . . . d'Tr
= 1 . · . 1 [ I - u( (S 1 + . . . + Sr») r x tb 1 . . . tb r
k 1 k ( r ) r l t / r i t / r ( )
+ £.J (- I) - C r - . . . U k( 'T 1 + . . . + 'T r) X d'T. . . . d'T r.
k=1 too
We use the elementary formula
( . . . l cp( 'T( + · ; . + 'Tr ) d'T l . . . d'Tr = l cp(A) ar(A)dA. (1.38)
Here the functions ar(A) can be determined from the recurrence relation
l nA/(n -I)
n an - I ( JL) dJL,
o '
f nA/(n -I)
an(A) = n an-I( JL) dJL, 1/ n A (n - 1)/ n,
( nA - I) / ( n - I)
n f 1 an _ 1 ( JL) dJL, (n - I) / n A I.
( nA - I) I ( n - I)
We note that ar(A) = rrAr-I/(r - I)! for 0 A I/r. Applying (1.38),
we obtain
o A I/n,
x = ([ I - U(At)rxar(A) dA
k 1 k I ( kt / r ( kt / r
+ £.J (-I) - C r r Jt1 . . . In U(SI + . . .
k = 1 ( kt / r) 0 0
+ Sr)X ds. . . . ds r .
(1.39)
I. A SMOOTIllNG APPROXIMATION PROCESS
299
If we use the notation of (1.35)-( 1.37), then this identity can be written in
the form (1.9), and the operator-valued functions L("A, t) and N("A, t) can be
assumed to be identically equal to I, da("A) = (M + l)rar("A) d"A, and the
function {3("A) is the sum of jump functions with jumps (_I)k - IC/ at the
points "A = kl r, k = I, . . . , r. Thus, (1.39) has all properties of (1.9).
Theorem 1.2 yields the following theorem.
THEOREM 1.8. Let the space EBr.Gr-s be constructed from the operator B r , the
smoothing approximation process [t- I f U( 7") d7"r, and an interpolation space G
between LI.. and Loo... If the operator -B is the infinitesimal generator of a
bounded semigroup U(t), then the space EBr.Gr-k is isomorphic to (E, EBr)fJ.G.G'
where () = sir.
As has already been mentioned, the operator B satisfies (1.15), and there-
fore Theorem 1.6 remains valid if on the right side of (1.27) we understand
the space EBr.Gr-s described in Theorem 1.8. The corollaries of Theorem 1.6
also remain true.
To make things more definite, we mention that in our case the space
EBr.Gr-s consists of those elements x E E for which
I/xl/Esr,c;r-s = /lxll E +llt-SII[ I - U(t)]rxIIEIIG < 00,
( 1.40)
according to the general construction and (1.6).
Repeating the arguments preceding the formulation of Theorem 1.6, we
may show that for mo < s < m l and x E EBr.Gr-s we have
tmo-sil [ I - U(t) ]ml-mo Bmoxll E C G.
Next we show by means of Lemma 1.10 that, conversely, from the last
formula it follows that x belongs to EBr.Gr-s. (In the proof B(tB + I) must be
replaced by t-IBf U(7")d7".) Thus Theorem 1.7 remains true for spaces with
the norm (1.40).
We mention two formulas, analogous to (1.28) and (1.29), for equivalent
norms in EBr.Gr-s. For nonintegral s the norm
Ilx/IE + IIt[s]-sll[ I - U(t)]B[S]xIIEIIG'
and for integral s the norm
( 1.41 )
Ilxl/ E + 1IIIt- l [ I - U(t)]2BS-IXIIEIIG (1.42)
is equivalent to (1.40).
7. A system of operators with commuting resolvents. Let a system of opera-
tors Bj (j = I, . . . , n) act in a Banach space E, the operators satisfying
300 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
condition (1.15) and having the property that the resolvents (Bj + A)-I com-
mute:
(B; + A)-I(Bj + JL)-l = (Bj + JL)-I(Bj + A)-I.
From each of the operators Bj we may construct the space EJJ.ir,G r - s U =
I, . . . , n) by the method described in subsection 5 and consider the intersec-
tion n E r,Gr-s of these spaces. It is convenient to define a norm in this space
by the formula
n
II xII n E"'J.if- s = Ilxll E + j 111 tT-SJl[ Bi tB j + I)-I r xliEl1 G"
( 1.43)
On the other hand, we may consider the intersection of the domains OJ) (Bj)
of the operators Bj and introduce the norm
IlxllnEs.r = Ilxil E + IIBjrxllE
J
in it. With this norm it becomes a Banach space.
From the Banach couple E, n EB.r interpolation intermediate spaces may
J
be constructed by the - and -methods. There arises the question of the
relation of these spaces to the spaces n EB.r,Gr-s.
J
THEOREM 1.9. Up to isomorphism,
(E, n E r)(J,G,G = n EB/,G r - s = n (E, E r)(J,G,G'
where () = sl r.
PROOF. The space n EBr is imbedded in every space EBr, and therefore, by
J J
applying the interpolation Theorem 2.4 of Chapter IV to the imbedding
operator, we conclude that
(E, n E /)8,G,G c n (E, EJJ.ir)8,G,G.
We show that the reverse imbedding also holds. Let x E n EB.r,Gr-s. This
J
means that the quantity (1.43) is finite for x. Using (1.25) and assuming that
t I, we represent x in the form
x = [I - j 1 (tBj + IfTq((tBj + 1)-I)]X + j 1 (tBj + IfTq((tBj + 1)-I)X
- j 1 ([tBitBj + 1)-I]'p((tBj + 1)-I)X( (tBj + IfTq((t + 1)-I)X
+ j 1 (tBj + IfTq((tBj + 1)-I)X).
l. A SMOOTInNG APPROXIMATION PROCESS
301
If we denote the first term by vo(t) and the second by VI(t), then, taking
into account that all factors commute, we obtain
n
r S IIvo( t)1I E ..;; C j l tr-sll [ Bi tB j + 1)-1 r xll E
and
n
t r - s IIv l ( t)1I nEB.' = t r - s II (tBj + 1)-r q( (tBj + 1)-I)X
j=1 E
n n
+t r - s B{ II (tBj + 1)-rq((tBj + I)-I)X
k=1 )=1 E
n
Ctr-sllxll E + C tr-sll[Bk(tB k + 1)-I]r xIIE .
k=1
In the first term we have used the assumption that t I. If t I, we
represent x in the form
x = [ 1 - t -r VI ( 1) ] x + t -r V I ( 1 ) x.
Then t-SII[1 - t-rv1(1)]xII E Ct-SllxllE and
tr-sllt-rvl(I)llnE r Ct-SllxII E .
Sk
Thus, we have obtained a representation x = vo(t) + vl(t) in which the
functions t-Sllvo(t)IIEo and tr-sllvl(t)IIEns.r are majorized by the function
J
n
Cj;1 tr-sll[ Bj(tBj + 1)-lrxIIE + C min{t r - s , rS}llxllEo
Both summands belong to the space G; the first one by assumption, and
the second one by virtue of the relation min{ t r - s , t- S } ELI.. n Loo.. C G.
Thus
IIxll(E.n ESf)o.G.G II t- S II v o ( t)11 Ell G + II t r - s II v l ( t) II Ensf II G c Ilxll n ESf.{7"-s.
The theorem is proved.
The following theorem can be proved similarly.
THEOREM 1.10. Up to isomorphism,
(n E mo, n EB;ml)(J'.G.G = n (E, E r)(J.G.G'
where (J' = (s - mo)/(m l - mo).
It is possible to introduce a space EBr smaller than n EBr, on which all
"j
possible products of the powers of the operators Bj with total degree rare
302 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
defined. It is natural to introduce the norm
Ilxll£ir = IIXII£ +
II B I . . . B;"xll £
rl + . . . + r" = r
in this space.
It turns out that the following theorem holds.
THEOREM 1.11. Up to isomorphism,
(E, E B r)9,G,G = n EB/,G r - s = n (E, E r)(J,G,G.
The proof of this theorem is somewhat more complicated than that of
Theorem 1.9, and we refer the reader to the original publications [153 and
246].
8. A commuting system of semigroups. We assume that a system
U1(t), . . . , Un(t) of bounded semigroups of linear operators commuting with
one another: If;(t) ('T) = ('T) If;(t) acts in the space E. If the resolvents of
the infinitesimal generators Dj of the semigroups (t) commute with each
other, then so do the semigroups by virtue of (1.31). We may also verify the
converse of this assertion by means of the inversion formula of the Laplace
transformation. Thus, the operators Bj = -D j satisfy the conditions of subsec-
tion 7.
From the system of the semigroups (t) an n-parameter semigroup can be
formed by setting
U(y) = U1(Yl) . . . Un(Yn)
( 1.44)
for Y = (Yl' . . . , Yn) E R: (+ means thatYj > 0).
We ose the problem of expressing the norm in the space n EBj,Gr-s =
(E, D(Br))s/r,G,G in terms of the semigroup U(y).
We assume that the space G is constructed from the spaces L1,.(O, (0) and
Loo,.(O, 00) by means of some interpolation functor. We consider the spaces
L1,.(R:) and Loo,.(R:) of functions on the half-space R: with measure
dy dyl. . . dyn
- (l y l = y 2 1 + . . . +Y n 2 ) .
Iyln = Iyln
We denote by G(R:) the space constructed from L1,.CR:) and Loo,.CR:) by
means of the same interpolation functor. We introduce the quantity
Illyl-S 11(/ - U(y ))r xii £11 G(R ).
LEMMA 1. 11.
Iit-SII(I - (t))rxIIEIIG Clllyl-SII(1 - UCy))rxll£IIG(R ). (1.45)
I. A SMOOTIllNG APPROXIMATION PROCESS
303
PROOF. Consider the identity
r r r
(_I)kCrk(Zk - W)r = (-I)kCrk (_I)r-1Zk r-ll;
k=O k=O 1=0
r r
= (_I)r-1C:W r - 1 CrkZ k1
1=0 k=O
r
= (_I)r-1C:Wr-1(1 _ Z l)r.
1=0
Leaving only the term with k = 0 on the left, we obtain
r
(I - w)r = (-I)kC/{(Zk - I)rw r - k - (Zk - w)r},
k=l
in which we substitute the commuting operators U1(t)U(y) and U1(t) for z
and w. Then we have
r
(/ - U1(t»r = (-I)kC/{(U1(kt)U(ky) - /)rU1((r - k)t)
k=l
- (U1((k - I)t) U(ky) - /)rU1(rt)}.
Hence
11(/ - Ul(t»rxIIE
r
<::M 1 Crk{II(U(ky + kte 1 ) - /)r xIIE + II(U(ky + (k - l)te 1 ) - /)r xIIE },
k=l
where e 1 = (I, 0, . . . , 0) is a basis vector in R n . We integrate both sides of the
inequality over the cube 0 yj <:: t. Then we obtain
11(/ - U1(t»r xii E
M. r 1 t 1 t { ) r
<:: -f C/ ... II(U(ky + kte 1 ) - / xil E
t k=l 0 0
+ II( U(ky + (k - l)te 1 ) - /)r xIIE } dyl . . . dyne
Replacing the domain of integration by a larger one in every summand, we
arrive at
r C 1 2rt 1 rt 1 rt
t- S II (/ - U 1 ( t» x II E It . . . II ( U( Z) - /) r X II E dz 1 . . . dz n
too 0
_ 1 2rt 1 rt 1 rt IZln+s -s r dz
- COO 00' 0 tn+s IZI II(U(Z) - I)xllE lzln o
( 1.46)
304 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
Consider the integral operator
- 1 2 ,t l ,t l ,t Izln+s dz
cp - ° 0'" ° t n + S cp( z) I z In '
If cp E Loo,.(R ), then
1 2' 1 ' 1 ,
I! :Kcplk,,,..(o,OO) , II cpllL.",..(R ) ° 0'" ° lul s duo
If cp E LI,.(R ), then
I I n+s ( ) (00 dt dz
00 dt 00 00 00 z cp z ') 1 t n + s + 1 I z n I
II :Kcpl! L,..<O,oo) ,1 :Klcpl t = i i ... 1 I {/2 }
o 0 0 0 r max z 1 , Z 2' . . . , Z n
= I 1 00 ... l°O ( Izl ) n+scp(z)
n + s 0 0 max{zl/2, Z2'...' Zn} Izln
CllcpIILI..(R ).
Thus, the operator acts from Ll,.(R ) to L1,.(O, 00) and from Loo,.(R:)
to Loo,.(O, 00), and therefore it also acts from G(R ) to G.
Inequality (1.46) can be written in the form
t-SII(1 - U1(t))'xII E C (lyl-SII(I - U(y))'xII E ).
From the indicated property of we obtain (1.45) for j = I. The case of
an arbitrary j can be treated similarly. The lemma is proved.
LEMMA 1.12.
n
Illyl-SII(1 - U(y))'xIIEIIG(R ) Iit-SII(I - l!;(t))'xIIEIIG. (1.47)
j=l
PROOF. By virtue of the definition (1.44) we have
(I - U(y)Y = L l (I - l!;(Y)) U1(y\) · . . l!;-\(Yj-I) r
Therefore
n
11(1 - U(y))'xII E C II (I - l!;(Yj))'jx . (1.48)
'. + . . . +'n =, j = 1 E
Now we consider another identity:
I = 2- P (z + I)P + (z - I) q( z ),
where q(z) is a polynomial of degree p - I. From this identity it follows that
(z - I)P = 2- p (Z2 - I)P + (z - I)p+l q(z).
1. A SMOOTHING APPROXIMATION PROCESS
305
Substituting the bounded operator (Yj) for the variable z in the case
p = r j , we obtain
( (Yj) - 1)'1 = 2-'j( (2Yj) - I)'j + ( (Yj) - 1)'1+ I q( (Yj)).
Multiplying these identities for j = 1, . . . , n, we arrive at a relation of the
form
n n
II ( (Yj) - I)'j = 2- r II ( (2Yj) - I)'i
j=l j=l
n n
+ II (Uk(Yk) - I)r k ( (Yj) - 1)0'j'
j= 1 k = 1
where the 0'j are polynomials in the operators UI(YI)'.. . , Un(Yn). In this
equality we replace Yj by Ayj and then carry out an estimate:
n
sup lyl-S II ( ("AYj) - I)'i x
A>O r) + . . . +rn=r j= 1 E G(R )
n
2s-rsup l2"Ayl-S II( (2Ayj)-/)'ix
A> 0 r) + . . . + r n = r j = 1 E G(R )
n
+Csup
A>O r)+ . . . +rn=r j= 1
n
l"Ayl-S II (Uk(Ayk) - I)r k ( ("AYj) - I)x
k = 1 E G(R"+)
As is easy to see, the first summand on the right differs from the left side
only by a factor 2 s - r < 1. We bring it to the right side and observe that the
dilation operators 0l/A are bounded both above and below in the interpola-
tion space G(R':) between LI,.(R':) and Loo,.(R:). This yields
n
lyl-S II ( (Yj) -l)'ix
r) + . . . + r n = r j = 1 E G(R )
n
C
r) + . . . + r n = r j= 1
n
lyl-S II (Uk(Yk) - I)r k ( (Yj) - I)x
k = 1 E G(R )
The total degree of the operators (Yj) - 1 is equal to r in all terms on the
left side, and to r + 1 in all terms on the right side. Besides, the structure of
the right side shows that the inequality can be integrated, and thus the total
degree can be brought to any number, for example, to rn. Then at least one of
the factors (Yj) - 1 in every term will occur to a power not smaller than r.
Retaining this power, majorizing the remaining factors by constants, and
taking account of (1.48), we get
n
Illyl-SII(1 - U(y))rxIIEIIG(R ) C Illyl-sll(1 - (Yj))rxIIEIIG(R:). (1.49)
j=l
306 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
Now we consider the operator assigning to a function cp(t) on (0, 00) the
functionyjSlyl-scp(Yj) in R . It is obvious that
y/ly\-slcp(Yj)\ Icp(Yj)l,
and so this operator acts from Loo,.(O, 00) into Loo,.(R ). Next,
oo . . . oo y/lyl-slcp(y)1 dyl i;l dyn
= (ooyslcp(Y.)1 (00 . .. (00 dyl · · · dyj-Idyj+1 . . · dyn dy .
)0 J J)o)o \y\n+s J
r 00 dy 00 00 du... du. 00 dt
= J" Icp(y)l-l ... ( I J-I = C ( Icp(t)I-.
o yj 0 J o (VI + L uj2 r+ s J o t
This means that the indicated operator acts from LI,.(O, (0) into LI,.(R ),
and so it also acts from G into G(R:). It is exactly this operator that occurs
on the right side of (1.49) applied to the function t-SII(I - U(t))rxIIE. Conse-
quently (1.49) implies the required inequality (1.47).
The lemma is proved.
The quantity on the right side of (1.49) is the main summand of the norm
in the intersection of the spaces EB;r,Gr-s. A combination of Lemmas 1.11 and
1.12 yields the following theorem.
THEOREM 1.12. If (t), j = 1, . . . , n, is a system of commuting bounded
semigroups with infinitesimal generators Dj = -Bj' then the norm in n EBj,Gr-s
is equivalent to the quantity
Ilxil E + Illy\-SII(I - U(y»rxIIEIIG(R )'
( 1.50)
where G(R:) is constructed from LI,.(R:) and Loo,.(R:) by means of the same
interpolation functor used to construct G from LI,.(O, 00) and Loo,.(O, 00).
9. Unbounded operators in a couple of Banach spaces. Imbedding theorem.
Let ED be a separated topological linear space and let E be a Banach space
imbedded in ED. If R is a linear operator acting in ED, then by R E we denote
the operator defined by the equality REX = Rx for elements X E E for which
Rx E E. I t is easy to verify that, since the operator R is continuous (closed, if
ED is a Banach space), R E is closed. If in this case R E turns out to be defined
on the whole of E, then it is bounded. Its powers also enjoy this property, and
(RE)m = (Rm)E.
Now in addition let R have an inverse R- I . Consider the operator (R-I)E.
It is defined for elements x E E for which R -IX E E and (R -I)EX = R -IX.
This implies that R -IX E GD (R E ) and RE(R -IX) = R(R -IX) = x. Thus
1. A SMOOTHING APPROXIMATION PROCESS
307
RE(R -I)EX = x. On the other hand, if x belongs to the range of R E , then
x = REy = Ry (x, Y E E), and consequently R -IX = Y E E, i.e.
x E GD((R-1)E) and y = (R- 1 )EX = (R- 1 )ERFY.
We have proved the equality (R-1)E = (R E )-I. The operator R m has the same
properties as R, and so
(R-m)E = ((R m )E)-1 = ((R E )m)-1 = (RE)-m.
Now let Eo and E 1 be a couple of Banach spaces imbedded continuously in
t9, and let B be a continuous linear operator in t9 such that for A > 0 the
inverse (B + AI)-1 exists and is continuous on the whole space t9. We briefly
write BE = B;. Assume that the operators «B + M)-I)E are defined on the
I I
whole space E;. According to the preceding (the role of R is played by the
operators (B + M)-I) we then have
((B + M)-I)E, = (B; + M)-1 and 6D(B;m) = 6D((B; + M)m)
= GD((B + M)'g). (1.51)
Here the operator (B; + M)-I is bounded in E;.
Consider an interpolation functor associating a Banach space (Ao, A l)j to a
Banach couple Ao, A I. Since (Eo, E 1 , (Eo, E 1 )j) is an interpolation triple, the
operator (B + "AI)-\ acts in the space (Eo. E\)/. and so Bf = (B)(E..E')j satisfies
(1.51).
The operators (B; + "AoI)-m ("Ao > 0), which are restrictions of (B + v)-m
to E;, effect isomorphisms between the spaces E; and 6D (B;m) (with the graph
norm). Therefore (GD(Bo), 6D(B;n))j is isomorphic to the image of (Eep E 1 )j
under (B + M)-m, and for x = (B + M)-my (y E (Eo, E 1 )j) it may be as-
sumed that
Il x ll(6D(B O ),6D(Bj»j = Ilyll(Eo.E1)j = II(B + V)mxll(Eo,E1)j (1.52)
(see the Introduction to Chapter IV).
On the other hand, the indicated image is precisely the domain of B j m , and
the norm (1.52) is equivalent to the graph norm on GD(B j m ). Thus, up to
isomorphism we have
(GD(B;Z), 6D(B;n))j = 6D(B j m).
(1.53 )
In particular,
[6D(Bo), 6D(B;n) J9 = 6D(B(Jm), where B9 = (B)[Eo.Ed9'
(6D(B;), GD(B;n))9,G,G = GD(B(J G,G)' where B(J,G,G = (B)(Eo.E 1 )9.G.G.
(1.54)
( 1.55)
308
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
THEOREM 1.13 (imbedding theorem). Let Eo and E 1 be Banach spaces
imbedded algebraically and topologically into a separated topological linear
space E9, (Eo, E 1 ) is a Banach couple and let B be a linear operator on ED having
the following properties:
1. The operator B is continuous, and the operators B + AI are continuously
invertible on E9 for any A > O.
2. The restrictions (Bj + M)-1 of the operators (B + AI)-1 to the spaces Ej
act in E j and satisfy condition (1.15):
IIA(B j + AI)-IIIE E < a (A > 0).
, ,
3. The operator (Bo + AI)-1 acts from Eo into E 1 , and
II(Bo + M)-IIIEo El cAP-I,
where c and v are positive constants independent of A.
4. The domains of the Bj are dense in Ej.
Then for 0 < 0 < 0 + v / r < 1
( 1.56)
( 1.57)
(Eo, E Bf )8+v/r.G.G C (E, EB'i)8,G,G.
( 1.58)
PROOF. First we assume that s = Or > 1. If x E (Eo, E Br )8+p/r,G,G then by
virtue of Theorem 1.5 this is equivalent to the relation
tr-(s+p)11 [ Bo(tBo + 1)-1)' xii Eo E G. (1.59)
Since s > 1, Theorem 1.7 implies that x E GD(Bo). It follows from property
3 that GD(Bo) eEl. Indeed, for x E GD(Bo) we have
x = A(B o + M)-IX + (Bo + M)-IBoX EEl.
We consider the function
ct>(t) = tr+l-sll[ BI(tB I + 1)-I)'+IXIIEI
= tr+l-sll[ BI(tB I + 1)-b l )'BI(tB I + 1)- lx II E ..
Since x E Eo n E 1 , the element B 1 (tB 1 + I)-IX = Bo(tBo + I)-IX =
(tBo + /)-IBOX also belongs to Eo n E 1 . By (1.57) we have
II(tB o + I)-III Eo- £1 ct- V
and therefore
ct>(t) = tr+l-sll[ BO(tBO + 1)-I]'< tB O + 1)-1 BoXllEI
.;;; tr-(S+P-I)JI[ BO(tBO + 1)-1 )'BoXIIEo"
2. TRACE THEORY
309
By Theorem 1.7 it follows from (1.59) that the function on the right belongs
to G, and so we also have cI>(t) E G. Corollary 1 to Theorem 1.6 then says
that x E EB'j,Gr-s = (EI' E B 'j)8,G,G. Hence the imbedding (1.58) holds.
Now let s = Or < 1. The operator (B + AI)-1 implements an isomorphism
between the couples E;, EB.r and E B ., EB.r+l, and therefore it also determines an
, "
isomorphism between (E 1 , E B 'j)8,G,G and (EBI' E B 'j+1)8,G,G' and also between
(Eo, E B [)8+v/r,G,G and (EBo' E Bo +I)8+JI/r,G,G.
By Theorem 1.6 we have
(EBI' E B 'j+1)8,G,G = (EI' E B 'j)8+1/r,G,G
and
( E B , E B r+ I ) = ( E o E B r ) .
o 0 8+v/r,G,G ' 08+(v+I)/r,G,G
up to isomorphism. According to what has been proved, we have
(Eo, E Bo )8+(JI+ 1)/r,G,G C (E 1 , E B 'i)8+ l/r,G,G
and consequently
(EBo' EBo+I)8+ v/r,G,G C (EBI' E B 'j+1)8,G,G.
This implies that
(Eo, E Bo )8+v/r,G,G = (B + M)(E Bo ' E Bo +I)8+v/r,G,G
C (B + M)(EBI' E B [+1)8,G,G = (E 1 , E B 'j)8,G,G.
The theorem is proved.
2. Trace theory
1. Generalized functions with locally slUlUlUlble derivatives, and traces. We
recall that a generalized function on the semiaxis (0, (0) with values in a
Banach space A is, by definition, a continuous linear mapping of the space
D(O, (0) of infinitely differentiable scalar-valued functions with compact
support in (0, 00) into the space A. A generalized function is said to be regular
if it is induced by a locally summable function u(t) by the formula
u(cp) = oou(t)cp(t) dt
(cp E GD(O, 00 ».
The derivative of order m of a generalized function u is, by definition, the
generalized function u(m) defined by
u(m)( <p) = (-1 )mu( cp(m»).
We note that (u(m-l))'(cp) = _u(m-I)(cp') = (_I)mu(cp(m») = u(m)(cp).
310 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTI0NS
Now we assume that u(m) is a regular generalized function, i.e., it is induced
by the locally summable function u(m)(t). We consider the function
vet) = ft u(m)(t) dt (a > 0) (2.1)
a
and for any cp E G]) (0, 00) we calculate
(00 v(t)cp'(t) dt = 1 00 ft u(m)(s) ds q/(t) dt
J o 0 a
= - l a u(m)(s) l s ep'(t) dt ds + foo u(m)(s) foo ep'(t) dt ds
o 0 a s
= _ 00 u(m)(s)ep(s) ds.
From this it follows that v(cp') = _u(m)(cp) = u(m-I)(cp') or [v - u(m-I)]'(cp)
= O. This means that v( t) = u(m - 1)( t) + ao almost everywhere, where ao is a
fixed element of A. Thus, the derivative u(m-l) is determined by an absolutely
continuous function. It then follows from (2.1) that v(t) = u(m-I)(t) -
u(m-I)(a). In other words, we obtain the Newton-Leibniz formula
ft u(m)( t) dt = u(m-l)( t) - u(m-s)( a). (2.2)
a
I t is obvious that all lower order derivatives u(J)( t), j = 0, . . . , m - 2, are
continuously differentiable in the strong sense.
If 0/( t) is an infinitely differentiable scalar-valued function, then the follow-
ing formula holds for the derivative of a product:
( o/u )' ( cp) = -u( ') = -u( ( )') + u( 0/' cp) = u' ( ) + u( 0/' cp )
= (o/u' + 0/' u) ( cp ).
If u' is regular, then, using (2.2), we obtain the ordinary formula for
integration by parts:
fb J/;(t)u'(t) dt + fb J/;'(t)u(t) dt = fb(t[;u)'(t) dt = J/;(b)u(b) - J/;(a)u(a).
a a a
By passage to the limit this formula generalizes to functions 0/ E C I(a, b).
Now we choose two ideal lattices Fo and FI which are interpolation spaces
between Ll,*(O, 00) and Loo.*(O, 00), and two real numbers 110 and 111. As we
saw in Chapter II, 8.5, these spaces consist of locally summable functions.
We assume u(t) is such that u E F6°(A) and u(m) E F?I(A).
By integration by parts we obtain
uU)( b) = (-1 )m+ 1 fb Pm,/ 'T)u(m)( 'T) d'T + fbp ":J( 'T )u( 'T) d'T
a a
(0 < a < b < (0),
2. TRACE THEORY
311
where Pm,J( 7) is the polynomial of degree 2m - 1 satisfying the conditions
p(a) = p'(a) = . . . = p(m-l)(a) = 0, p(b) = p'(b) = . . . = p(m+ J -2)(b) =
p(m+J)(b) = . . . = p(m-I)(b) = 0 and p(m-J-I)(b) = (-I)J. The coefficients
of this polynomial as functions of b are uniformly bounded provided b
remains bounded. Therefore
II u U )( b )IIA ..;; e( a, b) [ bll u(m)( T)IIAdr + bll u( T)II A dr ]
c( a, b) [ II ull F8O(A) + II u(m) II F)l1(A) J. (2.3)
Now we study the behavior of the derivatives u(J)(t) as t O. For this we
use the formula
u(J)(t) = ( _l)m-J f b (s - t(-j-l u(m)(s) ds
t (m - ) - I)!
+u(m-l)(b) (b - t)m-j-l + . . . +uU)(b). (2.4)
(m - j - I)!
Hence we obtain
II u U )( t) II A ..;; e[ f b Sm - j -III U(m)( s)1I A ds + : jl II u(i)( b) II A ].
By (8.43) of Chapter II we have
b sm - j-III u(m)( s) II Ads";; ell u(m)11 F('I(A)'
if m - j > 111. In this case u(J)(t) is bounded down to zero. Passing to the
limit in (2.4) according to the Lebesgue theorem, we obtain
( l)m-J
lim u(J)(t) = -. (b sm-J-Iu(m)(s) ds
t-++O (m - ) - I)! J o
m-l 1
+ L b i - Ju(i)( b). (2.6)
i = j (i - j)!
DEFINITION 2.1. The limit limt-+o u(J)( t) = u(J)(O), if it exists, is called the
trace of order j of the function u( t).
Relation (2.6) and inequalities (2.3) and (2.5) lead to the following asser-
tion.
THEOREM 2.1. If Fa and FI are two interpolation spaces between L I ,* and
L oo ,*, and u( t) E Fdo(A) and u(m)( t) E F?I(A), then for j < m - 111 the trace of
order j of the function u(t) exists and
II u(J)(O)IIA c(11 ull fdO(A) + II u(m)11 F(I(A») (2.7)
312
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
REMARK. We have used the interpolation property of Fo only in the
estimation of f llu(t)IIA d'T in terms of IlullfdO(A)' which of course holds for
larger classes of spaces.
2. Some simple operators in spaces of differentiable functions. We consider
how the convolution and Hardy-Littlewood operators act in spaces of dif-
ferentiable functions.
Let cp be an infinitely differentiable scalar-valued function with compact
support, and let u( t) be a locally summable function with values in A. The
function
1 00 ( t ) ds 1 00 ( t ) d'T
cp * u(t) = 0 cp s U(S)s = 0 cp(7")U 7" --:;:
(2.8)
is also infinitely differentiable, and its derivatives may be calculated accord-
ing to the formulas
(cp * uP)( t) = oo s-jcp( ; ) u(s) .
(2.9)
If u( t) itself is j times continuously differentiable, then the derivative of the
convolution can be written in the form
. 1 00 .. ( t ) l oo ( s ) J ( t ) . ds
(cp * u )(])( t) = 0 cp( 7" )7"-J u U) 7" d7" = 0 t cp s u U )( s) s' (2.10)
If the trace of order j of u( t) exists, then the first equality in (2.10) implies
that
(cp * u)U)(O) = uU)(O) l oo T-jcp( 7") d7" .
o 'T
(2.11 )
We have already applied the convolution operator in order to smoothout
functions, which we shall also do in what follows.
Now we study the Hardy-Littlewood operators
Blu( t) = t 1 1 t SIU(S) ds. (2.12)
According to (8.33) of Chapter II, this operator can also be expressed in
terms of convolution, but not with a function of compact support. If I > 11 -
1, this operator acts in the space FTJ(A) (see (8.40) in Chapter II).
If u( t) is continuous, then
1 1 + I
(B/u)'(t) = t u(t) - t B/u(t). (2.13)
If, in addition, u E FTJ(A), then (B/u)' E £71+ l(A). By using the equality
Blu( t) = 00 7"-1-2 u( ) d7"
2.TRACETHEORY
313
for a i-fold continuously differentiable function we obtain
(H1u)U)(t) = oo 'T- 1 - j - 2 uU)( ) d'T = Hl+jUU)(t). (2.14)
If u(j) E FTJ(A), then (HIu)<j) E FTJ(A) for I + } > '11 - 1.
If the trace w( j)(O) exists, then
(H u)(})(O) = u(})(0) f oo'T-l-j-2d'T = 1 uU)(O). (2.15)
I 1 1+}+1
Thus, the Hardy-Littlewood operator HI preserves the membership of u(j)(t)
in FTJ(A) for I + } > '11 - 1, increases the smoothness of the function by one, and
ensures that the derivatives of order} of the function t(HIu)'(t) belong to the
same classes FTJ(A).
3. Spaces of traces. Now let Ao, A I be a Banach couple, let Fo and FI be
two interpolation spaces between L I ,. and L oo ,*, and let '11ep '111 E R. We
denote by W m ('11o, Fo, Ao; '111' F I , A I) the space of functions with values in
AO + Al for which u E F6o(Ao) and u(m) E F?I(A I ), where the derivative is
understood in the sense of generalized functions with values in Ao + A I. In
this space we introduce a norm by the formula
II ull W m = II ull FQ7o(A o ) + II u(m)11 FpI(AI).
We leave it to the reader to prove the completeness of the normed space
thus obtained.
For functions u belonging to W m ('11o, Fo, Ao; '111' F I , AI) we obviously have
u E F6°(Ao + A I) and u(m) E F?I(Ao + A 1). Consequently Theorem 2.1 holds
for them relative to the space A = Ao + AI. Therefore the following defini-
tion is meaningful.
DEFINITION 2.2. Let} be a natural number with} < m - '111. The space of
traces of order} is the collection 1jm( '110' Fo, Ao; '111' F I , A I) of all elements x
belonging to Ao + A 1 of the form
x = u(})(O), where u E W m ( '11(P F(p Ao; '111' F I , AI)'
the norm being
II x II1jm = inf II u II W m ( TJO'F(pAo;TJI,FhA I).
x = u(J)(O)
In our case (2.7) implies the inequality
Ilu(})(0)IIAo+A 1 c(ll u llFQ7O(A o ) + Ilu(m)IIFpI(A 1 ») = cllull wm , (2.17)
which in turn implies that the subspace of functions u E W m for which
u(J)(O) = 0 is closed in W m . The space 1Jm of traces of order j is isomorphic to
the quotient space of W m modulo this subspace. Consequently 1jm is a
Banach space.
(2.16)
314 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
REMARK 2.1. In calculating the quantity equivalent to the norm (2.16), the
infimum may be taken over the set of all infinitely differentiable functions
belonging to W m (110, FO' AO; 111' F I , AI).
Indeed, if we choose the infinitely differentiable function cp(t) with compact
support so that fa T-}cp(T)T- I dT = 1, we can calculate the infimum in (2.16)
only over functions of the form cp * u(t), where u E W m (110, Fep Ao; 111' F I , AI)
and u(J)(O) = x. By (2.11) we have (cp * u)<J)(O) = u(J)(O) = x, and by (2.10)
and (8.38) of Chapter II we have Ilcp * ull w cllull w , where the constant c
m m
depends only on the choice of cpo This implies the validity of Remark 2.1.
LEMMA 2.1. If 110 <j < m - 111' then the space 1jm(110' Fo, Ao; 111' F I , AI) is
intermediate between Ao and A I.
PROOF. The imbedding 1jm c Ao + Al follows from (2.17). Let x E Ao n
AI. We consider a smooth function cp(t) = (l/j!)t j o/(t), where (O) = 1,
0/'(0) = . . . = 0/(/-1)(0) = 0, (t) = 0 for t a > 0, and I is sufficiently
large. We set u( t) = cp( t)x. Then the function
1 .
(lJollu(t)IIA = -:- , tTJo+J\o/(t)lllxIIA
° J. 0
belongs to Fo, since it belongs to L I ,* n L oo ,* by virtue of the condition
-110 <j. The function
t 1J 'lIu(m)(t)IIA, = ), t 1J '[ tjl/l(t) tm)lIxll A ,
has order tf-m+}+TJI at zero, and so it belongs to FI for I > m - j - 111' for
the same reason. Finally, u(J)(O) = x, i.e. x E 1jm( 110' Fo, Ao; 111' F I , A I).
The lemma is proved.
The number of parameters on which the spaces 1jm depend can be
decreased.
LEMMA 2.2. If m 2 and -110 < m - 1 < m - 111' then
T::- I (110' Fo, Ao; 111' F I , AI) = T= 21(110 + 1, Fo, Ao; 111' F I , AI)
up to isomorphism.
PROOF. Let x = u(m-I)(o) and u E W m (110' Fo, Ao; 111' F I , A I).
We apply the Hardy-Littlewood operator Hf with sufficiently large I to the
function u(t) and consider the function w(t) = (I + m)(Hfu)'(t). According to
(2.15) we have w(m-2)(0) = (I + m)(Hfu/m-I)(O) = u(m-I)(O) = x. Next, the
function tw(t) belongs to F6o(Ao), i.e. w E F6o+ I(Ao), and, finally, w(m-I)(t)
= (I + m)(H,uim)(t) E F?I(A I). From these relations it follows that x E
T: 21(110 + 1, Fo, Ao; 111' F I , AI).
2. TRACE THEORY
315
Conversely, let x have the last property. Then
x = W(m-2)(0), W E W m - I ( '110 + 1, Fo, AO; '111' £1' AI).
We consider the function
I+m-I
u(t) = m _ I tH/w(t).
Then H/w E F6o+ I (Ao), and consequently u E F3°(AcJ. Next, by (2.13)
I+m-I
u'(t) = m _ 1 [w(t) - IH/w(t)]
and by (2.14)
u U )( t) = 1 : ':.- 1 [ wU-1)( t) - IH/+ j _ 1 w U - 1 )( t) ].
In particular,
(2.18)
u(m-l)(o) = 1 + m - 1 [ w(m-2)(o) _ 1 w(m-2)(O) ] = w(m-2)(o) = x
m-I I+m-I
according to (2.15). Finally, from (2.18) for j = m and properties of the
Hardy-Littlewood operators it follows that u(m) E F?I(A 1)' and consequently
x E T:: _ 1 ('110' F 0' A 0; '11 l' F l' AI).
The lemma is proved.
LEMMA 2.3. If -'110 <} < m - '111 and} m - 2, then
1jm( '110' Fo, Ao; '111' F I , AI) = 1jm-I( '110' FO' AO; '111 - 1, F I , AI)
up to isomorphism.
PROOF. Let u E W m ('11o, Fo, Ao; '111' F I , A 1). Consider the operator
v(t) = 1 m 1+ j + 1 [ u(t) - IH/_m(t) ] .
} + - m
For sufficiently large I (for I - m > '110 - 1) this operator acts in F6o{Ao).
Using (2.14), we obtain
v(m-l)(t) = 17:: I+!; 1 [u(m-l)(t) - IH/_1u(m-l)(t)].
Upon integration by parts we find that
(m-I) ( ) _ I - m + } + 1 H (m) ( )
V t - . I t /u t.
} + - m
By a property of the Hardy-Littlewood operators we then have v(m-I) E
F?I-I(A 1). A calculation shows that v(j)(O) = u(j)(O). Thus, if
x = u(j)(O) E 1jm( '11ep Fo, Ao; '111' F I , AI)'
then x = v(j)(O) E 1Jm-I('11o, Fo, AO; '111' F I , AI).
316
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
The reverse inclusion can be obtained by means of the operator u(t) =
(I + j + I)H/v(t).
Indeed, if v E F6o(Ao), then u E F6°(Ao). Next, by (2.14)
u(m-I)(t) = (I + j + I)H/+ m _ I v(m-I)(t).
From this, by virtue of (2.13), we obtain
() . [ 1 l+m ]
u m (t) = (I + ) + 1) t v (m - 1)( t) - t H/ + m _ 1 v(m - 1)( t) E F? t (A I)'
provided that v(m -1) E F?t-I(A 1). Finally, u(J)(O) = v(J)(O). Thus, if x =
v(J)(O) E 1jm-I(Tlo, Fo, Ao; 111 - 1, F I , A 1)' then
x = u(j)(O) E 1jm( 110' Fo, Ao; 111' F I , AI).
The lemma is proved.
REMARK. According to Remark 2.1, in the proof of the lemmas we assumed
that all functions were as smooth as necessary.
From Lemmas 2.2 and 2.3 it follows that all spaces of traces which are
intermediate between Ao and A 1 can be obtained as spaces of type T d.
Indeed, the following corollary is true.
COROLLARY.
1jm( 110' Fo, Ao; 111' F I , AI) = TJ( 110 + j, FO' AO; 111 - m + j + 1, F I , A I)
up to isomorphism.
It turns out that the spaces of traces coincide with the spaces constructed
by the X- and -methods.
LEMMA 2.4. If -110 < 0 < 1 - 111' then
T J (110' F 0' A 0; 11 I' F l' AI) = (A 0' AI) . PI' I - I
up to isomorphism.
PROOF. Let x E TJ(110' Fo,Ao; 111' FI,A I ). This means that x = u(O), u E
F6°(Ao), u' E F?I(A 1). Consider the convolution cp * u = v of u(t) with a
smooth function cp(t) with compact support normalized in such a way that
J cp(S)S-I ds = 1. Then by (2.9) and (2.10) we have
v' ( t) = t -1 1 00 cp' ( ) u( s) ds = 1 00 !. cp ( ) u' ( s) ds .
OS S sot s s
It follows from the first equality that tv' E F6°(Ao), and from the second
that v' E F?I(A 1). Consequently, tv' E F?I-I(A 1). Finally, by (2.11) we have
v(O) = x.
2. TRACE THEORY
317
Next, by virtue of (8.44) of Chapter II, under the condition -1 > -111 the
integral
1 00 1 00 ds
v'(s) ds = sv'(s)-
b b S
is absolutely convergent in A I' and so v(t) has a limit at infinity in the norm
of AI' and thus in the norm of Ao + Al as well. Next, v E F6o{Ao), and hence
v E FdO(Ao + AI). On the other hand, for the function Ilv(t)IIA +A having a
° I
limit at infinity, the function t1JOllv(t)IIAo+AI can belong to Fo eLl,. + Loo,.
for 110 > 0 only if the limit is equal to zero. Thus,
1 00 ds
v(b) = - sv'(s)-
b S
in Ao + A I.
Since v E W I (110' Fo, Ao; 111' F I , A I)' the limit limb-+<> v(b) = v(O) = x ex-
ists in Ao + A I on the basis of Theorem 2.1, i.e.
v(O) = _ (00 sv'(s) ds .
J o s
In this representation -sv'(s) E Fdo{Ao) n F?I-I(A I)' and consequently
x E (A <p A l)ko,FpI-I.
Conversely, let x satisfy the last condition. Then x = f W(S)S-lds, where
w E Fdo(Ao) n F?I-I(AI)' and the function w(t) can be assumed smooth (see
the proof of Theorem 2.3 in Chapter IV). Consider the function
u( t) = f oo w(s) ds .
t S
By virtue of (8.41) of Chapter II we have u E F6o{Ao). Next, u'(t) = t- (t)
E F?I(A I). Finally, u(O) = x. Thus, x E TJ(110' Fo, Ao; 111' F I , A I).
The lemma is proved.
Lemmas 2.1-2.4 imply the following theorem.
THEOREM 2.2. If -110 <j < m - 111' then
1jm( 110' Fo, Ao; 111' F I , A I) = (Ao, A 1) +j,F?I+j-m
up to isomorphism.
(2.19)
COROLLARY 1. If -110 <j < m - 111' then
1jm(110' Fo, Ao; 111' F I , AI) = (Ao, AI)e,Fo,F 1
up to isomorphism, where () = U + 11o)/(m - 111 + 110).
This corollary immediately follows from Theorem 2.3 and the corollary to
Lemma 2.4 of Chapter IV (taking account of Remark 2.3 of Chapter IV).
(2.20)
318 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
COROLLARY 2.
1Jm( 110' Fo, AO; 111' F I , AI) = 1Jm( 11, FO' AO; 11, F I , AI)' (2.21)
up to isomorphism, where 11 = ((m - })110 + }l1I)/(m - 111 + 110).
The space on the right side of (2.18) is intermediate between Ao and A l'
SInce
.+
. + j 110 > 0
j l1=m
m - 111 + 110
According to (2.20) this space is also isomorphic
(J = (j + 11)/ m = (j + 110)/(m - 111 + 110).
m-j-l1
and m - 11 - ) = m 1 > o.
m - 111 + 110
to (A(p A I)8,F fft F J with
COROLLARY 3. For fixed 110' 111' Fo, F I , m and j the functor
1Jm( 110' Fo, Ao; 111' F I , A 1) is an interpolation functor.
This corollary follows easily from the definition of the spaces.
COROLLARY 4. Under the conditions -110 <) < m - 111 and 0 } m
-1, the operation of taking the trace u(j)(O) is a continuous mapping of the
space W m ( 110' Fo, Ao; 111' F I , A 1) onto the space (Ao, A 1)8 F. F' where (J =
, 0> J
U + 110)/(m - 111 + 110).
4. Complete theorem on traces. If in the preceding constructions 110 > 0,
then the function u E W m ( 110' Fo, Ao; 111' F I , A 1) has traces of orders} =
0, . . . ,) max' where} Max is the largest integer not exceeding m - 1 and smaller
than m - 111. According tc the preceding, we have u(j)(O) E (A(p A 1)8. F. F'
J' fft J
where = U + 110)/(m - 111 + 110). Thus, there arises the mapping yu =
{u(O), u'(O),. . . , uUm.J(O)} of the space W m ( 11(p Fo, Ao; 111' F I , A 1) into the
direct product II ( A 0' A I) ,F fftFJ.
THEOREM 2.3. The mapping yu = {u(O), u'(O), . . . , uUm.J(O)} is a continuous
mapping of W m ( 110' Fo, Ao; 111' F I , A 1) onto II =o(Ao' A I) ,FfftFJ.
PROOF. For the proof of the theorem it suffices to construct, for
every element x E (Ao, A 1)8. F F (0 i jmaJ, a function V; E
I' 0> J
W m (110' Fo, AO; 111' F I , A 1) such that v;(;)(O) = x and v;(j)(O) = 0 for} =1= i
(0 ) )rnaJ. According to Theorem 2.2 there exists a function u E
W m (110' Fo, Ao; 111' F I , AI) such that u(i)(O) = x. By means of the Hardy-
Littlewood operators we construct the functions W I (t) = HI u(t) (k =
Ie Ie
0, 1, . . . '}maJ with sufficiently large lk. These functions also belong to
W m (110, Fo, Ao; 111' F I , A 1). By (2.15) we have
wF)(O) = 1 uU)(O).
Ie lk + j + 1
2. TRACE THEORY
319
We now seek the function v;(t) in the form v;(t) = akwl/c(t). For the
determination of the numbers a k we obtain the system
j max 1
I . 1 a k = 8;j
k=O k + } +
(J = 0, . . . ,J max ).
Using the arbitrariness in choosing the lk' we can always make the determi-
nant of this system different from zero. Then the numbers a k , and together
with them the function v;(t), can be determined.
The theorem is proved.
We have proved that to every set of elements X; E (Ao, A I )8;,1;;,F 1 (i =
0, . . . , Jrnax) there corresponds a function u(t) belonging to
W m (110, Fo, Ao; 111' F I , A I) such that X; is the trace of order i of u(t). However
an analysis of the proof shows that we have not guaranteed the linearity of
the correspondence. In the following subsection we construct such a linear
correspondence in a more special situation.
We also note that the assumption 110 > 0 has been made for the simplicity
of the formulation. If 110 < 0, then Theorem 2.3 is true for the space
jmax
II (A(p A I)i?;,F()tF 1 ,
j = j min
whereJmin is the smallest integer greater than -110 and not exceeding m - 1.
5. Functions smooth in t and smooth relative to an unbounded operator. Let E
be a Banach space in which a closed unbounded linear operator B is given
with domain 6D(B) dense in E. For a function u(t) (0 < t < 00) with values in
E two notions of smoothness may be considered: smoothness in t and
smoothness relative to the operator B in the sense that the values of the
function belong to spaces constructed from the powers of B (smoothness in
the space coordinates, so to say). If we assume that the operators dj dt and B
are "of equal strength", then by total smoothness of order m of the function
u(t) we must understand that u(t) belongs to some space of functions of the
form B1u(J)(t), where I + J m.
Suppose that B satisfies the condition
II(B + A)-IIIE aj (1 + A)
(A 0),
(2.29 )
which is stronger than (2.15).
Clearly this condition implies (1.15): 11i\(B + A)-IIIE a, but, in addition, it
also implies that B has bounded inverse with IIB-IIIE a. Under this as-
sumption we have IIB1xli E ar-IIIBrxIlE for r > I, and therefore an equiva-
lent norm can be introduced in the space EBr = 6D (B r ) by the simpler
formula
Ilxll Esr = IIBrxIlE.
320 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
Now let G be an interpolation space between L I ,. and Loo,., and let TJ > o.
We denote by Wm(TJ, G, E, B) the space of functions u(t) (0 < t < (0) with
values in E for which the functions B1u(J)(t) belong to G"'(E) for I + j m.
By virtue of what has been said above, it is sufficient to require that the
functions B1u(J)(t) belong to G"'(E) for I + j = m. In the space
Wm(TJ, G, E, B) the norm
m
II ull Wm(.",G,E,B) = L II B1u(m-I)11 GTJ(E)'
/-0
can be introduced, after which it becomes a Banach space. For functions u(t)
belonging to Wm(TJ, G, E, B) we have Bmu E G"'(E) by definition, i.e. u E
B"'(EBm) and u(m) E G"'(E), and so the imbedding
Wm(TJ, G, E, B) c Wm(TJ, G, EBm; TJ, G, E)
holds.
Then it follows from Theorem 2.1 that the functions u E W m ( TJ, G, E, B)
have traces of orders j = 0, . . . , j max' where j max is the largest integer smaller
than m - TJ. By Theorem 2.2 we have
1Jm(TJ, G, EBm; TJ, G, E) = (EBm, E)8 j ,G,G = (E, E B m)I-8 j ,G,G' (2.23)
up to isomorphism, where OJ = (j + TJ)/ m.
Let x E (E, EBm)I- ,G,G. By Theorems 1.5 and 1.6 we have x E EBr,Gr-s,
where s = (1 - )m = m - j - TJ and r is any integer greater than s. By the
definition of the space E Br,Gr-s this means that
tr-sBr(tB + I)-r x E G(E).
We choose r = m - j > m - j - TJ = s. Then the last membership rela-
tion can be written in the form
Bm-j(tB + I)-(m- j ) x E G"'(E).
Consider the functions
(2.24 )
Uj,q(t) = B-j(tB + I)-qx,
where the integer q is sufficiently large. We calculate
B1u(m-/) ( t )
},q
(2.25)
= -q(-q - 1) . . . (-q - m + I + I)Bm-j(tB + I) - q - (m - l)x.
Since (tB + I)-I is uniformly bounded in t, it follows from (2.24) that, if
q m - j, then
B1u(m- I) E G"' ( E )
},q
and consequently Uj,q E Wm(TJ, G, E, B). Thus, (2.25) enables us to constuct,
for every x E (E, EBm)l- ,G,G' a function belonging to Wm(TJ, G, E, B), and
2. TRACE THEORY
321
then for i = 0, . . . ,jmax we have
U5,i (0) = -q( -q - 1) . . . (-q - i + I)B i - j x. (2.26)
(In this calculation we have used the fact that on the basis of Theorem 1.6 it
follows from the inequality jrnax - j < m - 1] - j = s that the membership
relation x E EBr,Gr-s implies that x E 6D (Bjrnax-j).)
Weare now ready to prove the following assertion.
THEOREM 2.4. There exists a bounded linear operator R mapping the space
IIjs:o(EBm, E)fJ,G,G into the space W m (1], G, E, B) (OJ = (j + 1])/ m) such that
_ J _ I UmvJ
yR - I, where yu - {u(O), u (0), . . . , U (O)}.
PROOF. It is sufficient to define the operator R for elements of the form
{xO' Xl' . . . , XJrnax}' where Xi = 0 for i * j and x j = X E (EBm, E)8 j ,G,G. Using
the notation of (2.25), we set
lmax
R {O, 0, . . . , Xj' 0, . . . , O} = L ak,juj,q/c (t) = V j ( t)
k=O
where the qk are distinct sufficiently large integers. As we saw above,
v j E W m (1], G, E, B). We choose the numbers ak,j so that v}i)(O) = 0 for i =1= j
and v}j)(O) = x. By virtue of (2.26), for this it is sufficient that the ak,j satisfy
the system
(i = 0, . . . , j max)
(2.27)
It is easy to see that the determinant of this system coincides with the
Vandermonde determinant constructed from the numbers -qk' and so it is
not zero. Thus, the ak,j are determined uniquely. Finally, the operator R is
constructed according to the formula
lmax
L - qk (-qk - 1) . . . (-qk - i + 1) ak,j = 8 ij
k=O
lmax lmax
R {xo, . . . , Xjrnax} = L L ak,jB-j( tB + I)-q/c x j
j=O k=O
where the ak,j are solutions of (2.27).
The theorem is proved.
Our constructions simplify if only traces of order zero are considered. In
this case we have
Tf:( 1], G, EBm; 1], G, E) = (E, EBm)I-.,,/m,G,G.
For X E (E, EBm)I-.,,/m,G,G and q 2m the function UO,q(t) = (t B + I)-qx
belongs to W m (1], G, E, B), and UO,q(O) = x. Since q does not depend on 1]
322
v. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
and is bounded only from below, for x E T;I(11I' G I , EBml; 111' G I , E) =
(E, EB)I- 71 G G' where 1 m l m and 0 < 111 < m l , we have U o q E
11 m 2' I' I ,
W ml (11I' G I , E, B) and UO,q(O) = x. We have arrived at the following assertion.
THEOREM 2.5. The operator Yo of taking the trace of order zero (You = u(O»
induces a right invertible mapping of the Banach couple
W ml ( 111' G I , E, B), W m ( 11, G, E, B) (1 m l < m, 0 < 111 < m l , 0 < 11 < m)
onto the Banach couple
(E, EBml)I-71I/ml,GI,GI,
(E, EBm)I-71/m,G,G.
Theorem 4.10 of Chapter I implies the following corollary.
COROLLARY. For any interpolation functor the operator Yo of taking the
trace of order zero maps the space (Wm (111' G I , E, B), W m (11, G, E, B» onto
I
the space
(E, EBml)I-71I/ml,GI,G I , (E, EBm)I-71/m,G,G).
Moreover, the mapping x (tB + I)-qx (q 2m) is a right inverse of Yo.
Similar assertions can be formulated for every trace of a fixed order.
3. Spaces of smooth functions of n variables
1. Sobolev spaces. We shall consider scalar-valued functions defined on the
space R n . Moreover, we shall use the following abbreviated notation: s =
(SI' . . . , sn) is a point of R n , Isl = (s: + . . . + S;)1/2, (s, = SI 1
+ . . . + sn n' x(s) = X(SI' . . . , sn) is a function on R n , and
1 ... 1 X(SI' . . . , Sn) ds l . . . ds n = f x(s) ds.
Rn Rn
In the space 4(R n ) (1 p (0) constructed for Lebesgue measure we
consider the group
UI(t)X(S) = X(SI + t, S2, . . . , sn)
(-00 < t < (0)
of isometric translation operators and determine the infinitesimal generator
D] of this group. By assumption, a function x E Lp(Rn) belongs to the
domain of DI if the limit
. x ( S I + h, S 2' . . . , S n) - X ( S I' . . . , S n ) ( ) ( n )
11m h = DI usE Lp R
h O
exists in Lp(Rn).
3. SPACES OF SMOOTH FUNCfIONS OF n VARIABLES
323
We denote by Gj) (R n ) the space of infinitely differentiable functions with
compact support (the space of test functions). Let cp E Gj) (Rn); then
f x ( S I + h, S 2' . . . , S n) - X ( S I' . . . , S n) ( ) ds
h cp S l' . . . , Sn
_ f cp( S I - h, S2' . . . , Sn) - cp( S I' . . . , Sn) ds
- X(SI'...' Sn) h .
Passage to the limit shows that
f D\x(s)cp(s) ds = - f x(s) ; ds
and consequently DI coincides with the differentiation a /as l in the sense of
generalized functions. Thus, DI = a/aS I on the domain of D I , and ax/aS I E
Lp(R n ). A
We introduce the operator DI equal to a/aS I for all functions x E Lp(Rn)
A
for which ax /aS I = 4(Rn). It is clear from the preceding that DI is an
"
extension of DI. We show that DI has resolvent for A > O. For this we
consider the equation
"
Dlx - Ax = ax/as l - Ax = y.
The function
f oo
_ A(SI --r)
x( s) - - e y( 'T, S2' . . . , sn) d'T,
Sl
is a solution of this equation. This function can be written in convolution
form:
(3.1 )
x(s) = Joo cp(s\ - 'T)y( 'T, S2' . . . , sn) d'T,
-00
where cp(u) = 0 for u > 0 and cp(u) = eAu for u < O. By the Young inequality
we have
i: Ix(s\, . . . , snW ds\ ..;; (i: Icp( u)1 du r i: Iy(s\, . . . , snW ds\
1 f oo
= A P ly(sl' . . . , sn)IP ds l .
-00
By integrating with respect to the variables S2' . . . , sn' we obtain
1
Il x I1 4 (R") < A II YI1 4 (R").
" "
Thus, (3.1) determines the resolvent R(A, D 1 ) of D I , and
II R(A, D I )11 4 (R")-+L p (R") < I/A (A > 0).
(3.2)
324
v. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
For A > 0 the operator DI has a resolvent according to the general theory
. A
of sermgroups (see 1.6), and for the operator DI we have constructed the
resolvent. Hence it follows that these operators coincide. Indeed, for any
x E Lp(R n ) we have
(D I - M)R(A, DI)x = (D I - M)R(A, DI)x = x
A A
because DI is an extension of DI. Applying (D I - M)-I to both sides of the
A
equality, we obtain that R(A, DI)x = R(A, DI)x. Then the domains of DI and
A
D I , and hence the operators themselves, coincide.
Thus, the infinitesimal generator of the group UI(t) is defined for all
functions x E 4(R n ) having derivative ax jas l (in the sense of generalized
functions) belonging to Lp(Rn), and for these functions we have Dlx =
ax jas i .
Estimate (3.2) shows that the operator BI = -D I has the property (1.15),
and for it a = 1. For each} the group (t) of translations along the }th
coordinate axis and its infinitesimal generator Dj = ajas j can be considered
similarly. It is obvious that these groups commute with each other, and we
find ourselves under the conditions of 1.7 and 1.8. Consider the space
which was denoted by n EB.r in 1.7. In our case this is the space of functions
J
for which akxja5.ik E 4(Rn) for all } = 1, . . . , nand k = 0, . . . , r. It is
denoted by wrJ'(Rn) and called a Sobo/ev space. The norm in it is defined by
n arx
Ilxllwr-o = IIxllL" + L _ a r
)=1 5.i 4
Sometimes it is more convenient to introduce an equivalent norm by the
formula
n a r x p
IlxllPwr,p = Il x ll4, + L ar . (3.3)
)= 1 Sj 4
In accordance with what we said in 1.5 (see (1.21», these norms are also
equivalent to the norm
II xII W'$ = {k Oj l ;; 4r .
'-.,;
We introduce some notation. If a = (ai' . . . , an) is a vector with nonnega-
tive integral coordinates, then lal = a l + . . . + an' D a = Dfl . . . Dna.,. and
a ' = a , . . . ,
. I . \An . .
The Sobolev spaces admit a description in terms of the Fourier transforma-
tion. We mention the necessary information from the theory of the Fourier
transformation (see [42] and [48]).
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
325
The collection of infinitely differentiable functions on R n for which all the
norms
r
Pr(x) = sup (1 + Isl2)r L IDax(s)1
sER n lal =0
are finite is called the Schwartz space S(Rn).
The system of norms Pr(x) defines a topology in S(R n ) in a natural way.
The Fourier transformation
(r = 0, 1, . . . )
u(g) = ] f e-;( 'S)u(s) ds = 'J(u)
(2'1T )n/2 '
is defined for functions belonging to S(Rn). It maps S(R n ) onto itself. The
inverse transformation is given by
o(s) = ] f e;(S' )v(g) dg = 'J-l(V).
(2'1T )n/2
We have the Parseval identity
f u(s)v(s) ds = J u(g)t3(g) dg or f u(s)v(s) ds = f um om df
The space 6D (R n ) of infinitely differentiable functions with compact sup-
port is densely imbedded in S(R n ), and so the reverse imbedding S'(Rn) c
6D '(R n ) holds for the dual spaces. The space S'(Rn) is called the space of
slowly increasing Schwartz distributions. The spaces lp(Rn) can be imbedded
in S'(Rn) in a natural way. If T is a distribution (generalized function)
belonging to S'(Rn), then its Fourier transform 6J(T) is defined by
6J(T)(u) = T(u) = T(u)
(u E S(R n )).
In particular, the Fourier transformation 'J is defined for all functions
belonging to lp(Rn), 1 P 00. We have
'J(Djx) = i j'J(x).
(3.4)
The problem of multipliers for the Fourier transformation is especially
important in applications. I t consists. in the following: a function ct>(g) is
given, from which the operator 'J -I (<I>(g)'J(x» is constructed. Under what
conditions will this operator act boundedly in the spaces Lp(Rn)?
S. G. Mihlin's well-known theorem (see [30] or [45]) contains sufficient
conditions.
326 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
THEOREM 3.1. If the function et>(g) is continuous in R n \ 0 and has derivative
a net> lag l . . . ag n at every point there, and all lower order derivatives are
continuous and
Igl k ag. ag. ..;; M,
'. 'Ie
where k = 0, I, . . . , nand 1 1 1 < 1 2 < . . . < i k n, then the operator
'?f -I et>'?f is bounded in Lp(R n ) for 1 < p < 00. Moreover, the norm of this
operator depends only on M and p.
Now we return to the Sobolev spaces. The fact that arxjas./ E Lp(R n ) for
x E wr,p(R n ) may also be written, invoking (2.4), in the form
'?f-I( (igj)r'?f(x)) E 4(R n ).
(3.5)
From this we draw a number of conclusions. Since x also belongs to
Lp(Rn), we have
'?f -l( (I + i /) (x) E Lp(R n ).
Now consider the function
[ , / 1 + g 2r ]
-I( y l + 2r (x») = -l + i £j £j-l((1 + i '")£j(X»).
I t is easy to verify that the function
Il>(g) = y l + 2r /1 + ig;
satisfies the assumptions of Theorem 3.1. Since I et>(g) I = 1, it is sufficient to
show that Igllaet> la 1 M, and this can be verified by direct calculation.
Then by Theorem 3.1 we have
11£j-l( y l + 2r £j(x) )114 ..;; CII£j-l((1 + ig;)£j(x»)k
c IIxll wr.p (1 < p < (0). (3.6)
Now we introduce the function
'?f-I((I + IgI 2 )r/2'?f(x)) = L '?f-I [ (I + Ig1 2 y/2 L V I + g: (X) ] . (3.7)
k L . ' / 1 + g.2r k
lV ,
It can be shown that the function
Il>(g) = (I + IgI 2 y/ 2 / L y l + 2r
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES 327
satisfies the hypotheses of Theorem 3.1, and therefore from (3.6) and (3.7) we
obtain
11'5- 1 ((1 + Igf)'/2'5(x»)lk .s;;; Cllxll w ''''.
We show that the reverse inequality also holds. For this we write
(3.8)
Ilxll = 'J- I ( 1 'J ) ( '5-1( (I + IgI 2 )'/2'5(x»))
lp (1 + Ig1 2 y/2 lp
and
a a : = '5- I [ (i Y '5 ] ('J-I((I + IgI2)r/2 (x)))
4 (1 + Ig1 2 y/ 2 Lp
Applying Theorem 3.1 again with the functions
(I + IgI 2 )-r/2 and (i )r( 1 + IgI 2 )-r/2,
we obtain
Ilxllwr,p CII'J-I((I + IgI 2 )r/2'J(x))IIL,;
Now we can also solve the problem of mixed derivatives of functions
belonging to wr,p(R n ). We have
IIDaxllL" = 11'J-I(ilalga'J(x))IIL"
= 'J-I ( . i1alg a (5 ) ('J- I ((1 + IgI2)r/2 (x))) ,
(1 + IgI 2 )'/2 4
where ga = gfl . . . gncx". If lal r, then the function ga(1 + IgI 2 )-r/2 satisfies
the hypotheses of Theorem 3.1, and therefore, taking account of (3.8), we
obtain IIDaxllL" Cllxll wr,p for lal r.
We sum up our discussion in the following theorem.
THEOREM 3.2. If 1 < p < 00, then in the space wrJ'(Rn) with norm (2.3)
equivalent norms can be introduced by the formulas
II xII w'''' = 11'5- 1 ((1 + IgI 2 )'/2'5(x»)114
(3.9)
or
{ } I/p
Ilxll w r . p = II Da x l1 4 .
lal<r
(3.10)
We note that the assertions of Theorem 3.2 are not true for p = 1 and
p = 00.
328 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
Theorem 3.2 has major significance for us, since in studying the spaces
wr,p(R n ) for 1 < p < 00 it enables us to get rid of the cumbersome scheme of
several commuting operators ( 1.7 and 1.8) and to apply a scheme with
a single operator ( 1.5). Indeed, we consider the operator B =
-1 (1 + 1 12)1/2 , and show that it satisfies condition (2.19), which is stronger
than (1.15).
For A > 0 the operator (B + M)-1 has the form 'J-l«(1 + 1 12)1/2 + A)-I)'J,
and to estimate its norm we may use Theorem 3.1 again. For the function
<I>( ) = (V I + 1 12 + At = (VU + Ar l
we have I<I>( )I 1/(1 + A). Next,
1 lk a a k <1>( = 2 k 1 lk i I . . . i. ( 1 )
i I . .. ile VU + A
(u = 1 + 1 12)
1 k -} a.
= 1 lk i .. . i L J .
I Ie (vu )k(VU + A)2 )=0 (VU y(VU + A)k-l- j
In denominators of the terms on the right the factors (VU )k+j(VU + A)k-j
neutralize the growth of i I . . . ile 1 lk 1 12k at infinity, and 1/ (VU + A)
1/(1 + A) for the remaining factor. Therefore by Theorem 3.1 we have
II(B + M)-I II L"-+L,, = 'J- 1 ( 1 ) ..; 1 :I A . (3.11)
V I + 1 12 + A L"-+L,,
Now it follows from Theorem 3.2 that wr,p(Rn) is isomorphic to EBr for
1 < p < 00, where E = Lp(Rn) and B = 'J -1«1 + 1 12)1/2)'J.
We establish some more properties of Sobolev spaces.
LEMMA 3.1. The collection Gj) (R n ) of infinitely differentiable functions with
compact support is dense in wr.p(R n ) provided that 1 p < 00.
The lemma can be proved by a standard method which we describe briefly.
A sequence of nonnegative functions Pn(s) E Gj) (Rn) with J Pn(s) ds = 1 is
considered, where the supports of the functions shrink to zero.
Then for x E wr,p(R n ) the functions
Xn(S) = Pn * x(s) = f Pn(s - r)x(r) dr = f Pn(r)x(s - r) dr
are infinitely differentiable, and
6}YXx = P X Dcr. x Dcr. x
n n
(Ial r)
in l.p(R n ). Hence the infinitely differentiable functions xn(s) converge to x(s)
in wr,p(Rn).
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
329
Now we choose a nonnegative function cp(s) E 6D(Rn) equal to 1 for Isl 1
and to 0 for Isl 2. Then the functions YN(S) = o/(s 1 N)xn(s) E 6D (Rn) and
converge to xn(s) in Lp(Rn), 1 p < 00, as N 00. Next,
o/(sl N)D{xn(s) D{xn(s),
and Dik[o/(sl N)]D{-kxn(S) 0 in Lp(Rn) for 1 i n. Therefore D{YN X n
in Lp(Rn) as N 00.
The lemma is proved.
REMARK. In the first part of the proof we have established that the
collection of infinitely differentiable functions is dense in Wr,p(R n ); more-
over, the arguments remain valid for p = 00. If a function x belonging to
wr,oo(Rn) is of compact support, then X n = Pn * X E Gj) (Rn), and X n x in
wr,oo(Rn).
LEMMA 3.2. Let 'T = cp(s) be a diffeomorphism of R n onto itself such that its
derivatives up to order r are bounded in R n and its Jacobian is bounded below
and above by positive constants. Then for 1 < p < 00 the operator cp. x(s) =
x( cp(s)) determines an isomorphism of wr,p(Rn) onto itself.
If we introduce a norm in wr,p(Rn) according to formula (3.10) and take
into account that generalized derivatives change according to classical for-
mulas under a coordinate transformation, then the assertion of the lemma
becomes obvious.
REMARK. If for p = 1 and p = 00 we define the spaces wr,p(R n ) to be the
spaces with norm (3.10), then Lemma 3.2 remains valid for them.
2. Spaces with intermediate smoothness. Let G be an interpolation space
between L},*(O, 00) and Loo,.(O, 00 ). We write
B ,p(Rn) = (Lp(Rn), wr,p(Rn))o,G,G' where (J = Or, 0 < 0 < 1.
According to Theorem 1.9 we have
n
B ,p(Rn) = n (Lp(Rn), Lp(Rn)DJ )o,G,G
j=l
up to isomorphism.
REMARK 3.1. According to Theorem 1.9 we obtain spaces isomorphic to
BG,P(R n ) if we introduce a norm in wr,p(Rn) (1 p 00) by (3.10).
By Theorem 1.8 the norm in each of the spaces on the right can be written
in explicit form. For this we write
Llj(t,s; x) = ( (t) - I)rx = L(-I)k Cr k x (s + kt )
330 v. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
( is the }th unit basis vector) for the rth difference of x(s) with respect to Sj.
Then by Theorem 1.8 an equivalent norm can be introduced in B ,p(Rn) by
the formula
n
II x II B G '" = II x IlL" + L II t --0 II j ( t, s; x) 114 11 G.
j=l
As follows from the arguments of 1, the space B ,p(Rn) does not depend on
r > a up to isomorphism. We may also use (1.41) and (1.42) for the equiva-
lent norms. Then we obtain the following: if a is not an integer, a = m + a,
o < a < 1, then
n
IlxllBG,p = II x l1 4 + L Ilt-aII J(t, s; D j m x)11 4 1IG;
j=l
if a is a positive integer, then
Ilxll BG'P = II x l1 4 + II t-ll1 J( t, s; D j o-lX )11 4 11 G.
In particular, we may set G = Lq,.(O, 00) and denote the corresponding space
by B;,P(R n ). Then the last two norms take the form
Ilxll ,p = II xII L" + j I {{XJ r-aq-IIILlJ( t, 5; Dt x )II dt f/q
(a=m+a,O<a<l) (3.12)
and
IlxIIB;,p = Il x llL" + j I {{XJ rq-11ILl;(t, 5; Da-Ix)ll dt f/q
(a-integral). (3.13)
According to Lemma 1.7, in all formulas for the equivalent norms, instead of
norms of differences of x(s) in l.p(Rn) we may write the corresponding moduli
of continuity of x(s) in Lp(R n ):
w j r ( t; x; p) = sup II j(h, s; x)11 4 .
O<h <1
Formulas (2.10) and (2.11) for the norms with just this alteration have been
proposed by O. V. Besov, and the spaces B;,p(Rn) are called Besov spaces. For
q = 00 we obtain the Nikol'ski; spaces with norms
n
IlxIIB '" = Ilxlllp + L sup {t-aw/(t; Djmx;p)}
j= 1 1
for a = m + a,O < a < 1, and
n
II xII B '" = Ilxll L" + L sup { t- 1 w j 2 ( t, D j o-lX; p)}
j-l 1
for a > 0 integral.
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
331
We also note that if p = q = 00 and 0 is not an integer, then we obtain the
space B:;oo consisting of the functions whose derivatives of order [0] satisfy
the Holder condition of order a = 0 - [0], i.e. the space C[ ](Rn).
We also use Theorem 1.11, which enables us to write the equivalent norm
in B;'P in invariant form not depending on the choice of coordinates in R n .
F or this we wri te
r( T, s; x) = L (-I)k Cr k x (s + kT).
By Theorerrl 1.11 the norm
II x II BG'P = II x 114 + III T 1-<7 II r ( T, s; x) 114 11 G(R'?.)
is equivalent to the preceding ones. In particular, for G = Lq.. we obtain
{ } I/q
II xii Ba" = II x 114 + f l'TI--<7 Q - n lI r( 1", s; x) 11'4 d'T .
For p = q the corresponding spaces were introduced by L. I. Slobodeckii.
They are often denoted by WD,p(R n ) (= BpD,fJ(R n )) and are called Sobolev-
Slobodeckii spaces.
From Theorem 2.4 of Chapter IV we obtain the following theorem.
THEOREM 3.3 (interpolation theorem). The triple
(Lpi (Rnl), WrIJJI(Rnl), B I,pI(Rnl))
of spaces is an interpolation triple of type (J relative to the triple
(L p2 (R n 2), wr2JJ2(Rn2), B 2JJ2(Rn2)),
provided that oil r l = 021 r 2 = (J (0 < (J < 1).
If we consider the two spaces B G °oJ'(Rn) and B I,p(Rn), then for an integral
o I
r > max 0i we have, by definition,
B r: = (Lp' W r ,p)9o.Go.G o and B :,p = (Lp' W r p)9 1 .G 1 .G 1 '
where (Ji = oil r. Then we can apply the reiteration Theorem 2.8 of Chapter
IV, assuming that Eo = EI = G. We then have Fo = FI = G and we obtain
( BDOtP ( R n ) BOI,p ( Rn )) = ( L ( Rn ) wrJJ ( Rn )) = Bo,fJ ( R n ) ( 3.14 )
Go ' G 1 9'.G.G P' 9.G.G G ,
up to isomorphism, where 0 = (1 - (J ')00 + (J' 0 I.
From this the reader may easily infer (Theorem 2.4 of Chapter IV) that a
triple of spaces of the type B 'P with the same p and distinct indices 00, o ,
and 0' of smoothness is an interpolation triple of type (J' relative to another
332
v. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
such triple with another p in general and indices a o , ai', and a'lI of smooth-
ness provided that
a' - a'
o
a' - a'
I 0
a" - ao
a" - a" .
I 0
Moreover, the spaces G occurring in the subscripts may be arbitrary.
Theorems 1.5 and 1.10 lead to the following isomorphism:
(Wro,p(Rn), Wr.,p(Rn))o.,G,G = B ,p(Rn) (ro < r 1 ), (3.15)
where (j' = (a - ro)/(r l - ro).
We apply Theorem 2.10 of Chapter IV to a couple of spaces with the same
index of smoothness. We obtain
(B O,p B o,p ) - (L W r,p ) - B o,p
Go' G. O',Fo.F. - p' olr,H,H - H'
where H = (Go, GI)O',Fo.F..
In particular, if Go = Lqo.*' G I = Lq.,* and Fo = FI = L q ,*, then
1 1 - (j' (j'
H = ( L , L * ) (J'L L = L-, where - = + - .
qo.* q., , q..' q.. q,q q qo ql
From (2.12) it follows that
( BO,P BO,P ) = BO,p
qo' q. O,Lq..,Lq.. Lq.q
and, finally, if q = q,
1 1 1 - (j' (j'
where - = - = + - .
q q qo ql
As we mentioned in Remark 3.1, in studying the spaces B ,p(Rn) it may be
assumed that the norm is introduced in wr,p(Rn) (I G P (0) by (3.10). It
is obvious that, with this norm, the operator D a is a bounded operator from
wr,p(Rn) into wr-1al,p(Rn) for lal < r. It follows from interpolation theorems
and (3.13) that the operator D a acts boundedly from B ,p(Rn) into
BG-1al,p(Rn) for every a > lal.
3. Imbedding theorem. We use results of 1.9 to obtain an imbedding
theore m for the spaces B!;'p. As the operator B we again take
6J -I (y l + 1 12 )'J. This ope rator is continuous in S'(Rn) and so is the operator
(B + AI)-I = 6J« y l + 1 12 + A)-I)'J -I for A > O. Thus, this operator satisfies
condition 1 of Theorem 1.13. If we assume that Eo = Lp and EI = Lq
(0 < p, q < (0), then it follows from (3.11) that inequalities (1.56) (i.e. condi-
tion 2 of Theorem 1.11) are satisfied for B. Condition 4 is also satisfied, since
the restrictions of B to Lp and Lq are automatically defined on S(Rn), which is
dense in- 4 and Lq. It remains to verify condition 3, i.e. to obtain (1.57). We
( BO,P BO,p ) , = B 'P
qo' q. o,Lq..,Lq.. q'
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
333
perform this verification relying on the following generalization of the multi-
plier Theorem 3.1 due to P. I. Lizorkin [217]:
THEOREM 3.4. If the function <I>( ) is continuous together with the derivative
a n<l> la 1 . . . a n and all the preceding derivatives outside the coordinate planes
j = 0 (j = 1, . . . , n), and if
ak<l>
.. . fJ ./ M
a ,
; . ;k . . . . a l:. .
'. 'Ie
where k = 0, . . . , n, 1 i I < i 2 < . . . < i k n, (3 E [0, 1) and 13 =
r . . . !, then the operator qf -I <l>qf acts boundedly from Lp(Rn) into Lq(RlI),
where 1 I p - 1 I q = {3 (1 < p, q < 00). Moreover, the norm of the operator is
proportional to M.
We apply this theorem to the operator qf-l( tv I + 1 12 + A)-I)qf. Repeating
the arguments on p. 327 we arrive at the estimate
ak<l>
l:. . . . l:. l:. f3
; . ;Ie a l:.. . . . a .
'. 'Ie
fJ
M I
V I + 1 12 + A
( = A1]).
1] 13
MIA nfJ-I
1 + 11]1
If we now require that n{3 < 1, we have
ak<l>
l:. . . . l:. l:. 13 M A nfJ - I
; . ;k a . . . . a . I .
'. 'Ie
It follows from Theorem 3.4 that for p < q and lip - I/q < Iin the
operator B = qf-I(I + 2)1/2qf satisfies condition 3 of Theorem 1.13 with
11 = n/p - n/q.
THEOREM 3.5. If 1 < p q < 00, then B ,p(Rn) is imbedded in BG'q(Rn) for
(J = s + n/p - n/ q.
PROOF. First we assume that 11 = n/p - n/ q < 1. Then from the defini-
tion of B ,p(Rn), Theorem 3.2, and Theorem 1.13 it follows immediately that
B ,p(Rn) = ( T (R n ), wr,p(Rn)) = ( L (Rn), W r ,P(RlI) )
s/r,G,G P s/r+v/r.G.G
C (Lq(Rn), wr,q(Rn»s/r,G,G = B(jq(R n ). /
If now 11 > 1, then we partition the interval [p, q] with points Po = P < PI
< . . . <Pk = q so that n/p; - n/p;_1 < 1 (i = 1, . . . , k). According to
334 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
the preceding we have
B ,p(Rn) C BG-n(l/p -1/PI),PI(R n ) C . . .
c BG-n(l/p -1/Pk),Pk(Rn) = B<1 q (Rn).
The theorem is proved.
It is natural to call the number K = nip - (J the dimension of the norm in
the space BG'p. If p and G are fixed, it is clear that a space with smaller
dimension of the norm (with greater smoothness) is imbedded in any space
with larger dimension of the norm ('Nith less smoothness). Theorem 3.5 says
that p < q the space BG'P is imbedded in BtJq if the dimensions of the norms
are equal, and thus in the case where the dimension of the norm of the first
space does not exceed the dimension of the norm of the second space.
4. Function spaces on regular domains. The Sobolev function spaces on an
arbitrary domain Q c R n can be defined as the collection Wr'P(Q) of func-
tions having derivatives D ax (in the sense of the theory of generalized
functions) for lal r and having finite norm
{ } l/p
Ilxll wr.p( ) = Ilxll ( ) + IIDaxll ( ) .
lal r
(3.16)
Since the operators of generalized differentiation are closed, this space is a
Banach space.
If the function x(s) belongs to wr,p(Rn) and the norm in wr,p(R n ) is
introduced by (3.10), then the restriction S x = xl obviously belongs to
Wr,p(Q), and IIS xll wr.p( ) IIxll wr.p(Rn). Thus, the restriction operator Sf1, has
norm 1 in all spaces Wr,p.
If 'T = cp(s) is a diffeomorphism of the domain Q onto a domain l having
bounded derivatives up to order r and Jacobian bounded above and below by
positive constants, then the operator cp* x(s) = x( cp(s)) is an isomorphism of
Wr,P(QI) onto Wr,p(Q) (see Lemma 3.2).
o
In Wr,P(Q) we may single out the important subspace Wr,p (Q), which is the
closure, in Wr,p(Q), of the set of infinitely differentiable functions belonging
to Wr,p(Q) and having supports inside Q. It is obvious that the subspace
o 0
Wr,p (QI) is mapped onto the subs pace Wr,p (Q) by the diffeomorphism
described above.
On the set of infinitely differentiable functions with support in Q the norms
o
of Wr,p(Q) and Wr,p (Q) coincide, and therefore every function belonging to
o
Wr,p (Q) extended to zero outside Q belongs to wr,p(Rn). It is clear from the
proof of Lemma 3.1 that if a function from wr,p(Rn) (1 p < 00) has
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES 335
support in Q, then its restriction to Q belongs to r,p (Q). (If Q is bounded,
this is also true for p = 00.) The following question arises: Does the last
assertion hold for functions from wr,p(R n ) with support in Q? A positive
answer to this question will be given for a special class of domains later (see
Lemma 3.4).
We also note that r,p (Q) (and thus Wr.P(Q) as well) is densely imbedded
in Lp(Q).
DEFINITION. A domain Q is said to be r-regular if there exists an operator
lIr which extends functions from Q to the whole of R n and is a bounded right
inverse of the restriction operator Sg, in the spaces Wk,P(Q):
Sg,lIr x = x (x E Wk,P(Q)), IIlIr xii Wk,p(Rn) C Ilxll WIc,P(g,).
If a domain Q is r-regular for every r = 0, 1, . . ., we call it regular.
In an r-regular domain the space Wr,p(Q) consists of restrictions to Q of the
functions belonging to Wr,p(R n ).
It follows from the definition and the remark to Lemma 3.1 that the
restrictions of infinitely differentiable functions from Wr,p(Rn) to an r-regular
domain Q are dense in Wr.p(Q).
From the fact that Sg, is right invertible in the spaces Wk,p provided that
the domain Q is r-regular, it follows that Wk'P(g) is isomorphic to the
quotient space of Wk,P(R n ) modulo the subspace of functions equal to zero in
Q (Sg,x = 0). In Wk'P(Q) an equivalent norm can be introduced by the
formula
IlxllwkoP(O) = inf Ilxll Wk,p(Rn).
x E snx
The operator lIrSg, projects the whole space Wk,p(Rn) onto a subspace
isomorphic to Wk.p(Q).
By means of Theorem 3.2 it is easy to see that for 1 < p < 00 and for an
r-regular domain Q the norm (3.16) is equivalent to the norm
( n a r P ) lip
II x II w,.p(12) = II x 11"4,(12) +.L a
J= 1 4(g,)
in wr.p( ). Now we introduce the spaces Bl;'P(Q) by setting
Bl;,P(Q) = (Lp(Q), Wr'P(Q))olr,G,G.
(3.17)
For an r-regular domain, Sg, is a right invertible mapping of any couple
Wko.po(Rn), Wkl'PI(Rn) onto the couple Wko.po(Q), Wko.po(Q) (ko, k l r), and
so we can apply the results of 4.5 of Chapter I. In particular, applying
Theorem 4.10 of Chapter I to the functor ( , )olr,G,G' we conclude that Bl;'P(g)
336 V. INTERPOLATION IN SPACES OF SMOOTH FUNCfIONS
consists of restrictions to Q of the functions belonging to B ,p(Rn). In B ,P(Q)
the norm is equivalent to
II x II BGJ'(fl) = inf II x II BcJ'(Rn).
SgX = x
In other words, B .p(Q) is isomorphic to the quotient space of B JJ(Rn)
modulo the space of functions from B ,p(Rn) equal to zero for S E Q.
If we take account of the isomorphism (3.15), an application of Theorem
4.10 of Chapter I to wrOJP(Rn), wr,p(Rn) and WrOJP(Q), Wr.p(Q) (ro < r) leads
to the isomorphism
(Wro'P(Q), Wr,p(Q))o',G,G = B JJ(Q) (ro < r), (3.18)
where (j' = (0 - ro)/(r - ro).
From what has been said it is clear that the interpolation Theorem 3.3
remains true if the Rn, are replaced by ri-regular domains Q; in RIIt (i = 1, 2).
The imbedding Theorem 3.5 with R n replaced by an r-regular domain with
r > 0 also remains valid.
A simple example of a regular domain is the half-space R:+ = {s: S =
(s', sn)' sn > O}.
THEOREM 3.6. The half-space R + is a regular domain.
PROOF. First we prove that the restrictions to R:+ of infinitely differentia-
ble functions from wr,p(Rn) form a dense set in Wr'P(R;:+). Let x E
wr,p(R +). We consider the functions xh(s) = X(Sl' . . . , sn-l' sn + h) (h >
0). By virtue of the continuity of translation in the spaces Lp and the fact that
DQXh(S) = DQX(Sl'. .., Sn-l' sn + h), the functions X h converge to x in
wr,p(R +) as h O. Now let cp(t) (t ERn) be a nonnegative infinitely
differentiable function equal to 1 for t -h/2 and 0 for t -h. Then the
function y(s) = cp(sn)xh(s) for sn -h and 0 for sn -h belongs to wr,p(Rn)
and coincides with xh(s) for Sn O. By the remark to Lemma 3.1 the function
y(s) can be approximated with arbitrary accuracy by an infinitely differentia-
ble function z in wr,p(R n ). Then the restriction of z(s) to R + approximates
xh(s) (and consequently also x(s) for sufficiently small h) with arbitrary
accuracy in Wr,P(R:+).
We define the desired operator IIr by the Hestenes-Whitney method first
for functions x(s) infinitely differentiable on R + by setting
IIrx(s) =
X(Sl' . . . , sn)
r+l
AkX(Sl' . . . , Sn-l' -ks n )
k=l
for Sn 0,
for sn O.
(3.19)
3. SPACES OF SMOOTH FUNCfIONS OF n VARIABLES
337
The numbers Ak are chosen so that
a i a'
----:-IIrx(sl' . .., Sn-l' 0) = ----:-X(SI' . .., Sn-l' 0) for i = 0,1, . . ., r.
a a
For this it is sufficient to determine them from the system
r+ 1
L (-k )iAk = 1
k=l
of linear equations with determinant different from zero (Vandermonde
determinant).
The function IIrx(s) has continuous derivatives up to order r inclusive, and
therefore, if x E wr,p(R +), then IIrx E Wr,p(R n ). An elementary calculation
shows that
(i = 0, 1, . . . , r)
IIIIrxl1 Wk,p(Rn) Cllxll wk,p(R:..) (k r).
It follows from this and the preceding that IIr admits closure by continuity
to the whole space Wr,P(R:+). From the continuity of Sg, it follows that the
relation Sg,IIrx = x remains valid for all x E wr,p(R +).
The theorem is proved.
REMARK 1. It follows from Lemma 3.1 and the above proof that under the
condition 1 p < 00 the collection Gj) (R +) of restrictions of infinitely
differentiable functions on R n is dense in Wr,P(R:+).
REMARK 2. It is clear from the definition of the extension operator (3.19)
that if the function x(s) has support in the half-ball Isl a, sn 0, then the
function IIrx(s) has support in the ballisl a.
We also mention another important property of Wr,P(R:+).
LEMMA 3.3. The space r,p (R +) consists oj restrictions to R + oj all
functions belonging to wr,p(Rn) with support in R +.
PROOF. As we indicated in the definition of the spaces r,p ( ), every
element of r,p(R +) falls in wr,p(R +) after it is defined to be zero outside
R +. Conversely, let x E wr,p(R n ) and let x have support in R +. Then the
functions x_h(s) = X(Sl, . . ., sn-l' sn - h) (h > 0) have supports in R:+, and
X_h x in wr,p(R n ) as h O. By the remark to Lemma 3.1 the function X_h
can be approximated by functions infinitely differentiable in R n and the
approximating functions can be chosen so that their supports lie in R: +. The
restrictions of these functions to R: + will approximate the restriction of x(s)
in Wr'P(R:+), i.e. SR" x E Wr,P(R:+).
" ..
The lemma is proved.
338
V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
Theorem 3.6 and Lemma 3.3 imply the following remarkable decomposi-
tion up to isomorphism:
o
wr,p(Rn) = wr,p(R +) EB wr,p(R _).
Indeed, since R + is regular, the space wr,p(R +) is isomorphic to the
subspace of wr,p(R n ) complementary to the subspace of functions equal to
zero in R +, i.e. with supports in R _. The last subspace is isometric to
o
wr.p(R _) by virtue of Lemma 3.3.
Now we pass to a large class of domains Q with boundary aQ such that
Q = Q u aQ is a Coo-submanifold of R n with compact boundary.
We describe our assumptions more precisely.
We assume that there exists a finite covering of the boundary aQ of Q by
bounded domains VI' . . . , V N in R n such that the following conditions are
satisfied:
a) The mappings cp;( s), i = 1, . . . , n, are given, which are diffeomorphisms
of the domains V j onto the open ball V of R n with coordinates T =
('T I , . . . , 'Tn)' and the Jacobians of the diffeomorphisms are positive.
b) Under the mapping cpj the set V j n Q, i = 1, . . . , N, is taken to
V n + = V n R +, where R + = {'T: 'Tn > O}.
We shall consider the covering { V j } and the mappings {cp;} fixed. We add
one more open set V o to the system {V j } so that U o c Q and the system Uep
VI' . . . , V N forms a covering of the whole domain Q.
We fix a partition of unity corresponding to the covering { ll;} of Q, i.e. we
fix nonnegative infinitely differentiable functions {1I;(s)} such that support of
1Ij(s) is contained in V j and L 1I;(s) = 1 for s E Q . By means of this partition
of unity, to every function x(s) defined on Q there corresponds a collection of
functions {1I;(s)x(s)} defined in the domains V; n Q. For i > 1 the mapping
1Ij(s)x(S) 1Ij( cpj-l( 'T)x( cp;-I( 'T»))
converts these functions into functions defined in the half-ball V n +. These
functions are equal to zero in the neighborhood of the nonplanar boundary of
V n +. We extend them to zero in R + outside V n + and denote by X;*(T) the
functions thus obtained. It is easy to verify that the mapping making the
collection {xt} of functions correspond to the function x E Wr,P(g) is a
bounded operator from Wr'P(Q) into wr,p(R +) X . . . X wr,p(R +). Now
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
339
we introduce the operator
N
IIr,Qx(s) = L IIrx;*(<Pi(s» + 17o(S)X(s),
; = I
N
S E U Vi'
;=0
N
= 0, s fl U Vi.
;=0
By Remark 2 to Theorem 3.6 the function IIr X i * ( T) has support in V, and
so IIrx;*( <p;(s)) is defined and has support in V;. Its extension by zero outside
ll; belongs to wr,p(Rn). Thus, IIr,Qx E wr,p(Rn), provided that x E Wk'P(Q)
(k r); moreover,
IIIIr,QxII wk.p(R n ) C II xii Wk.p(Q).
It is clear from the construction that
N
IIrQx(s) = L 17;(S)x(s) = x(s)
,
;=0
for s E Q. We have arrived at the following assertion:
THEOREM 3.7. Any domain Q with properties a) and b) is regular.
REMARK 1. The reader can easily find what minimum smoothness must be
required of the mappings <Pi in order that Q be r-regular with given r.
The following lemma is easily obtained from Lemma 3.3.
o
LEMMA 3.4. If a domain 2 has properties a) and b), then the space Wr,P(Q)
consists of restrictions to Q of all functions from wr,p(Rn) with support in Q.
PROOF. If x E wr,p(Rn) and x has support in Q, then the functions x;*
(i = 1, . . . , N) constructed from the restrictions of x to V; n Q as above and
extended by zero outside V n + belong to Wr,p(R n ) and have supports in V n +.
By Lemma 3.3, they can be approximated in wr,p(R +) by functions y;( T)
infinitely differentiable and having support in V n + n R +. Then the func-
tions y;( <p;(s)), extended by zero to Q outside V; n Q and infinitely differentia-
ble with support in ll; n Q, belong to Wr'P(Q), and approximate the functions
17;(S)X(s). By virtue of Lemma 3.1 the function 17o(X)X(s) can be approximated
by infinitely differentiable functions Yo(s) with support in V o . Then the
functions Yo(s) + L y;( <p;(s)) have support in Q and approximate x in
Wr,p (Q).
The lemma is proved.
340 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
Denote by Qe the domain complementary to Q. If Q has properties a) and
b), so does Qe. Theorem 3.7 and Lemma 3.3 imply the following decomposi-
tions up to isomorphism:
o
wr,p(R n ) = Wr,P(Q) EB Wr,p(Qe)
and
o
wr,p(R n ) = Wr,P(Q) EB Wr,P(Qe)
In the first decomposition, for example, the subspaces of wr,p(Rn) isomorphic
to the summands on the right are obtained by means of the projections II,-nSQ
and I - I1 rQ S Q , respectively.
We write
P( ) = ( Lp( ), r.p( » ) . (3.20)
alr,G,G
By virtue of the interpolation theorem applied to the operator of imbedding
o
Wr'P(Q) in Wr'P(Q) we have
o
B P(Q) c B ,P(Q).
o
By Lemma 2.14 of Chapter IV the space Wr,P(Q) is densely imbedded in
o
B P (Q) (a < r), and since the infinitely differentiable functions from Wr,P(Q)
o 0
with support in Q are dense in Wr,P(Q), they are also dense in B P (Q), i.e.
o
B P (Q) can be defined as the closure in B 'P(Q) of the infinitely differentiable
functions from B ,P(Q) with support in Q.
By Theorem 4.10 of Chapter I, for a domain having properties a) and b) we
have
o
B ,p(Rn) = B ,P(Q) EB B a,p (Qe)
and
o
B ,p(Rn) = B P(Q) EB B(';'P(Qe)
up to isomorphism.
Next, the operator I - II rQ SQ projects wr,p(Rn) onto a subspace isomor-
o
phic to W P (Q), and the isomorphism is established by the restriction
o
operator SQ. Therefore the same operator projects B a,p(Rn) onto B a,p (Q), and
the isomorphism is also established by the restriction operator SQ. Thus, for a
o a P
domain with properties a) and b) the space jl (Q) consists of the restrictions of
all functions from B ,p(Rn) with support in Q.
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
341
5. Function spaces on the boundary and trace theorems. We need to view the
space wr,p(R +) in a new way. If x E Lp(R +) for p 1, then by Fubini's
theorem we have
flx(SW ds = oo {Ln)X(S', snW ds'} ds n
(S = (s', sn))
where the inner integral on the right exists for almost all sn. This means that
the function x(s) = x(s', sn) generates the abstract function x(sn) of Sn with
values in Lp(Rn-l) defined for almost all sn' and
IIxlIL,,(R n ) = {£OOlli(Sn)II"4,(Rn-,)ds n } liP.
Conversely, every function x E Lp(O, 00; 4(Rn-l)) induces some function
x(s', sn)' if for every Sn we fix some function of s' in the class of equivalent
functions which is the value of x(sn) in Lp(R n - 1 ). There arise the (technically
difficult) problems of strong measurability of the function x(s) constructed
from x(s, s') and of the measurability of x(s) constructed from x(s). We shall
not discuss these questions, but refer the reader to the book [11], where they
are treated in a rather general situation. Thus, as far as the indicated
problems are concerned, we have shown that the space 4(R +) can be
considered isometric to 4(0, 00, Lp(R n - 1 )). Finally, the last space coincides
with Lpl/P .(0, 00, Lp(Rn-l)).
We note that this isometric correspondence is lacking for p = 00, although
the formal arguments remain valid.
Now let the function x E Lp(R +) have generalized derivative axjas n E
4(R +) with respect to sn. Ibis means that for every function cp belonging to
Gj)(Rn) with support in R + we have
f a ax (s)q:>(s) ds = - f x(s) a aq:> (s) ds.
sn sn
We choose cp(s) = o/(s')p(sn), where 0/ E Gj) (Rn-l) and p E Gj) (R 1), and p has
support in (0, 00). Then
1 o/(S') (00 a ax (s', Sn)p(sn)dsnds' = ( - 1 o/(S') (00 x(s', Sn)p'(Sn) cis n ) cis.
Rn - I J 0 s n Rn - I J 0
Since 0/ E GD(Rn-l) is arbitrary, it now follows that
1: 00 a ax (s', sn)p(sn) ds n = _ 1 00 x(s', Sn)p'(Sn) cis n
o 0
almost everywhere.
It can be verified by means of the Holder inequality that the left and right
sides of the last equality belong to Lp(Rn-l), and so they coincide as elements
342
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
of this space. This means that x as an abstract function with values in
Lp(Rn-l) has generalized derivative axjas n .
Conversely, let y(S', sn) E Lp(R +) be the generalized derivative of x(s', sn)
E Lp(R +) as an abstract function with values in Lp(Rn-l). This means that
for every function cP E Gj) (R I) with support in (0, 00) we have
CXJy(s" sn)q:>(sn) ds n = - {X> x(s', sn)qJ'(sn) ds n .
The equality is understood here as equality of two functions in Lp(Rn-I). In
particular, it is satisfied for almost all S'. Now let CPa(sn) be a family of
functions belonging to Gj) (R n ) with support in (0, 00) which is compact in
C 1(0, 00). Then
1 00 y (s', Sn) q:>a ( Sn) ds n = - (X> x( s', Sn)q:> ( Sn) ds n (3.21)
o )0
almost everywhere for all a. Indeed, choosing a countable net everywhere
dense in the set CPa (in the metric of C 1(0, 00 )), we can construct a set of
measure zero in R n + 1 such that (3.21) holds for all points s' not in this set and
for all functions CPa of the net. Passage to the limit shows that the equality
holds for s' outside this set for all a.
Now let cP E Gj)(R n ) and let cP have support in R +. The family of functions
CPs,(sn) = cp(S', sn) is compact in C 1(0, 00), and so there exists a set e of
measure zero in R n - I such that
1 00 1 00 a<p-,
y(S', Sn)CfJs'(Sn) ds n = - x(s', Sn)- a s (sn)ds n ,
o 0 sn
for s' fi e and for all S' E R n - I. In particular, it also holds for S' = s', i.e.
(ooy(S', sn)CP(s', sn) ds n = - (00 X(S', sn) a aq:> (s', sn) ds n
)0 )0
for s' fi e. Integrating with respect to S', we obtain
J y(s', sn)q:>(s', sn) ds = - J x(s', sn) a aq:> (s', sn) ds,
sn
which means thaty = axjas n in the sense of generalized functions.
Thus, the assumption that the generalized derivative ax jas n belongs to
Lp(R +) is equivalent to the existence of the derivative of the abstract
function
X(S', sn) E Lp(O, 00, Lp(Rn-I»)
in the sense of generalized functions.
It is clear from what has been said that the assumption that x belongs to
wr.p(R +) (1 p < 00) is equivalent to the assumption that the abstract
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
343
function x(s', sn) of sn takes its values in wr,p(Rn-l) and its derivative
x(r)(s', sn) (with respect to sn) takes its values in Lp(Rn-l) and
/lxll wr,p{R:+) = IIxll .1(wrJ'{Rn-I» + Ilx(r)II4J.1(L p {R n - I »
Following the notation of 2, we may write
wr,p(R +) = Wr(llp, L p ,*, wr,p(Rn-I); lip, L p ,*, Lp(Rn-I»). (3.22)
If I <p < 00, then wr,p(Rn-l) coincides with the domain of the rth power
of the operator B' = '!f'-I«I + 1 '12)1/2) ', where the prime indicates that the
Fourier transform is taken in R n - I . According to (3.11), the operator B'
satisfies condition (2.22). Equality (3.10) shows that in this case
W r ( lip, L p ,*, wr,p(R n - I ); lip, L p ,., Lp(Rn-I»)
= W r ( lip, 4,*, Lp(Rn-I), B') (3.23)
and we can apply Theorems 2.3 and 2.4.
THEOREM 3.8 (on traces). If I < p < 00, then the operator defined on the
infinitely differentiable functions from wr,p(R +) by the formula
Tr: x ( x(s', 0), (s', 0), . . ., :; (s', 0»)
admits closure to a bounded operator from W r.p(R +) onto
IIj:bBr-J-I/p,p(Rn-I). There exists a bounded linear right inverse of Tr.
PROOF. It follows from (3.22) and (3.23) that in our case 110 = 111 = TJ =
lip,
F. o =F I =G=L
p,.'
A - E - W r . p (R n-l )
o - B'r - ,
and
Al = E = Lp(Rn-I).
Next, jrrutX is the largest integer not exceeding r - I and smaller than
r - I I p, i.e. jrnax = r - I. The operator y defined in Theorem 2.3 coincides
with the operator Tr on smooth functions, and according to the theorem it
maps wr,p(R +) continuously into
r-l
II ( wr,p ( Rn-I ) , 4( Rn-I »)
. 8 j ,4... I "..
}=o
r-l
= j!.I o (Lp(Rn-l), Wr,P(R n _ 1 ) )1-8,.4..,4..'
344
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
where OJ = (j + 1/ p) / r. By definition, the last space is
I1j:6Br-j-l/p,p(Rn-I). The first part of the theorem follows from this and
Lemma 3.2. The second part follows from Theorem 2.4 immediately.
The theorem is proved.
COROLLARY 1. Let D a be differentiation of order lal < r. The operator D a
admits closure from the set of infinitely differentiable functions to a bounded
operator from wr,p(R +) into B;-lal-l/p,P(R n - I ).
Indeed, D a = (acx,,/asncx,,)D a ', where D a ' contains derivatives only with
respect to the variables Sl' . . . , sn-I. The operator D a ' acts boundedly from
Wr'P(Q) into wr-1a'I,p(Q), and according to the theorem so does the operator
a a,. /asncx" from wr-1al,p(Q) into
Br-la'I-a,. -1/ p ,P(R n - 1 ) = B;-lal-l/p(R n - I ).
COROLLARY 2. If B = Llal 1 ba(s')D a is a boundary differential operator of
order k < r on R n - I with bounded infinitely differentiable coefficients, then its
closure is a bounded operator from wr,p(R +) into Br-k-l/p,p(Rn-I).
We note that if the traces
ajx/as = Yi(S') E B r -j-l/p,P(R n - 1 ) (j = 0,.. ., r - 1)
of x E wr,p(R +) are known, then from them we can calculate the value of
the boundary differential operator of order k < r at this function. Indeed,
Bx s = L ba(s')Da(x) = L ba(s')D a '
lal <.k lal <.k sn
= ba(s')Da'ycx,,(s').
Moreover, from what we said at the end of subsection 2 it follows that
Bx E B;-k-l/ p ,P(R n - I ).
Soon we shall have to use this remark in the following situation. In the
half-space R + let a new system a = (aI' . . . , an) of coordinates be given in
such a way that on the boundary plane R n - I we have a i = Si' i = 1, . . . , n,
and let functions {3j(s') E B;-j-l/p,p(Rn-l) be given. We would like to
construct a function x E wr,p(R +) such that
aix
a i
an S = (J -0
12 12
= {3i( s i).
The derivatives ajx/as = "'Ii(S') can be represented in the form of differential
operators of order j in the variables ai' and therefore by the above we may
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
345
express them in terms of the functions f3;(s'). If we assume that the transition
functions from coordinates 0 to coordinates s and conversely have bounded
derivatives up to order r - I, then the functions 'Yj(s') thus determined will
also belong to the spaces B;-j-I/P,P(R n - 1 ). According to Theorem 3.8, we
can construct from them a function belonging to wr,p( +) such that
ajx /as = 1)(s'). It is obvious that we then have a;x /ao = f3;(s').
By Theorem 2.5 the operator Tro x = x(s', 0) effects a right invertible
mapping of the Banach couple WI,P(R:+), Wr,P(R:+) onto the Banach couple
B)-I/p,P(R n - I ), B;-I/P(R n - 1 ). We apply the corollary of Theorem 2.5 to the
functor ( , )o'.G.G' where 0' = (0 - I)/(r - 1) and I < 0 < r. From (3.17) and
(3.12) we find that
( W I,p W r,p ) - B O,P (R n )
(R +)' (R +) O',G,G - G n+
and
(B) -1/P,P(R n - I ), B;-I/p,p(Rn-l)o'.G.G) = B -I/p,p(Rn-I).
From the indicated corollary we obtain the following theorem.
THEOREM 3.9. If I < p < 00, then for 0 > I the operator Tro x = x(s', 0) is
a right invertible bounded operator from B 'P(R:+) onto B:-I/P(R: I).
REMARK 1. Theorem 3.9 holds for 0 > 1/ p. This fact can be proved by
means of more complicated constructions.
REMARK 2. If we apply the preceding results to functions whose supports
are concentrated in some half-ball VI = {s: Isl a;, Sn O}, then the right
invertible operator occurring in the results can be constructed so that its
range consists of functions with supports in an arbitrary half-ball V ::> VI.
Indeed, for this it is sufficient to multiply an arbitrary right inverse by an
infinitely differentiable function equal to 1 on VI and 0 outside V.
Now consider a domain Q with properties a) and b) (see subsection 4). The
partition of unity {11;} considered in subsection 4 induces a partition of
unity { J relative to the covering {U;}7 of Q \ U o , and, in particular,
relative to the covering of the boundary aQ with the sets U/ = U; n aQ
(i = I, . . . , N). We denote by V' the intersection of V with the space
R n - I = {'1": '1"n = O}. With every function o/(s) on aQ functions 1/1;* can be
associated, defined by 0/;*( '1"') = 17;( CfJ;-I( '1"'))0/( CfJ;-I( '1"')) for '1"' E V' and extended
by zero to all of R n - 1 outside V'. We denote by Wr,p(aQ) the space of
functions 0/ defined on aQ for which all 0/;* belong to wr,p(Rn - I). In this
space we introduce the norm
{ N } I/p
110/11 wr"'(aQ) = ; l 1I00;*IIPwr"'(Rn-') .
346 V. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
Similarly we may define the spaces
Bc,p(aQ) = (Lp(aQ), Wr,p(aQ) )o/r,G,G.
We need the following auxiliary assertion.
LEMMA 3.5. Assume that the function x( 7"') belonging to :Jvr,p (V') has support
in V'. The function x(s') = X( <Pi(S')) (s' E V/), extended by zero on aQ outside
U;', belongs to Wr,p(aQ).
PROOF. We consider the function (s') = 1lj(s')X(s'). Its support is con-
tained in V/ n V/, and so the support of the function
*( T') = 1lj( <pj-I( T') )X( <Pi 0 <pj-I( T'»)
lies in <Pj( V/ n V/). The function *( T') is obtained from the function
11.;( <p;-I( T'»)X( T') by means of the diffeomorphism <Pi 0 <pj-I of the domain
<Pj( V/ n ') onto the domain <p;( U/ n V/). Since
o
1lj.( <Pi-Ie T') )X( T') E wr,p( <p;( V/ n V;)),
o
we have xj E Wr,P(<pj(V/ n V;), and so *, extended by zero to R n - 1
outside V', belongs to wr,p(Rn-I).
The lemma is proved.
Denote by VI a ball containing all images of the supports of the functions
l1;(S) under the mappings <Pi and such that VI C V. Let V; denote the
intersection VI n R n - I . If 0/ E Wr,p(aQ), then by construction the 0/;* have
o
supports in V{, and so they belong to Wr,p(V{). Now let {x;(T')}f be a
o
collection of functions belonging to Wr,p (V{). By Lemma 3.5 the functions
X;( <p;(s'», extended by zer.o to all of aQ outside <p;( VI), belong to Wr,p(aQ),
and so
N
L X;( <PieS'») E Wr,p(aQ).
i= I
It is clear that if 0/ E Wr,p(aQ) and X; = 0/;*, then Lf-I X;(<p;(s'» = o/(s').
Thus we obtain a right invertible mapping of :Jvr,p (V{) x.. · X :Jvr,p (V{)
(N factors) onto Wr,p(aQ). From (3.20), (3.24) and Theorem 4.10 of Chapter I
it follows that the space BG,r(aQ) consists of all functions 0/ defined on r for
o
which the restrictions of the functions 0/;* to V; belong to B p (V{) or,
similarly, the functions 0/;* themselves belong to Br,p(Rn-I). An equivalent
norm is introduced in BG,p(aQ) by the formula
II 0/ II B .p(aQ) = {L II 0/;* 1I .p(R.- ') } 1/ p.
3. SPACES OF SMOOTH FUNCTIONS OF n VARIABLES
347
We note that Lemma 3.5 remains valid if Wr,p is replaced by B 'P in its
formulation.
We need a special coordinate system in the neighborhoods Vi (i =
1, . . . , N). The mappings CfJ; induce a coordinate system T = (T I , . . . , Tn) in
U;. In particular, on the boundary aQ we obtain the coordinate system
T' = (T I , . . . , Tn-I). Denote by n(s) the distance of s from aQ and by
(al'. . ., an-I) the coordinates of the point of aQ closest to s. We shall
assume, without loss of generality, that the covering {Vi} is chosen so fine
that the numbers (a l ,. . ., an-I' n) yield a sufficiently smooth coordinate
system in ll;.
Now let x E Wr'P(Q). We have xt E Wr,pCR:+), and the supports of the x i *
are contained in VI. For the smooth function x the traces on the boundary of
the domain are defined in a natural way:
x( r', 0) = 1/t0( r'), (r', 0) = 1/t'( r'), . . ., : (r', 0) = 1/tr-'( r').
By the Leibniz formula we have
a J ( 71; x )
anJ
n a k TJ .
- L Sk 1/J-k(T')
aD k=O an aD
(3.25)
and, furthermore,
a} = a J ( TJ , .x . )
1/J( r') = £.J .,
an} aD i= I an} aD
Remembering that x;*( T) = 71;( CfJ;-I( T))X( T( CfJ;-I T )), we obtain that the deriva-
tive a J (1J.;x)/an J laD' upon application of the mapping CfJ;, will be expressed in
the form of a differential operator BJ of order j in the variables T with
infinitely differentiable coefficients, applied to x;*( T). By Corollary 2 of
Theorem 3.8 these operators act boundedly from wr,p(R +) into
Br-J- l /p,P ( R ) Thus if x E wr,p ( ) then x'!'" E wr,p ( Rn ) and B.x'!'" E
P n-I ., " n+ } I
B;-J-I/P(R n - I ), and the support of the last function is contained in V;. Since
Lemma 3.5 is valid for the spaces B;-J-I/P,P, this implies that
(3.26)
a J ( 71JX )
anJ
E Br-j-l/p,p(aQ).
aD
Then it follows from (3.26) that 1/J E Br-J-I/p,p(aQ). Thus the mapping
x (0/°, 0/ I, . . . , o/r- I) admits extension by continuity to a bounded operator
from Wr,p(Q) into IIj:bB;-J-I/p,p(aQ).
N ow we construct a right inverse of the mapping just obtained. If a system
{1/J}O-1 of functions is given and 1/J c B;-J-I/p,p(aQ), then from it we can
calculate the right side of (3.25), which will also belong to B;-J-I/p,p(a ) and
348
v. INTERPOLATION IN SPACES OF SMOOTH FUNCTIONS
have support in U;'. After the mapping cp; this function turns into a function
/3ij E B;-J-I/p,P(V{). Now we are in the situation described on p. 344. In the
half-space R + we can introduce a coordinate system (aI' . . . , On = n) by
"transferring" it from ll; by means of cp;. It will have the properties described
on p. 344, and therefore we can construct a function y; E Wr,P(R:+) such
that ay;/an J = /3;J (j = 0, . . . , r - I). By Remark 2 this construction can be
done so that the supports of the functions y; are contained in V n +. The
functions y;( cp;(s)) = y;(s) belong to Wrp(Q) and satisfy the relation
aJ - n a k .
Yi = cf lJti-k. (3.27)
ani an k=O an an
If we now consider the functiony(s) = = I y;(s) E Wr,o(Q), then by (3.26)
and (3.27) we have
aJy
= 0/'.
anJ an
It is easy to check that the construction of y from { k } - I is effected by
means of a bounded linear operator. We have arrived at the following
assertion.
THEOREM 3.10. The correspondence
x { X I ,H2' a a x , . . ., a r - I X }
n a an r - I a
is a bounded right invertible mapping of Wrp(Q) onto IIj:bBr-J-I/p,p(aQ).
Similar arguments, using Theorem 3.9 and taking into account Remark I
(which we did not prove), lead to the following assertion.
THEOREM 3.11. If I <p < 00 and a > lip, then the operator x xla is a
bounded right invertible operator from B 'P( ) onto B -I/p(a ).
NOTESONTHELnERA
Chapter I
1. As has been mentioned in the Foreword, the notion of imbedding of Banach spaces first
appeared in a paper by Sobolev [355]. The important notion of relative completion has been used
implicitly by S. G. Krein and Petunin [184]; it has been introduced explicitly by Gagliardo [139]
and studied in detail by Aronszajn and Gagliardo [55].
2. Lemmas 2.1-2.3 and Theorem 2.3 are contained in [55], Theorem 2.1 in [184], and
Theorem 2.2 in Berens' book [4].
3. Intermediate spaces for a couple of Banach spaces were first considered by Lions [202].
Formula (3.6) and its consequences are due to Sedaev [41]. Lemma 3.4 is taken from Calderon
[95]. The dual spaces of a sum or an intersection of spaces were studied by Aronszajn and
Gagliardo [55] (see also [232]); they also introduced the sum and intersection of a family of
Banach spaces.
4. The notions of interpolation triples and interpolation spaces have been introduced in one
form or another in all publications devoted to the abstract theory of interpolation of linear
operators. The most important results of 4 are due to Aronszajn and Gagliardo [55]. In
connection with subsections 4 and 5, see [37]. The important Theorems 4.9 and 4.10 were
obtained in [55].
Chapter n
Introduction. Ideal Banach lattices are also called Banach function spaces; their properties are
described in [29] and [49] (see also [95], [188], [25] and [47]). Concerning Lebesgue space, see
[305].
1. Theorem 1.1 was obtained by Peetre [284] (concerning (1.7), see [122D. For Lemma 1.1, see
[107]. For more detail on logarithmically convex functions, see the book [39]. Lemma 1.4 was
obtained in [59].
2. Theorem 2.1 was obtained by Krein and Semenov [191]. The books [8], [17], and [50] have
much information on rearrangements of measurable functions. Many properties of rearrange-
ments have been established in [87], [88], [95], [172], [317], and [312]. Inequality (2.40) is
published here for the first time.
3. Formulas (3.4) and (3.5) were obtained by Peetre in [262] and [261] (see also Oklander
[250]). The important inequality (3.14) was established by Lorentz and Shimogaki [227]. Orbits of
the semigroup of contractive operators were studied by Ryff [311].
349
350
NOTES ON THE LITERATURE
4. Symmetric spaces with the additional assumption that the norm is semicontinuous were
introduced by Lorentz under the name spaces invariant under permutations in the book [28] (see
also Luxemburg [231]). Without this assumption they were studied by Semenov [331]; the main
results of subsection 1 were obtained by him. The main theorem describing all interpolation
spaces between LI and Lao was proved by Calderon [96] by another method. This theorem has a
long history: for integral operators in Orlicz spaces it was first proved by Orlicz [256], for integral
operators in spaces invariant under permutations by Lorentz [28], and for arbitrary operators in
separable spaces or symmetric spaces dual to separable spaces in [241] (see Theorems 4.9 and
4.10). The theorem has been generalized to the case of nonlinear operators satisfying a Lipschitz
condition in [257], [227], and [349]. We note that Browder [85] has obtained a general theorem
enabling us to obtain interpolation theorems for Lipschitz operators from interpolation theorems
for linear operators.
The hypothesis of Theorem 4.3 can be formulated in the following way: the assumptions that
y E E, x E LI + Lao, and K(t, x) < K(t,y) imply that x E E and IIxllE < lIyllE. In connection
with this there arose the following conjecture. Let (Ao, AI) and (Bep B I ) be two Banach couples.
For the triple (A A I' E) to be an interpolation triple relative to the triple (Bep B I , F) it is
necessary and sufficient that the assumptions y E E, x E Bo + B I , and K(t, x, Bep B I ) <
K(t, y, Ao, A I) imply that x E F and IIxliF < CIlYIlE. This conjecture has been confirmed for
the following couples: 1) (Lq, Lao), (Lq, Lao) (Lorentz and T. Shimogaki [228]; Sedaev [324]);
2) (/fO,/fl), (Ira, If I) (ao and a l are weights) (Sedaev and Semenov [327]); 3) (4WO, Lpw 1 ),
(4 WO , 4 W1 ) (Sadaev [324]); 4) (Aep A I) (L:O, L:I) (Peetre [37] and Sadaev [41D; 5) ( , Lao),
(LI' Lao) (Y. I. Dmitriev [117]); 6) (LiOLi l ), (LI' Lao) (Y. I. Dmitriev [119]); 7) (4:°, 4 1),
(4 O, 4 1) (Sparr [361]). In the general case the conjecture could not be confirmed (see [37] and
[117]). V. I. Dmitriev [119], [10] singled out a general class of couples of spaces (difference
couples) for which the conjecture is true.
The action of dilation operators in symmetric spaces has been studied in [348], [350], and [83].
In terms of the norm of the dilation operator, the problem of interpolation of the property of
complete continuity of linear operators in intermediate spaces for the couple (Ll' Lao) has been
solved in [348]. The upper and lower dilation exponents of symmetric spaces were introduced by
Boyd [81] under the name upper and lower indices. Lemma 4.7 is due to Semenov and plays an
important role in what follows. The notion of fundamental function was introduced in [331].
There are examples of spaces for which inequality (4.29) is strict, and, even more, the left and
right sides have different asymptotics at infinity (see [350]).
A5. Lorentz spaces were introduced by Lorentz in [225]; he established that their dual spaces
are the Marcinkiewicz spaces. Some properties of Lorentz spaces were obtained in [191]. The
spaces were considered by Semenov [330]. The imbedding theorems were obtained in [331].
Theorem 5.9 is contained in [342] under different assumptions.
The first example of a noninterpolation symmetric space was constructed by Russu [307].
6. Inequality (6.1') is actually contained in [347]. The main interpolation Theorems 6.1 and
6.1' are extensions of Marcinkiewicz' theorem [234], whose proof was published by Zygmund
[407]. Many authors have dealt with the generalization of this theorem; see [64], [74], [94], [96],
[106], [148], [161], [163], [190], [214], [250], [262], [342], [371], [406], etc. Here the versions of Krem
and Semenov are given from [192]. For the case of the 4 spaces an analogue of Theorem 6.1 can
be found in [334], where the conditions on the space E are given in terms of the fundamental
function E. However, the proof contains an error and is true only in the case where the norm
II <1.,. liE of the dilation operator coincides with M fPE (concerning (4.29), see above). A correction
and strengthening of this result is expounded in 6 (see [192]). The first theorems on the
optimality of interpolation triples of concrete symmetric spaces were obtained by Dikarev and
Macaev [Ill] and Calderon [96]. Calderon's ideas lie at the base of the proofs of the optimality
theorems of [192].
NOTES ON THE LITERATURE
351
Hardy-Littlewood and Hilbert operators and majorant functions for symmetric spaces have
been studied in [332], [347], [255], [144], [78], and [218]. A generalization of the Hardy inequality
(6.41) has been obtained by E. A. Pavlov.
The spaces 4,.r are special cases of the Lorentz spaces A p ,1/I [225]. Interpolation theorems for
them are contained in [96]. Theorems 6.12 and 6.13 are new. Applications of Theorem 6.1 to the
convolution operator are indicated in [193]. Theorem 6.17 is a sharpening of a result of O'Neil
[253].
7. In discussing the properties of the Hilbert singular operator we have followed Zygmund's
book [51]. The main formula (7.9) is due to Stein and Weiss [371]. Theorem 7.2 was proved by
Boyd [78].
8 has an auxiliary character. The operation of taking the dual in symmetric spaces has been
studied in [133], [144], [254] and [333].
9. For the classical Paley theorem, see [50]. The generalization of it in the present form is
published here for the first time. The articles [313], [314], [316] and [333] are devoted to the
generalization of the Hardy-Littlewood theorem on series with monotone coefficients by means
of interpolation theorems. For properties of the Haar system, see [20]. Theorem 9.5 for the space
4 was proved by F. Riesz (see [2] and [50]), for symmetric spaces it was proved by Semenov
[333]. Theorem 9.6 for the 4, spaces was obtained by Marcinkiewicz [235], for Orlicz spaces by
V. F. Gaposkin, and for symmetric spaces by Semenov [336].e) Concerning Theorem 9.8, see
[304].
Chapter III
I. The properties of scales of Banach spaces are expounded in [186]. The notion of normal
scale was introduced and studied by Krem [181]. The problem of related Banach spaces was
studied by Krem and Petunin in [184]. The condensation of a normal scale by means of relative
completion is considered here for the first time.
2. Maximal scales have been studied in [181], and m-in-imal and regular scales in [185]; their
properties are discussed in detail in [186].
3. The scale of Holder spaces was considered in detail in [186]. V. Friedrich has kindly
indicated to us that there are inaccuracies in [186], which we correct here. The new exposition of
interpolation properties of the Holder scale is based on the work of Petunin and Plicko [297]. We
note that in a more general case a detailed exposition of the properties of the Holder scale and its
dual with application to the transportation problem is contained in Friedrichs' book [14].
Chapter IV
Introduction. Theorem I was obtained by V. I. Dmitriev [116].
I. The complex method of interpolation in the form expounded here was suggested indepen-
dently by Calderon [95] and Lions [204]. The main results of subsections 3 and 4 are contained in
Calderon [95]; the useful Remark 2 was made by Stafney [363]. Theorem 1.4 is also due to
Calderon [95]; its proof has somewhat been simplified by I. Ja. Snei'berg. In [95], besides the
spaces [Ao, A da' the interpolation family of spaces [Ao, A Ir* is also introduced, and the
space ffiA A I) of functions in the strip 11 having the following properties is considered:
I) IIj(z)IL-4o+A 1 <; C(l + Izl); 2) j(z) is continuous in the norm of Ao + Al in 11 ; 3) j(z) is
analytic in II; and 4) the difference j(1 + it - j(1 + it l ) belongs to Al and the difference
e) Concerning [333] and [336], the remarks to 6 must be taken into account.
352
NOTES ON THE LITERATURE
f( it - f( it I) belongs to Ao; moreover,
{ f(it2) - f(;t l ) f(l + it0 - f(l + it l ) }
max sup , sup = IIfll@ < 00.
t 2 - t I Ao t 2 - t I A I
The space [Ao, AIr consists of all x E Ao + Al for which x = f'(a),f E ffiAo, AI)' and
II x II[Ao.A J]0 = ' ( inf ) IIf1l3.
f a -x
It turns out [95] that ([Ao, Ada)' is isometrically isomorphic to the space [Ae" Air (if Ao n Al
is dense in Ao and A I). In subsec..tion 6 the dual of [A A da is described in terms of relative
completion (Theorem 1.6, I. Ja. Sneiberg). From what has been said we obtain the equality
[Ae" Air = [AQ, Ai]a. Changes have also been made in the proof of Calderon's reiteration
Theorem 1.7. We note that in [320] it is indicated that the condition that Ao n A I is dense in
Aa n Ap is superfluous; however, the proof of this fact contains an error.
The notion of an analytic scale of spaces was introduced by Krem [180]; he established the
connection of this notion with the complex method of interpolation [186].
The theory of Hilbert scales of spaces was constructed independently (in different terms) by
Lions [200] and Krein [180]. An important role is played by families of topological linear spaces
obtained from a Hilbert scale by means of projective or inductive limits. By means of them a
number of delicate properties of spaces of analytic functions has been studied.
Theorems 1.11 and 1.13 were obtained from other considerations by E. Heinz, and were
sharpened by T. Kato (see [23] and [24]). Heinz has proved an inequality more general than
(1.49), in which the fractional power of the operators J and J I is replaced by more general
functions. Let us consider the class of functions positive on the semiaxis [0, 00), admitting
analytic continuation to the complex plane with the negative semiaxis removed, which results in a
function mapping the upper half-plane into itself. Let the function q>(t) be such that q?2(t 1/2)
belongs to the indicated class. Under the conditions of Theorem 1.13 we have the inequality
(Tx,y) <; 1Iq?(J)xIlHollq?(JI)YIIHo' where q?(t) = t/q?(t).
All interpolation spaces with interpolation constant I between a couple of Hilbert spaces have
been described by means of functions of the indicated class (Foi and Lions [134], and
Donoghue [120]; see also [160]).
The family of spaces XJ -aXi was introduced and studied by Calderon [95]. Here we do not
discuss its connection with hyperscales (see [l88D. Lozanovskll [230] constructed an example in
which tl}e triples (X o , XI' XJ-aXi) and (Yo, Y I , YJ-aYi) of idea1lattices are not interpolation
triples. Sestakov [340], [341] showed that in the general case [X<>, X.]a is the closure of Xo n XI
in XJ-aXi, and consequently is a closed subspace of it. For further study of the family XJ-aXi,
see [401] and [229].
We note that Schechter [320], [319] constructed a generalization of the complex method of
interpolation based on the idea that the intermediate space is constructed not from the values of
functions analytic in a strip or their derivatives but rather from the values of some generalized
function with compact support defined on these functions.
Interesting results concerning the unambiguous solvability of linear equations and the spec-
trum of linear operators in the family of spaces [Ao, A da have been obtained by Sneiberg [353],
[354] and Stafney [364].
2. The methods of constants and averages originate in a paper by Lions and Peetre [214],
where they are constructed for the case that Eo and EI are 4 spaces with power weights. The
generalization of these methods to the case of arbitrary ideal lattices discussed here was proposed
by Peetre in [35] and [264], and developed by Dmitriev in [114], [116] and [118]. The appearance
of spaces of the Calderon scale in the reiteration Theorem 2.8 was unexpected (Y. I. Dmitriev
[ 59]).
NOTES ON THE LITERATURE
353
The X- and -methods were proposed by Peetre [262] in the case where E is an 4 space with
power weight (see subsection 8) and have been the most widely disseminated of all real
interpolation methods. These methods were generalized to the case of more general ideal lattices
by Peetre [35], Bennett [67], and other authors. For a special case of Theorem 2.10, see [262] and
[73].
Theorcms 2.11 and 2.12 were proved by Lions and Peetre [214].
Theorem 2.13 on the number of parameters was obtained by Peetre [261]. It has been
generalized to the case of arbitrary ideal lattices Eo and El by V. I. Dmitriev [118].
Extreme spaces (subsection 9) were studied by Hayakawa [156].
The duality of the methods of constants and averages (in the simplest case when they coincide)
was studied by Lions and Peetre [214]. Here we have expounded the results of V. I. Dmitriev
[116].
Applications of the methods of constants and averages to quasinormed spaces have been
studied in [168] and [157].
The connection between the methods of constants and averages and the theory of scales and
the almost interpolation properties of scales were studied by Petunin [186].
Lemma 2.20 and Theorem 2.23 were obtained by Lions and Peetre [214]. The problem of
conditions on the commutativity of the functors corresponding to the method of averages and the
complex method has been studied by Grisvard [153].
Chapter V
I. The approach to the construction, by means of an abstract approximation process, of
intermediate spaces between a Banach space and the domain of an unbounded operator acting in
it is expounded here for the first time. Concrete realizations of it have been studied by many
authors. Closely related but different constructions are found in Berens' book [4]. It was
apparently there that the connection between the behavior of an approximation process and
relative completion (Lemma 1.1) was noticed for the first time. See [24] for more details on the
subordinate operators of subsection 4. The approximation process constructed from a power of
the resolvent was first studied in Grisvard's fundamental paper [153], the results of which are
discussed here only partially. In particular, in it the relation of the intermediate spaces con-
structed there to the complex interpolation method is studied. Lemma 1.10 is due to Ljubic [220].
For operators satisfying condition (1.15) fractional powers are defined (see the books [23] and
[24]), and therefore the theory expounded in subsections 5 and 7 can be carried over to spaces
constructed from fractional powers of the operator. The connection between interpolation spaces
and domains of fractional powers of operators was studied by Lions [209] for accretive operators
in a Hilbert space, and in the general case in a long series of articles by Komatsu [177], and by
Sobolevskll in [356]-[359] (see also [72], [246], [392], and [398D.
The construction of interpolation spaces by means of bounded semigroups of operators was
first done by Lions and Peetre [213], and since then it has been studied in many publications-
see [3], [69]-[71], [91], [92], [151], [153], [222] and [392]. We note that a fairly complete exposition
of properties of the spaces constructed from resolvents and semigroups of operators in the case
G = 4". is included in the book [7] by Butzer and Berens. Theorem 1.9 was obtained by
Grisvard [153]; this theorem and Theorems 1.10 and 1.11 were generalized to the case of
fractional powers of operators by Muramatu [246]. The proof of Theorem 1.12 given here is also
due to Muramatu. Interesting but apparently not complete research has been carried out in the
case where the operators do not commute and are infinitesimal operators of some representation
of a Lie group (see [281]).
354
NOTES ON THE LITERATURE
Grisvard in [153] began to consider unbounded operators in a couple of Banach spaces. The
imbedding theorem given here, namely Theorem 1.13, is due to Yoshikawa [394]. It has been
developed and applied in [245], [396] and [397].
2. Spaces of traces were introduced and studied in a series of publications by Lions (see [201],
[202], [205] and [207D. The study has been continued by Grisvard in [152] and [154]; the
exposition given here is based on his articles.
3. As we mentioned in the Foreword, Sobolev-Nikol'skll-Slobodeckll-Besov spaces and
imbedding theorems for them served as a guideline for the construction of the corresponding
abstract theory expounded in Chapter V in an incomplete form. In almost all publications
mentioned in I and 2 there are applications to the theory of the indicated spaces. In addition,
we mention the series [379]-[386] of articles by Triebel. We have discussed only the facts which
can be obtained from results of I and 2; besides the abstract theory, we have here used
Mihlin's [30] and P. I. Lizorkin's [217] theorems on multipliers for the Fourier transformation, the
Hestenes-Whitney extension of smooth functions, and other techniques which have become
standard in the theory of partial differential equations. Here we have used the monograph [27] in
an essential way.
We entirely omitted other classes of smooth functions, in particular, Lebesgue spaces or spaces
of Bessel potentials, which are connected with the complex method of interpolation. Their theory
is expounded in Nikol'skll's book [32] without this connection, and in connection with interpola-
tion theory in Triebel's new book Interpolation theory, junction spaces, differential operators
mentioned in the Foreword (see also [93], [212] and [218D.
On other classes of functions, see [15], [86], [87], [90], [95], [99]-[101], [129], [171 ]-[174], [194],
[198], [211], [212], [216], [218], [249], [279], [321], [365], [366], [373] and [374].
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278. , Interpolation functors and Banach couples, Proc. Internat. Congr. Math. (Nice,
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279. , Interpolationsriiume, Elliptische Differentialg1eichungen, Band I, Schriftenreihe
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280. , Zur Interpolation von Operatorenraumen, Arch Math. (Basel) 21 (1970), 601-608.
281. , Non-commutative interpolation, Le Matematiche (Catania) 2S (1970),159-173.
282. , Interpolation of Lipschitz operators and metric spaces, Mathematica (Quj) 12(35)
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283. , A new approach in interpolation spaces, Studia Math. 34 (1970), 23-42.
284. , Concave majorants of positive functions, Acta Math. Acad. Sci. Hungar. 21 (1970),
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285. , Sur ['utilisation des suites inconditionellement sommables dans la theorie des
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289. Arne Persson, Compact linear mappings between interpolation spaces, Ark. Mat. 5
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290. Ju. I. Petunin, The relationship of three Banach spaces, Dokl. Akad Nauk SSSR 170
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370
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SUBJECT INDEX
absolutely continuous norm, 45
almost imbedded family of Banach spaces,
279
analytic scale of Banach spaces, 234
associated space, 45
averaging operator, 80
B. Levi's theorem, 40
Banach couple of spaces, 9
Banach function space, 40
basis, 178
unconditional, 179
Besov space, 330
Bochner integrable function, 209
bounded operator in couples of Banach spaces,
18
bounded semigroup, 296
Co-condition, 296
compact scale of Banach spaces, 187
complemented subcouple of a Banach couple,
29
complex method of interpolation, 214
concave function, 46
convergence in measure, 39
conjugation operator, 169
convex function, 46
contraction semigroup, 84
convolution operator, 147
covariant functor. 35
dilation exponents of a function, upper and
lower, 54
dilation exponents of a space, upper and lower,
99
dilation functions, 53
dilation operator, 96
dimension of the norm, 334
discretization of an ideal lattice, 44
distribution function, 58
dual family of Banach spaces, 195
dual scale of Banach spaces, 196
Egorov's theorem, 39
elementary function, 92 (Lemma 4.2)
equimeasurable functions, 58
equivalent functions, 48
Fa tou property, 44
Fatou's lemma, 40
Fourier transformation, 325
fundamen tal function. 101
generalized function, 309
derivative of a, 309
regular, 309
trace of a, 311
generalized Hardy-Littlewood theorem, 224
generalized Paley theorem, 224
Hardy inequality, 167
Hardy-Littlewood operator. 138
Hilbert operator, 140
singular, 150
Hilbert scale, 237
Holder scale, 201
ideal Banach lattice, 40
ideal lattice, 40
imbedded family of Banach spaces, 279
373
374
SUBJECT INDEX
imbedding constant, I
imbedding of Banach spaces,
compact, 2
dense, I
normalized. 2
incomplete scale of Banach spaces. 188
incomplete scale wi th base. 188
infinitesimal generator of a semigroup. 297
integral, 210
intermediate space. 15
of type (J. 257
interpolation constant. 20 (Lemma 4.3)
interpolation functor, 35
[ A o. A d ex' 221
(Ao.Alh .p' 271
of type a. 36
interpolation property, 195
almost, 280
normalized. 195
strong, 195
in terpolation space. 20
of type a. 22
interpolation theorem. 22
interpolation triples of spaces. 20
good. 21
interpolation triples of type a. 22
normalized. 22
intersection of the spaces of a Banach couple,
9
isomorphic Banach couples. 13
isomorphic measure spaces. 46
Lebesgue space, 46
Lebesgue's theorem. 40
logarithmically convex function, 51
Loren tz space. 107
majorizing normalized scale. 197
Marcinkiewicz space, 112
maximal scale of means. 278
maximal symmetric space. 104
method of constants (X-method), 246
method of means (}method), 248
minimal scale of Banach spaces, 197
Nikol'skii space, 330
normalized interpolation space of type a, 22
normal scale of Banach spaces. 188
continuous. 189
maximal. I 93
regular. I 96
normative linear manifold. 7
operation * * . 124
operator of fractional integration, 149, 150
operator of strong (weakened, weak) type,
130, 13 I
optimal interpolation triples, 27
orbit of a point, 89
problem of multipliers, 325
quasiconcave function, 49
Rademacher system, 183
rearrangement of a function, 59
reflexive couple of spaces, 9
regular domain, 335
regular ideal lattice, 45
reiteration theorem, 23 I, 261
related space, 189
relative completion, 3
restriction of an ideal lattice, 43
Riesz- Thorin theorem, 22
right invertible mapping, 33
r-regular domain, 335
scale of Banach spaces, 187
scale of means, 278
Schwartz space, 325
separable measure, 45
simple function, 39
smallest concave majorant, 47
smoothing approximation process, 283
Sobolev-Slobodeckii space, 33 I
Sobolev space, 324
space complete with respect to another space,
6
sp ace 0- ( A 0' AI), 216
space of slowly increasing Schwartz distribu-
tions, 325
space of traces, 313
spaces invariant under permutations, 350
strictly simple function, 39
strongly continuous semigroup of operators,
296
strongly measurable function, 209
subadditive function, 51
sub multiplicative function, 52
subordinate operator, 290
support of an ideal lattice, 45
symmetric linear subset, 94
symmetric space, 90
total linear manifold, 8
truncation, 102
right, 102
two-fold, 103
upper and lower indices, 350
weighted ideal lattice, 43
Young inequali ty, 167
NOTATION INDEX
C , imbedding, 2
A , Lorentz space, 98
M"" Marcinkiewicz space, 98
Xe( t), characteristic function of a set e, 39
x N , truncation, 102
xN, right truncation, 102
x z., two- fold truncation, 103
I
C , imbedding with imbedding constant I, 16
'1T)(AB, CD), unit ball in L(AB, CD), 23
L ( A B , CD), linear space of bounded opera-
tors from a Banach couple A, B into a
Banach couple C, D, 19
x*(t), rearrangement of x(t), 59
n x( T), distribution function of x, 58
M (s), dilation function of 1/;, 53
cP E' fundamental function of the space E, 101
UV(R n ), space of infinitely differentiable func-
tions with compact support, 323
GJ, Fourier transformation, 325
E, closure of E (used occasionally)
ABCDEFGHIJ -CM-898765432
375
.,