/
Text
Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
1034
Julian Musielak
Orlicz Spaces
and Modular Spaces
Spri nger-Verlag
Berlin Heidelberg New York Tokyo 1983
Author
Julian Musielak
Instytut Matematyki, Uniwersytet im. A. Mickiewicza
Matejki 48/49, 60-769 Poznan, Poland
AMS Subject Classifications (1980): 46E30
ISBN 3-540-12706-2 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-12706-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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PREFACE
The first version of these notes was published in Polish by the A.
Mickiewicz University in 1978, under the title "Iv10dular spaces". It
contained Sections 1,3,4,15-22 of the p sent text, which is enlarged
by the theory of generalized Orlicz spaces. We limit ourselves to the
case of spaces of scalar-valued functions, which is most fundamental
for the applications. The general, not necessarily convex case is
stressed as is modular convergence, not only norm convergence.
The reader who intends to study generalized Orlicz spaces only,
may restrict himself to Sections 1,2 and 5 of Chapter I and proceed
then directly to Chapter II.
The author would like 1;0 acknowledge the help of all who have con-
tributed to the present text. Doz. Dr. A. Kozek, the referee of "1'i10-
dular spaces" to "Wiadomosci Matematyczne", was so kind as to communi-
cate to the author his de.tailed critical remarks. "Modular spaces" was
also read critically by Dr. H. Hudzik and Dr. A. Ka:minska, who contri-
buted their valuable remarks. The content of some parts of the presen't;
Section 12 was communicated to the author by Doz. Dr. L. Drewnowski,
and Dr. H. Hudzik is the author of the first draft of Section 14. The
author would like also to express his gratitude to the many others
who helped in any way in preparation of this book.
Julian IvLusielak
Poznan, March 1983
CONTEN'£S
Chapter I, lilodular space s, .............................
1. Modular spaces, . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Conj ugate mod ulars ,
3. F-modular spaces,
4. Bimodular space s,
5. 1\10 d ul ar convergence,
6. Modular bases,
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..;I
.. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
Chapter II, Orlicz spaces, . . . . . . . . .
7. Generalized Orlicz spaces, . . . . . . . . . 33
8. Embeddings of generalized Orlicz classes and Orlicz spaces, . 43
9. Compactne ss in space s E{f', . . . . . . . . . . . . . . . 55
10. Generalized Orlicz-Sobolev spaces and spaces of functions
of finite generalized variation, . . . . . . . . . . . . . . . . . . .
11. Uniform convexity of space s L'P, . . . . .
12. Disjointly additive modulars,
13 . Complementary functions and con tin uous line ar functionals,
14. Interpolation of operators in generalized Orlicz spaces,
Chapter III, Countably modulared spaces, . . . . . . . . . . .
15. Countably modulared Gpaces Xc; and X o ' .. . . .
16. Spaces of infinitely differentiable functions,
17. Spaces of f-integrable functions vvith supremum modulars,
C apter IV, Families of modulars depending on a parameter,
18. Various families of modulars depending on a parameter,
19. Families of Orlicz classes,
20. Spaces of analytic functions,
Chapter V, Some applications of modular spaces,
21. Application to integral equations,
22. Application to aQproximation theory,
Comments,
Bibliography,
Index,
7
10
15
18
25
33
66
74
77
82
94
112
112
113
120
129
129
136
141
151
151
156
164
198
217
CHAPTER I
IvIODULAR SPACES
1.Modular spaees
1.1. Definition . L t X be a real or complex vector space. A func-
tional :X-LO,<D] is called a pseudomodular , if there ho"ds for arbi-
trary x,y eX:
1 0
2 0
f;lOI = 0,
(-:X:)::I (X) in case of X real
it
(e x)::I (x) for every real t in case of X complex,
o
3 £?(oCx+py) ftX)+ l;(Y) for d.'(.>'l0' OI,+(?,= 1.
If in place of 3 0 there holds
3 0 ,
(Q.'x+ yJ s (X)+ f:'s (YI for o-,f.> 0, o:,s+ (3s ::I 1
with an s €(O, 1) , then the pseudomodular r:: is called s-convex . 1-con-
vex pseudomodulars are calle d convex . If besides 1 0 there holds also
1 0 , ().x)=Oforall 11)0 impliesx""O,
then is called a semimodular . If moreover,
1 0 " lx)= ° implies :x: = 0,
then is called a modular .
1.2. Examples . I. If X is an s-normed space with an s-homogeneous
norm /I lis, then lx)= IIXI/ s ii3 an s-convex modular in X. Similarly, if
II t is an s-homogeneous pseudonorm in X, then 9tx) sllx/l s is an s-con-
vex pseudomodular in X. However, in general a modular is not a norm;
among others, it may assume the value +00 .
II. Let X = L P over the interval [a,bJ, then (X) ...S iXLtJIPdt is a
p-convex modular on X for 0< p < 1 and a convex modular on X for p 1.
III. Let X be the space of real numbers and let (X/? ° for every
x E X, (OJ "" 0. Then 2 0 means that (f is an even function, 3 0 is equi-
valent to the statement that the function C? is nondecreasing for x ?-O,
and 3 0 , with s = 1 means that is a convex function.
1.3.Properties of pseudomodulars
. (d.xJ $ (xl for
loll 1 ,
2
(; n n n
II. L <:J: i X i ) L <?(X i ) for <Xi 0, L o(i = 1,
=1 i=1 i=1
n n
III.if f; is s-convex,o<s 1,then l2"oL i x i ) 2c( (X i ) for d i O,
\.1=1 i=1
n
2: d: = 1.
1::1
1.4. Definition . If fl is a pseudomodular in X, then
X = {xC:::X: lim ().x)= oj
"..,. 0
is called a modular space .
Obviously, X is a vector sUbspace of X.
In the following, we shall need the notions of an F-norm (F-pseu.-
donorm) and of an s-norm (s-pseudonorm). Let us recall that a funo-
tional I I: X_[O,oo) in a vector space X is called an F-pseudonorm , if
it satisfies the following conditions: (a] 101= 0, (b) J-xl-lxl (orleitxl""
IXI for real t in case of a complex spa.ce X) , (e] Ix+yj Ixl+jYI ,(d) if
clk f/, and 1 -xl""O, then Idk - xl-O. If, moreover, (el Ixl"" 0 im-
plies x :::; 0, then I I is called an F-norm . Every F-norm (F-pseudonorm)
I I such that /ctx I is a nondecreasing function of (;t 0 for every
x "X is a modular (pseudomodular) in X. It should be noted that for
every F-norm I J one can define an equivalent F-norm satisfying the
above condition (see S.Rolewicz [2], p.16). If I I: X-.,,[O,oo) satisfies
the above conditions (a)-(c) and the condition(d*) Id.X'- 1<Lls'xl,O<s 1,
then I I is called an s-homogeneous pseudonorm or briefly, an s-pseu.-
donorm , and adding (&), we obtain an s-norm in X. .A:a s-norm we shalJ..
usually denote II lIs. If s - 1, we obtain a norm which will be denoted
" II. A space X with an F-norm (s-norm, norm) is a metric vector space
with distance d(x,y)=Jx-YI.
, .5 . T heorem . 1ft: is apse udomod ular in X J then
I)C.' = inf {u )'0: ( ) u i
is an F-pseudonorm in X , having the following properties:
1. if (>'x,) (A ) for every >-')0, where x"x 2 € X , then
X,lg lx2le
II. if x (: X , then I tI:. x I is a nonde cre asing
III. if Ixl < 1, tnen (x) I x .
function of
0,
3
If 'f is a semimodular in X, then I '<;> is an F-norm in X . If <? is an
s-convex pseudomodular, 0 <: 10 1, then
fiX"; '" inf u;>O: ( u / s) 'S 11
is an s-pseudonorm in X, and the properties I-III remain valid if we
replace I I by 1/ I s. If is an s-convex semimodular t then " II is
1
an s-homogeneous norm in X (if 10 '" 1 t we shall write also /I II = 1\ fJf)'
. It is clear that the set {u /,0: (' ) u} is nonem ty for
xEX , and. o O JXI (OO and 'Ol = O. Also the symmetry conditioni-xf =
'xl (or le XI) '" 'xl ) is obvious. Moreover, if is a semimodular and
IXJ = 0t then ( ) u ;for every u,O, and so 1f().x) e'''ux J.1 X for
o <u <;;.. and every :\,.Oc Thus ()..x)", 0 for all /I;> 0, and so 'x'" O.
Now, let x,y (i X and a /,0t u = IXf f +a, 'V '" iYI + b. Then ( ) $ u and
( ) Vt whence
. G: ) za (;v + U:v ) (>( }+ ) Ui-v.
Hence !X+YI Ul-v+2a, and we obtain iX+YI( ixi +IY/r' because a>O is
arbitrary. Properties I and II follow from the definition of I I and
from the Property 1.3, immediately. Now, let <Xk --'» 0( and i xl - O.
Writing =: d.k-a and Yk '" -x , we have _ 0 and iY1rI O. Now,
x ( akJj"
given an E70, we get ( ) -;>0 as k oo, and sO TJ E for suf-
ficiently large k. Hence J XI( Oc Consequently, taking as N a positi-
ve integer such that I ' N for all k and lal N, we have
I -oixt I ( -x)j + I( -.x)xl< jNY k '4 + l xl N lykJ f + l x ""O
as k-oo. We prove now that tX) lx J for Ix I < 1. Indeed, let IXlf< u< 1,
then ( ) u, and 100 tx)= (u;i) G).$ u. Since u is arbitrary, this
implies (x) lxlr
The proof of the second part of the theorem with an s-convex pseu.-
domodular runs along similar lines to the above:, taking 0(::> 0, we
have
10 ( Ol X ) . ? 10 r. u x ) < 1 s. 10
I,iixll = inf {u)'o: t:; / 1j=d: inf't '/O: ) 1/10 ,,1 "'rX IIXA .
u 10 u/o{s
1.6. Theorem . Let be a pseudomodular in X. If x E X and Eo X
f.or k = 1,2,..., then the condition IXk-xl -O as k ([) is equivalent
to the condi tion (;>..( -x)) 0 as k --!' 00 for every ,';\ > O. If
for k '" 1 ,2,..., then lXI() is a Cauchy sequence in the space If with
4
respect to the F-pseudonorm I i , if and only if, t::(A( - )). Oas
k,l..".oo for every A? O. If c; is an s-convex pseudomodular in 'X,
then the same statement holds, replacing i I by II u;.
Proof . We limit ourselves to the proof of the first part of the si&-
tement, with x '" O. Let (i\ ) -? 0 for eve ry > 0, then for an arbi t-
rary A O there exists an index k such that .o(: x,}<.: + for k k .
. 1 0) K,/\ "0
Hence I f A for k ko' and so 1 lf O. Conversely, let I I _O,
then JAXJ.tlf-» 0 for every ).. >0. Taking 0<.: € < 1, there en lOtS a k 1
such that 1.).XJ.tI < € for k k1' Hence (A )$' 1 \XJ.tI <!:' for k k1'
and so (A )-+O.
1.7. Definition . A pseudomodular in X will be called
a) right- continUous , if lim c; (). x)'" f (x \ for all x f: X f '
A-" 1+
b) left-continuous , if lim <:,CA xl: (x) for all x E X<, '
>. 1-
c) continuous , if it is both right-continuous and left-continuous.
1.8. Theorem . Let be an s-convex pseudomodular in X. If rr is
right-continuous, then the inequalities jiXJl; < 1 and (x)< 1 aLe equi-
valent for every x E-X . If is left-continuous, then the inequalities
lbel'; 1 and (xJ 1 are equivalent for every x, X .
This theorem follows from the above definition, imro.ediately. Let us
still remark that the equivalences mentioned in 1.8 are not generally
valid without continuity assumptions on fj , as show the examples where
X is the space of real numbers and f1 (x): 0 for 0 x 1, 1 (xI'" 00 for
10
2lx): 00 forx f 1, because IIxl/ 1:
x71, or 2{.x)", 0 for 0 x<1,
. s . I s
!lXJ/ 2 =: Ix .
1.9 .Examples . A function If defined in the interval [0,00) , nonde-
creasing and continuous for U? 0 and such that \f(0) = 0, \flu) 0 for u;>o,
If(u) 00 as u CD , is called a If- function . Let i!L., ,,.....) be a measure
space , i.e..Q.. is a nonempty set, :2: is a <J"-algebra of subsets ofJl and
f- is a nonnegative, complete measure in "2 , which does not vanish iden-
tically.
1. Let X be the space of all real-valued (or complex-valued) Z-mea-
surable and I'-almo lOt everywhe re finite functions on..!l, with eq uali ty
f-almost everywhere. Then
(X)=Sif(JX(t)l)d for Xt X
.A
5
is a continuous modular in X. If <f is additionally an s-convex funo -
tion , i.e. If(oC'Ui-r-v) o(s\f'(u)+ s\f(V) for u,v O, oc,f-' 0, ct s +f.>s=1,
where 0< 10 1, then f is an s-convex modular in X. The respective
modular space X is called an Orlicz space and denoted LIf(...Q., , ) or
briefly L'i'; we shall deal with Orlicz spaces and their generalizations
in Chapter II.
II. Let be a. 6"-algebra of subsets of a nonempty set..a and let
1: be a nonempty set, which will be called the set of parame1a:'s. Let
I':'"'
us suppose that to every I('e t there corresponds a nonnegative, comple-
te measure r't" on:£ , which does not vanish identically. Let X be the
spaoe of all real-valued (or complex-valued) .[-measurable functions,
1-t.;-almost everywhere finite for every 1:'€- '[ , with equality fT.-almost
r.-
everywhere for all tt'G- c.... . Then
(x)= sup S (tx(t)l)df'l1:-
L 't -..a
is a modular in X. If 'f is additionally s-convex, then C1 is an S-CCl,l\-
vex modular in X. In case when the set l' consists of one eJs1ent only,
Example II is reduced to Example I.
III. Let us yet consider the following special case of Example II.
We take both IL and r as the set of all positive integers, md as 2:
the (1-algebra of all subsets of..Q.. Given an infinite matrix [ani] of
nonnegative numbers such that for every i there exists an n for which
a - F O. For any A c:..a. we define M (A) = 2: a.. Then the modular
ro. r n iroA n:L
from Example II be come 10
00
lX)= sup 2: a .to(lt.1)
n i=1 n )
If a . = 1 for all i and n, we obtain
ro.
for x =(t.).
CD
x) = 2: If( It.!),
1
and the re sul ting modular space X is the Orlicz seq uence space =- .
IV. Let 'T and X be defined as in Example II. Moreover, let. be
a Hausdorff topolOgical space, 4?; and = 'f () {t'o f. A topology in
f is defined, taking as neighbourhoods of 'L the complements with res-
o 0
pect to of compact sets in f . Then
lx;:2 lli f \f (jxtt)/)dt< :2 inf sup S <P((x(t)1) d '
'C" 't"o - ... U 't"E, U 'lTv$ .:2.
6
where U runs over all neighbourhoods of 'i:' in T, is a pseudomodular
o 0
in X. If If is additionally s-convex, then is an s-convex pseudomo-
dular in X.
V. Let X be the space of all real-valued (or comPlex-valued) func-
tions defined in an interval [a, b] and vanishing at t = a. The value
(posSibly also 00) m
V(a (x)= sup L. W( Ix (t.J - x (t. _ 1 )1) f
:r IT i=1 I ).
where the supremum is taken over all partitions IT: a=t (t < .. .<t '" b
o 1 m
of the interval La, b], is called the cp- variation of the function x f X.
The f-variation VI.f is a modular in X, and in case of if s-convex, is
an s-convex modular in X.
VI. T u.rning back to Example 1.2.1 of an s-normed space X with an 10-
norm 1111 10 , 0 < 10 1, the s-norm II lI s defined by means of the modular E;(xr
UXl's as in Theorem 1.5 is equal to ",,10, Le. IIxll "'uxUs for all x t X.
)
We go now back to the definition of the s-paeudonorm II II; in X?' gi-
ven in 1.5, and we show that there exists another s-pseudonorm in XIS
equivalent to " lit .
1.10. Theorem . If is an s-convex pseudomodular, then the functio-
nal 1+ (;' ( 1/ loX) u(1+ e (j-/s))
10 inf inf
/IIxllf = ;
i)O wO
is an s-pseudonorm in X) and
10 10 10
IIxll Illx," 2 11 X ll e
for every x X .
Proof . In order to prove the triangle inequality for III ",10, let X'YEX
and a niwber a>O be given. Then there exist U,V) 0 such thaZ
10 10
( ulis) <: "'XUl u+ a 1, ( ) < I "YIK / a - 1.
Hence
( x+Y )
(u+v)l/s
and so
IIIX+YII/ s (U+v) [1 + ( . x+Yl/ S)] < '''XIII s +/11 YIII,s + 2a.
u+v } 10 10 10
Thus t/lx+y",s ,"x IN,s +illy IN. . The condition ,,,ocxl/l = loll !IIXlfl is
\
verified similarly as in the proof of 1.-5. In order to prove the ineq u-
s
alities for norms let us first observe that lz) > 1 implies z) )lzll
10 10
..J:L to ( --1S.- ) + .:!- fJ ( -L- ) '" x I/f +"'y/ll + 2a _ 1,
" u+v ') 1/10 u+v \ 1/10 <. U+V
u v.
7
for eny z . Indeed, in the converse case we would have
1 < ( (X l/S ) } (X)= 1,
a contradiction. Thus, supposing Ll/S 1 ;. 1, u;. 0, we obtain
( Ul;S ) > I lu1/S fIe 10 :2 IIXI S ,
end so
U(l + Cl/S ) )) U(l + IIX" s) /lXI' S.
lvIoreover) if Cl/S J :f 1, then I/xl' u, and again
u(1 + C1/S j ) . u 4 IIXI( '
Hence 'IX/f; II'XWt. Now, let us take en arbitrary a:>O and let us wri-
te u'" /lXI'; + a, then (ul/S) 1, and so
u0 + Ll/S ) ) 2u = 2 !I X 1/; + 2a.
21'XI + 2a, end since a is arbitrary, we obtai II/XII/ s
10
Hence ftlx /II
10
211xl/ .
In case of 10 :2 1, the norm II If( If 1I in.I f is called the -
burg !!.Q.!1!! and the norm III 1//( III '/I in Xe is called the .Amemiva !!Q.m.
Applying the notion of the conjugate modular It we shall introduce in
2 still enother norm II II; , called the Orlicz norm.
2. Conjugate modulars
Let fj be a convex pseudomodular in a (re al or comPlex) vector space
and let x be the space of all linear, continuous functionals (real or
cOlilvlex) over the normed space <X ,j( /I > with IIxl/ = influ,;>O: l ) 1;.
Following H.Nakano (1J, we shall define a semimodular ,. on x; and in-
vestigate the resulting modular space. First, we prove that
2.1. Theorem .A linear functional x1t over X is continuous with respe-
ct to the norm /I II ' if and only if, there exists a constant !;;> 0
such that ,x xl ( (X)+l) for every Xt X .
. Supposing the above condition to be satisfied,we have for
8
every E.;>- 0
Ix ("x,,;+e. )) dd "X/(sx+€ ) + 1) 2 ,
and so x* X;. Conversely, if x € X;, then taking f/Xl/ 1 &'1d writing
(-,Ix"il , we have }x*xl::; d""xlle t n (X)+'). Taking 'IXlI 71 we see, as
in the proof of 1.10, that (X) I/XI . Hencelx*xl o n Xl/ o- (x}:;:
' ( (X\+1) .
2.2. Definition . The conjup:ate modular fl' to is defined by the
formula
(( ) =< sup (/x'E-xl_ f;(x t) for x* X;.
x
2.3. Theorem . If (j is a ,:::onvex pseudomodular over X, then q* is a
convex, eft-continuoU3 semimodular over X;.
. It is easily seen that f*loJ = 0 and «-x)= 1:X) , 'tx) 0
for every xllc . Moreover, taking X"',yAf: x1 and cI., 0, oc+ (6= 1, we
have
*()(:?"+ I""} $ sup /x.xl- ol f(xl + ,1>/ xl- f(xJ J «f!t-(x1\1 + f (y"J .
. xE- .
Finally, let (t'Ax!1=: 0 for every .A.> O. Because J:lYI 5 (y)+ rtyft)
for all y E- X , y"'f= X;, so taking an arbitrary x (f and an '''1.> 0 such
that (ttX) < 00, we have
A1'[lx*xl=J'\x ("lx)j 5 ( x)+ ?':(AXjt)= ( xl.
Hence pc/t x / ( X)/A . Taking )..... 00', we obtain x""x '" O. Hence
x ::z O. The left-continuity of (jK follows from the fact that lim
{CAX*)- sup sup U.lx*xj _ fLx>J= Jt-{x*). A -..1-
0<"\( 1 xex
From the above theorem it follows that one may define in X; a norm
generated by means of the semimodular t;fr :
/fx*I : inf{u? 0: ( : ) 13 for x*€ x; 0
Besides this norm, supposing f to be a convex semimodular in X, one
may define in X; a norm
IIx/f-"f"" sup [tX*XI : IjX 1, x E X !.
In order to investigate the connection between norms 1/:x1 andllx!tl .If
we prove first the following
2.4. Lemma . If 'I is a convex pseudomodular in X, then HX/ >l imp i-
e 10 LX) 4 1/ x II · 1
p Let l'(k < i/x/ , then ( } >' 1 and we obtain 'kf(x)
9
f »l, Le. (X) k. Hence we conclude flX) /lXI .
2.,. Theorem . If is a convex, left-continuous semimodular in X,
then ;fXItI'f xjf"; 2 /I x"I 1\- for every xl\- EO xt.
Proof . By Theorem 1.8, the inequalities /lXII {.l and (xi 1 are
equivalent. Hence II#I sup ( (x\+ »)= 1 + \x' "j. Supposing
. Hxl l,x
Ilxltlf * u, we have I' t:} 1 and so ffxJ\of 2u, which implies the right-
hand side inequality. In order to prove the left-hand one, let us first
remark that if x"'G x; and /lx*lI/l- l, then Jt-(x*1 = sup l/xA"X\- q(X»).
e flxJ 1,XE
Indeed, we have
sup (Ix xl- lx») sup (/lx"'ll; /lX/l
(X»l (x»1 '>
by Lemma 2.4. Hence
*(x* ... max[ sup (lxtx1- (xl), sup ( lx lf':x1- lx)l]= sup (Ix x\- (x)l.
(x) 1 ftx)71 H l
Now, since IIx*lflx*t II '" 1 for x/: 0, we have
l ( x 1i", ) ... sup ( 1x!t , _ f(x)\ sup ( I/X*I IfXI - lxll suP(llxf -f(X)) 1.
Ilx*'lI t.x:){l I(x*/{; ) lx) l Ilx*I'; ') fLx) l
But this implies the req uired inequality.
Applying the conjugate modular ff: we define now the Orlicz
1/ ' in by means of the formula given below:
2.6. Theorem . If l? is a convex pseudomodular in X, then
/lXfl '= sup {Ixkx/: x x;, <';*{Xflc) If
is a pseudonorm in x and /lxlli ,/I xfl/ for every x E- x . If is a convex
semimodular, then 1/ II is a norm in x .
Proof . Let x6 X . Since l:X!xl ((jtJ+ (xl for every x*E x;, we ob-
tain IIxl/ 1+ 4Slx)and so O<llxlfi < 00. Moreover, it is easily seen
that Illi-xll; '" lollI/xI'i for every number ot . Hence, taking >O, we have
!IX"; ::: II x i 1 + ( x).
Thus IIxlf -1 (1+ ( x» for every ?O, and conse q uenUY,lI x l/; S /I' Xfl .
Since the triangle inequality for 1 f' is obvious, we conclude that /I I
is a pseudonorm in X . Now, let us suppose that II xII '= O. Then x x = 0
. II- ../'r.
for every x* X; satisfying '1x""l 1. Let us take an x*f with l:x!1>1,
then taking y*= xf<,/ k(x1r), we have >t y ) 1, by convexity of *" . Hen-
ce y x '" 0, but this implies x.x '" O. Consequently, we obtain x := O.
- f(X)) sup (/Ix/If - f(x)) 0,
lJQ>1
10
3. F-modular spaces
As we have seen in Theorem 1.5, the F-pseudonorm I I) in case of a
general pseudomodular and the s-norm II , in case of an s-convex mo-
dular are given by different formulae and both theories are going para-
llel, but not identical lines. There will be given now a unified theory,
embracing both cases. For this purpose we define an operation on the
real halfline R+ "" [6,(0), which will be denoted by means of the symbol
F lu,v) or u@v for u,v fR .
+
3.1. Definition . An F- operation in R will be defined as a function
- - +
F: R+i< R+ - R+ having the following properties for u,v €- R+:
1 0 F(u,v>", F(v,u), 2 0 F(u,F{v,w}):: F(F(u,v) ,w), 3 0 F(u,O)= F(O,u)=u,
4 0 F(u,v)is nondecreasing as a function of each of the variables,
se parately,
50 F is continuous from R X R to R .
+ + +
We shall wrii>e also u<&v = F(u,v). The operation GP may be extended to
any finite number of terms by means of the forlJJ.ulae
1
@ u. == u 1 '
i==1
2
@ u i '" u 1 $I '
i=1
u. = F ( $1
i=1 i=1
u.,u ) for n/2.
n
3.2. Proper:ties . Obviously, we have F[u,v) O and F(O,O)= O. The
following properties of F-operations are immediate:
1. If 0 u1 and 0 v1 v2 ' then F(u1,vl) F(u 2 ,v 2 }.
II. If F are F-operations for n ::: 1,2,... and F tu,v ) F (u, v) as
n n
n CO uniformly in R x R , then F is an F-operation.
+ + ( p P ) 1/ P
3.3 .Examples . I. F1(u,v)= u+v, II. Fp(U,v}= u +v , p 1,
III. F oo(u, v) = ma:x: (u, v) ,
{ u+v for u+v < 1, u < 1, v < 1
IV. F(u,v) = 1 for u+v 1, u< 1, v<.1
max(u,v)for u 1 or v 1.
3.4. Remark . F is the smallest possible F-operation, i.e. F(u,v}
00
Foo(u,v} for any F-operation F and all u,v O. Indeed, we have F (u,v)
F(u,OJ= u and F(u,v) F(O,v}= v, and so F(u,v) ma.x(u,v)= Foo(U'V)'
3.5. Definition .Let h:R+ R+ be a homeomorphism of R+ onto itself
and let G = h*F be defined by G (u,v):: h-:- 1 F(h(U) ,h(v)) for u,v R+,
where h-1 i.s the inverse function to h. If such a homeomorphism h
11
exists, we say that the F-operations G and F are equivalent and write
G,.....F.
3.6. Properties . Denoting by 0 the operation of superposition of fu-
nctions and by i the identity map of R onto itself, we have
f< It ,) + ft
(h 1 0 h 2 ) F = li 2 (h 1 FJ and iF", F.
If F is an F-operation and G/VF, then G is an F-operation. The rela-
tion AI is an equivalence relation.
3.7. Examples . 1. If h(uJ= uP, p 1, then rfF 1 = F. ILIf h(ul=
u f< (uv. P-li
e -1, then h F 1 (u,v)= InLe +e -1). III. If hlu)= In(1+u), then li F1/!.l,v)
= u+v+uv. IV. h1\"F = F for an arbitrary h.
(X) (X)
3.8. Definition . Let k:R --R be such that klu}> 0 for u»O and
+ +
let F be an F-operation. The function k will be called - superadditiva ,
if F(k(u), k(v)) k(u+v) for u,v O.
It is easily seen that the function k{u)= u is F 1 -su p eradditive.
Any nondecreasing function k with klul/O for u> 0 is F -superadditive.
CD
In particular, k(\.1.)= 1 for U?O is F -superadditive.
co
3.9. Theorem . If k is F-superadditive for aome F-operation F, then
k is nondecreasing for u6- O.
Proof . Let us suppose that 0 u<v, but k\U) k(v}. Writing = u,
v 1= v-u, we have u 1 +v 1 = v and F(k(u},k (V-U) k\.v)< k(41;. Let u' '" kt u ),
v' = k(v-u), then F(u',v')(U'. But F(u',v'l F(u',O)= u', a contradic-
tion.
'£he following statement is easily proved:
3.10. Theorem . Let h be a homeolJ!Orpmsm of R+ on R+, k(U):> 0 for
u>o. The function k is F-superadditive, if and only if, the function
h- 1 uk is G-superadditive with G = h*F.
In the following, we shall extend the F-operation F to the closed
halfline R+ = [O,ooJ including 00, taking F(u,v)= (X) if u == (X) or v::: 00.
3.11. Definition . Let X be a real or complex vector space and let F
be an F-operation, 0 < s $1. A functional : X-,. (0, (0) is called an
LF.s )- pseudomodular , if
1 0
)/
(O)= 0,
(-x) = lx lin case of X real
't
(e xJ:: (.x)for every real t
f(d-x+ f' y) F l Qc), l:r)J for
in case of X complex,
o(, 0, d. s+ (b s= 1.
"
-'
12
If besides 1 0 there holds also
0'
1
then
( x)"" 0 for all A,70 implies x = 0,
is called an - semimodular . If, moreover,
1 0 "
lx)"" 0 implies x = 0,
then is called an tF. s l- modular . The vector space
X" = x X: lim (AX)= 01
'\ '\-'70+
will be called the - modular space .
It is easily seen that (). xl is a nondecreasing function of 4 0
for x (;: X and that
n n n
(t;tixi) (xi) for oli 0, f;1 /X' "" 1 .
3.12 .Theorem . Let f be an(F,s)-pseudomodular in X, 0 <s 1 ,and
let k be an F-superadditive function. Then
Ixtf,k"" inf {u>O: C /s) k(u)J
is an F-pseudonorm in X , satisfying the inequalities
i>\lsIX1;,k P,x/ ,k Ixl;,k for JA/ 1,
IXI ,k I>.xl ,k I:>./s I xl;,k for /'>'\7 1 ,
and such that if ().x1) (Ax2) for every ).>0 then )X 1 /;,Ifl x 2 1;,k'
and /I)(xj;,k is a nondecreasing function of r).) 0 for every x,X 1 ,X 2 E'X f "
If at leasi one of the following conditions:
li) lim ktu)= 0,
u-:!>O+
(ii) for every 1'),'7 0 there exists a J" 7 0 such that for e.w;ry x X
and an arbitrary i-. with 0 A f there holds the inequality
(A x) 17e()c)
is satisfied and is an (?,s)-semimodular, then I I k is an F-norm.
s ),
If k(u}= 1 for all u, then I I ,k is an s-pseudonorm.
Proof . In the proof of the first part of the theorem we limit our-
selves to proving the triangle inequality and the inequalities for
lAxl ;,k' Taking x,y GX and any a)O and writing u -Ix I ,k+a, v""j I ,k
+a, we obtain 1/s 1/s
( (:')1/ .) = :v ) u / s + t':v ) -jI.] ( P [ O/ s) . (-jI.)h
F(kLu) ,k (vII k (Uf-V) ,
13
and we conclude that Ix+y) s i xl s + Iy IS, because a /0 is arbi t-
,k <,k ,k
rary. Now, let 0 < >. 1, XE X . Then
{u>O: u Vs ) k(u)fc )o: C is ) .klU) {U?0: ( (v/A:)1/ S) k(X\)J
and taking the greatest lower bounds of the above sets we obtain Ixl
s s s \,k
'?1AXI ,k A Ixl ,k' The other inequalities are obtained applying
the above ones. It is also easily seen that taking klu)= 1, we obtain
I I s s s
AX .c k == 1>"/ lXI, k for all xGX.. and all .A . Now, supposing (i) and
s"" ), ( X ) \
IXI ,k'" 0, : have k(u) for every u>O. Taking any '" /0
and 0 <.. u <. we obtaJ.n
Ox}= (Au1/ Su /s ] $. F C1is) ,0)= L1is ) k(u! O as u O+.
Hence €O, x)= 0 for all i\"/ Q, and so x = O. Finally, let us suppose
(ii) and Ix I ,k = 0 and let us choose 1'}? 0 arbitrarily. Taking f? 0
;from the condition (ii), 0' <. 1, we have for an arbi trary ;70 and for
o < u <: (1)S :
(>.x):: e u1/s u1/S h ( u1/S ) k(U) "1k(;>"-S).
Since 'VJ > 0 is arbitrary, this yields o. x)= 0, and we obiB..n x :: O.
3.13. Examples . I. If is a pseudomodular in usual sense (see
Definition 1..1) and ifF(u,v)= u:+v, k(u)= u, then IXI: k=lx/" as de-
I)' "
fined in 1.5. Also, the condition 3.11(i) is satisfied.
II. Let be an s-convex pseudomodular in X, 0 < s 1.. Tre n, taking
d, q 0, o(s+ (!Is == 1, we get
(olx+ y)" s LX)+ (.>S ty) ma.x ( cx), ly)J= Foo( lX), flY)}.
Taking klu}= 1 for u O we observe that Ixl ,k ""IIX/l . JiJ.Qreover, there
is satisfied 3.11 (ii} with :s min(1, /S)..
We shall investigate now the pro blem of defining an IF, sl -modular
in a quotient space by means of an (F, s) -pseudomodular in X. We shall
need here the notion of a la- closed set X c:.X, defined by the condition:
) - 0
xnE Xo and (>..(xn-xJ>......O as n co for some A";> 0 imply xoE Xo;thiS
notion will be investigated in detail in 4.
3.14. Theorem . Let be an (F,s}-pseudomodular in X and let Xo be
a -closed vector subspaoe of X. Let us denote by [x) the coset from
the quotient space Xix containing the element x E X and let US write
o
o(lx)J= inf { (y): ye(x) if l:Y)<oo for some y<=(X,teo(X)= 00 if
14
lY)= 00 for every YE(X). Then o is an (:E<',s)-modular in x,/X o '
Proof . Obviously, o([xJ)? 0 for every x t X, 0a:0]) '" 0 and o is sym.-
metric. It is easily seen that the cosets [x) are -closed. Supposing
o([:x:JJ '" 0 let us take Ynf[x]in such a manner that (Yn) -.0. Since [x]
is -closed, we conclude that OG[x], and this implies (x]= Xo' i.e.
the zero element of x,/X . It remains to prove the condition 3.11.3 0 for
o
ea. Let X,JlG:.f.., (j.., p 0, o(.S+ foB:::: 1, b<s 1< VIe m.ay Sup )ose that
oUx:)} < (J;) and o([y) < w. We hav1 )c<.[x]+ f>[Y1)= inf {fl ) t zef (?;i].
Taking. x1dz] and Y1€[ir] arbitrar:Lly, we obtain ol.X 1 + f.> Y1 G [.xx+ () yJ. Hen-
ce there hoJ.ds Xo =c{x1+ Y1-z .x:o for z € [ x+f>yJ.. Thus any ZE [O(Xi- Y.l
is of the form z "'Gtx1+ Y1-xo with Xo€.f..o' H.ence
i d.[x)+ fy = inf { (o( x1+ Y1- x o) Xo fX o {oix1+ f.>Y1) F{ (X1) , ty1))'
]'or any E;:> 0 there erists a 6'.> 0 such that
F( 0([x1"6 , o(rj.J)+d) < F( o([xlJ, o«(Y]») + E
and elements x1E[ Y1€[Y]for which (X1J< <So{[ijJ+b and 5(Y1)<fo([YV+ ·
.tience
o(oc[x)+ t-' [y)) F( x1)' for 1») F ( a([X)J+d , f a( [ )+6)' F( o([x) ([yJ\)+€ ·
Since b 0 is arbitrary, this ;hows that ic<[X)+ tJ[Y1) F( o(LX]J, 'So((y))'
3.15. Theorem . If C; is an (:p,s)-pseudomodular in X such that ftJq-O
implies (2X)" 0 for all xEX, then the set Xo ={xcX: (x)", 0I is a
o/-closed vector subspace of X.
. First, we show that x,yE Xo imp.Lies x+y €Xo' This follows
from the inequalities
(x+y)= ( 21/Sx2 / 1/Sy ) ]' ( l21/sx), (21/Sy)) F{f(2x), (2yJl"'0.
Ubviously, x EX implie 10 ol'x E X for all numbers c(. l'jOW, let x Eo X
o 0 n 0
and (>\(xn-xd) O for some- A::> 0, then (2-1/ xo) ]'(fC>..xJ' fl).(x n -
Xo))):s O.(xn-xd)-,?O. .tience Xo E..{o'
'iheorems 3.15 and 3.14 show that supposing <7(x)= 0 implies f(2x}-
0, the (1i',s)-pseudomodular in X generates always an (F,s)-modular
in the quotient space XI £'0' where Xo ={x X: q(x/'" OJ. The definition
of o may be simp.Lified in case when is right-continuous (see 1.7):
3.16. Theorem . lJet be a right-continuous (F,s) -pseudomodular in
oX. such that (x/= 0 impJ.ies (2X)= 0 for every x E X. Let Xo '" .fX€ X:
(x):s oj. If x,y €X and x-yE Xo' then ftxJ:sf(YI. Consequently,
15
'f o((xlF (x)for any x€X and Il X )jS k '" Ixl k for xE-X .
, )'
Proof . Since x-y f Xo we have, by 3.15, € Xo for every (?>" o.
.Let 0<0(<.1, (!>;'?0, s+ s=1.Then
LYI= (<<' + (07) F (<y(!) ,0)= r(i).
'J:aking c1.. 1-, we obta:ln (y) fex). '.roo convese inequality f(yl
lx) is obtained similarly, and we get gcl'" (y).
4. Bimodular spaces
4.1. Definition . .Let X and Y be two real or complex vector spaces
and let : XX Y -fo,ooJ. The functional C; is called a bipseudomodu-
lar on X X Y, if the following conditions are sati sfied:
a) is a convex pseudomodular in X for every Y G Y,
b) is a convex pseudomodular in Y for every x e-X,
c) (..\x,y): <f(x,).. y) for all (x,y)c Xx Y and for every number.A .
If, moreover, (x,yl:s 0 for all y Y imp.Lies x '" 0, then is oal.led
a bimodular.
Let us denote
(lx)"" q(x,y)with fixed ytf;Y, fxl Y )= (x,y) with fixed xe X ,
then two modular spaces X eX and Yl) c:Y are defined:
fY '>x
X =lXf-X: lim x,y)", oj, Y x ={ylOY: lim (x,>.y>J= o.
y r\ 0+ A O+
It follows from 4.1.0) that if x €X, YfY, t:hen x(:X is equivalent
y
to YGY x .
4.2.Definition.
The set of (x,y)GXX Y such that x e.X ( i.e. that
(!Y
y Y x) will be called the bimodular set and will be denoted (Xx YJ(
The common value of IIxJ/ and IfYi/A for lx,y} (Xx Y),. will be denoted by
y)X )
II(X,y¥/(
There hold the relations:
/I(O,y)lt '" /I(x,O)fl '" 0, 1I()'x,y)U = /I(x, A y)// '" IAIII(x,Y)u ,
lI(x,+x2,Y)I/ I/(x, ,ylU f + II( 'Y)/I , l/(x'Y1+Y2)1 /I(x'Y1J/1 + U(X'Y21/(f '
if 1I(x,Y)/I < 1, t:hen (x,y) II(X'Y)f '
We extend now the bimodular set (AX Y)f to the tensor product X @Y,
defining a bimodular space (X@Y} CX@L .Let [x,y) E- x@y, X€X, yE Y
n n
and let z:s 2: o(i[ 'Yil €:.L0Ci x i @ Yi .
i=1 i=1
16
4.3. Definition . The bimodular space (X@Y)< iB defined as the vec-
tor sUbspace of the tensor product spanned over elements x(iJ.. y such that
(x,yJ fC (xx YJ(
n Obviously t every element of (X@Y»)'!IJ£i Y be written in the form
Xi(/SlY i ,where (Xi,Yi)E (XXY}.., i", 1,2,...,n.
i=1 )
4.4. Theorem . The functional
n n
IIwl/ = inf { 2. lI(xi,Yi)/lo : .z: [x, ,y.] E w, lx, ,Y i ]C (Xx Y)),
, i=1 ) i= 1 1. :c 1. )
where WE (X 0Y) , is a pseudonorm in the bimodular space eX@Y)f .
Proof . We limit ourselves to the proof of the triangle inequal.ity.
21aking W 1 ,w 2 € lx 01 Y we have w 1 +w 2 (: (X \81 Y) and
r n n
IIw 1 +w 2 /1"= inf.;' 2: I/(x i ,y,)/I,o: 2: [ X.,y.JE W 1 +w 2 ' lX 1. "Y 1. 'JE(xXY),cL ,
) li=1 1.) i=1 1. 1. ) j
. p q p q
inf { 2: IIlxi,Yj)n.+ L. lI(x ',y ')/ : L L:Xi,Y ]Gw1' 2.lx "Y J ' w2'
i==1 I) j=1 J 3 ') i=1 1. j==1 J
lxi ,Y )€ LX x Y) , (xj',y;') (xx y) /lW 11! +(/W 2 11 (
4.,. Definition . Let X be a vector space and let Y be a normed space
\vi th norm" 1/. Then we write
!IX/(o == sup {1I(x,Y)I/ : /!YII 1, (x,y) E (Xi( Y)f)' X O ={x X: jtxl!o <. 00).
The following statement is easily proved:
4.6 . T heorem . X O is a ve ctor subspace of X and iI IF is apse udonorm
in X o . If is a bimodular in X X Y! then II 11 0 is a norm in X O .
4.7 . Example . Let E and F be two Banach spaces and let Y = J( (E ,F) be
the space of all linear, bounded operators from E to F with topology of
uniform convergence on bounded sets. Let (...Q,5: ,ft.) be a measure space
(see Example 1.9). Let Z be the space of strongly measurable functions
z: .lL-.1>F, with equality f-allIlOSt everywhere. Let X be the space of fun-
ctions x: ..a..., E such that yx <:: Z for every Y E Y; we shall write x = 0, if
yxltl= 0 r--alm.ost everywhere for every yeY. A map XxY -Z will be
defined by the formula z ltl == yx(t), and the elements pf the iensor Wo-
n
duct are functions of the form c;- I)( .y,x. ( t ) , where x. I: X, Y 1. ' Y ;thus
. 1.1.1. 1.
the tensor product X Y is cont Jed in Z. Now, let us suppose that a
convex pseudomodular is defined on Z and that is given by the
17
formula
tx,y)"" ( x) for (x,y) XXY.
Under these assumptions, the following theorem is true:
4.8. Theorem . (a) is a bipseudomodular in x.x Y and
/lxl/o:a suptUYXI/(: /ly/l q.
(b/lf, moreover, r is a convex semimodular on Z, then is a bi-
modular in XXY, /IfF is a norm in X O , (X Y) C Z( and liZ/If' I/z/l<for
ZE(X n) .
lo/Under the assumptions of (b), an element x <:X belongs to X O , if
and only if, ?fL)..YXJ o as :\ _0 uniformly in the ballllYfl 1.
Proof .(a) follows from the equality
n(x,y)/l '" inf {u>O: f( ) 1! =bxll r
.Let us now suppose that r is a convex semimodular and let (x,y)- 0
for all ye:Y. Hence ()..yx)= 0 for all yeY and .\..>0, and this im-
plies yx "" O. Hence x "" O. Consequently, Cf is a bimodular end "If is
a. norm in X O . Now, let Z G- (xay) ' that is 2. (tl "" Yixi t), where
::1
xi ex, yiE:Y and xi<:X Yi for i:: 1,2,... Thus, (>'Yixi)",, Yi(Axi)
-? 0 as A......,:. O. Hence
n n
( Az)"'e{l L Y.X. ) 1 2:. '{()..Y i x.) O as --.,? O.
\n 1=1 n i=1
Consequently, zEZ....... moreover, we have
n n n
If Z /t",,=/I.2: Yixill.... $ IIYixill",,::O lt(xi'Yi)n",.
= 1 e = 1 = 1 )
This implies
n n
/lZlrf inf[{;111(Xi'Yi)tI): z(t/"" f;/ixilt/, (xi'Yi'c (XXY)<r3=lfZ// ,
o
and we proved lb). In order to show (c) let us first assume x €ox. ,
what means that K "" sup{lIYXll:tIY/I q <00 , Le. /lYXI {K for /lY/I 1.
Hence" y:x/2KIL' < 1 for IIY ' 1 t and we obtain
f( ) ] JI for IIYII 1.
Thus, taking 0< '" < 1/2K and IIyfr 1 we get
(AYX) 2}. K l ) "K,
and we conclude that (fC'>' yx) 0 as >. -,> 0 uniformly in the ball
11 y/I 1. Conversely, let US suppose that (>.. yxl 0 as >t - 0 unifor-
mly in the ball n Y Ir 1, then f (),. yx) 1 for sufficiently smll >. 0
18
and Ily" 1. This means that Ilyxll l/).. for I/YII 1 and sufficiently
o 1
Gall )., :> O. By (a), we have then/lx II .. sup{IIYx/( : IIYI/ 1 r '\ < 00,
o
whence xE X . Thus we proved lcl.
4.9 .Example . Let tp be a convex cp-function (see Example 1.9),
1. .. E*.. the conjugate of E. .Let (z)=S r.p({Z (tH)d?-. Then X(ioX o means
that ilf(JeltxttJJ)dt-L-;;>0 as A4' 0 uniformly in the balllle il 1 in
E"*. fhis leads to problems of Orlicz spaces of weakly -integrable
vector-valued functions.
5. lliodular convergence
As we have seen in Theorem 1.6, if q is a pseudomodular in a vec-
tor space X, then convergence of a sequence ( ) of element 10 of X f to
x C X in the sense of the norm I I is equivalent to convergence (A(:X:IC
xl) 0 as k..... 00 for every J\ '7 o. This condition may be weakened,
supposing the above convergence holds for some ).. '7 0 and not necessa-
rilly for all A '/ O. This leads to the notion of modnlar convergence
in X
5.1. Definition . Let be a pseudoillodular in X. A sequence( ) of
elements of X is called modular convergent to x €- X l briefly, -c.9!!::.
vergent to x X f ' if there exists a :7 0 such that , (:x:k-xJ) _ 0
as k-+ 00. We denote this writing x k x.
5.2. ,properties of modular convergence. 1. If xk-4 x' and xIc'.J...:x:",
then xk + x;,' ,j.x'+ x" .
II. If xk x and c is a constant then cxk.i;. cx.
Ill. If is a modular, then every sequence (x k ) of eJ.anent 10 of
X, has at most one modular limit.
IV. -convergence in X follows from norm convergence in X .
Norm conve rgence and -convergence are equivalent in X , if and only
if, the following condition holds:
(B) if c X ' lxkl- 0, then (2:X:k!"9 O.
5.3. Remarks . We shall see later that there exist modular spaces
(Orlicz spaces), where (B) does not hold, i.e. -convergence does not
imply norm convergence (see Remark 8.15(2»). So it makes sense to in-
vestigate the problems connected with <f _convergence, separately.
19
T his is important in the development of the the ory of modular spaces,
because if there would be only norm convergence in X" then the whole
,
use of a modular f; would be only a way of defining a norm (or an F-
norm) in a vector space; however, -convergence being in general not
reducable to a norm convergence, the modular notions lead to problems
which cannot be formulated in the language of the metric vector spaces.
.A..lso the notions of topological vector spaces are not applicable heJ:(j;
as we shall see in 6, a !lot on of 0/ -topology must be introduced in
a vector space to be compatible with modular convergence.
5.4. Definition . Let €? be a pseudomodular in a real or complex vec-
tor space X. A set A c:X will be called
1) - bounded , if for every sequence of elements "1c 6 A and any se-
quence of numbers £k -40 there holds €k Ji,.o,
2) - closed , if "1c E A and x imply x f:: A,
3) relatively - compact , if every sequence of elements E:i A con-
tains a SUbsequence, -convergent to an element x E X f '
4) - compact , if every sequence of elements E A contains a sub-
sequence, -convergent to an element x fA.
The smallest -closed set containing the set ACX< will be called
the - closure of A and denoted by A . If A :: X , then A will be cal-
led - in X . A modular space will be called - separable , if the-
re exists a f -dense, countable set in X .
We shall examine now the connections between the following condi-
tions of a set A Co X<:
(a) A is -bounded,
(b) there exist positive constants M and a such that (ax) M for
all x A,
(c) there exists a positive constant a such that (ax) 1 for
all x fA,
(dl there are a posi ti ve number K and an 10, 0 < 10 1, such that
s
I II K for all x<= A,
for every sequence x k c A and any seq uenc e of numbers k -p 0
there holds (fk ) ...,.0.
5.5. Theorem . If is a pseudomodular, then there hold the impli-
cations le) \al ;::pCb) (cl . If, moreover, is -1-co n vex, then all the
20
conditions (a/-le) are pairwise equivalent.
Proof . la)=<X.b). Supposing (a) is satisfied but (b) does not hold,
-2
there exists a sequence ( ) of elements of A such that (j ( k ) > 1
( -1
for k '" 1,2,... Taking a> 0 in such a manner that k )_0 as
k -.,. 00 and k so large that ak;> 1, we get
( -2 ) ( 1 -1 ) -1
1 < k ". ak k ( k ) 0,
a contradiction. The implications te) -=>(a)and lblc,7(c)are obvious.
Now, let be s-convex. Let us suppose lc) and let E. k -> 0, "1cf-A.
Then
because
valence
( Ek ) S £k
(Ek"1c) -;: (axk) -; -';> 0 as k CXJ ,
Ella < 1 for sufficiently large k. Hence (c)=>(e). The equi-
(b ).-:a/Cd) follows from the definition of the s-pseudonorm II II;,
immediately.
The following theorem is immediate :
5.6. Theorem . Let f? be a pseudomodular. ta)A set Ac.X is -closed,
if and only if, A::I At:. (b) If A is 9'-closed, then it is closed with
respect to the pseudonorm in X . (c)There holds AC.Ac.'A(f" where A is
the closure of A with respect to the pseudonorm in X .
Let us remark that a set A closed with re spect to the pseudonorm
does not need to be t; -closed. Indeed, let X be a modular space with
a modular not satisfying 5.2. tB), then there is a sequence ( I of
elements of X which is <?-.convergent to an XE X , but contains no sub-
seq uence convergent in the norm, for otherwise, taking .t? x € X and
an arbitrary subsequence (x kn ) of l l, one could extract a norm-con-
vergent sequence from (xk n I and this would imply x k - x in the norm
of Xe' Now, let l ) be a sequence -convergent to an x X but con-
taining no norm-convergent subsequence. Taking as A the set of elements
of tne sequence (Xk), we observe easily that A is closed with respect
to the norm but is not -closed. Consequently, we see also that none
of the inclusions AC.ACA needs to be an identity.
5.7. Theorem . If t; is a pseudomodular and if a set AC.X is -_bou,..
nded, then its -closure A is also -bounded.
Proof . Let xkr=A and Ok O. For every index k there exists a
number k'7 0 and an element Yk A such that (Ak( -Yk\)<'l/k. We
21
extract a subsequence ( ) from (¥'k J for which 2a. < A , k = 1,2,...
1 K k
Then ( xk) l2 ( -Yk)) + ({ (2 Yk) < k + (2 Yk)' SiIlLCe A
is -bounded and Yk € A, one may extract from the sequence l ) a sub-
l . ( 1
sequence b k ) such that 2bkYk) < k for k =; 1,2,... Hence <?(bk
0 as k - 00 Since ((k) was arbitrary, we conclude that I is -
bounded.
5.8. 'rheorem . If is a pseudomodular, then every relatively t<-
compact set is -bounded.
Proof . Let "1c Ii: A, Ok} 0, and let us write '" (I;k ). Let (k n ) be
any increasing sequence of indices. :£here exist a;:>O, x Xf! and a sub-
sequence (kn i ) of (knJ such that (a{ . - xIJ -0 as i _ 00. Taking i
so large that 2 Sk < a, we obtain
ni
a._.= €;(£j,- .x,_ ) (a("1c - x)) + (2 Ek n 'X) O as i _00.
"'n .""n "'n ni
Thus 0, and so A is -bounded.
We shall consider now the problems of boundedness and of continuity
of a linear map T of a modular space X in a modular space Y('1'" i.e.
T: X -YO-, where and fJ are pseudomodulars in vector spaces X and J;
respectively.
5.9.Definition. T is called
( ,c;') - contiLluous ,
( ,G') - bounded , if
set in Y.,.,
(c) ( ,<rl- contraction , if O"(Tx) (:](x) for all x E Xe'
{d' restricted (,?,O')- contraction , if (rtTxJ lX} for x E X satisfying
the inequality (Jc) 1.
h10reover, we shall say that T satisfies the condition
10
(eo) if there are constants a,1VI:>O such that <5"'[Eirx) Mljx' for eve-
ry x E: X satisfying the inequality IIX U ; 1 , wre 0 < 10 1 is given,
(e 1 > if there exist constants a,M> 0 such that '3"(Eirx) M lX)for all
x € X satisfying the inequality li'!-X) 1,
(e 2 > if there are constants a,M'/ 0 such that a- (EirX):$ lli Lx)for all
x Ii< Xe.
Operators, satisfying (e 2 ! are called also illongly ( ,<TJ- bounded .
It is easily seen that {e 4 } implies T to be both { ,cr)-continuous
la}
(b)
if x e X", , x .i,. x, imply Tx Tx ,
n ') n n
it maps any -bounded set in X q on a
'1"-bounded
22
and continuous with respect to the F-pseudonorms I /1 ' I '(1"' . The con-
verse does not hold in general even if we take l( and v to be convex
modulars, as may be checked on examples of Orlicz spaces. It is also
easily proved that if <ff is an a-convex, left-continuous pseudomodu.-
lar, then (e 1 J implies (eel (see 1.8). Also, taking as Y the space of
real numbers with G"ty)"" I yl and as X a suitable Orlicz space not sa-
tisfying 5.2. tB), we see that leJ does not imply in general (e 1 ) .
Concerning (eo)' there holds the following
5.10. Theorem . If and () are s-convex pseudomodulars, then a li-
near operator T: X is ( ,a?-bounded, if and only if, it satisfies
(e ).
o
Proof . Let AC be .;-bounded and let (eo) hold. Let us take €A,
£k O, 0 <. Ek< 1, then applying 5.5, we obtain c?(J6k )-O. Hence
, U;< 1 for sufficiently large k, and so
O"'(a f, x) & 2 6'" ar (ffk ») E 2M Ilffk /I; E 2. M -?O as k....oo.
Consequently, E 0, and so T is ( ,(J)-bounded. Since the set A ""
{x/6/ x ,,; )1/10: xe X ! is -bounded, so T(A} is v-bounded. By 5.5, the-
re are a,M> 0 such that O'(arx) $ M for all x E A. Thus
v(arx) .$ IIX"s O ( ar ( 10\/10 .)) J.'IJ.IIXI
10 (ffXI/ ) )/
for all x€ such that I/X" 1. Thus we proved (eo!'
From 5.10 it follows that if and (j are s-convex pseudomodulars
and T is ( ,tn-bounded, then T is continuous with respect to the s-norms
II 11 s and U II; .
5.11. Theorem . If and 0 are left-continuous, convex pseudomodu.-
lars, then every linear restricted ( ,<I1-contraction T is a contraction
with respect to the pseudonorms 1/ /I J If ff .
.Proof . Taking x € ' IIxlI 1, we ge.t c.XI 1 and so O"(Tx I q(X} 1.
Consequently, IITxII0"' 1, Hence IITX"(j "x" for all xfX<,'
Given a linear operator T: X -i':t;,.., where and v are convex mo u-
lars in X and Y, respectively, one may define the adjoint operator T :
Y;""" X ' where X; is the conjugate space to <X "I ,, ) and Y:' is
the conjugate space to <Yep" 11($'>' Obviously, T is linear. As we know,
(Cx"') "" sup Lx*x _ f(x]) and ()*( ::& sup ({ty - o-(y"J) are convex, left-
xe- Yf
23
continuous modulars in X*
1
re holds the following
5.12. Theorem . Let T: X -;> Y be a linear map and let T*": 7".....,. r be
f
the adj.oint of T. Then
(a) 1fT is strongly ( ,t1')-bounded, thenT"" is strongly(O"'A', ";}-bounded,
(b) if T is a. ( ,O")-contraction, then T"" is a (O" f*)-contraction.
Proof . Let T be strongly ( ,O"')-bounded with constants a;>O and M O,
i.e. ()"'( x) M (?Clfor all xc . Then
«( T*"Y:) =z sup (i y!t"(Tx)_ f(X sup (i yk(Tx)_ <r( +
X&xr x X
'" sup (yJ.- y _ (J"(y)) ()JI.( y"") .
y"T( ) a.
which proves that T is strongly (() ()-bounded with constants -; and
1 T' ho
;. hJ.s 10 ws both (al and (b).
lJet us still remark, that if we denote by J'dd b (re lOp. lI'ld 1 ) the cate-
gory of all modular spaces X with convex modulars ' where morphisms
are strongly ( ,a')-bounded operators (resp. ( ,a')-contractions), then
the map X _X and T 4":r'" is a contravariant functor in any of the ca-
tegories li1db and Md,.
At the end of this Section, we shall present an approximation re-
sult concerning filtered families of linear operators in modular spa-
ces. First, let 1.ts recall that a nonempty family 'to of subsets of an
abstract set IT is called a filter , if ,e) 1)() , V 1 ,V 2 E'J() implies
V1,,,,V2 1() JI and V,E:-'J{) , V,cV 2 implyV2t9IJ. A function g; V--R=
(-00 ,(0) to yQjill respect to '}(,) , g(vl 0, if for every £ > 0
there is a set V EO ')() such that fg (v) I < E for all v E v . A function
o 0
G; 'W R tends to !!!tl! respect to ')() , GW).2Q,..o, if for every
<c. ';> 0 there is a set V€ I){.) such that IG (v') V£) 1< t for every Ve-1() .
Now, if x ,x t X , VE V , where X is a modular space, we shall write
v M '/{]
x v !6..x, if tnere is a >,;>,0 such that ().(xv-x)-O. xv'-x will
'H?
mean that IXv-xl - O.
5.13. Definition . Let f be a modular in a real or complex vector
space X and let '}() be a. filter of subsets of an abstract family 1/ of
indices. A family T :=tTvl v,...,. of linear operators Tv: X ....;oX will be
called ?() - bounded , if there exist positive numbers k 1 ,k 2 and 9. func-
and Y;' , respectively(see Theorem 2.3). The-
24
tion g: IJ-R+ =[0,(0) such that gcv)' o and for every xtX there is a
set Y E: 1,(,) for whi ch
x
(TvX) k, 1k2x)+ glV)
for all v tE:.V x '
Let us remark that if is convex, then the constant k 1 may be
taken always equal to ,.
5.14. 'J:neorem . If X is a normed space with norm 1/ Ii and
then a family T =(T) .'>- of linear operators T : X _ X is
V VE v v
ded, if and only if, there exists a constant M:> 0 such that for every
x X there is a V !}(J for which liT XII lVIIIXi! for all v €: V 0
X V x
Proof. If the above condition holds, then the definition of H1-boun-
/?(X)=llxli ,
'k') -boun-
dedness is satisfied with glV)= O. Conversely, let f be 'J.() -bounded and
let us suppose that the above condition is not satisfied. '.chen there
exists a sequence (x n ) of elements of X, sets Y XnE: 1{> Yiith Jg (v)J< 1
for v € V , and indices v f: V such that n = nUx 11< liT x II k , j/k 2 x II
Xn n Xn n v n n n
+ g (V n ' k,k 2 + " a contradiction.
The following theorem is a general tool in various approximation
problems:
5.'5. Theorem . Let Q be a modular in X and let T =(T J be a
'i" v VEll
')l)-bounded family of linear operators 'l'v: X X . .Let A be the set of
all finite linear combinations of a set Ao C Xe' Then:
ta) If Tvx 2Q,. x for every x EO Ao and A is t:-dense in X , then TvX' x
for every :x: € X .
(b) If T x x for every x €: A and A is the closure of A in X.. with
v 0
respect to the F-norm I I ' then T x x for every x EO- I.
v <)0
. lJet us suppose that for any x €' Ao' q (ax(Tvx - x)).-O for
some » O. Then the same holds for every x EO A, because if x = c,x,+...
+ cnxn wi th G Ao and we put c :::I I c11 +.. .+/ c n ' , then (a(Tv x - x)}
£: (ac(TvX i - xi)) 0 for 0< ac ruin aX:L' Now, let E;:> 0 be
i ' ' i n
arbitrary and let x be a -limit of a sequence of elements of A. '.chen
there is a b » 0 and an element y f A such that
(3bk 2 (x-y)) < 6 , and (3b (i'v y - y)) O ,
where we may assume k, ,k 2 "7" :Let v E: V 3b (X-yt the set V 3b (X-y) being
chosen according to the Definition 5.13 correspondingly to the element
25
3b (x-y). Then we have
[b (Tv X - x)) f3t:JJ:v (x--yn + C; [3 b (TvY - y)) + 5 [3b (y-X))
k 1 [3 b k 2 (X-y)]+ g(v)+ f3b(TvY - y)]+ [3b ly-x» 2k 1 e; [3bk 2 (X-y)) +
gt,vl+ f3b(TvY - y/J c + glV) d3 b (Tv Y - YU.
No , let V 1 ,V 2 fl}() be so that glv){3't. for V€V 1 and r3b(TvY - yJ]
<3't for v Y2' Taking V = V 1 '" V 2 f'1V 3b (x_y)' we obtain [b(TvX - x})
<€. for all vEV. Hence iC[b(T x - xn. O, i.e. T x x, and part
'> v v
(a) of the theorem is proved. It is easily observed, that the same ar-
gument proves also part (b), because by the assumptions of (b) , there
holds [b(TvX - x)) 0 for every b> o.
5.1 6Definition . Let T ::a(T v )ve\7- be a family of linear operators
T: X """"X in a. modular space X and let (){J be a f:j.lter of subsets of
V. Then the T-modulus of smoothness of an element x E X is defined by
means of the formula
WT(x,VI= sup (Tvx", x) for every VeiJ.()
It is easily se ftVthat WT(XtV) 0, if and only if, (Tvx - x)
0 0 Hence Theorem 5.15 may be reformulated in terms of T-moduli of
smoothness in the following manner:
5.17 o T heorem . lJet be a modular in X and
?() -bounded family of operators Tv: X -"J X . lJet
combinations of a set Ao c. X .
(a) If A is l?-dense in X'S and if l'(Ax,V} O for every x f::Ao and
each A;::' 0, then for every x € there is a ;\ > 0 such that (>.x,V)
O.
(b) If CJ T (Ax,V) 0 for every
for arbitrary x EA.
'rheorems 5.15 and 5.16 will be applied to generalized Orlicz spa-
ces in two cases: if Tv are translation operators and if Tv are convo-
lution operators (see 7.11 - 7.24).
let T = (Tvlv 1! be a
A be the set of linear
>'"7 0 and x {: A , then the same holds
o
6. l.iodular bases
6.1. Definition . lJet be a pseudornodular in a real or complex
vector space. The sets
26
U(E) =: iXl:x : f(X)<cJ t.70,
will be called - neighbourhoods of zero in X . The set of all -neigh-
bourhoods of zero in X we shall call the .o-base in X and denote J3{6}.
,,-
The set of all -neighbourhoods of the form U(2-k+1), k = 1,2,...,
will be called the sequential -base in X and denoted by J3 s (fl.
I t is easily seen that 0, if and only if, there exists a >.:> 0
such that for every U A{O there exists a K such that A U for
all k )K. Moreover, we have
6.2. Theorem . Norm convergence and -convergence in X are equi-
valent, if and only if, there holds the condition
( 2) for eveXO'J U € i?J ( ) there exists a V €: P.:,( I such that 2VC U.
If (6d is satisfied, then .f>( ) is a fundamental system of neighbour-
hoods of zero in the norm topology in .
Proof . The first part of the theorem follows from 5.2.IV, immedia-
tely. Now, V('1) =fx€X f : tx' '}'d={XEX : ( ) 1'/.f ' where 'It-> 0, give
a fundamental system of neighbourhoods of zero in the F-norm topology
in X'? Without applying ( ) we see that for eve ry E;> 0 there is an
"'l"/O such that Vl"1)c UtE) , for otherwise there would exist E J!f, k :=
1,2,..., with ( ) and L }?,f, for an > 0 and every index
k, a contradiction, Conversely, we prove that for any ,:> 0 there is an
'/ 0 such ttlat UtE) Co V(' ll . Otherwise there would exist a number 'i{:> 0
and a sequence of elements E X for which ( ) < and Xk ( ))'1 for
every k. Hence (Xk O, and applying (4 2 ) we obtain (11) -,II> 0, a
contradiction.
T he following statement follows from the definition of the -base,
immediately:
6.3 . Theorem .Let be a pseudomodular in X, X I: (OJ. Then
La) 't is a modular in X , if and only if, for every XE X r ' :t F 0, there
exists an E. 70 such that U(E)€:JO(f) and x4 U(f) ,
(b) is a semimodular in Xc: ' if and only if, for every x, X , x F °t
there are a I: 0 and ;> 0 such that U(f)E-J3te) and x aU(€) .
We shall need in the sequel the following customary notation
6.4. Definition . Let UCX . The set U is called absorbing in X ,
if for every XE:X there is an a I: -) such that axeU. The set U is cal-
f;
led bala.nced in , if fQr every x f U and tal 1 there holds ex E U.
27
The set of all elements of X of the form ax with lal 1 and x € U is de-
noted by bal U ; of course, the set bal U is always balanced. If.J3 is
a family of sets U c:: X , then bal.2 will mean the f'amily of all sets
bal U with UEj) . Thesst of' all. elements of X of the form o<-x+(?>y with
10(1 + ' I 1 and x,y E- U is denoted by r(U).
6.5. Properties of -bases. 1. If U .f!(f) , then Uis absorbing and
balanced.
II. For every U 1 ,U 2 E-JDtf) there is a U 3 e'.8lf) such that rc( 3 ) CoU 1 t1U 2 .
III. For every U1,U2c J\<f) there is a tJ 3 E-J3 s lf) for which rcu3)c
U 1 n U 2 .
IV. For every UtC:-..Bs( ) there exists a vc;..fl{ l such that VCU.
Properties I and IV follow from the properties of pseudomodulars,
immediately. III is proved showing that if U
1 1
where 0< E 1 e 2 , then r(u(2'£1))C U(f 1 )." U(E 2 ).
II.
'" Ul £1) and U 2 = U ( '
This inclusion prove$' also
The above properties are the starting point for defining a modular
base in a vector space X.
6.6. Definition .Let X be a real or complex vector space. A non-void
family Jb of subsets of X is called a modular in X, if all sets
U <C are absorbd.ng in X and if for every U l' U 2 E J?> there exists U36-
such that r(U 3 ) c U 1 ("I U 2 . A modular base .13 2 in X is called !1Q.Q- w e aker
than a modular base -2 1 in X, if there exists a number a/;O such that
for every U 1 € J?>1 there is a U 2 € J.:>2 for which aU 2 c U 1 ; we shall deno-
te it writing $1-:S .P.>2' If £1 2 and .f>2 .Y.\1' then we call the mo-
dular bases 1 and ..,1?)2 equivalent . A sequence [Un) of sets in X is
called a se q uential modular base in X, if U are absorbing and balanced
- n
and r(U 1 )CU forn 1,2,...
n+ n
It is easiJ.,y seen that a sequential modular base in X is a modular
base in X. If (> is a pseudomodular in X such that X = X, then the pro-
perties 6.5 show that B (f) is a base is X, 13 10 ( ) is a 1iJequential base
in X and bases .B(f), s(f) are equivalent. Hence it follows that spa-
ces with a modular base form a generalization of modular spaces. Since
the equivalence of bases is an equi¥alence relation, it is reasonable
to be interested not only with spaces X with an individual modular base
J!J , but with pairs (X,Lf.I)), where [j?I] is the class of all modular
28
bases in X equivalent to the base .
6.7. Definition . If [J:,J is an equivalence class of modular bases in
a vector space X, then the pair (X, [P..) is called a generalized modular
sPace .
6.B. Theorem . If J3 is a modular base in X, then bal.0 is also a ba.-
se in X and bal.P> is equivalent to 53 .
. Let be a modular base in X. Obviously, the sets bal U with
U SO are absorbing in X. Let V 1 ,V 21: ba.1.J'?> , Le. V 1 ... bal U 1 , V 2 ... bal
U 2 with U 1 , U2G . Let U 3 J3 be such that r(u 3 ) c U 1 '" U 2 . Then, wri-
ting V 3 ... bal U 3 we have PlV 3) = rl U 3 ) c U 1 ("\ U 2 c::.V 1/"1 V 2. Hence bal J1 is
a modular base in X. Evidently, balJH.I?I . Let us take any U6-.13
then there exists a VG-R> such that U z U (I U ::> rCVbbal V, and so .134
bal . Hence S and balJ3 are equivalent.
6.9. Defir tion . Let( ) be a sequence of elements of X and let 10
be a modular base :i.n X. The sequence ( ) is called J3 - convergent to an
element :x: X, if there exists Iii number ..\;;-0 such that for every U"J:1
there is an index K such that >. -x) G U for all k.> K. The J?> -conver-
gence of ( ) to x will be denoted ..1; x.
Evidently, if is a pseudomodular in a vector space X, then the.
conditions x and x are equivalent.
6.10. Theorem .Let J3 1 and .E 2 be equivalent bases in X, E X,XE: X.
,
There holds XX: x, if and only if, XX: x.
. We may suppose without loss of generality that x = O. Let
x k :!'-i' 0 and let U &-S1 be arbitrary. Then there is a number a I: 0 and
a set V"' 2 such that aVcU, because £ 1-1.B 2 . Then we may take a
number i\ ;;- 0 and an index K for which A V for k> K, with:A in-
dependent of U. Hence Aaxk U for k7K, and so O. The converse
implication is obtained changing the roles of.13 1 and ..8 2 '
From the above theorem it follows that the notion of 13 -conver-
gence in X defines also a convergence in (X,[J'») uniquely, where
x k x means that x for a .B 1 E [J:3] . In particular, applying
both 6.8 and 6.10, we may always define J3 -convergence in X supposing
the t 10 from J'> to be balanced. We shall see now that in case of cou-
ntable modular bases one may limit itself always to sequential modu-
lar bases.
29
6011. Theorem o If S3 is a countable modular base in X, then there is
a sequential modular base .P.> 1 =(U n ) in X equivalent to J3 . If, moreo-
ver, £ consists of balanced sets, one may always choose 33 1 <::..'13 .
Proof . According to 6.8, bal,\?) is also a count able modular base in
X equivalent to.Pc> . Let bal,f> =(U n ). Let V 1 :::: U 1 . For any index m 2,
applying the fact that bal 13 is a modular base in X, one may find a
set (U nm ) from (Un) such that r(Unm)CUmC:V _1; we put V m == U nm . It
is easily seen that -8 1 =tvm), m == 1,2,..., is a sequential modular ba-
se in X equivalent to bal , and thus equivalent to .B . Moreover, we
have .B 1 C bal Jb .
As we have seen
above, if is a pseudomodular in X and X = X, then
.$ s( \ is a sequential modular vase in X. hioreover, it is easily seen
that the convergence x for ,x e-X is equivalent to -$lt) x 0 We
shall show now the following converse statement :
6.12. Theorem o If J3 =(U n ) is a sequential modular base in X, then
there exists a pseudomodular in X such that X = X and the bases.f>
and .B( ) are t>quivalent.
Proof 0 We shall construct the pseudomodular in such a manner
that for every index n there exist indices m,k such that U(2-m+1)c Un
and UkC U(2-n+1) , where U it) are defined as in 6.1. Taking' 110 -= X, w.
shall wri te
-n
{ 2 if x e U 2 n:' U 2n + 2 , n ;:; 0,1,2,...,
IYlLx!= 0 if xe () U 2n "" n Un'
n.=1 n=1
Let x 1 ,...,x m f X andol 1 ,...,olm')0 satisfying the condition oe1+"0+
'" 1 be arbitraryo Applying induction, we shall prove the inequality
( ) (o('1x1+" .+limx m ) 2. ( (x11 +00 0+ 4.lxml).
For m = 1 it is evident. Let us suppose it to be true for a positive
integer m and any choice of xi and «i' i :: 1,2,... ,m. Let x 1 '... ,xm+ 1 E-X,
where "llx:1)("1 ) u, lXmr1) , (t,1,...,"'-m+1'>0, 1+"'+o(m+1 :: 1 and
let us write i'" "l (x 1 +.. .+xm+1 ). If t = 0 or t 4 12 ' then (If) holds
wi th m+1 in place of m, so we may suppose 0 < < 2 . Let us choose an
index k m in such a manner that
1 1
"Llx1 +".+ ( (2r and '1(xr-..o+,t \+ t +1) 2'r 0
Then
30
, 1
l +2) +...+ "tl x JIH.1) '2t;
if k =: m, the left-hand side of the above inequality will be understood
to be zero. By the induction hypothesis, we have
( «'1 x 1+" .o(,k )
'1 1+" .+ 2( tx1)+"'+ ( ) r ,
( oL k + 2 x.rc+2+.' .+QtJIH.1 x JIH. 1 )
rt. 2 (1L ( +2' +...+ &t (xJIH.1») 0
k+2+" .+ JIH.1
Idoreover, lXk+1) . Let us choose a positive integer n in such a
-n-1 -n
manner that 2 0<2 . Then
( ot 1 X 1 +.. '+<XkXk ) < -n-1 -n-1 ( ' o('k+2 +2+" '+'Thr1XJIH.t ) 2-n-1
'1. <£.1+"'+1: ,2 , l +1) 2 ,h[ '\+2+'''+ 1 r .
Hence the elements
o(1 X 1+" .+<\:
ol.1+"'+ ' "'k+1'
+2"'k+2+" '+ 1xJIH.1
0Sc+2+" .+olJIH.1
belong to U 2 2 . Consequently,
d n+
1 x 1+' ..+"lk+1 +1 oC 1 +.. .+ o<"1 x 1+" ' "'k cl k + 1
ol1+"'+<Xk+1 = oc;+"'+"k+1 d 1 +"'+e\: + ol1+'''+<\:+1 +1 r:
6- r(U 2n + 2 1 c: U 2n + 1 and so
Ol <X o(1 x 1+"'+<\:+ 1 "'k+ 1
o<.1 X 1+...+ oL JIH.1 X JIH.1 == ( 1+"'+ k+1) 0(1+"'+<\:+1 +
) <ik+2"'k+2+" .+olJIH. 1 xm+ 1
+ ( k 2 +...+ot 1 € P(U 2 1 )c. U 2 .
+ JIH. oL +0<- n+ n
k+2+'" JIH.1
Thus lot 1 x 1+" .+oCJIH.lxm+1 ) 2- n 2?f, and the inequality !I?/ is proved.
Let US still remark that (If/ remains true, if we suppose only leX, I +. . .
+ \o(m l 1 in place of the assumptions 0('1"" 'o{m )0, 0l1+" .+d m :: 1.
Now, we define the functional t; by the formula
lx):: inf f i IY} ): x =: i olkxk' i ld k I:S 1 , x k € xl.
k==1 k:=:1 k=1
Fi'rst, we prove that is a pseudomodula in X and that X ( X. Eviden-
tly, LX)1 0, lo)= 0 and t"x)== LXJ tx)== flX) for real t, if X is
a. complex vector space). Now, let x,y+:X, ot, -:? 0, + == 1. Taking
P q P q
x = 2:'cik and y = 2' g,kYk with '2:'lc(kl 1, !:.l kl 1, 'YkEX, we
k=1 k=1 k=1 k=1
31
obtain
p q
.).x+ {':>y "" L: rt'1t + 2: 0P> k
k:= 1 k=:: 1
p q
and r. loL o( + 2 Ir'> kl 1.
k:::=1 k:::=1
Hence
(0( x+ f.> yJ r:. ( ) + t. lY)(y ).
k=:: 1 k=:: 1 L k
Taking the greatest lower bound at the left-hand side of this inequa-
lity, we obtain (ctx+f.>y) (Xf+ (y)" We show now that (AX)"'O as
,.\ 01- for any x E X. Since U 2n are absorbing and balanced, t1l':re exists
a 0:>0 such that (>.x) 2-n for 1?-1<6 . Hence (AX)$'''1{AX)'''''O
as }..... 01-. Thus, is a pseudomodular in X and X = X" Now, we have
(X) fYtlX) 2 I!$(x) tor every x€ X. Hence U(2- n + 1 ) U 2 and U 2 1 ? U 2
-n-1 n n- n
::/U(2 ) consequently,.13 and J.3( ) are equivalent.
6.13. Let (X,T) be a topological vector space with topology T.
Then there exists a family J3 T of absorbing subsets of. X such that for
arbitrary U1'U 2 f:.P.J T there exists a U 3 E' J3 T such that Ll(U 3 )CU 1 I\U 2 '
where (U) denotes the set of all elements of X of the form o<.x+(5 y
with lotl 1, It?J 1 and x,y U. The family J3 T which we shall call a
topological in X, is a fundamental system of neighbourhoods of ze-
ro in X, Le. every neighbourhood of zero in X contains a set from ..t3 T
and every set from J3 T contains a neighbourhood of zero in X. Moreover,
the above conditions on J1 T define the topology T, uniquely. Clearly,
r(U) c 6.{U) for every UC:X, and so a topological base in X is also a mo-
dular base in X. The converse does not hold generally, Le. a modular
base does not necessarily define a linear topology in X. Namely, we ha.-
ve the following
6.14. Theorem. A modular base JJ in X is a topological base in X, if
and only if, there holds the condition (D 2 ) from 6.2.
" Let jj be a topological base in X. Then for every U c 13 the-
re: exists a V E such that (V) c: U 1'\ U '" U. But 2Vc: (V) for every
VeX. Hence 2VcU, Le. we have ( 2)' Conversely, let us suppose that
a modular base fb in X satisfies ( 2) . Let U 1 ' U 2 G 13 be arbitrary
then, by (42), there exist V 1 ,V 2 E-£' such that 2V 1 CU 1 and 2V 2 <::.U 2 "
Hence 2 tv 1'\ V ) cUI') U . Since .12> is a modular base, so there is a
\ 1 2 1 2
V EO- J3 for which P(V) c V 1 () V 2. However, we have /),(V) c. 2. f'(V) for
3 3
32
every Vex. Hence
.L\(V 3 )C 2r(V 3 )G 2(V 1 AV 2 )GU 1 1" U 2 .
Consequently, J3 is a topological base in X.
6 0 15. Theorem . If two modular bases $1 and .15 2 in X are equivaJ.ent
and .13 1 is a topolOgical base, then .13 2 is also a topological base.
Proof' . By the assumptions, there are numbers a 1 ,a 2 /o 0 such that
for arbitrary two sets U 1 E J3 1 and V2 .$02 there exist sets V 1 E' J3 1
and U 2 € J3 2 for which 8 1 U 2 C U 1 and V 1 c V 2' Since ..e 1 is a topological
base, it satisfies (.L\2). We shall conclude first that for every a /0 0
and every U J?J 1 there exists VE-J3 1 such that aVcU. Indeed, let n be
so large that Jal 2nthen, takingW 1 '" U, there are setsW 2 ,W 3 ,...,
Wn+ 1 c<J3 1 such that 2Wk+1C::Wk for k = 1,2,...,n. Because .13 1 is a modu-
lar base, there is a set V E J3 for which rev) c W l ' and since bal V
n+
c:: rcV), we obtain U:I W 1 :'::> 2W 2 ? . . .:::> 2n., 1 '::> 2 al V::> aV. Hence, given
n+ -1 -1
U 1 (:- $1' there exists a set V1 ..13 1 such that 2a 1 a 2 U 1 C::V'1. Taking
-1 -1 -1 \;-
U 2 E.J6 2 for which a 1 U 2 cU 1 , we obtain 2U 2 C2a 1 U 1 :1 a 2 (2a 1 s2 U1,- V1
c V 2" Applying 6.14 we conclude that J3 2 is a topological base"
The last theorem shows that the possibility of defining a linear to-
pology in X by means of a modular base is a property of the generalized
modular space (X, [£>1) . Moreover, making use ctf theorens 602 and 6014,
it is evident that
6 0 160 Theorem o Let be a pseudomodular in X such that X f = X. Then
the following two conditions are equivalent :
(a I no rm conve rgen ce and -conve rgence in X are e qui valen t ,
(b) J;(e) is a topological base in X.
CHAPrER II
ORLICZ SPACES
7. Generalized Orlicz spaces
7.1. Definition . Let (.a., ,f'A-) be a measure space, Le...Q..is a
nonempty set, 1: is a <r-algebra of subsets of..o. and?- is a nonnegative,
complete measure not vanishing identically. A real function 4' defined
on ...o..X R+, where R+ .. (0,00), will be said to belong to the f , if
it satisfies the following conditions :
(i) \fI(t,u)is a 'i-function of the variable U?O for every tE-..Q , Le.
is a nondecreasing, continuous function of u such that f(t,O>= 0,
(t,u);:>O for u.>O, (t,u)-:I'OO as u-:!"oo(see also 1.9),
\?-i} (t,u) is a I-measurable function of t for all U?o.
Moreover, let X be the set of all real-valued [or complex-valued),
I-measurable and finite f't-al!1lost everywhere functions on4, with equ.-
ality r-almost ever<JWhere.
It is easily seen that (t, jX(tJI> is a -measurable function of
t for every x E X and that
(x)=S<P(t, jx(t)!)d('A-
...a.
is a modular in X. Ivloreover, if If tt, u) is an s-convex function of u
for all t .Q. (see 1.9.1), where 0 < s 1, then is an s-convex modu-
lar in X.
7.2. Definition . The modular space Xe will be called a generalized
Orlicz space and denoted by LIf(..Q., ,It)(or briefly, L'f):
LI¥ ={X€ X : S r(t, A !x It)/) df'-"O as ..." o+}.
..A
I'I1oreover, the set
L "{X€X :l.If(t,lx(tJVd/l1< ool
will be called the generalized Orlicz class . A function x E X will be
called a finite element of L'f , if ).. x € L-P for every >.» o. 'rhe spaoe
o
of all finite elements of X will be denote d by Elf .
If X is the s}lace of sequences x = (t i ' with real
00
t:iJt(=N J, \0. are tp-functions and (x)= _J:.. i{lt ), we
I =1
or complex terms
shall write l<f
34
and Itp O in place of L'fI and L1f , respectively; Ilf and Itf' are called the
o 0
generalized Orlicz sequence space and the generalized Orlicz seq,uence
class .
703. Remark . If 'f(t, u) '" If(U) is independent of the variable 'i, we
say that Ltf and Ld are the Orlicz sPace and the Orlicz class , respec-
tively lsee Example 1.9.1). The distinction between Orlicz space and
Orlicz class is generally essential t for example, taking as r- the Le-
besgue measure in the (j'-algebra of Lebesgue measurable subsets of
the interval...Q..=(0,1J and \flu)= e 1u1 _ 1, we have for the function
) .1 -n -n+1
xlt =< 2 n for 2 <t 2 , n:= 1,2,...,
( ) n - n
(x)= L. ; - 1 < 00 and (2x)= L ( ) - 1 = 00 0
" 1 1
Hence 2x Lit, but 2x ct Lb' Let us still remark that in the case of 'f
independent of t, the modular and the norms I , ' II II; (see 1.5)
are rearrangement invariant, that is if x,y € are equimeasurable fun-
ctions, then e(xl= (y), IXI =1Y'le and "xII; =IIY/I; . This property, very
useful in the theory of Orlicz spaces, does not hold in general for ge-
neralized Or icz spaces.
7.4 .Elementarv properties . la) Llf is the set of all x € X such that
(). x) < 00 for some >t.> O. (bl L is a convex subset of L'f and Lf is the
smallest vector sUbspace of X containing L . lC)EIf is the largest vec-
tor subspace of X contained in LI¥ 0
o
R1:2.21. la) follows, applying the Lebesgue dominated convergence
theorem. Convexity of LI.f is obtained, making use of the condition 1.103 0
o
in the definition of a pseudomodular. From (a) follows that L c: Llf and
that L4' is the smallest vector space with this propertyo 1.1.3 0 implies
also Elf to be a vector subspace of LI.f , in fact, the largest one.
o
7.5. Definition . A function If€ will be called locallY' inte rab e ,
if SIf(t,u)dr< 00 for every u,>O and At-I; with ft(AI<OO'
t .
7.6. Theorem . Let S be the set of all simple, integrable func ons
on.l1. and let tfE: <f be locally integrable. Then Sc.EIf. Moreover, sup-
posing t'- to be 6"'-finite, Elfi is the closure of S with respect to the
F-norm. I I and S is -dense in L If .
Proof . The inclusion SCEIf follows from local integrability of'f '
immediately. Elf is closed in L , since if xnG Elf and Ix- I -? 0, the'n
35
the inequality o.:X:':( (2A(X-:x:J) +g(2J\x n ) implies o.Jt}< CD for
every ;x.,. O. Now, let x€E<P, xltl o in..Q. . Let(xd be a sequence of
inte grable simple functions such that 0 x n (t) t x (t) for t . Then
If(t, 'A !xnlt)- xLtH)->o and (t,), Ixnlt/ - x(t)/).{!f(t,,\x(t) for te-....'1 ,
>.,. 0 j moreover, l lf ( t, ? xl t)1 df-t '" (>. J9 < (x). Hence, by the Lebesgue
dominated convergence theorem, we obtain (A(xn-x))"",O, Le.\xn-,--+O.
If we drop the assumption xlt)?O in...Q., we may split x into positive
and n\3gative part which belong again to E'P, and apply the above result.
Thus, S is dense in E(f with respect to the norm. The fact that S is -
dense in Llf (see Definition 5.4) is proved using analogous arguments.
Let US remark that if \f(t, u)m \fLu) is independent of t, then If is
always locally integrable. Moreover, ()-finitene s of fA- in Theorem 7.6
is then not needed, since from the inequalities 0 sx (t) x(t) and x€ L'f
n
fOllows directly that x are integrable.
n
7.7. Theorem . Let fA- be (j-fini te. Then the generalized Orlicz space
Lf is complete with respect to the F-norm I I .
. 1) First, let us suppose (J2}<oo. Then the measure f"- is
absolutely continuous with re.spe ct to the me asure v( AI"", S. <P(t ,E J d '
A
A f- , for every E"7 O. Hence for every E;> 0 there erists a 6'> 0
such that if AE-2: and \f(t,f) dfA-< 0' , then (A)<f . Now, let (xd be
a Cauchy sequence in rlf and let ;\;:> 0 be arbitrary. There exists an
index N such that (>.(xn-Xm»<d for m,n>N. ThuS, writing Am,n -ft€-J2.:
}.Ixmlt)- xnlti/ t], we have SA 'f (t,€)dfl<: $ , and consequently,
u(Akf. for m,n)N. Hence tw nse(f_uence (,>.x) is convergent in mea-
r- m,n n
sure }A- to a function .>. x € X in.Q . By Riesz theorem, l contains a
subsequence (Xni) convergent to x f-.-almost everywhere. Henae
If(t,,>,I (t)- Xni(t)j) lf(t,'>'lxnLt)- x(t)!) as i CD r-a.e. in...o.
for every n. Applying Fatou lemma, we obtain
Q('Alx -x» lim dil(;. -Xn.J)<£ for n;>N.
"'1 n -::>(X) 'f n
Thus jX n -xl O. Evidently, Xf t'f.
2) Now we drop the assumption ft{..5l)<oo. Let (.sz):: CD , .s2:::..Q1 u
JL 2 u "" J2 1 cSt 2 c ... t e4 lP k' < (X) for each k. Ap:9 1 ying the previous
part of the proof to.12 k in place of...Q. we see that there is a functi-
on x X such that Lx) tends to x in measure f on each of the sets..Q.k
n
and
36
l <f(t, A /x n (t)- xlt)l)d _ 0 as n _ CD
for k .. 1,2,... Taking n fixed, let us write Yk(tJ.. Lp(tt illx n tt}- xttW
if t , Yk (tl- 0 if t 4 , y(t).. <pet, .x )xn(t) - x ltJl). Than we get
lYk(t}d "' \f(t, lxn<.t)- Xlt)Vdt"
S tp(t, A/x ltJ- Xni(fJl)lil"" " lli LA\xn-Xni») E
i Q) n i oo
for sufficiently large n (independently of k). Since Ykttl-ytt)a.e.
in...o.., we obtain (>-,(x n -x))-l ylt 1 df- Yk ('i) dt'- E. for suffi-
ciently large n. Conseqtlently, )Xn_ _O;(X)..$l
Let us still remark that taking ..p(l;, u) "" (u) independent of t, we
do not need to have tA tr-finite in order to get completeness of Llf .
This follows from the fact that then the first part of proof of Theo-
rem 7.7 may be repeated without the assumption of finiteness of rt- .
Thus we have
7.S. Theorem . The Orlicz space Llf is complete with respect to the
F-norm I fe .
7.9. Remark . Arguing as in the proof of Theorem 7.7 one may prove
easily that if /-t(..Q.)<00 and (",xn)"""O for some .x» 0, then the sequen-
ce (X n ) is convergent in measure fA- to zero.
Now we shall inveatigate the problem of separability of Elf and L4f .
at US recall that a measure I"'" is called separable , if there exists
a sequence LAn) of sets Anf: 2: having the following properties
1) f'1lAJ < 00 for n == 1,2,...,
2) for every set Af. with !4(A) < ooand for e'fery ;> 0 there
exists an index n such that /«A':' A ) < f ;
. n
here, A .:. B means the symmetric difference of the sets A and B. We ha-
ve the following
7.10. Theorem . Let Iff: <P be locally integrable and let r- be o-fini-
te and separable. Then the space Elf is separable with respect to the
F-norm I I and the generalized Orlicz space Llf is -separable (see
5.4).
. Let us take a sequence (An) of sets from the definition of
separability of the measure fA- . Let So be the set of all simple func-
37
n
tions of the form y(t)... z: b i y (t), b i being rational numbers and v
. 1: 1 '\Ai "'Ai
be ng the characteristic function of the set Ai' According to 7.6 it is
sufficient to show that the set So is dense in S w;i th respeot to the F-
normll . Let XES, Xlt'.:£1 ai iltl withBie- pairwise disjoint
'and /A tB ;} < 00 for i ::s 1,2,...,n be given, a '" max lail . Let US take ar-
1 i n
bi trary >.. 0 and E> O. Then we may choose sets Ak1'." ,A kn from the
sequence lAn) in such a manner that
S V'(t,4 a )dfA- < : for i = 1,2,...,no
Ak i !. B. n n
n
Taking B ::I Q B i , we have S <p(t,n1/) d _ 0 as ...>O. Let 0,/ 0 be
=1 1 B
such that <f(t ,2A6') dft< '2E . Let us choose rational numb;) rs b 1 ,..., b n
in such a :manner that lb. -a_ I < 6" and Ib _ I 2a for i = 1,2,... ,n. Then
J,;
we obtain
Hence
n
jxtt)- y(til DXBltl+ 2a L j X B (t)_ X.A k t) I.
i:1 i
n n
(i\(x...y)} J <.p(t,2>'O) dr- + S \flt,4.Aa 2:. I B; ltl- A ki lt)Vdl(.(.
B ..a.. i::c1.L
n
S tp(t,2;\b) d + 2: If (t,4 i\an)df < E.
B i=1 Aki - B i
Let us still remark that in case of l(I(t,u)= <.fCu}independent of t,
the assumptions of local inte grabili ty of lp and (}-fini tene 1010 of rv'- in
Theorem 7.9 are superfluous.
Applying Theorem 5015, we shall now prove some approximation theo-
rems in spaces L'f... Ltf([O,b), 2 ,m) and in l , where O<b<<D, m is the
m
Lebesgue measure and is the (j"'-algebra of all Lebesgue measurable
subsets of' the interval (O,b); this notation will be valid until the end
of 7.
7011. Definition . Let tpE-<P with...Q.=(O,b) be extended b-periodically
with respect to the variable t to the whole real line R for every u€R+,
i.e. \f(t+b,u)::s <.f(t,u) for tER, 1>LfR+, Let "t"=('t"JvGR be the tranla-
.:ti.Qg operator , i.e o 't"vx It,= x (t+v) , where x is extended b-periodically
to R. The function If will be called '('- bounded , if there exLst cnsta-
nts k 1 ,k 2 > 0 such that
f(t-v,u) k 1 1f'(t,k 2 U}+ f(t,v) for v,t fR, UER+,
38
where the function f: RxR _R is measurable and b-periodic with res-
+ b
pect to the first variable and such that writing h(v):: S f tt ,v) dt for
o
as v - Ot- and v -? b-.
every v E R, we have H = 10M hlv) < co and hlv) 0
v J:t.
Let us remark that if If is s-convex with respect to u, 0 < s 1,
then we may take in the above definition k 1 = 1. A trivial example of
a 't'-oounded function is obtained taking <f independent of the parame-
ter t. 't"-boundedness in non-periodic case will be used also in Theorem
9.6, concerning convergence of 1:" x to x in the norm of L if .
v
7.12 o Theorem . Let tfE be <C-bounded and let ')() be the filter of
neighbourhoods of zero in '0"= R. Then the family 1""= (-r') R of transla-
v :go
tion operators is '}{)-bounded(see 5.13). If, moreover,S !f(t,a)dt <. <D
for every a) 0, then 't' x x for every characteristic 0 function x of a
v
Lebesgue measurable subset of [0, b) .
Proof . 1(1-boundedness of " follows from the inequalities ({U: x)=
;p v
blf'(t,/x(t+V dt k 1 E?(k2X)+ g(v), where glvl= h(V)=S f(t,v)dt. Now,
o 0
let x be the characteristic function of a measurable set AC[O,b), then
for every v E R, (a ('t'vx-x))= JAy 'f(t, a)dt , where Av = (A-v)..!. A. Since
o \f(t,a)dt < 00 for all a> 0 and it is easily seen that mlAv l - 0 as
o
V '70, so [a(t'"vx-x)) O as v -.,. O.
From 7.12 it follows easily, applying 7.6 :
7.13o Theorem . If if is 'j;'-bounded and X O 'f(t,a)dt < 00 for every
.>0, then for every x ELr.f there is a c>O uch that
Wr(cx,S)"" sup b 'P[t, c Ix (t+v) - xlt)1] dt -!>' 0 as 6 -P Ot-.
ITj 0
Let us still remark that the same holds with respe ct to the F-norrn
in Lif , if we restrict ourselves to x EIf in place of xf'LIf (see 7. 6 ).
We are going now to investigate the convolution operator Tw' where
w E1{ ,1J' is an abstract set and'}')() is a filter of subsets of 1ir .
7014 .Definition . Let l\v : [O,b) R+ for we1V be integrable in
[0,0) .lKw)wtV' will be called a singular kernel , if
b b-o b
(j(w,= S Kw(t)dt 1 , Q6(W7= S K It\dt O for every %10(0'2)'
o 0 w
b
0- sup S Kw(t)dt < 00.
wE1t 0
b-periodically to the whole R, let us write
b
TWX tS)" S Kw (t-s) x It) dt .
o
Extending
(-IE')
39
7.15. Theorem. Let iflE- be convex and «"-bounded. If(K is a
wE
singular kernel, then T : L L for every weVT and T =(:r ) is
w w weW'
7JD-bounded.
Proof . According to 7.4 (b), it is sufficient to prove that T is
W-bounded ; henceforth follows that T i LIf_LIf. Applying b-periodici-
w
ty of l' ,u}and x(-) , Jensen's inequality and t"-boundedness of <f with
k 1 .. 1, k 2 .. k? 1, we obtain for x (: LIf:
(lJ!wx)- S <f(s, oi..wJ S KwLt} 6"(w) x (s+t)dt)ds
1;> 0 b 0
(j W) S Kwlt)J <P(tu-t,CJx(uJ)du dt (k x)+ g\.wl,
o 0
1 b
g(w). O(w) J Kw ltl h(t) dt .
Let us split the integral definingOg w into three integrals over inter-
vals (0,6' , [6,b-S] ,(b-a,b], where 0<' < . Let e? 0 be given. We
may choose $ so smell that h It)( for It I < (j and then we may take
W EO ty)() such that b- 6
1 S E
(1Uv) o Kwlt)dt< 3H for W6W.
Then we obtain g(W)<E for weW. Hence glwl O.
Now, we are able to prove the following
b 7 .16. heorem . Let ifI€ be convex, 't"-bounded and such that
S W(t,a)dt <.. 00 for every a.> 0, and let Uc} "f,,, be a singular kamel.
o 1 w W€ N
Moreover, let T be tlefined by (1E-). Then T x I.t.. x for every x (; L <fl.
w w
Proof . By 7.15, TwX€Lif' for x€L'f. Let XtLIf' then, by 7.13,
wJ.cx.,c5) O as 6"-1"0+- for sufficiently small c>O. Now, let us choose
a> 0 so small that 2aG'" c and (4 O"kax) < 00, where k is equal to the
constant k 2 from the definition of «,-boundedness of If, k 1 being taken
eq ual to 1 ; we may suppose k). 1. We estimate now
b b "/
Ca(jwx-x») S f(s, cr ) y KwLt) 2 G(y.r) e.(x(s+tl- x (10») dt)dS
o 0
b
+ 5 (s,2a(01)v)-1)X(s» ds
b b 0
2 Lw) KwLtli (s,2o-a(x(s+t) - X(SJ)ds dt + (2a (G''(w)-1)x).
Now, we split the first of the integrals on the right-hand side of this
inequality into three integrals over intervals [0,6J, [ ,b-J] , [b-6 ,bJ,
Wihe re
40
where o..( < is arbitrary. The first integral is estimated as follows:
G b b
S Kw It I f If(s,2 0- a ex (s+t) - x (s)>>:ts dt... SK (t) f) (2 (T"a (r x-x»dt
o 0 OW ') t
{. cr6v1 (2 G""ax, b)
and the third one, by substitution t = b-u,
b b q
S Kwlt)S If(s,20''a(x(s+t)- X{s)))ds dt K (b-u) (2o-a ('!:_ u X-x»dU
b- 0 0 w
6"t w )W-r(2<Tax,J') .
Finally, the third integral
b-5 b b-&
S KwltlS tf(s,2()a(x(s+tl- X(sJ»)ds dt .r Kw(t)[ (4(jattx)+
D 0 6
b-8
L4o--ax}]dt t S Kw[tl[ (4Q"'akx)+ h(t)+ (4<rax)]dt
6
b-o
[ (4 6"'akx)+ t HJ % Kw (t Idt
(4tY"akx)+lH b-o
la(Twx-xl) wJ2t5"ax,6) + 6Lw) 2 KwLt}dt + t (2a(a@llr1)zl.
Let us take 8..."l arbitrary E. O. y 7.13, taking 0> 0 sufficiently
small we may make the first term on the right-hand side of tJ::e above
inequality smaller than . Then, by singularity of (Kw)' the second term
may be made less than ,taking W E W 1 with an appropriate W 1 E-1)')O . Sin-
ce x trL'f and U(y.r) (YII);:. 1, the third term becomes less than for w €-W 2 with
some W 2 EIOO . Thus, (alTwx-x»)<t for WEW 1 (1 W 2 .
Let us remark that taking as 1V the set N of all nonnegative inte-
gers and as tr){} the filter of all sets of the form N"-A with A finite,
ACN, we obtain an approximation theorem for a strong summability method
defined by the kernel (K n (u)), n = 0,1,2,... Taking as --W an interval on
R and a point w 0 in the closure of 1AT (may be also 00) and as 'Y}() the fil-
ter of all (may be also one-sided) neighbourhoods of W o in '-W- we get
an approximation theorem for strong summability method defined by the
kernel K(u,w), where u€-[O,b) , wt!:""W""
We are going now to apply Theorem 5.15 to generalized Orlicz seque-
nce spaces tf; for the sake of convenience we shall takesequencesx :I (t j )
OC>
with j:::: 0,1,2,... and (xl=2:.tD,(lt,I). We shall investigate two
, I
=o
Hence
41
families of operators : & sequence of translation operators and a fami-
ly of convolution operators in 111'. Here, 11 will be the set N of aJ.1.
nonnegative integers and the fd:J..ter 4D will consist of all sets Vt:V"
which are complements of finite sets. The set -W- and the filter'f>O of
its subsets remain abstract.
7. 17 .Definition . The translation operator 't"'m' m"" 0,1,2,..., is
defined by the formula 'Lmx "" (( x )i)' where we take for x "" (t j ) ,
{ t i for i m
It'mx)i "" t. for i >m.
+m
Evidently, the 't'-modulus of smoothness of x ..it ,)is equal to W (x,V)
J '('
.. i vfiO\+m-t ). If V =[r+1,r+2,...}, we shall. write w (x,r) in pla-
ce of w..(x,V).
7.18. Definition . dJ=C.!D. ) will be called t' - bounded , if there
'f :'{ -0 -
exist constants k 1 ,k2 1 and a double sequence ("1n,j) such that
fn LU) k 1 . (k 2 U)+ 'YI , for u'i! 7 0, n ?J? 0,
n+J "tn, J
where th ,>,. 0, Ih '" 0 and f ,< 00 uniformly with respect to
"I.I1,J, '(n,o n,J
m. Cf will be called - bounded , there are constants k 1 ,k2 1 and a
double sequence (e j > such that
n,
If ,(U) k 1 cO (k 2 U) + ' for u9-0, n,j '" 0,1,2,...,
n+J 1n n,J
00
whe re f . 0, e. n '" 0, t. "" 2. <C . - 0 as j 00, £:z Ii! up E. < 00 .
n,J n,,,, J n.:s. n,J O J
We shall write in the following e l =( C. l ) , where 5, l is the
, -o ,
Kronecker symbol.
7.19. Theorem .la) If If is <['_-bounded, then the family't'""=(t'm):oof
translation operators is I){) -bounded.
lb1 The set of linear combinations of sequences e o ,e1'e 2 ,...is
-dense in 1 If .
leJ (a( mel-el)) o as m.-.oofor &>0 and 1 '" 0,1,2,...
Proof . (a) is obtained, because for x(, l ,
m 00 00 00
(!'mx)= C{'.Ot.l) + k 1 crj(k 2 lt i l) + E" l"fIi m k1 {k2x)+. 2"1 i ,m
i=o j=2m+1 i=m+1' =m+f
CD
and I)'J, _ 0 as m _ (x), by the assumption of uniform convergence
. 1 1 l ,m
=m+
of of .' (b) is evident and (c) follows from the fact that Tm 6 1 -e l
I1=0 n, J
42
-(0,...,0'(2IIH-1,1- 0IIH-1,1' 02IIH-2,1 - SIIH-2,li''') "" 0 for m?l.
From Theorems 7.19 and 5.17 follows immediately
7.20. Theorem . If 4> is 't'_ -bounded, then for every X€ llf there is an
&7 0 such that 101:' (a:x:,r)_ 0 as r (J) .
In order to investigate the convolution operator T in 1'1' ,where
i w
T w x :s((T x) i )?O with (T x) i = 2:'K _ _to , x =(t.), WE:1iT ,
w ""O w _ w, -J J J
J::::o
and ty)() is I!t filter of subsets of an abstract set 14T , we adopt the
following
7.21. Definition .Let K : N -R for w E:--ur . eK) L-t,,,", will he cal-
w + w Wo;;- ""
led an s- singular , C <: 10 $ 1, if
0'" :s ( fx: s j) 1/; ()" < CD, K 1,
w . w, "" w,o
J"'O
7 .23 . Theorem . Let (K ) be s-singular and let llJ = Llf; J -o be
w w"',.,.. 1 .L.L
'C+-bounded with <Pi s-convex for some 10 (;;(0, 1] and i = 0,1,2,... Then
T : ll/{_l(f for every wE;; 11' and T =(T ) is 1}()-bounded.
w ww
Proo . As in the proof of 7.15 , it is enough to show that T is
I}t)-bounded. By Jensen's inequality applied to s-convex function Ifi'
K , ry\(J
-YiJ.J. 0
Qw
for j = 1,2,...
we obtain i 1/10
00 ( ( K S )
(TwX)'" [ i j"'o ; j
=o
f
i=o
i
K S .
;]=0 w.;]
l)s
w
t. K _(Jto_ )
.. = w. J w -J 5:
. 1/10 ""
K S ,
. =0 w J )
( K ,():t. - 1 00
rD. -0 w w - $ --:::!J L
1 ( """ ) 1/10 (J' 0
Ks w =o
J=O w, j
CD k 00
'W. ,(() t. J . -l c::;- K S ,
T +J w O"'!.-?- W,J
=o w J=O
i
2:: K S ,<p i ((] t. ,)
. W,J I w -J
J=O
00
_.-L 'K s
- ()S ?- w,j
w J=O
0>0
2: <D, (k 2 t.) +
. T w
=o
1 00 CD
+ <Js 2: K S e- , k 1 (k2 c-x) + c lW) ,
'=0 w,j i=o , J
w
1 CD 10 ,'r1O
where olw). --:t"L K .e, . We need only to p rove that c(wJ-? O.
0"16' ,. w,J J
J=o €-< 1
But taking any fY} '> 0 one may choose an index r such that su¥ 2 '
J') J
Then
43
r KS _
o c (w ) 2: -!!l.l t: + -L
j=1 0-: Q'
00 -tI
L. K S , -" 2 $
- 1 w,J
J =:rf-
r K S I\(
6 l ) + 2'
J=1 <fw
for arbitrary we-1V' . Now, let W€-ty}Q be chosen in such a manner that
Kw J - 0-;1 < ,,1/S(2 rr1/s for WEW and j = 1,2,...,r. Then c(w)<"1
, l}')()
for w E: W. Conseq uently, c (w) -;> 0 and so T is 'hD-bounded.
( . CD
7.24. Theorem . Let if) = If.) _ be a 'L -bounded sequence of s-con-
,. =o +
vex functions lfi' where 0 < 10 1. Let tKj w 1J' be an s-singular kernel,
l\v :S(K w J -)CD J '- O ' where the family of elements xl =(O,...,O,K 1 ,K 2 ,...)
, ' -, w w, 1 w,
mth zeros on the f rst 1 places satisfies the condition ( xw) O
for some >O for l:s 0,1,2,... Then for every XEliP there is an a." 0
such that (a(Twx-xIJ O.
Proof . By 5.15 and 7.23, it is su;fficient to show the theorem for
x = e l , 1 = 0,1,2,... However, it is easily calculated that
1
(a(Twel-e )= \(il(a(Kw , 0-1)) + 2: tp,(aK '1)= l(a(K -1))+ (ax )
i=1+1 w, - w,o w
for a>O. Choosing a>O so small that a.x ) o and taking into acco-
unt the assumption that K 1, we obtain the theorem.
w,o
8. Embeddings of generalized Orlicz classes and Orlicz spaces
8.1. When considering inclusions L6 C:L6 of generalized Orlicz cla-
sses and L\fCL of generalized Orlicz spaces, the following conditions
are of importance :
(1) ,¥It,u) x\f(t,u}+ hlt\for all u O and tk-almost every tfJ2 ,
where h is a nonnegative, integrable function in..cl and K is a positive
constant ;
(2) Ht,u) K1\flt,:rs.u)+ h(jo) for all u,?O and -almost every te-.Jl,
where h is a nonnegative, integrable function in.J2 and K1,K2 are posi-
tive constants.
The condition(z) will be also d noted writing \¥.::ttf ('f nonweaker
"t').
First, let us remark that in case of functions <y, "t'€ independent
of the variable t, both conditions can be given an equivalent form
not containing the function h. Namely, we have the following
44
8.2. Theorem . Let f'''f'E be independent of t, Le. tp(:t,u):s £f().1],
tt,u)= *). If ll4(JZ)(CD, then the condition 01 is equivalent to the
following one
{11) there are X, U o > 0 such that \f lu) K tu) for u ".fu o '
and condition l2) is equivalent to
(21) there are K1'IS, u o / 0 such that 't(u) K1lf(K2u) for u no'
If jL«.Jt):: CD, then the condition t11 is equivalent to
C1a)there is a Ie/O such that 'flu} K<.fLu) for all u O,
and condition (2) is equivalent to
(2a) there are K1'IS. ,/0 such that \.flu) K 1 (K2uJ for all u O.
Proof . We shall limit ourselves to the condition (1) ; the proof in
case of l2j is analogous. First, let f"t l.J2.) <00 . Supposing 01) and writing
flu):s 'fLu} - K £flu), we have o ma:x: f (u):: M < CD, by continuity of f.
o<u Uo
Hence ,/,lu) Ie Lu) + M for all u 0 and taking h til) '" M for tE-.Jl we obtain
(1) . Now, let US suppose that (1) holds but (11) is not true. Then there
is a sequence 1' 00 such that 'KuJ>2n\f(uJ. Hence, by (11 , htt) 't'LuJ-
K\.f('\1Pt (2 n -KNC u J- 00 8.'$ -tJ. oo, and so hlt)"" CD for almost all te-...Jl , a
contradiction. Secondly, let,u{.a.)= CD. Evidently, (1a) impliesl11 with
hlt)= 0 fortlOJ1.. Now, supposingL1) to be true but (1 a) not, one may
find a V;70 such that hlt} VLul- Klflu) for all u»O, but \f"lv) '7 K tv
Hence h (t) 1'lv) - K\f(Y)70 for tE...Q. . Integrating this over -'l. , we
obtain S hLt) dj\-t= CD , a contradiction.
.Sl...
In order to obtain embedding theorems for generalized Orlicz clas-
ses and spaces, we shall need the following
8.3. Lemma . Let the measure fA- be atomless, and let a sequence (P(
of positive numbers and a sequence (a i ) of measurable, finite, nonne-
gative functions in..Q be given, satisfying the inequality
:h a i It) dl""':t 2 i c('i for i == 1,2»-..
Then there axis'i an incJ:"i)asing sequence (i k ) if integers and a sequence
L l of pairwise disjoint sets from 2: such tlat
S 8:1. It) df.. :s oC:i k for k:: 1,2,...
Ak :It
Proof . We shall outline the construction of both sequences li k ' and
(AJd ' which follows hy induction. Since is atomless, there exists
a set D E'i: such that S a ltl dfA '" a:: . T hen there exists a subsequen-
1 D 1 1
1
45
. 1
ce( ) of the sequence (a.) with i = 2,3,... such that either
r 1 1 1
(a.) 1 lt)d '2cek for k = 1,2,...
or
(b)
S 1 1 1
..a.....D1 (t) dt't '2 Q)k for k = 1,2,...,
where l<t ) is the sUbsequence oflDC.) with i = 2,3,... corresponding to
1
the subsequence (ak). In the case (a) we take A 1 = D 1 and i 1 = 1. In the
case (b) we choose A 1 c .a.....D1' A , so that SA a1lt) dt-A-= «'1 and i 1 =1.
In order to define A 2 and :i,2' we argue as above 1 replacing -!l ,La I and
1 ( 1 1 ) i
(o('i' by !l..... A 1 , La,..) and 2' ot k ' respectively, thus obtaining subsequan-
( 2..1 ) .Z' 1
ces IOfl , kl of k) , a set A2E and an index i 2 >i 1 . Next,
we replace..Q.by ..o.,(.A1VA2)' etc., thus obtaining and i3' Obvious
induction leads to sequences (AkJ and (ikl satisfying the required condi-
tions.
8.4. Theorem . Let If ''4'f .
la) If It',"r satisfy the condition 8.1. (1), then L%C L .
lb) If L'-P L't' and the measure u is (j-finite and atomless, then the
o 0 r
condition 8.1. (11 is satisfied.
Proof . Since (a) is obvious, we pro cede to the proof of (b). Let
JL=.2. 1 vS1.. 2 V ... , where.!li are pairwise disjoint and of finite measuxe
Ai. Let x . Lt) = r for t e....<1., x ,(t) = 0 elsewhere in...!l , where r is
,.- r, r,
a nonnegative, rational numbe r. Let us write
(t)= sup ('t'(t,u)- 2 n \{J(t,u)).
u O n
First we prove that h l' ) = sup { ,tJt,x ,ltJ)- 2 Iii (t,x . (tJ}1 r O, r ra-
n Tl r, r, ,
tional, i = 1,2,... J. Let us fix tE' , and let tE:-..C2. i . We take a sequa-
n l -k
nce l,\:1 of nonnegative numbers such that (t, uIJ - 2 tf{t, ) h n t) - 2 .
Since 'rlt,u) and Iflt,uJ are continuous with respect to u, there exist
rational rk 0 for which I\y'(t,rk):r '\f'lt, '\:) - 2- k and tf(t,rkl «I(t, '\:) +
-n-k
2 . Then
r(t,xrk,ici» - 2 n tp(t,x ilt)j:s '\f'(t,r k )- 2 n tp(t,r k )-r
r k ,
f- 't ,'1J- 2- k - 2nl\'(t, }- 2- k -;, hn(t)- 3.2- k .
Hence
!1:J.lt) ,+,(t,xIk,iLt\)- 2nt.p(t,xrk,ilt)) + 3'2- k
46
(sup{tlt,xr,iltl)- 2 n \f(t,xr,ilt))1 r 9 0, r rational, i:s 1,2,...}+ 3'2-
Taking here k......,.oowe obtain the required formula. This may be written
in the fonn
hn(t). sup C"",(t,Xg;(tJ)- 2 n If(t, lt))
k:=1,2,...
wheretxk)is any rearrangement of Qcr,i) with x 1 "" xo,i . From this for-
mula. follows that are measurable functions. Obviously, h n (t)-30 in..Q..
It is easily observed that (b) will be proved, if we show that h
n
is integrable in.n for some n. We shall prove this indireotly. Let US
suppose that S Lt)df'l-= 00 for n = 1,2,... Let us write
..n. n
bm,n(t}:s ma:x: ('t'(t'"'klt1j- 2 (t, (t) .
1 k m
Since X 1 ltl. 0 in..Q., we have ° (tl O. Moreover, b are measurable,
m,n m,n
and (b (t) ) is a. nondecreasing sequence tending to h (t) as m _ (J) for
m,n n
every 1I ..Q... Hence for every n there exists an index m n such that
1. °:\11n,n (1;) dfl- 2 n . Writing b n .. 0In,:J.tn , we have .1 on (t) dfA- 2 n for n ·
1,2,... Let us denote
.Rn,k -ttE'A I (t,Xg; Ltl)- 2 n If (t, tt}).. on (til, B A n,1 .. .VBn,IDn) ,
then ftCBn). O. Let US put
f o if t EB 1 VB
It). n, n k-1
Xg;Lt) if t Bn,k"" U
j=1
B "k:s 2,3,...,m.
n,J n
Then
bnlt).'f(t,Xnlt))- 2nlflt,xnlt)) 0.
Hence
S 'f(t,x n (t)df-- 2 n S <p(t ,x n Ctl)df + b n (1;)df'<- 'J Sb n ltJ d 2 n .
..Q. A ..n.. -a.
Hence, applying Lemma B.3 with a i Lt). f(t,xi (t/J and eC i · 1, we conclude
that there exist an increasing sequence ( I of indices and a sequence
(AIJ of pairwise disjoint, measurable sets such that
iAk't'(t''X nk (t)}djl-t · 1 for k == 1,2,...
Denoting
Fn Lt) for tEAk' k=
x (t) =t 0 lsewhere in J:l
we obtain CD
o (x). S'Y(t,Xlt)'d .2. 't(t,xnkltl)d . 00
"')'¥..a. k-1 k
and
1,2,...
47
+X)- (t,x It))d . £ 2- nk (4 l'tt,x Lt)}d - b t) df")
f 2- nk J 1(t,2' (j;»dl (1.
1 K
Hence x f- Lo ' but x 1 L ' a contradiction.
B.5. Theorem . Let tf.1.f'E P. (Q.) If 't' 'f ,
itltt&. L'IcCf'aftOt e.-\fc E't . If, moreover, "f' is locally integrable,
then modular convergence Crespo norm convergence) in L(jI is stronger
.'r
than modular convergence (resp. norm convergence I in L .
(b) If Llfc. Lifo' and the measure r- is (J-finite and atomless, then '/I 'f .
Proof . (a) It is obvious that '+' <f implies L(fCLlf'and ElfcEo/, by
7.4La). Now, let us suppose that'f' is locally integrable, XnELif and
tpL2I xJ. 11:f(t, IX n ltll)d ""O as n ""'" 00 for a fixed '>. '7 O. Let E'> 0
be arbi trarily given. T a,king the function h from the condition B.' l2)
we may find a set A61: such that.A A hlt)djlA.< and f,tLA)<;.CDI' Since
'i' is locally integrable, there exists a number a/, 0 such that
1 (t, );K;' a)dfA-< 1 . By 7.9, (x n ) tends to 0 in measure in the set A.
Wrl ting An · tt € A I Ix l tll &1, B ::s A"'" A , where K 2 is the constant
n n n., £
from B.1. t2), we have thus jl1(B ) _0, and sc \' h (11) djL'L < _ 4 for suffi-
n "En
ciently large n, say n>N. lVloreover, we may suppose K1 tf'( xn)<: 4 for
n > N. Hence, by B.1 <.2) ,
1).K 1Xn)"" S'r(t, >-IS:',x n ltJl) djlA fA (t, A 'a)dl"1. +
T -'2- n
K1fBp 1f(t,>"lxnLt) f +5 B hlt/dr- + K'.Q!A (t,.\lxnlt)l}d +
n
e
.Jl hlt)dj\'t i'f ' >'K;'a)df + K, CAXd+ 2'4 < € ·
Thus, 1.jI(AK;1xn)<£ for n>N. This proves the second part ofla).
Now, in order to prove (bl we put
It) "" sup ('t'tt ,2- n u)_ 2 n tp{t, u)).
ufO
Taking the sequence (x k ) of functions as in the proof of B.4, we find
easily that
h n l t\.,. sup (i'(I;, 2-n Lt 1)- 2nlftt, (t))),
k::s',2,... S
and so are measurable, We need only to prove that ..{1.. h n (t) d < CD for
some n. Supposing this to be false and arguing as in the proof of 8.4,
48
applying Lemma 8.3 with a i (t)... y(t,2- i x i It)) and i =: 1 , we may defi-
ne a sequence l k) of functions such that for any .A::> 0, the function
x defined as in the proof of B.4 satisfies the inequalities
00 00 00
AX)= L S yet, ).. k ltl)d f 2. f '0/ It,2- nk X'nk(t»d/IA 2::: 2 nk =00,
k==1 :Ak 1\:=1 A k=l
-n k
where 1 is so large that 2 1 .A ,
00
tO(x) ;[ 2 -nk S r (t ' k (t)) d f<- 1.
T k==1 Ak
Hence xfLIf, but xEtL'I-', a contradiction.
We are going now to examine the problem of embedding of generali-
zed Orlicz classes and spaces in case of spaces of sequences, the mea-
sure \1.4, being pure y atomic. There will be given a general approaoh '0
his problem.
B.6. Definition . Let us: suppose that to every positive inte r n
there corrosponds a nonvoid set An of real (complex) numbers, and
let
t:,. ={(n, y): n EN, Y if: Ll n l ,
N being the set of all positive integers. Let two functions U,V :
(O,oo\xA_[O,oo]be given such that U and V are nondecreasing functions
of the first argument for every fixed value of the second argument,
and that there exists a sequenceG J, Y A for n e-N, satisfying the
00 n n n
condition J;1 U(o<',n,yn)<OOfor every OC;;,O. Finally, let a = (a(n)) be
a sequence such that a(n\tl: A for n eN. Then we define I
n (J)
U ={a : there is an index k such that kk U(oc,n,aln)) < 001,
v ={a : there is .an index k such that k V(('.> ,n, a(n)) < ooJ'
En(d.,('>, ,6) ={ye-.t. n : U(<<',n,y) <min{o, 4' V( ,n,y))f,
Fn , ,6) = sup{V( ,n,y): y€EnltJ,., ,!,an
for tt.t ,r,q > O. If Entd- , ,(,61 "" /J , then we put Fn ,f '0,6") = O.
8.7 .Elementary properties . (a) If O<<:tl<ci" , then II c I; more-
- 0
over, \Yn)E U or for every rA) o.
(b) Fntol, ,r,d"1 is nonincreasing with respect to oc, and nondecrea-
sing with respect to t 6' .
(01 If U(.rJ..,n,y)<6' , then V ,n,y) rU(ol,n,y)+ Fn(tJ,., ,(,[).
8.8.Theorem. Let A and B be nonvoid sets of positive numbrs. If
49
rj..Q u c v '
then there exist d. G A, f.> e B, ( O, 6'" 0 and an index m such that
00
L Fn ld' ,r-> ,(,<0 < 00.
n=m
Proof .Let US suppose the theorem to be wrong. Let elk e A and k (; B,
k := 1,2,..., be chosen in such a manner that for every dE: A and € B
there are indices k' and k" for which a'k '?ot for k k' and fk (3
for k k H o Let us write
bn,k = Fn( t ,k,k-2J.
If our theorem is not true, then ;: b k "" 00 for all k and m. Let
llcm n,
n 1 ,n 2 ,... be an increasing sequence of indices such that writing N'1 :::
{1,2,...,n 1 f' N k ={ _1+1,...,nki, k = 2,3,..., we have
(i) 2: b k" ' 2:. b k for k = 1,2,...,
m€N k n, meNk'£ 1 n,
where we put I: b k = O. Let US write
n'.0 n,
N D{n cN k : bn,k::> 01.
By the definition of F and b k and by the first of the inequalities
n n,
li), for every n€N there ensts Iii ZnE:En(o(k,(\,k,k-2j such that
(ii1 2+ v((\;,n,zn» 0
nE-N k
Applying the definitions of E ,F ,b k and the second of the ineq uali-
n n n,
ties <.1), we obtain
(.iii) + u(Gik,n'Zn}< +5:. V( k,n zn)+ -t
neN k nfN k ,{ 'k
.1 L b + --1- /L
k k 2 2'
nEN 't ' n, k k
Thus, for every n e N+ k there exists a Z e Ll surlJ. that (iil and (iii) hoH.;
n n
if n eN k " N ' we put zn ::: Yn. Now, let b = (zn)' We show that
b E: n u o " U v '
o..E A \10. f>€B
which provides a contradiction. Let c(€ A be arbitrary and let k' be
so large that Gi- for all k'+ k'. Then we have
t>O
2: Ut9(.,n,zn) ::: 2: :E uta( ,n,zn) <
.....
n= n - 1 k;::k'nl:N
k'-1 k
50
[L+ U(st.k,n'Zn)+ + u ,n'Zn)]<OO,
k=k' ncN k nEN \ N
k k
by (iii) . Hence b E DA U . Moreover, for an arbitrary €B apid m f; None
may find a k" such that ?-(\ and _1+1 m for all k k". Hence
00
2: V ,n,zn) f. L. Vl ,n,z)= 00,
n=m k=k" n€N+ n
k
by (;ii) 0 Hence b" "'YB V .
8.9 o Corollaries . (a) If J);. U c:. f> V , then the following con-
dition holds:
l+)there exist cI...€-A, e.>tB, ( O, 0>0 and a seq-uencela) suchii\a,t
CD n
man < 00 for some m and that U(ot ,n,y)< b implies
V( ,n,y)" ¥ UCcI...,n,y)+ a for n = 1,2,....
o n
lb) If l+) holds, then U oL Co V .
() 0 \J 0
le) If 'reA UcI.: c: "' B V"', then there exist cI.€A and GB such that
o orA "
U(i..CVf.>'
. 0 0
Cd) There holds Uotc VA' if and only iI, there exist ([70, f> 0 and a
'CD
sequencela.JsUCh that m an<oo for some m and t t U ,n,y)<o im.-
plie 10
V'0,n,y) U( ,n,y)+ an for n = 1,2,...
(e) Let Ud:={.a.: 'B. 6 U ,n,a(nJ)<oof' If UolC.0 B V;, then there exists
a (:JE B such that U ct C V ; in par"ti cular, if Uo(. C V , then c: V .
Proof . (a):rollows from 8.8 and 8. 7lC), immediately. In order to pro-
o 00
ve(b}let us supPose that l+) holds and aE;U oL , i.e. U(cX,n,aln))<"oo,
where we may suppose k'fID and Ulol,n,aLnIJ<6 for n k. Thus,
00 00 00
1:. V ( ,n, at.M) r L U ,n.aln\}+ 2:' a n .( 00,
n=k k n=k
Leo a€V;. Corollarieslc)and(d) :follow from(a/and(b), immediately. Fi-
nally, in order to show(e)let us first show that if Uot. C.A V , then
o V 0 0 00 I'
also UotC: eB V . Indeed, let a UoC' Le. k U(oI-,n,atnl)<oofor some k
and let us put a(n)= Y n for n<k, alnl= al for n k, whereGn)is the
sequence given by the Definition 8.6. Then aE' Ud,. Hence, by our assump-
tion, there is a EB such that aE:V . But this implies a€V . Now, ap-
. 0 0
plying (c 1 and the inclusion U c(p'tB V we just proved, we obtB..n UotC V(':I 0
We apply the above results in order to obtain necessary and su:ffi-
cient conditions for inclusions of generalized Orlicz sequence classes
51
l and Orlicz sequence spaces l'f (see Def1ni tion 7.2). Here, will be
the set of sequenoes q-Ll!k)of lop-funotions n' n - 1,2,... (compo 7.2).
8.10. '1!heorem . Let \f,'o/ e . The inclusion l (:1: holds, if and only
if, there exis numbers 6,0, It> 0 and a sequence LanJ of nonnegative
numbers with J;1 < 00 for which
fnLuJ<o implies "t'ntU! K\fntU)+ an for u. 0, n. 1,2,...
Proof . Taking in 8.6, .. the spaoe of real [or complex) numbers,
U(g.,n,u)_ n lul} and Vl ,n,U)"""I'n lulJ, we have 1 = u;', 1 "" V 0 The-
refore, the inclusion 13 C1! is equivalent to u C V , any by 8.9ld),
this is equivalent to the following oondition : there exist numbers
r> 0, 0"70 and a sequence (a ) of nonnegative numbers satisfying the
(I) n
condition z:. a < 00 for some m, such that to (lul)< t implies \JI ( Iul )
Il;:Im: n "f n 1 n
¥<fnl1ul)+ an for n = 1,2,..., U}O. Taking an '* ma.x[ ("/'ntU)-(l.fnlU)),c8
for n < m, where S is the (compact) set of all u -; 0 such that n lU I d
for n "" 1 ,2,... ,m-1, and writing K = 3' ' we get the required oondition.
8.11. Theorem .Let 'rE-<£ . The inclusion Ilfc.1'tholds, if and only
if, there exist numbers (;>0, K 1 > 0, K 2 > 0 and a sequence lan'of nonne-
gative numbers with l: a "00 suoh that
n
CFnLu1<O implies 'f'nlu)(K1lfntK2u, + an for u O and n =- 1,2,...
I'iloreover, then modular convergence (resp. norm convergence) in Ilf is
stronger that modular convergence (resp. norm convergence) in 1'V .
Proof . Taking n' U(ot,n,u) and V( ,n,u) as in the proof of 8.10,
we observe easily that 11.f==oI..Vo U and 1'1' -f>Yo V . Hence, applying 8.9l e )
we may conclude that Itfc:::.lll', if and only if, for every 0(./0 there exi-
00
st positive 'r, and a sequence (an)' an 0 with m an < 00 for
some m, such that for arbitrary u O and n == 1,2,..., the inequality
IfnC9lu)<. 0 implies \fn( U) 'fnto(u)+ an' But substituting v == (6u
and writing K 1 = ' K 2 = ot/f-> ' we may rewrite this condition as the
following one : there exist K 2 :> 0, ['70 and a sequence tan' as above
such that 'fnlK2v)<o implies rnlv) K 1 tpn(K 2 v) + an for v O and
n ::I 1,2,... Arguing as in the proof of 8.10, we may take m ::& 1.
In order to show that convergence in lq' implies convergence in 1'1'
let us suppose that xnf: 1", n:: 1,2,... and 1f("xJ O as n...,oo for
some >''70, where x =lll n k ) Then t _O as n oofor everyk and
n . 1 n
'fk()lK; Itkl)<:S
52
for every k and sufficiently large n, 10 ay n» n , where n is inde pen-
o 0
dent of k (here weQ) take Kt > 1). Taking an arbitrary t70 and choosing
m so large that k= trc < '3 we have :for n > no :
'lI O.K- 1 x ) 2: 'rk ( ,\ K;1! t 1) + K 1 ,() (). xnJ + %'
2 n k=1 '><f
N w, we may take n 1 '7 n o so large that K 1 '?<f(.\X n )< 3 and_ 1 'fJc.K;1 (t I)<
3m :for n>n 1 and k", 1,2,...,m, thus obtaining 1f(').K2 xn)<g ;for
n:> n 1 .
B.12. Remark . It is easily observed that in c alOe when \fn ::q.f/ and
n "" 'ji for n :s 1,2,..., then
la, l c:.ll( , if and only if, there eXist numbers u '? 0 and K > 0 such
000
that
"'VLU) Kcp(u) for 0 u< U o ;
(b\ l\PCl'f, if and only if, there exist numbers u O/K1/0,and K 2 /' 0
such that
'\jI(ul K 1 <f1(K 2 uJ for 0 u (u o '
this embedding being continuous.
Theorems B.4 and B.10 which give necessary and sufficient conditi-
on in order that LC(>c:. LI¥ and IG> c;llf' , respectively, may be used in or-
o 0 0 0
der to establish conditions under which L\fI ,. LlV or lCP "" Ilf. Namely, it
o 0
is obvious that L<¥ '" L"i' (1'1 "" l\f), if and only if xE:L'f implies 2x€].1.f
o 0 0 0
C.xEl implies 2x lip), i.e. L\fcL\f' (llpc.llf") withl\ll ( t,u}""rG{t,2u}(liIlU)
o -0 0 0 0 0 1 'f In
= tf n l2ul}. Thus, we obtain the following
B.13 . Theorem . (allf is o-'-finite and atomless and t f , tb.en
Elf"" L\fI ::01 LIf, if and only if, the following condition holds:
o
(Q2) cp(t,2u) K\f(t,u)+ h(tl
for alJ. u 0 and almost every t E-...C2.. , where h is a normegati ve, inte-
grable function in Jl and K is a positive constant,
(b) If \pE- , where <f=(<fn}' then I! '" l<e, if and only if, the followi-
ng condition holds : there exist positive numbers &',K and a sequence
(an) of normegative numbers with { 1 an <. 00 such that
(6 2 ) tfnlu,<& implies 4'n(2u) K(yn(U)+ an for u O, n "" 1,2,...
Let us still remai.'k that the implication ( 2\ L "" Lip holds with-
out any assumption on the measure for any Cj>€ .
Theorem B.13 may be extended to the following statement, making
53
claar the role of the condition (Ai resp. l {2) .
8.14. Theorem . CalIf r- is O"-finite and atomless and <pEP, '{lo-
cally integrable, then the following conditions are mut ually eq ui va-
lent :
l1) 1 = Ltf, (2) Elf :::: L , (3) condition (Ll 2 > ,
(4\ modular convergence and norm convergence are equivalent in L£.f ;
(b) If cpE<£ , where <f =(<fn J , then the following conditions are mutual-
ly equivalent:
(1) l = lCf, (2) Elf :::: l<f. ()) condition ( 2) ,
(4) modular convergence and norm convergence are equivalent in ltf.
Proof . Equivalence (1) !3) is given by B.13 and (1) t2) is evi-
dent, in both cases lal rot{ (PI. In order 1:0 prove (3) ) in Qase la)we
choose a set A € 2 such that Q.J A h \..t) d < ' (AI < 00, a number a> 0
for which J If(t,2 "a' dfA- < * and we repeat the argument in the proof of
B.Slal with 'Yl t , u) == Cf(t ,2U), K 1 :::: K, KZ :::: 1, obtaining the inequality
(2:Axn) l Cf (t,}. a)d + Kg<f(AXn) +
for n == 1,2,..., which impliesL4/, In ord r '0 show(4) {.})in ease (a)
we observe, that supposing ( 2}to be not true and taking o/(t,uj= tt,2u'
and the function x from the proof of B.4(b), writing (tl" xttl for
t€ V +1V ..., Xg;lt)= 0 elsewhere in.I2., we have l2Xg;!:I ex:> and
lxIJ = i 2...nk 0 as k;>oo. Hence X:lc 0, hut Xg; does not tend to 0
in the norm of Llf. The implication(3) (4' in case (bHs obtained in a
similar way as above, using the argwoonts from the proof of 8.11 con-
cerning convergence , with 'Plt, u).. qtt ,2u), K 1 :::: K and '"" 1. The con-
TerSe implication may be obtained, tracing the proofs of 8.B and 8.10.
B.15. Remarks .(1) Let us note that supposing the measure f'A- to be
separable, from the above theorem and from 7.10 and 7.6 follows that
in both cases (a) and lb \ , the condition lLl 2 ' (resp. (°2 1 ) implies the
space L (resp. l\f) to be separable and the set S of simple, integrable
functions in..Q to be dense in L<f (resp. the set S of sequences posse-
ssing only a finite number of terms different from zero to be dense
in lq». We shall turn back to the conditions (1.1 2 ) and (Q2' tater, when
considering uniform convexity and strict convexity of spaces L If and
linear continuous functionals over spaces 1\f 1 resp. 1" (see 11 and
12).
54
(a) From 8.2 we concl ude that 8.13 (6 2 ) in case of Cflt, uJ independent
of t is equivalent to the condition (f(2u) K <fI!J.1 for all u 0 and a
K)' 0 if fo((,A.,) =: 00, and to the condition ft2u) K II'Lu) for u U o with some
u o » 0, K"7 0 if !«Jt) < 00. Since there are functions I..p independent of 11
such that L% 1= LC, (see 7.3) so, by 8.14la), for such a I.f (tJ 2 ) does
not hold and modular convergence is not equivalent to norm convergence
in LW.
(3) It is easily observed that the condition ( )implies, without
any assumption concerning fA- '} the modular (x) "" &. q;(t, Ix It}Qd/1-t.. with 'fE f
to be continuous (see 1.7), because from L "" L follows that 1im
o .\..,.1+
O< x)= qlx) for all xfL\f> Hence, assuming <f to be convex and to satis-
fy (A 2 \ , we have then /lX/ == 1 ill[ and only if lXI = 1. It is, moreover,
easily observed that the equivalence of II xII"" 1 and Lx) "" 1 for all
x Llf , is then equivalent to the condition LIf = Llf. Let us still re-
o
mark that assuming (.6 2 1 resp. (6) and convexity of 'f ' together with
respective assumptions on the measure , for every f. "'? 0 there exists
an I'ft 0 such that for each x 'L If, the inequality If x,, » E implies
lXb' . Indeed, in the converse case we get a contradiction with The-
orem 8.14 la), l41, resp.lbJ, (4) .
8.16. Definition . Let ,'Ve . The functions tf1 and"f' will be called
equivalent , <f"'''¥ ' if both \fI 't' and "f'-3<f (see 8.1), Le. if there
exist nonnegative, integrable functions h 1 , h 2 on....Q and posi ti ve con-
stants K 1 ,K 2 ,K 3 ,K 4 such that
K3 \jJ(t,K 4 u)- h2<'tl lf 'u) K 1 q>lt,K 2 u)+ h1lt)
JA..- almost everywhere in 1. In case when.1l= N is the set of positive
integers, <f=Uh'E ,'V-0,yilE- , If and 't'will be called s-equivalent .
\{I "f ' if there exist sequences la n ), ('pd of nonnegative numbers and
positive constants b',K 1 ,K 2 .K 3 ,K 4 such that Ian < 00, ,!b n <: 00 and
for every u O, n == 1,2,..." \Pnlul<f implies "t'1) ) K1 CfnlK2u)+ an
and '\f n (u).c::( implies K3 Cf n (K 4 u) - b n "f' n Lu) .
From Theorems 8.5 and 8 .11 it follows, immediately
8.17. Theorem . Let , 't'€ <f . Then
(?) If the measure f-is G""'-finite and atomless and the functions <f',t(
are locally integrable, then the following conditions are equivalent
1 0 \f""''Y
55
2 0 Lip = L'f' , modular convergence in Llf is equivalent to modular con-
vergence in L't', and norm conver&ence in L<t is equ:1valent to norm
convergence in L .
(p) If..Q.= N ={1,2,... ,2= the G'-algebra of all subsets of..Q., fl-((nf)a1
for n = 1,2...., then the following conditions are equivalent:
l' f '
2' l'f = l'f , modular convergence in 14> is equivalent to moduJa: con-
vergence in l , and norm convergence in l is equivalent to norm
convergence in 10/.
As an example of application of the above theorem, we shall derive
the following
8.18. 'rheorem . Let 0 < 10 1 and let "t' be a tOlvex function of the
variable u, lpE . Let XCt,u):::\jI(t.Us)with some s. Then
(a) If r-is\J-finite and atomless and ,X are locallyinteg ra.ble,
then the condition \f '" X is sufficient in order that an s-homogene-
ous norm may be defined in Llf equivalent to the F-norm i I) , generated
by lx) = S (t, Ix It)/) d .
(b) If ..!l""':-:r., are as in 8.17lb!, then the condition o/ is sufficie-
nt in order that an s-homogeneous norm may be defined in l equivalent
(J)
to the F-norm I I &enerated by the modular lx)= i 1ri(ttij), x = (t i ) .
Proof . Applying 8.17 it is sufficient to prove that the modular
O"()c) '" 1 X:[t, Ix (tU)d (res p . 0"lX) = 1 i (ltiO) is s-convex. However, this
follows from the definition of X ' immediately, because taking q..., ?- 0,
d... 10 10
+(? = 1, we have
6"(cl x + y) , cf Ix Ltll S + s 17lt)\S) dft- 0('10 r (t, Ix (:J;)I d Jk +
..II.: ..0..
t sS\jl(t,ly(!iJlS) d = (j.s 6Lx)+ (:>106"01).
..fl..
9. Compactness in spaces E'I
9.1. Definition . Let tfE<E (see 7.1), be defined by 7.1 If) and let
I be the F-norminLlf=Lt.V(....Q.,Z,,u) (see 1.5). We shall say that a
no emPty family acE<f has equicontinuous norms , if for every 7 0
there are a set AE! of finite measure /Iv and a number t" 0 such
that for an arbitrary x E a there hold the conditions
56
I x x..a.'A Ie. <: e.
Ix X:al«f for B CA, BIOi: with (B)<:.c
Here, Xo is the characteristic function of a Uti C c;;..Q.. .
Let us remark that if If(t, u) is a- convex with respect to u 0 for
all t , 0 <. 10 1, then the F-norm I I may be replaced in the above
conditions equivalently by II 0; .
9.2. Lemma . Let 6 E!f. Thera holds IXnl -O, if and only if, for
every set A":i: of finite measure, xnlti-O in measure fA' on A and
the family Q, == fX 1 ,x 2 ' . . .} has eq uicontinuous norms.
Proof . Sufficiency. Let 0 < e. <.2, >. 0 be arbitrary, then there
xist- A '" and Ii> 0 such that tt(,A) < 00 , IX n 'X .!2'AI < E and e)-.xnX.B)<'
2 for BC:A, BE with (-tLBi<S" and for n == 1,2,... LetexnIJ be an ar-
bitrary subsequence of the sequence(xd and let(yx =(Xnkd be a subse-
quence of (Xnk) convergent to 0 f-a,e. in.!2.. Then tf'(t, >'1 Yi (t\I)+O f-a.e.
By Egoroff thecrem, thera axists a lOst A.b€ , A A, SLlQh that /A-(A6)<
and (t, A1Yi It'I) O as i..,.oo uniformly in A .....A . Thus,
(AYi"XA) C\Yi\ ) + (>'Yi"(A\A6kf
for sufficiently large 1. Hence ( YiXAJ-O as i-l>iDfor every .A.:> 0,
and so IYiXRf"';O as i....oo. Thus, IYiXAI < te for sufficiently large i.
But IYfX..Q.'AI < '2C , and so 1Yil(-+0 as i_oo . Consequently, IXnl - 0 as n O).
Necessity, Let us suppose that !x:d _O' tlBn in particular, (xd O.
Let (A\<:oo_and let ,>O be given. Since the measure}A- is absolutely
continuous with respect to the measure v(:B) = \flt, ) dr- for BcA.,B6£
(compare the proof of 7.7), so there exists a number 0 such that
if B , BcA and v(B)c::8" , then ,u(B)< . Let An ={tGA : \Xnlt)\ t:r'
Since (xnl-O, thera exists anN such that txn)<cf forn»N. Hence
\f{t,e,)df .hq?(t, IX n ltH)df'== (xn}<O for n>N.
Consequently LAn)<t ;for n7N. Thus, lXJ converges to 0 in measure f-
on A. We are going to show now that the family q,=!X 1 'Xz,...1 has
equicontinuous norms. Let '7 0 be given, then there exists an N such
that IXnl <E for n 7N. Hence IXnX.i2.'A.I <t and )Xn\BllE for n.,N and eve-
ryA,B6-'£.. Now, we consider indices i = 1,2,...,N. Since Xk&Ef, so
S \f(t, i 1 i Xg; (t)O df-t<. 00 for every E ';> O. Hence there is a set Ak 1:.
:.a.
of finite measure and a numoo r 6'k;> 0 such that
57
( -1 >\ '.AJ:: \j1(t, -l' (t)1I d <f ,
--k .a'Ak
(e-1 ):: q> (t, e- 1 i (t}l)dr < for Be BE-I, tttJ3k b k .
Hence Ixk'tJl,Aklf<e and IXJJBI .c:: for the above set B. Now, tald.ng
A:: A1V...v and D= min(S1'.... ) we observe that IXn J2\Alf<f
and IXnXBI <e. for BCA, B€2'.. 'f(.tBk&" and n = 1,2,...
9.3. Theorem . A family Q.CE<f is conditionally compact with respect
to the norm 1 I in E , if and only if, the following two conditions
are satisfied :
(1) Q.. has equicontinuous norms,
(2) every sequence lX n ) of elements of a.. contains a subsequence (Xn k ) ,
convergent in measure /IA.. to an element x (; Elf on every set A L. of
fini te measure f'
Proof . Sufficiency. Let (xd be a sequence of elements cf Q., . Apply-
ing 9.2 to the sequence (x - x) we find that IXn - :xl. _0, thus proving
nk k }
a to be conditionally compact in EI.f .
Necessity. Conversely, let a..CE'I' be conditionally compact in Elf.
Let xn6o. for n = 1,2,..., then there is a subsequencelXnld of the
sequence Den) and an elen:ent x eE'I' such that IX nk - xl """O as k (I) . App-
lying 9.2 to the sequence (xnk - x) we obtain the condition (21 , immedia-
tely. In order to prove tl) , let us fix an f. "'? 0 and let us take an
e.
'3 -net {X1""'Xp for the family Q, XiE-ct for i:: 1,2,...,p. Arguing
as in the necessity part of the proof of 9.2 we may find a set AE
. 1
of fim te measure and a number J' ':;7 0 such that IXiX..Q.\AI < '3 c and
IXiXB' ' t for BcA, BG-2. ,ttlBl.d" , i = 1,2,...,p. Let xt'Ct be arbit-
rary. There exists an element x. for some i = 1,2,...,p such that
1 1
IXi-XI ( '3 f 1 Hence IX'X..Q.\A'e<lxi .!t\AI ... '3&':; E and \XXBI < IXi BI +
IXi .!t'A) + '3f <. for CA, BE-2.. , fdBk r . This shows (1) .
Now, we are going to obtain a criterion of Kolmogorov type for
conditional compactness of a set C C.E(j1. This will require investi-
gation of the translation operator. In order to deal with this case we
shall assume that il. is a measurable set in R n and fA- is the Lebesgue
measure in the G'"-algebra L of. Lebesgue measurable sets in..a. . Siace
the functions under consideration are not periodic, we adopt a defini-
tion of of translation operator slightly different than in the perio-
58
dic case 7.11.
9.4 o Definition . With the above assumptions, we define the transl a-
tion operator "h' h J2 , by
tr hx)(t)
= { X (t o +h) for t (f ...a () (n-h)
elsewhere in Sl.,
for every measurable fUl1.ction x.
9.5. Remarks . Let Us remark that
({"hx)= f (t,lx(t+bJl)dt= S t.p(t-h,lxLtl/ldt.
Let-h) (..'1+ h)(I..!1.
Thus, h(x)"" lt'hx) is a pseudomodular in the space X defined in 7.1.
By means of the family of pseudomodulars Cfh) l h ' where 0';:> 0, one may
t a-
define a modular (,F)= [!i h lx). Since o (xl= lx), we have (x) .
("Dc) , and so X (5fX = Llf f we shall wri. te LS = X e (6t thus having L6CL<r .
We shall prove now that r-boundedness (see Definition 7.11) for a
nonperiodic If is a necessary and sufficient condition for Rt"hx-xll<-O.
9.6. Theorem . Let be the Lebesgue measure in a Lebesgue measura-
n
ble set ...Q.C:R, and let If€- be locally integrable (see 7.5),lp{t,u)
convex with respect to the variable u O for all t E..!l . Then the fol-
lowing conditions are mutually equivalent:
(1) /l'rhx-x/l O as h O,
l2) there exist constants o,k;:> 0, a family of nonnegative, integrable
functions f h inQ th(:-.J7..) , and a set AIf- ,fiLA);: 0, such that
tp(t-h, uJ tf'(t ,ku) + f h (t)
for every u 0, Ihl b and t €-llll(Q+bJ'A, where sup f f h (t) dt < 00,
<f thl' .P-
(3) L 6 "" L and there exist constants c,[ '/ 0 such that
II x lip up /It' hxnp c )Ix/!( for x E: L CP .
"Iht,,/i "
. (1)-=9' (2) . Let..Q =.Q1vil2u ...,.!lj pairwise disjoint, r-«. j)<'OO'
and let (;x:,) be the sequence of simple functions of the form x J ' It)= r
r,
for t .Jl" x ,(tl "" 0 for t SJ." r - a nonnegative rational number t
J r, J J
we take x 1 (t)= 0 in..!2.. Then the set {x 1 Ltl ,x 2 (t) ,...\ is dense inlO,oo}
for every t ",...0. . Arguing in an analogous manner as in the proof of
Theorems 8.4 and 8.5, we define
n n
f h It)= sup (If(t-h,u)- Iflt,n2 u))
ufO
for te-Jlo(1t+h), f Lt)::I 0 elsewhere, n"" 1 ,2,... It is easily observed
that
59
for te.o.o{.2+h) ,
(t)= s:,-P(lf(t-h,2-nx i Ct)) -\P{t,nx, (t))}
n
and so f h is measurable. Now, supposing (2) does not
positive integer n there is an h & such that
-n n n
!hd 2 and If t) dt n.
br:- h {t) '" max (<f(t_h,2- n x. (t)) - tp{t ,nx. (tl))
J, Hi:!>j
b ,h(t)= 0 elsewhere. Then b ,hlt)t Lt) as j..."oofor
hold, for every
Let
for tE: !In(Jl+''1,
t r:.a . Hence
and there exists
obtain
Sb h{t)dt J (t)dt asj oo,
A J , .!l.
indices N(n) such that writing bnC t ) = bN(n) , t), we
(a)
Let
5bnlt)dt n for n::: 1,2,...
...a.
Bn,k ==[t€.Cl.('(..Q+hd: (t-hn,2-n ltl) - {f(t , (t)) ::: b n Ctq
for k = 1,2,...,N(n), and let
X (t) ""
n
{ Xl (t) for t Bn, 1
k-1
X k ltlfor t EB n,k ' U B . , k::: 2,3,...,NlnJ,
i:::1 n,
o elsewhere .
Then Elf and
(b)
for
bnltl= \f(t- , 2- n x n It)) -'f(t,nX n Ctl)
t c l1.(\ (.!l +hd ' b n It)::I 0 elsewhere. Hence and from (a) we obtain
S -h , 2-n (t dt n for n::: 1,2,...
.all (.a.+ hJ n n
Now, we apply Lemma 8.3 to the sequence of functions
f\f\t-h , 2- n X' ltl) for tf.Q!I(.Q+h"1
an C t) :::t 0 e sewhe;.
We conclude that there exist a seqUilnce( 1 of pairwise disjoint, mea-
surable subsets of Jl, and an increasing sequence of indices (n k ) such
that
(c) Akc .D.n(..Il+h nk ) and S Lf(t-%k' 2- nk XnkCt})dt == 2- nk
Ak
for k ::: 1,2,... We take
{ Xnkltl for tfA k , k= 1,2,...
x I t ) =
o elsewhere in S2.. .
60
{ XCt) for t t'A 1 v...U A i
y,(t)...
0 elsewhere in..n..
Let A'1 0 be arbitrary and let k be so large that A n k . Tmn, app-
lying <.b\ and (c), we get
(ACX-Yi)) f (t, I)dt
k=i+1 Ak
ro
=?: ip(t-hn k , 2- nk k(t})dt:s L 2-nk..-. 0 as i <Do
k=H1 Ak k=i+l
Hence xirE't. We show now that (1) does not hold. It is sufficient to
prove that (}.x)= 00 for all /\;:>0 and [> O. Hcwever, taking .>., t '/ 0
(61 -i
foxed one may choose indices i and k 1 such that 2 <. A , lhn k 1<. 0 and
nk-i 1 for all k?-k 1 . Then
00
(A x) = sup "> S q:>lt-h, ).. k Ltl)dt <p(t-hn k ,2- i XnIfti)dt
(6") Ih/ 6 k=1 AkA (.Q..+h) Ak
2 nk - i i k \f(t-%k' 2- nk 'Xn*t))dt = 2-i for each k k1'
by (C).. Consequently,(1)deos not hold. Thus, we proved thatl1)=7(2
l2) ':::9(3) . We may suppose that d '" sup f f h Ct) dt)' 1. Taking S defi-
Ihl o a.
ned in the condition l21, we have -.>
(f;)C u ) 2 (6{ )
2 sup (t,u-lkxlt))dt + 2 1 d sup S fh(t)dt
Ihl b nnLat-h) lhl cf .QI\(1l.+1:V
1 ( -1 \ "
'2 u loc, + '2
for eve ry u'1 0 and every x E X. Hence X€; L<f implies x eo Ld"' and so L(= L Cf .
Now, let x f< Lif. Taking u so large that kIIXI' <U, we haye (o) ul < 1 ;
thus, ( : ) fh (2 U ) < 1 for ihl b . Hence i(1:'hXU 2du. This shows
that it ( h xi!r' 2dlc/lx/1 for Ih I . Since the ineq uali ty I/XII. sup /I'fhxl/ is
) ) 5 Ihl 6
obvious, we obtained (3) .
(3) =$> (1) . According to the assumption (3) and to the fact of density
of the set of simple, inteGrable functions in the ppace E'f (see :'rheoram
7.6 J ' it is sufficient to prove (1) in case when x = I\A is a characte-
ristic function of a set Acft of finite measure. However, ve have
for any A.70, CA.l KA-XA)1 :;:fA lp(t,.A)dt, where Ah =Jl<i(..Q-h)('{A-h) A.
Because fA.lAn' 0 as h.....O, so (}'(f'hU.-( )-O as h-'O>O. Since ;>, '/0 is
arbitrary, we concl ude (1) .
61
9.7 . Defini tion . Let..Q be a Le be sgue measurable set in R n and let
x be looal y integrable in.!l.. Then the function
Xr(t)=a....i- S xls)ds, "i ...sl, 1',.0,
nr K \t)
r
where KrC1:;) is the closed ball in R n with centre at t, radius r and VO-
lume m r , and XlS/'" XeS) for se-..Q , Xes). 0 for s Sl. , is called a -
klov function .
9 .8.Defini tion. A function f cP is said to satisfy the (I) -condi -
&2!:!, if for every k,. 0 there exists a nonnegative function f k , inte-
grable over every measurable set Ac..sl. of finite measure, such that
1
(00) u it 4>lt,u) f k l1:;) for every u;:>O and a.e. t .J2.
The following property is obvious a
9.9. Propositi.on . If Iff; satisfies the (I) -condition, then for eve-
ry set A e with IA (AI < 00 and for every measurable function x such
that x'XA, E- L'f there holds x A E' L 1 , where X A is the characteristio fun-
ction of the set A.
In particular, if f satisfies (00) and x Llf(A.) for every compact
set A c.D..c: R n , then the Steklov functions x , r,. 0, are well-defined.
r
9.10. Theorem . Let lfE satisfy (oo)and let <fCt,u) be a convex fun-
ction of u? 0 for t €SLc R n . Then
IIx;rl1 , sup H't'hxl
Ihl r
for every x (i Lif ,where 't"h is the translation operator defined in 9.4
and we takel\YI! = (I) in case when y. Llf .
Proof . Let k '7 0 be arbitrary. Applying Jensen's ineq uali ty, we get
(kx ) S lp(t,..JL S Ix (t+slldS) dt
r m r K (0)
r
-L S 5 qJ( t ,k Ix \.t+s*t ds sup (f;'t'SX) ·
!IIr K1' (0) .a. ISI r
Since k /0 is arbitraryi we obtain the desired inequality.
9.11 .Theorem . Let f satisfy (00), \(l(t, u) convex with respe ct
to U? 0 for t E-Sl. c: R n and let 1< C.LIt(.sl.) be such that "XII L for all
x E- J(.. , with some constant L > O. Then
La) the functions XE-'X have equi-absolutely continuous integrals,
lb) if A is a compact set in.a.. and :(r(.A.) is the family of functions
x 'I. with x -X , then -X lA) is conditionally compact in the space
r'\A r
62
C (A) of continuous functions over A,
(0) if the assumptions of (b) are satisfied and <.f is locally integrable
( see Definition 7.5) , then the family 1C (A) is conditionally compact
r
in L<f (A) .
Proof . (a) Applying the notation from 9.8, let us write (X)=tt<f_'1:
ilx(t)l fklt IJand let E be an arbitrary Lebesgue measurable subset of
finite measure of .D.. Then, by(oo), we obtain forxf;1('
i SIx ttl\dt S tf ,i Ix It) dt + S f k It) dt + 5 {Kitlctt 5
E E "Ak LX) E Mk (X) E \A K (.(.)
( ) + f k (tj dt + I fk(t} dt ·
If E>O is arbitrarily chosen, then we take k = 2L/E and we assume
S 7 0 to be so small that if the Lebesgue me asure MlE) < .r , then
fklt) dt <. . Then fEIX(t)jdt-<. t. for (E)<d , with 6 independent
of the individual function x.
(b) Let Kbe a closed ball containing all balls Krlt) with tEA. Apply-
ing the assumption (<D) and the inequalitYIIXil L for xf-Jc , we obtain for
any t A ,....;
t "/ L r Ixts)1 d < L r Ix(s») d
'x l 11 - j - 10 - J - 10 <
r Il!r K Lt) L "IDr K L -
r
.. Lf (10, tXis)' )dS + 2- J" fk(s) ds
"-r '1. IDr K
( + fk(SldS) <- 00,
and so the functions from "-;Kr(A) are uniformly bounded. Now, let us wri-
te K t ..iK (t\UK (10) ) ' ( K (t)"K (IOn. Evidently, K t C K for t,s" A.
,10 I.: r r r r ,10
Let E» 0 be chosen arbitrarily. By part (aj of our theorem, there exi-
sts a f> 0 such that E c...Q and ""t E ) < $ imply J E \x(u)jdu<. g . Let
us choese 6170 so small that lKt,s)<6 for It-sl< 6 1 , t,SfA.
Then
Ix (t) - x (10\/= -L! S x lu)du - J x(ul du \
r r IDr I K (th,K lsl K (SI"K It} I
r "r r r
-L \Xlull du <.
II1z. K t r
,s
for t,s G A such tlat It-s t..::: 51 and for all x E'"j( . Hence the functions
from 1C (A) are equ:Lcontinuous on A. Hence, (bl follows from Arzela's
r
63
theorem.
(c) Let (x m ) be an arbitrary sequence of functions from 'J( . By (b), there
exists a function x E:- C (A) and a subsequence (x n ) of the sequence (x ) such
k n
that the respective Steklov functionsCXnkJr converge to x uniformly in
A. Thus, given t.> 0, we have IlxnkJr(tJ- x(t)l< E for all tEA and
sufficiently large k. Hence (A([xnkJr - x)XA1 fA 'f(t,.xI[Xnk)ftl-xtt dt
{ <Plt,,>. ) dt for sufficiently large k and BIlY A..> 00 But, by -local
integrability of f ,SA t.f(t,>"E.Jdt_O as 'C- o. HencelllbcnkJr -x} A/I _o
as k c:o..
We shall prove now a Kolmogorov-type criterion of compactness in
the space E<r .
9.12. Theorem. Let <pE satisfy the condition loo! (see Definition
9.8) and be locally integrable in..o.C R n and let (t,u) be convex
with respect to u-qO for t E-.Jl. . Let 1\:. C E<r. If too following condi-
tions are satisfied :
(al there is an L)o 0 such that IlXff L for all x E:- 'K. ,
lb) for every E;:> 0 there is a compact set A 1 c:.J2 such thatllx X..Q..'A1"
<-E. for every x G "K.
(0) for an arbitrary compact set A c.12. and an arbitrary
exist 10 an r 1 ,. 0 such ttlat lI(x-x lA II < £ for all x (: 'j(,
are the Steklov functions defined by 9.7
then the family '1C. is conditionally compact in Elf . Moreover, if we
assume that lp satisfies the condition (2) from 9.6 and that -K is con-
di tionally compact in Elf, then the conditions La) , Lb) and (c) are also sa-
f;:> 0 there
and r < r 1 ( x r
tisfied.
Proof .Sufficiency. Let e> 0 be given. Bylb), there is a compact
set A 1 C..Q. such that !lx'>\ 'A111( € for all X€J . APPlyin (a)and.
9.11 we obtain that the family (.A1) is conditionally compact :Ln L\.P (A 1 '.
Now, let 1{;lA 1 ) be the family of functions of the form x \A1 with xe-J( .
We show 1C 11 to be conditionally compact in EIf1. Indeed, let x m £:-1((A1) ,
m ::II 1,2,... Then L x J (:-"'j( lA) and so there exists an x Llf and a
mr r 1 ,If,
subseq uence LX mk ) of (xul such that [xmIJ r x :Ln L lA 1 ). S1.nce [x mk ] r
continuous and A 1 is compact, so lXmkJ :M.A1 Elf ; consequently, X{A1 E E
By condition (c) , for any ,.O there is an r 1 ;:> OSlA4h that
IILLJ1n.klr- xmk)XA111 <' E/2 for r<.r 1 and all k.
64
Taking r <r1 fixed, there exists a k j such that
H f:r X)XA111 < for k,k 1 .
Hence h{xillk- x}X A II "" f for k >k 1 . This proves ')((A 1 J to be condi tiona-
lly compact in E$. Let x1 A1,x2 A1"",xmKA1 be an iE-net for the
family X (A 1 ) in Elf, with X i € 1C for i ::: 1,2,... ,m. We show that
x 1 ,x 2 "" ,x m form then an E-net for the family <:K-c Elf. Indeed, let
1
xex and let i be chosen in such a manner that /ltx:- ) A IL <. _ 3 e .
o 1 1 0 1'
By lb), we have ax .Q.\A1J1 < 3 E. and U oX ..Q'A1U < 3"€. . These inequali-
ties prove that IIx- o fl < E . Thus, x1' "" ,x m is an €-net for 1C
Consequsntly, 1( is conditionally compact in E .
Necessity. Without loss of generality we may limit ourselves 'too
n
the case.!L::: R , applying the result to the family :1C -{x a XE''Xt in
place of "J( , 'X being defined as in 9.7. So, let us suppose that 7C is
conditionally compact in Etf::: EIfCR n ). The condition «J.) is tten obvious
and condition Lb) follows from 9.3. It remains to prove lc). For any set
BC.R n , let Bro be the closure in R n of the set t B (j;) (see Def.9.7) .
Let us take an arbitrary compact set A CoRn of' positive measure and let
';7 0 and r ? 0 be given. Let 1«A r ) be the family of functions X'YA
o 0
with x e-'1C. . Since 1\.f/l. r o) is also conditionally compact in Elf, so the-
re is an net x1'x 2 "",x m for -:1((A ro ) in Elf, where .. e/3 max(1,c)
and c is the constant in 9.6 (3) . Applying 7.6 and Lus;1.n' 10 theorem i't
is easily observed that the function. xi may be chosen continuous on
Ar and vanishing outside Ar ,1.::: 1,2,... ,m. Hence there exist 10 a
o 0
&1 '70, &'1 ro such that
. < £
Ixilt)- xi(s)j 3I1X A I I( for i::: 1,2,...,m,
if only t,s E- Aro and It-sl<' 0 1 , Hence we have
E.
IDt i J r lt) - x; (tIt < ft
.L 3I1XAft
for tEA, r.s 01 and i::: 1,2,...,m.
Consequently,
li) 1l((x i J r - xJXAII < for r d 1 and i :::: 1,2,...,m.
Now, let us remark that
(ii) 1/ y BII IICyXB ro ) rll
for eYery ylE:ECf , BC....o... measurable and r ro' Indeed, if t B and
r r , then K (t)('\B :::: K It), and so
"" 0 r ro r
65
( YrX:s\=S , ; 5 Y(SJdS)dt SnCP(t,2.. S Y(Slds}dt
:s :r K (t) R m r K l tll\B
r r ro
= (>-(y B J r)
:.ro
for every A,;:,O. This implies lii) 0
Finally, since x, ,x 2 '." ,x m is an Ill-net for 1((Aro} in Elf, for eve-
ry x E- J( there exists an index i such that
i
HX ro - xill 11 = 3 ma:x: (1 ,c) .
Let us write r, = min (° 1 ,6) , where D is the constant from 9.60) 0
Then,applying(ii), Theorems 9.10 and 9.6, and the inequality (iii) , we
obtain for r r1 :
II ((;x:i 1 r - xJXAII \ 1/ [(xi -X)X A ]rl/,a sup II 'lh( i -x)XA ) I c !I (xi -XI A /10
'> ro "Iht ro <, ro "
€. €
o' 3 max {1, c) '3
and moreover, again by (iii) ,
e £
!I(x-x " H(Xi-X)\ArJI 3 max(l ,0) '3 0
Finally, by li) we have
IICx i -[Xi]r)rA!I ' '
Collecting the above inequalities, we obtain
. E
l\lX-x:J4" l/(x-x UIl + II (xi - Lxi) r) AII + II [xi 1 r -x A" 3'"3 "" E
for r r, and xE-4<; 0 This proves the condition (c).
9.13o Remark . Let us note that the condition (c) in 9.12 may be re-
placed by the following one :
(c')for an arbitrary / 0 there exists an r 2 ,O such thatIPc,..xrIl9,f
for all x E- '1( and r <r 2 .
Since (c')implies(c), we need only to show that\al,(b) andlc) imply
(c'lo But indeed, choosing an arbitrary /(,>0, taking the compact set
A 1 from (b 1 and applying lC) to the set A "" (A 1 ) r ' we obtain from 9.10
inequalitytii) and from 9.6(3)
IIX-xrll II(x-x MH +)lx Q'AII + Ilxr'X...a\AI/ 2E +II[X J1.,\A1rll 2£ +
.suPII'rh{XY.Q.\Alllp -so: 2£. + cllx"J1.\AIID.( E(2+c).
Ihl y- '\ '7 '\"
This proves lc').
Now, we give still an F.Piesz-type theorem on compactness in E(/J:
66
9.14. Theorem . Let us
integrable in..Q..c R n and
'jC c. Elf
suppose that E satis:t;ies oo) and ia ocal y
convex as a function of u 0 for t E...Q.. Le't
. Moreoyer, let the conditions tal and f.bJ from 9.12 and the
condition
td) for every compact set Ac..Q and for every E,> 0 there is an r 1 ,,?, 0
such that il .Il'f'llcrhx-X) A V < for Xlr'1(, and r < r 1
be satisfied. Then the family 1<. is conditionally compact in Elf.
Proof . It is sufficient to shoW! that the conditioned) implies the
condition(.c) from 9.12. Let a compact Ac.D- and a function xe:1(, be cho-
sen arbitrarily. Then, by Jensen's inequality, we have
((x-Xtv.) S rp(t, : S jxltJ- xls)\ dS)dt
A r Krlt}
5 (S (t,/I [x It) - 'Lhxlt)l)dt) ds sup ((x- x)x J :for every .-\>0.
Kr (0) A ISI r
Hence II(x-XtXAI/ I ( /I (X-1:'h:x) AII . This shows that Cd } implies (e) of 9.12.
10. Generalized Orlicz-Sobolev spaces and spaces of functions
of finite @6neralized variation.
There are developed many kinds of modular spaces of functions star-
ting with a -function f and applying a procedure more or less immita-
ting this used when defining Orlicz spaces. As examples of such spaces,
we shall outline now some basic facts concerning generalized Orlicz-So-
bolev spaces and spaces of functions of finite @6neralized variation.
10.1. Defini tion . Let k be a nonnegative integer,..Q. - an open, non-
void set in R n and let \.{>6 <Q (see 7.1 ) be a convex function. Let X be
the space of Lebesgue measurable J real-valued (or comPlex-valued) fun-
ctions x possessing generalized derivatives rYx of orders Ic:tl 11: belon-
ging to the Orlicz space LIf(.£2.). Finally, let
(X)=S Ct, Ix Lt)l)dt , f) lX)= ct'x), (X) =2: d.(X) .
...12. A:t tc(1{,.k
Then the modular space is called the generalized Orlicz - Sobolev .ill2!r
.Q2 and is denoted by Wf...Q)
10.2. Theorem . If €- is convex, locally integrable lsee 7.5} and
such that inf !,,{j; 1 ) "?'0 then the space W k (..C1.) is a Banach space with
te.A'" ,
67
respect to the norm IIxl ::a inf{u>O : C ) 1i.
. It is evident that W (..Q) is a normed space with respect to
the norm /I U . We have to prove that W (.!l) is complete. Let us suppose
(x m ) to be a Cauchy sequence in W (..Q) , Le. q((.l -Xi) O as m,l ())
for every /1.'70 (see 1,,6). This means that (>.,(;If'Lxm - DO( }-+O as
m,l-'P 00 for every /I. ';;> 0 and IO'\I k. In particular, (X m ) is a Cauchy se-
quence in Ltf(..Q) , and so x -;>X E::Llfl.!l) in the sense of the norm in LIf(..Q)
m 0
(due to the completeness of L'f(...a.), see 7.7). Let us observe th.t the
functions x are locally integrable in.Q.. , Le. SA Ix Lt)ldt < 00 for
m m
every compact Ac.Sl., m::: 0,1,2,... Indeed, the condition c"" infi.fl t ,1)
t'G-,A
> 0 and convexity of (t, u) as a function of u 0 imply u 'f (t , u)
for u 1, where we may cuppose c 1. Let
f c IXm tt)1 J
E "" t € A: IIXullf 1 .
r (. \X m ltll )
J \f\t, Il"1CII dt 1 ,
A --ill
Since we have
we obtain
(1)
g <Pc (t)\ 1 S r. cjx: (tJI ) S
A II Il- dt c; A \f , 1:/1_ dt + A lt,1)dt< 00.
m m
:xm defines a regular distribution
T x y ::: (' X It) y{t:)dt for yf: C oo (...i2> ,
m j m 0
4l.
m'" 0,1,2,...
Hence
and
T D d. y=(-1)14/Sx It)Do!.ytt)dt forYGCoof.,Q..), m= 0,1,2,...,
x m ..a. m 0
where Iqj=l(ol1,...,oLn}I=«-1+..o+ltn0 Thus, we have
IT...t y - T D (/.. y\ KS Ix (t)- x (t)ldt ,
lJX m Xo A m 0
where A is the support of y and K "' t m.ax k IJfy(t)l. Applying (1) to
E-A, l<lt
x -x in place of x , we obtain II
mom K ( c IX m l t) - Xo l t )
\TJ)'txm Y - T:IJX.XoY) T 1. tf t, IIxm-xolI dt .ttxm-xoll +
+ 1<f(t,1) dt 'lIxm-xJl ( c + )llf(t,1)dt HXm-xdl - 0
as m-t 00 since x _x in Ltf(...Q.) . On the other hand., applying comple-
, m 0
teness of L\jI(..Q.) to the Cauchy sequence (Dd.. Xnt with let! k, we obtain in the
same manner as above that SA!D(;\Xm(tJ- x"'lt)ldt..........O with some x Ol GL'f(J:2.),
tl - D x for
whence To'",.. y _TJ<u. Y as m...,.oo, 1«lik. Conseque n y, XO( - 0
68
tett k. Thus we proved that x G W t.Q.) and <?0(P - Do(x ))..0 as m-i'CXJ
o k ill 0
for every A. '7 0, Le. x _ x in W<lJl-'2.) with respect to the norm 1111-.
mOT if
10.3. Theorem . Let lp,'t'1:- be convex, locally integrable functions
such that inf 1ft!;, 1) > 0 and inf If'(:t, 1} :> 00 Moreover, let us suppose
t ,- t6-...I'1.
that
t2) 1'lt,u) K1\flt,K2u)+ hlt) for all u O and a.e. tE-..>.'1. ,
where h is a nonnegative, integrable function in....'1. and K, ,K 2 are po-
sitive constants (compare 8.' (2) . Then W (.i1.)CW (J2}:for k = 0,1,2,...
and the embedding of W (.n) in W L..rr.) is continuous with respect to the
norms (see alIOS 8.5 ) 0
. We shall prove that if XEW L.!l) ,then xG-W (..a.) and the
norms \l llw.If<,.J2.) ,1\ I(w (.su generated in the above spaces by the respec-
tive modulhs "( satisfy the inequality
D) l\XIIW t.n.) , max(K, ,K2) (1 + Cn,klh(t)dt)UxHw¥(..Q.)
for Xc W t..Q.) , where C k:S 2: , . If xltl = 0 a.e. in...Q., then C?) is
n, la, k k
obvious. Supposing the converse, we have for x€: W'f(..Q)
L ) tf(t, ID(cltJI ) dt 6 ,.
rol.l k..a. \: /Ix IW .Q)
Evidently, we may suppose K, 1. Then
't'{t, [Dt{xlt)l -klt L { K 5 tf(t, tD t)l ) dt + Shlt)dt?
lol' k..a \ K,K2 /I XIIW l Id' k '.ct \ K,U ..a..J .a. )
I c( I
2: \If(t, DxLtl )dt + C k)' hlt)dt 1+
IO(I k I!xIlW¥L..sl.) n, .a
Since O} 1, we have from the 1 ast ineq uali ty
L f tft I DcA. x (tl\ ) dt 1 ,
loll k 1. \:' OK1K2HxlIW (..a.)
k
and this give 10 x E w." (.0.) and
II x Il W k l .£L) 'CK,K 2 IIxl'w ..2.)
'11
We are going now to outline some general properties of spaces of
functions of finite generalized <p-variation.
1004. Defini tion . Let tp: [a, b]XR+ _ R+, where -oo( a < b < <D) satisfy-
ing the following conditions:
10 q>(t, u) is a continuous, nondecreasing function of u 0 for every
Cn,k h(t) dt = 0 <.00 .
n.
69
tf:[a,b], If(t,u)-.oo as u_oo,
o
2 tplt,O}= 0 for every t [a,b], and tf(a,u)= 0 implies u", O.
Let Tf be a partition a '" to < t 1 < ... < t m '" b of [a,b] with inte!rmediate
points Si€[i;;i_1,tJ, i::l 1,2,...,m. Then for every function x: [a, "'R,
m
V'f(x)= s p f 1 f(si,tx(t - xi t i _ 1 )J)
is celled the generalized - variation of the function x in [a, b] (see
also Example 1.9.V) 0
Evidently, there hold the following properties:
10.5. Properties o (a)Denoting by tLX) the generalized If-variation
of the function x in a subinterval [0, d]C [a, b) , we have
c b ..b,
,?l(LX)+ c..xJ ¥IfLxl for every c e (a, b) .
(b)Vtp is a modular in the space X of real-valued functions x in (a, b)
such that x(a}'" 00 If Cf(t,u) is a convex function of u for all tc[a,bJ,
then VIf is a convex modular in X.
10.6 J:>efinition . The modular space = Xv. is called the .Q1
functions of finite generalized If'- variation . 'f
We are going now to prove further properties.
10. 7 o Properti es )Every function xEAJIf is bounded in la, b) .
Lb) If x n e-'\T<f' n = 1,2,00" and S}W Vtf(!aJ < 00 for some k') 0, then the
functions x n are uniformly bounded in [a, b] .
oJ If x n E llf' n = 1,2,... and x n is v<p-convergent to 0, then the fun-
ctions -11. are uniformly convergent to 0 in [a, bJ .
(d) If x n (a)- 0 for n :: 1,2,... and x n l t)-;> x[t) for evary "'G/!3.1 bJ, then
V 1n LX) 1.im. V'f'J (x ).
'\: n....oo '\' n
(e) Helly's extraction theorem. If x n £:1J<p for n = 1,2,... and there is
a k;:>O such that sup V,D (kx )<00, then there exists a function x f17<f'
n .,. n
such that x It)_x(t) as n-.,.oo for every tE-(a,b).
n
Proof . (a) Since X&1J f ' we have f(a, k Ix (tJl)'M for some constants
k)O, M O and all a t p. Supposing x to be unbounded and taking a
sequence (t i ) such that a ti b, Ix lt t co, we get a contradiction.
lb) Again we have (a,klxn(tll) M for some k>O, Iv? 0 and for tEi(a,b) ,
n = 1,2,... ShouldlxJbe not uniformly bounded in la, b) , then there
would exist a sequence ltiJ, a t i b , and an increasing sequence of
indicesfn, ) such thatlxn- (t-)!.....ro, which gives a contradiction with
I..J. J. J.
70
the inequality lp(a,k!xnilti)I) M for i =: 1,2,...
(c) By the assUJT,ptions, ip(a,k IXn (t)ll VI.f (kxnl- 0 for some k70, unifor-
mly with respect to tE[a,b). Supposing(xn! to be not uniformly conver-
gent to 0 in [a, b), there would exist a number c > 0, a sequence (t i ) )
a ti b, and a.'l increasing sequence of indices(n such that IX ni (tll c
for i == 1,2,... Consequently, (a,kc) 1{J(a,klxni (ti)I) Vtf(Xni)-?O as
i CXJ, a contradiction.
Cd) Let v = lim V;o (x ) < 00 and let us take an incre asing sequence of
n...;. 00 "'[ n
indices ( such that Vtp(Xnkl-v as k-roo. For every E.:::> 0 there exists
a K such that Vl.f'lXnk!<" vH for k>K. 'l'hus, we have
m
2: w(s.,jx (t.) - x (t. , )1) < v + €
i=l I n k -
for every partition rf . Passing here with k to 00, and then taking the
sup, we get V>p (x) v+ t, which gives the desired result.
leI Let v n (tl denote the generalized t.p-variation of the function kx in
la,tJ, where a t b, vn(a)= O. Writing III = sii P V {kx:d we observe that
(vnJ is a sequence of nondecreasing functions uniformly bounded by M.
From the well-known Helly's extraction theorem we conclude that the
sequencelVn) contains a subsequenceCvn convergent to a nondecreasing
function v at eve J point tE[a,b). Let E be the set consisting of poi-
nts a. and b and of all rational points of the interval la, b}. Since t by
Property 10c 7lb) t xn. are unifonJ.l.y bounded in [a, b), so applying the
diagonal method we may show that(Xni) contains a SUbSequenceCXnij) such
that (xniltl) is convergent at every t E-E. Thus there exists a function
x : E ....,.R J such i::t<,a.t X j (tl x(tl for all tEE and j It) -,?vltl for all
t E-/?-, bJ ' where we denote x, == x n ' , ';{, = v n . . We shall prove now that
J j J j .
that the sea, uence 0C j It)) is also convergent at every irrational po nt t
of continuity of the function v in (a, b) . Indeed, let t be such a point
and let '7 0 be arbitrarily gi venc Then there en lOt a rational number
w <::-(t, b) and an index j such that
O Vlwl- vLt)< \f(t, }, ! jlW)- v w'I< lflt, f.k) and
IV(t)- V'jltJl< I.f(t, £k) for j jo'
Applying Property 10.5 la), we obtain
IL' [ t klx.ltl- x.(w)i ) 'XY,rik:x:,) ,(WI- ,{t) lv-(wt- v (V'l1I+{v lw)- v (t));.
I 'J J tor J J J J
IvLtl- v j (tll"'- \.f(t, E.k)
71
and sojXjlt)- X'jlw)k 6 for j jo' Since the sequence(x/w)) is conver-
gent, there exists an index jl jo such that /xi(wl- 2'/.w)I" E for
,j9jl. Hence !xilt}- x{tH jxilt)- Xilw)1 + l"XilWJ- XjlW)\ + lX'jlwJ-
xjltl/<€ for i,j.? j1' This proves the sequence(X'/tl) to be convergent
at every point of continuity of v, and we may denote its limit again by
xlt) . It remains an at most countable set Gc(a,b) of irrational points
of discontinuity of v, where{x.lt)) may be still divergent. Applying the
\ J .
property 10.7(b) and diagonal method, one can extract a subsequence C;xjJ!
of the sequenceLx j ), convergent at every point t€[a,b] to a function x.
By 10.7 (d), we have x '\Tc.p .
10.8. Theorem . The space ttJ'f is complete with respect to the F-norm
'V .
roof . By 1.6 , it is sufficient to prove that if xi ,,1J\f and
VlfI(k(Xp-Xq)).....O as p,q oo for every k>O,
then there exists an x 1\)1f such that V<p(k(xi-x»-f) as i.."oo for every
k»O. Arguing as in the proof of 10.7(c)we observe that supposing
Vq>(k(Xp-XJ) O as p,q_oo, k;:>O, there is a function XE-X such that
x p (t) _x tt) as p-+ 00 uniformly in (a, b] . Let E> 0 be arbitrary and let
US choose N so large that VcpCk (xp-XJ)<'E for p,q>N, where k)O is
fixed. T hen for any partition IT , we have
q>(s" klx ttJ - x (tJ - x (t, 1) + x Ct. 1>1)<. E
i=l P q p - q -
for p,q )N. Taking q--+oo, we thus obtain
t (si' klecplti>- X(J:;1)-(X p l t i _ 1 J- x(ti_l))j) E
=1
for p,q>N and arbitraryTr. Consequently, V<p(kLXp-X») E for p>N.
Hence V<p(k(Xp-X))_O as p_oofor every k>O. Condition Xf.'\j<f' follows
from the inequalities Vtplkx) VIpC2k(X p -X)}+ V (2kxp)<OO for sufficien-
tly small k )0.
We are going now to investigate the problem of continuity of fun-
ctions from spaces '1J1f . Evidently, functions from I\J do not need to
be continuous in [a,b]. For example, taking x(t)= 0 for a t c, x(t)= 1
for c < t b for some c E-(a,bJ, we have V'D (kx)= sup If'(s,k) which is
1 a<,s b .
fini te for small k > 0 supposing Iflt, u) to be a b01..L."1ded functlon of t
for sufficiently small u . Howevere. we have the following
10.9 . T heorem . Let us denote "Ylu) = inf <.plIO, u) and let us suppose
a s b
72
that
Then
mits
there is satisfied the following condition: if 't'luj = 0, then \1=::0.
every function x f\J<f has at each point t €-(a, b) both one-sided li-
and the set of points of discontinuity of x in [a, b] is at most cou-
ntable.
. Let us suppose that xe'\Jlf but th left-side limit of x at a
point to (a,b]does not exist. Then there exist an infinite, increasing
sequence ti-t o and a number £"'70 such that \x(t 2i ) - xlt 2i _l::>E for
i :s 1,2,... Taking for an arbit rary posi ti ve inte ge r m a partition
a < t 1<' . .( t 2m < b with arbitrary inte rmediate points si 8.."1d choosing
k> 0 so small that Yip l < CD , we have
00) YqJ (a) '"? If (S2i-l' k I x (t 21 - x (t 2i- .(1) m\f'(k:E:) ,
J:::l
which gives a contradiction as m..,.oo . Similarly,x is shown to have rigbt
-side limit at every t (a, b). Now, let An be the set of points t of
left-side discontinuity of x in{a,b]suchtlat jx(t)- x(t-OJI> . Supposing
An to be infinite, for every m there would exist points t 2 <. t 4< .. .< t 2m
with t2i An for i = 1,2,...,m. Obviously, there exist points tl) 'o"
,t 2m _ 1 such that a<t 1 <.t 2 <t 3 < t4<...<t2m_l<t2m b andJx(t2 - x l t 2i -i\
)1. Arguing as in the preceding part of the proof with 1 in place of
n n
6., we ge mf ) Y<9la)( 00, obtaining a contradiction as m (I) . Hence
the set U A of points of left-side dis continui ty of x in la, b) is at
n=l n
most countable. The game may be proved for the set of points of right-
side discontinuity of x in La, b] ,
10.10. Exaillples . I. Let lop be a convex If-function without p8,rametel'
and let 11 : a::: u (. U 1 (. . .( u == b be a given partition of the inte r-
o 0 n
val (a, . Let x be a function defined in[a, , monotone in each interval
[u, 1 ,u.l and continuous at u' for i = O,l,.oo,n. Let us denote
.J..- J.. J.. m
S (n',x)= ?: 11J(l x lt,) - x(t'_l)1)
\f J= 1 I J J
for every partition Tf' : a = to <. t 1 <.. .,t m ::: b of [a,bJ. 'fie show that
Yif lX) '" sup S (rr ,x) .
flCrr"
In order to prove this it is sufficient to show that for every parti-
tion 11' of [a, b) there exists a subpartition iT of no such that Slf(f{/,x)
S'f(JT ,x). First, let us prove that the function <p is supe.radditive,
Le. 'f0)+ If(y) 'P(u+v) for every u,v O. Indeed, by convexity of\.fJ
and by the condition \flO}= 0, we have
73
\flu) ;v \.felli-v) and '{'(v) U:v <.felli-v)
and addinG both inequalitieG we obta:Ln the inequalit;l If(u)+\f(v) <.f(lli-v).
Let j be the least positive index such that t. does not belong to if.
J 0
We consider two cases :
a) (xlt j l - xltj_l»)-(xltj+l/ - xltJh. 0 , then, b;r superadditivity oftp,
If Ox It j) - x (t j-1/1) + (Ix It j+l) - X (t il) f(lx (t j+l J - x (t j)1) and we have
Smrii',x) S,1i' l ' ,x), where 11' 1 : a'" t (...<'t. < t. <...<t == b.
T \f'" 0 J-l J+l ill
b) (Xlt j ' - xltj_iJ'(xltj+i - x LtJ)< 0 , then there exists an index io
suchihat t J - 1 ( 0 <' t, 1 ' Ix( )- xlt, l )I?l x lt,/- x(t. 1 )\ and
- J+ 0 J- J J-
Ixltj+1) - XlUio)1 I x lt j + 1 ) - X(t j )\; so we have SifrTI',X) StpUf2,xk
where IT 2 : a == t <...<t, 1 < <to <...<t == b. Thus, after a fi-
o J - 0 J+l n
nite number of steps, we obtain the desired subpartition iT of rr: such
o
that S If ((t l , xl S<f (rr ,x) .
II. Let the function tptJ,u) satisfy the condition u-1t(u/ as
u."O, where I. is defined as in 10.9, and let x E-\J1f . It is seen irmne-
diately that the function x is then of bounded variation in [a,bJin
the usual sense. Moreover, we shall show that x is constant in each
interval of continuity. Indeed, let us suppose that X€1J'f is continu-
ous in an interval [a' ,b C[a,b1 but not constant in [a' ,bj. Then there
exist ((, E(a',b']suchiiaat x ) == c< d == x(M, say d.< . Now, we ta-
ke for every index n a partition <t= to'" t 1 <...<t 2n =(1 ofL (?»with
x (t i ) '" 2- n Cd-C)+c for all i. Then, taking k) 0 so small that Vlf(kX)(oo,
we have
2 n ) n ( -n II
OO)Vlp(kx) ;L (ti_l' klx(ti - x{ti_il 2 '0/ 2 k(d-c),.
=1'
right-hand side of the above inequalities tens to OJ as n.-,.oo,
But the
a contradiction.
III. Let tp be a function without paralIE ter satisfying the con-
ditions u-l )l1.i)"'too as u....O and Iflul+",+uJ K(lf().ui + ... + <['auJ)
for n "" 1,2,.. . with' some constant K, II » O. Let US denote
10 lt1= x(e:l-O}- x (a) + t. (x (t,+O)- x(t,-O»)+ xlt)- xlt-O) for a<t b,
x t.<t
sx(a)= 0,
for every XE::'\JIf ' where t 1 ,t 2 ,... are all points of discontinuity of x.
We shall show that xlt) = sxlt) for every t E{a,b). It is sufficient to
prove that fl x E0 1.Jlf ' since then x-sxeiJtp, but X-loX is continuous in the
74
whole interval [a, b] and applying Example II we get
x(t)- sx(t)::
Now, taking an arbitrary partition1T':
S$ small that V (kx) <00, we obtain
m ( k ,j m k
f /sx(Wj)- SX lW j_l)!/= 1: tJI (1 2:. (xtt.+O)- x(ti-O)) +
J=l J=l I w. t:<w. 1'\
J-I J J
(X(Wjl- xC.Wj_J+ (X(w j _ 1 -0) - X(Wj-Oj)/)
m
K{L: q>(klxtt,+OI- xlt.-O )+ 2.. m(klxlw.)- xC w ._ 1 )1)+
i " 1 1 J J
J=
m
CP(k/X(W.-O)- X(W'_1 -O)I)J 3KV lD (kx',
J=1 k J J 1
and so V<p b. lOX) :S 3K1ft(kx). Hence loX E (T f ·
IV. Let us still remark that taking <P(t,u)= f(t),/ul with an ar-
bitrary nonnegative function f, V (X)is for every nond creasing fun-
ction x equal to the upper Riemann-Stieltjes integralS: f(t)dx(t).
10.11. Theorem . If there exist positive constants K 1 ,K 2 and U o
such that "f' (t, u) K 1 <f(t ,K 2 u) for t [a, b) and 0 u u o ' then '\lifC 1.i-r
and this embedding is continuous with respect to modular con"\8r'gence
as well as with respect to the F-norm convergence.
Proof . If x 1f then x is bounded infu,b), by 10.7C.a), and we may
suppose Ix(tV U o inta,b]. Hence ixlt - xlt i _ 1 '1 $ U o for every
t i _ 1 , t i [a, b) and we obtain V'f' (kx) K 1 V (kK2x) for every 0 k 1-
This proves the theorem, immediately.
const = x La) - 10 (a'= O.
x
a=w<w 1 <...<w ==bandk,>O
o m
11. Uniform convexity of spaces L
First, let us recall the definition and SOL properties of uni-
for8ly convex spaces.
11.1 .Definition . A Banach space X is called uniformly convex , if
for every >O there exists a SCE.»O such that for all x,y c- X
satisfyingljxlf=/IY/!= 1, the ineq uali tYllx-YII"' implies/I X? 1/ < 1 - o(t.).
As is well-known, spaces L P with 1..( p < 00 are 81110ng eX81 ll ples of
uniformly convex spaces; every uniformly convex Banach space is ref-
lexive. We are going to determine sufficient conditions for uniform
convexi ty of generalized Orlicz spaces Llf. First, we shall observe
the well-known fact that
75
11.2. Proposition . A Banach space X is uniformly convex, if and only
if, for every 't. '70 there eXists a &'1(£) > 0 such that for all x,y €oX
for whichI!XII l, I!yn 1, the inequalitYilx+y/l 2 - f1( ) implies I!x-y/l H .
Proof .Evidently, if X satisfies the condition from the Proposition,
then X is uniformly convex. In order to prove the converse implication,
let US suppose that X,y t::X, x;' 0, y;' 0, IIxl/ l, IIYII l and IIx+yll4 2- &;(€),
where &1 )= min(6@€-), ). Then flx/l 1-o1(f) andnYII 1-61( ) , by assum.-
ption. Writing a = l!11X/I, b =: 11,IYI" we have 11!8X+bY/l-lIx+;yi l)llxlI+
(b-l)hl/ 2 01LE.) . Hence 118X+byll lIx+YII- 2 0 l(E,) 2-3cJl(£) 2-6( £) . Bu'i
11 8X /I=/jbyll= 1, and SOIla.x:-bYII( e . This gives
IlIx-YII-IIa.x:-byl/l II(1-a)x + ll-b) yll la-l)llxl/+ (b-11/!y/l 2 "b 1 CE.) <;; € .
Hence lIx-y//</lax-byll+ :S; £ + E. = E.
11.3. Defihiti.on . A function €-<f (see 7.1)A.b called uniformlY ..QQ!!-
vex, if there exist e. function l'lapping the in 8rval (0,1) into it-
self and a set Aec-5-: of measure ,f<().!= 0 such1l1at for every u>O,
O<..a( 1, O b a, there holds the inequality
(t, l b u) (1 _ a-L J) (t.u)+2tf(t.b
for all t 1:.Jl' A.
11.4. Remark . Let US remark that even if \jJ(t,u) is independent of
the variable t and strictly convex with respect to the variable u,i.e.
(o/..\..H-{\vl<'d.'<f(u)+(,>f(y) for <::I,, ,.O, oc+(!>= 1, O u<v<oo , it does not
need to be uni formly convex ; a cowtter-example is provided by the fun-
f P
ction \flu)= u2 + 1 - 1. Obviously, powers (u)=dul with P?' are
uniformly convex, with ala)= 1 - 2-1*1 (1+a)P(l+a P )-l . Every unifor-
mly convex fU!1.ction lfE: <e is strictly conveX with respect to u, for
a.e. tcD. .
11.5 . })efini tion . A convex modular c:; will be called uniformly con-
, if for every €.> 0 there exists a number q( ),. 0 such that for
all x,y € X, the condi tions lx) = (y) = 1 and lx.-Y»E imply Y) <l-q(f).
11 c 6 . '£ heorem . Let cp€<£ be uniforml: r convex and let it satisfy the
condition 8.13(,62) for all t, i.e. there exist a constant K:>O and
a nonnegative, integrable function h such that
lp(t ,2u) KqJlt, u)+ h (t) for all u 0 and tE-.Q" A,
whe1.e A is a set of measure fiLA! = O. ivioreover, let /1(.. be a iJ-finite,
atomless measure on..o.. ..chen the generalized Orlicz space Ltf with norm
76
II is uniformly convex.
Proof . First, we prove the modular <VLx)" S lt, [x (j;W to be uni-
.$I..
formly convex. Without loss of generality we may suppose that 0< £. < 1.
Let A ={tQoJl: (xCt)- Ylt)I ma.x:(!x<.tJ/, ly(tWJ,with Lx,::& (YI'" 1 and
(x-y) > €. By uniform convexity of cp, we have
1f( J V ) 6 - S0_fJ) Cf(t,u} Cf{t.v) for u,v;O, tE-..Q.'A.
Hence we obtain
+ S tp(t,lxltIUdf<.+
.!t' A
- f(7) ::: t ( J' cp(t, Ixlt)Vdf:} + S (t, lYltJUdl1t-
A A
J \f(t, Iy (tWdf) - S 41ft, I xl:tl+ y(tll )df
JL'A A l 2
1 _ LX Y) ::& lxl (y)
- S If ' IX(tl v(t" )dr-< t S 'P(t, Ix (tllJdf+ j (t, Iy(t»}dl\,t
A A A
- S (t, tx(t' y{t}I )df t r\f(t,lxlt) I + f 4'(t,ly(tJI)d/\A.
A A A
- 1 - f(l- ) a (t, )x(t )df + I fCt,11(i1Vdr)= f(1-E)C9(x,u>+ (yNJ)'
where XA is the characteristic function of the set A, Now, let t €-.rt\. A,
then !x (t) - yet)1 {Ix (j;)/+ Iyltl!) and so
cRt, Ix (t) - y (tJl) t If (t, (X(i11 + Iy (tiV .
Hence, intiegrating over .0...... A, we get
(Cx-y)X.o.'A 1 Ct? (x 'X:.R\J>! + ( X;1t'A)) % CfQc\ + LYI) ; ,
where XJl.\A is the characteristic function of the set ..Q' A. Thus,
£ €
lQc-yl A? '" (x-y' - ClX-y) XJl \A) '? (x-y) - '2 / '2 '
(2lx-:A A) 2 ((x-y}XA» E
Consequently,
< [bc- ) t <?XXA 1 + (2YXA) K ( (xtp} + (! fll.A)
K O(l e) (1 - f tX Y)) .
Hence
(x y ) .( 1 - o(l- ) e
and unifoxm convexity of follows with q(€-)" K t (..1-e.).
Now, let Hxl :s IIYI = 1, flx-y ;;> t. By Be-mark B. 15 (3) we 0 bt ain 1(xI '" (yJ 121
and moreover, there exists an 11'0 dependent on E such that
77
lX-Y}/" Hence, being uniformly convex, we obtain (X Y ) < 1 - ql )'
From this we shall conclude that there is a f,o dependent only on E
such that Il X y t!!, < 1 - b, thus finishing the proof. In the converse case
there would exist a number q) 0 and a sequence of functions Z E-L lf such
n
that f4(zn) <. 1 - q and a:2lJ t1. Writing an =IIZn(l, we have kanZd = 1 ;
moreover, 1 (Bz1< 2 for sufficiently large n. By Remark 8.15(3), we
conclude tha.t (anxnl = 1. Hence,
1 = (anzn)::& ((a n -1) 2z n +L2-a n )Zn) la n -l) l2Zn)+(2-an) n)
$la n -l}(KfLzJ + lt) )+ (2-a n )r(zn)
for sufficiently large n. Taking in the above ineq uali ty n ([) and apply-
ing the inequality lzJ-< 1 - q, we obtain 1.::: 1'O-q), a contradiction.
From Theorem 11.6 there follows immediately:
11.7. Corollarv . Under the assumptions of 11.6, the generalized Or-
licz space L with normU U is reflexive.
12. Disjointly additive modulars
Let ' 'fl.) be a measure space with a <r-fini te measure f- on the \J-
algebra of subsets of the set J2., and let S be the vector space of
all extended real-ve.J,.ued functions on...o., 2:' -measurable and finite
f'--a.e., where we identify functions equal f.-a.e. S is a linear lattice
under the order relation x y equivalent to xlt' y(t)f.-a.e. We shall
show that any disjointly additive modular over a linear sublattice X of
S possessing some majorant property may be represented in the form of
en in te gral.
12.1. Definitions . La! A linear sublattice X of S is said to possess
the majorant property , if for every x EX and A e-z:, there hoLds X'(A e- X ,
where \A is the characteristic function of the set A.
lP) A functional : X -[0,00] is called a disjointly additive pseudomo -
on X, if the following conditions are satisfied:
1. l9}::: 0,
2. if x,YE-X andlxj IYI, then lx) (y),
3. if x,xnE'X for n::: 1,2,... and o xnt xf--a.e., then lxJ1 Uc),
4. if x,yeX and xl y, then (x+y)::z lx)+ f CY ), where xl y means
that x(t) y It)::: 0 a.e.
78
(c) A function 10: ..Q. X R ...... Ii = [0,00] will be said to belong to the class
_ i + +
, if it satisfies the following conditions:
(i) tf It, u) is for f--a. e. t G,J2. a nonde cre asing and left- continuous
functiDn of u 0 such that LPtt ,0) = 0,
lii) tt, u) is a 2 -measurable function of t for all u O.
1202. Remarks . 0) A disjointly additive pseudomodular on X with a
majorant property is a pseudomodular in the sense of the Definition
1.1. Indeed, condition 2. implies L-x)= c.x). hloreover, taking o(., 0,
d-.+(':>= 1, x,YEX, and applying the inequalities 100UJ-f.>vlio(lul+()ivl
maxQu/, I vi) we obtain, by 2. and 4., (oI.x+ f>Y) (max{lx/, /yQ)= (lxl\ A +
IYIXB) (lxl}r (ly/) = qlx}+ (y), where A ={tG...il: Iy tt)l IxLt" , B =Sl..'A.
(2) It is easily observed that if xt:X, then the function If(t, !xl..t){)
is -measurable in...Q and so lx)=S (t, Ix (tJUdfA- is a disjointly addi-
tive pseudomodular on X. ...Q
'rhe :representation theorem 12.4 shows that a converse statement to
Remark 12.2 2 is true. First, we shall prove a lemma.
12.3. Lemma . Let X be a linear sublattice of S having the majoran'i
property and. let S(x}={te-.n.: xlt),4 01 for xGX. Then there exists a
sequence lu} of functions O,"u I:- X with the following properties :
n n 00
S(un)are pairwise disjoint for n = 1,2,... and writing Sx == n l S(ud'
we have !AlS lx)......SX)= 0 for every x foX.
. First, let us suppose that ,.w.L.Q\ < 00 and m == ft(S (x)).
Let O vn X, n = 1,2,..., be such that (SLvJ)tm as n1'"ooand let w n ""
ma.x(v 1 ,v 2 "",vJ; obviously, O wn"X, S!..wJC.Slwn+1) for n = 1,2,...
and tt(S lwnOf m as n 1'00. Let u 1 = wl S (WV' un = w n1S twIi'S lWn_l) for
n = 2,3,... hen O un(:X, Stull = S(w 1 ), StuJ= S(wd'S(w n _ 1 ) for n =
2,3,..., and so S (uJ are pairwise disjoint. Moreover, for any x€'X we
have fA-CSLX)'Sx1= O. Indeed, supposing that there exists a O Xoi"X
such that fleS (x o ) , SxJ = mo:7 0, taking n so large that f-t(S (wJ);;:.m - mo
and writing z :. max lw ,x \ , we would have
n oon (j
S (X o )' Sx == S(xJ "£S(wn)cS(Xo)'S(w n ),
S(zJ = S lwrl uS (xJ::> S LWnivl S (;x:J"' SX) ,
whence f-'1. (S (Zn)) ft (S lWd) + mo > m + 1 m o ' a contradiction with the
definition of m, because zn G X.
Now, let us drop the assumption (,Q}<oo and let ...Q.=J2 1 v..R 2 v ,"
79
with pairwise disjoint, M(.llk)(OO , k = 1,2,... Then, applying the
above proved case of the theorem, for every k there is a seq Ir:e of
f ctions 0 {'1c,n XX.a. K such that S('1c,n) are pairwise disjoint and
P-\.s X \Sx ) = 0 for all XtX, where X ={Xt...1c: x X , SX =
= i l S ( ,n) · Arranging all ,n in a sequence (un)' we obtain the re-
quired result.
12.4. Theorem . Let f'. be ()-fini te. For every disjointly additive pse-
udomodular f? over a. linear sublattice X of S possessing the majorant
property there exists a. function Iff CE such that {:Jq = S.pft, Ix ltlldr-
for every x EX. ::Q.
. Let (u ) be the sequence of functions from 12.3 and let
n
X n = {.x;Ys (Un5 x, X 1, n ty)-= (y) for y €- X n .
Let ua suppose the theorem to be true for each of the couples (X ,Ii ),
n )n
Le 0 there exist functions 'fine P such that
.qn (Y)= S fn (t, Iy It)1) d
..tl
for every y € X n , n = 1,2,... Then we have for any x t X
(2(X)"" ( lim .r. IX1'Y SlU '» ) "" lim t;(.f \X I 1" S l . ) = ;. ..0 (txlY S luy)=
'n ro =1 I. J.. n-;.oo \J:::l ) =1 ') 1
00
= L S i (t, \x ttlV s l St) d = f \f(t, I x (tH)dr-.("
kl
00 -
where \j7(t,u)= .L lpi (t,U) S(Ui)Lt) for tE..Q , u O. Evidently, <f€ .
Thus, we n e! to prove the theorem only for each of the couples
(Xn' n)' characterized by the property that there is a 0 unE X n such
that S lu n ) = SXn . Dropping the index n, it is sufficient to prove the
theorem in case when there exists a function u(i,X such that ult);>O
for every t I:-..A .
First, let us suppose that u(t)= 1 for all tE...Q.., Le. 1 =X.Q6X;
then X contains all simple functions. Let x € X. From the assumption
12.1 (b) it follows that the extended real-valued set function vLA)::
E?lXXA,) is countably additive and absolutely continuous with respect
to the measure fA'. Applying the Radon-Nikodym theorem, we _may find a
measurable, extended real-valued function T4 = 'Yx : ...(2._R+= [O,ro};uch
that (x ) =SlTX)(t)d for every A6-2:" . The IJap T of X in the space
S of all :A:r-me:Aasurable, extended real-valued functions with equality
80
f-a.e. has the following properties :
ll) TO :::: 0, l2) T lxv.) :=lTx)XA ' 1..3} if x,y E X and IXI Iyt, then T:x: y.
(4) if x,x n e X and 0 x l' x, then Tx t Tx,
n n
(5) if x,yeX and xlt1ylt)::s 0 ('I-a.e., then T(x+y)= Tx + Ty.
The property (1) is obvious. (2) follows from the equalities
S(TX)(tJ A(t)dr-:::: S (Tx1(t)dr-== (x1A B)= S (T(x;v))(t]dr-c
B AM B
valid for all A,BE-2:. In order to get 13 )let us observe that if x,y X,
Ixl IYI, then (XXp) (yXA) for every Ae2: , i.e. (Tx)(t) df-<.=
IT;y)tt) dfoA.- for all AG2: ; hence Tx Ty. Now, supposing x,xnE- X and
o xn-t x, l3hmplies trxn) to be nondecreasing. Let TX n t y -a.e., "then
(Txn)(t) d,M- t 1 ylt) df- . But SA (TxJ(t) djlt= (xr.XA) f ( A>:: L )l.t) d
for every Ae- . Hence y :::: Tx and we proved (4\. The property(5)follows
from the equalities 1 T (x+y)tt) dp...:::: ((X+y)'A) :: ( + (y =SA (?:xlt) +
Ty(t»)dfA- for every A E 2: .
Now, let us define a function \11: ...o...x R If by the formula
1++
'flt,u)= T (u'X.J(t)=tu .a.. It).
Fror:J. properties(1\-(5)of T we may conclude immediately that -tlt,ul is
a measurable function of tE-Jl. for every u O and that
(1') tlt,O\= 0 (tI---a.e.,
C2' I if 0 $ u v, then '-r(t, u) \jilt, v) f't-a. e. ,
(3') if o untu, then 'f(t,un)t (t,u) -a.e.;
however, the sets of measure f"' zero in the above conditions depend on
the values u,v,v and in general a common set A c...;,'L of measure zero
n .
such that ij') -(5'J hold in..Q" A does not exist, and so 41 does not
need to belong to . First, let us observe that taking an arbitrary
m
simple function JC:::: 2: U,'l E ' where u. to R, E.!iS 2: , E. pairwise dis-
'-1 i
joint, we have m - m
y(t, Ix It) ,L "V , lui'}XE _ It)= ,L "Y11] I.y E ' (t):::: 'hx,(t)
1 1 '\
for -a.e. t f: , and so
lx) = (Pc/)= S lj-(t, Ix (t)1) df-'
In order to extend this equali to the whole of X, we have to re-
place the function If' by a function fE:-f ' equivalent to 'r .
Let Q be the set of all nonnegative rationals and let
+
81
Ao ::: r t.Q+Ar,r' ,where Ar,r,={tE-..Q: "f(t,r')<'f'(t,r)l.
r<r'
From l2') follows that p..(A r , r?- 0 for svery 0 r< r' and so JlLAol::: 0 .
Clearly, if r(r', r,r'EQ+, then 't't\;,r) 't'lt,r') for every tf...Q...A o .
Let lD: x R _ it be defineli as fOllows :
1 -+ +
{ 0 if u::: 0, te-S2. or u:>O, t fi:;A
Wet u)::: 0
l' lim '-K.t,r) if u)O, te..Q.....,A .
Q i r u- 0
+
It is clear that for every u 0, t,u)= "(tCt,u)ft-a.e. and tf .. Let
XEX be arbitrary and let (xJ be a sequence of normegative simple fun-
ctions such that x l t)11 x (tJ1 for all t E-J2 . Hence a set M:'J: of mease-
n
re /ollA)::: 0 may be found such that If(t,x n It))..'Y (t,x n ltl) for all tE-.!t'A.
Due to -the left-continuity of tp, -a.e., there holds
'i"(t,x n It))::: \f(t,x n It))t (t, txlt)l)f-a.e.
Thus,
(Xn}::: k'f(t,x n (t))dflt l'l t, lXltl/jdl"'.
However, f'rom O xnf x we conclude that (Xn) te(x).. Hence
(xl = S ({1(t, Ix (tllldf'<-'
We are going to omit the ddi tional assumption that ! = \.4t € X.
Let us suppose only that there exists a fu.."lction uE-X with u(t).:>O for
every t e..Q. . Then we may replace the linear sublattice X by X ={ :XE-xJ
and the pseudomodular by {]; ) = lx}for xfX. Then 1 fX and 1:
r- _ \.u ,..,
X-2>-R is a disjointly additive pseudomodular on X. By that what we
+
proved above, there exists a function E , such that (x) :::
J(t, 1!J[lt)I) f or every x . ) T 1x(tJ;n ) :s <f0, U SI} for t e J2. , v O,
we have G and lX) ::: 0 :::: J l' ,t, d for all x EX.
12.5 .Remarks . It is easily'-%bserved that if (x)<oofor all xeX,
then fin 12.4 may be chosen R+-valued, and if f1 is a modw.ar, than
one can choose so that t, u/)O for all u.:> 0 and r=-a. e. t e..Q. .
lvloreover, if if is convex, then tp may be chosen also convex with res-
pect to u O for f-a.e. t e:J2 .
12.6. Definition . Two functions x,y' S are called ft- equimeasura...
ble , if f({t Sl: x(t)7a})::: f<-CIt J2.: YLt)7aJ) for every afRo
12.7. Theorem . Let be a o_finite, atomless measure and let X be
a linear sublattice of S possessing the majorant property. Let {( be
a disjointly additive pseudomodular inX such that if x,yEX are
82
f'-eq uimaasurable, then xl = (y). Then there exists a function <f :
R+ _ R+ vanishing at 0, nondecreasing and left-continuous for 1..1> 0,
10 uch that
lxl= h {x.ttJO for every xeX.
. Arguing as in the proof of 12.4 we may limit ourselves to
the case of a finit..:. measurer- such that 1 =\Q..€ X. By 12.4, there exi-
sts a function IfG such that qlx)= ff(t, IxttJ/)dfL for every xfX. It
..Q ,
remains only to prove that f--almost everywhere. in a, i.f(s, 1..1) is a con-
stant function of u> O. Supposing it is not so we may find a 1..1;:.0 and
o a < b < co such that /4.I"(.f,t€-.a: If'tt, 1..1) a }O and f{ftE-.l1-: tf(t, uj * b )) O.
Since JA-- is atomless, there are sets A,B€"2: , such that A c {tG-J2 : I.f'(t, 1..1)
aI' B c {t J2.: lt, u} bf and lA\ =follB) O. Hence it is easily checked
that uXA is f-equimeasurable with urB' However, {u L2 =SA t.f (t,u)dll,(.
af..A)<bf'L(J3) 'Plt,u)df=> Q(u 1B ) , which contradicts (u:x..tJ=<f(u{B)
assured by the assumption of our theorem.
13. Complementary functions and continuous linear functionals
Let us first remark that if t.p (see Definition 7.1) is a con-
vex function of the variable uE R for every t r:...11 , then t.f' is of the
form
11..1/
('k) 'f'lt,u)== S plt,'t)d't',
where plt,u)is the right-hand dedvative of\f(t,u)for a fixed t .JZ . We
shall restrict ourselves to a subclass N of
13.1 . Defini tion . We shall say that a function IfG <p is an -func,:"
;!;ion , or 4' € N, if is a convex function of 1..1 for every tE-"p. and
there hold the conditions
(0) lim = 0
u 1..1
for every tEll .
From this definition and that what was said above there follows,
too)
lim
1..1..>0)
P{t . 1..1)
u
=> 00
iJ:DInB dia tely :
13.2. Theorem . I.fiE is an N-function, if a.YJ.d only if-, ifis of the
form O'-} J where p It ,'(""1 ;> 0 for '("7 0, P (t ,''L) is a right-continuous and
f 1""'7 0, P ' t,O) = 0 ' pl t , r)- 00 as 't"--.y 00
nondecreasing function 0 'v
for every t ...sl. .
83
Moreover, it is easily proved that
13.3. Remark . Let <f€:N be of the form(*) and let p*'Ct,G"'):=: supt't": p(t,ij
1. If p satisfies the conditions expressed in 13.2, then pk satisfies
the same assumptions.
13 .4. Det'ini ti.on . Let \fEN be of the form (}oj and let p*' be defined by
13.3. Then the function
tul
((t,u)::: S II tt,v) d6"
o
is called complementary 19 f in the of Young .
Evidently, If is again an N-function.
13.S. Examples . I. If \f'(t,u)::: IUl pLtl , 1 -( pLtl(oo , then (tt,V):=
q (tJ 1 1
IVI , where P(t )+ uq(t l =: 1.
II. If \f(t,u)= e - u - 1, then ef'Cv)=l1+v)ln(1+v)- v.
13 .6. Theorem . Let qJ ,(f:: N and let cp1t- be complementary to If in the
sense of Young, Then they satisfy the Young inequality
uv If(t,uJ + t,v) for U,V? 0, tE-..$l ,
and
(a) (It,v)= sup (uv - lt,u)}, (b) If(t,u)== sup Qxv - ,V));
u O V)O
consequently, lfis complementary to in the sense of Young. Moreover,
in case of If'{t,v1 the supremum is attained for u t =: p*(t,v), ald ip.
case of If(t,uj - for v t = p(t,u), Le.
to) (tt,v)= vp ,v)- (t,P'tt'V\)J(d) \(I(t,u):.u uplt,u)- 4(t,p[t,uj).
Proof . Since the statement has to be proved for every tE...!l separa.-
tely, we shall omit the symbol t when proving this theorem. Let us sup-
pose first that Plu) v. Then
v
S d""'1 (v-Plu))rl\P(ul)-q(v-P(u))u = uv -u p(u).
ptu)
Hence
u p(u) v u p ul
\f )+ v). S P(t)dt" + S :ift d(}+ ) p ct)d('j S pt()d'i + J P'i1i)dCl+
o 0 p lu) 0 0
l ) '\- UV - up (u) .
p u u
But f ItG'Jdo:= u pLul- S p('t)d<r 0 Thus, tflu)+ (tv)"J UVo If P(u)7 v , then
p tv) < u and we argue as gbove changing the roles of u, v and p, If ·
By Young inequality, we obtain flv) B (uv - lu)). Taking
u ::0 ptV1 , we hav .
p_"v! .Jf v .r'/<; rt:
v)+ tV)=: pLt)d'C'" + 'fLv):It vP :V' - Sop1G')dU + 'flv):= vP v).
84
This proves (c) and shows that c1h)= rflv}v - tft/LVI) B (uv - \f();tl),
and so we proved La) 0 Formulae (d\ and Cb) are proved analog!3usly.
Let us reJlla;rk that the formula 13.6 tal may be applied to define
the complementary function'f t to a function \ff-<}.
13.7. Definition .The complementary t)Jnctio to a function'fE-
is defined as follows :
W\t ,v) = sup (uv - tDlt, u)) for v 0, t E-..!2
Wo 1
13.B. Theorem . Let (fE<P be such that there hold the conditions
1301 t>} , (<D). Then the complementary function LV*' belongs again to N.
Moreover, (l.f\t,u}t Cf(t,u) for all u o, te...Q , and if lp is convex
wi th respect to u, then ( {t, ul)""= I('lt, u) .
Proof . It is obvious that *\.t,v) is measurable with respect to t,
nondecreasing with respect to v and that *(t,O)= O. Let us suppose
that <p'tt,v)= 0, then v q.'( ,u ) for ftvery u:;>O and from(o\we obtain
v :a O. Now, let us remark that the supremum in the definition of rflt,v)
is attained for auch u that ) v. Applying (00) we observe that the-
u
re is a. number 1l.vlt»O such that tflt,v} '"' sup (uv - I.('tt,ul), and we may
o Su Uytt)
assume Uv1ttJ u.,J:t;)for v1 v 2 . Hence we have, for 0 v1s, v 2 '
o \f'tt,v2l- t(tt,v 1 ) = sup (uv 2 - Iftt,uJ)- sup (uv 1 - lftt,u))
O u( ) o u Uv
10 up C uv 2 - uv 1) "" (v 2 - v 1) u., 2 ( t) ,
0 U!'Uy t)
and this proves continuity of Ifl.- with respect to v. Convexity of '{'II:
and the relations lo) and (00) for W in place of if are obtained easily
from the definition oftp . The inequality for (flt,u)t" is obtained,
since (t(Ct,u)f= sup (uv - l.\I t,v}) (t,U), because uv lt,u)+ k Lt,v)
v20 1<"
for all u,v o. The fact that (<jI*fu,u)/= ft,u)if If is convex with res-
pect to u is evident.
13.9 .Definition . If t.pE <P satisfies (0) and (oo)of 13.1. then the
function if(t,u)= (((1\t,u))*" is called associated 1!ti.h If .
Obviously, If is always convex with respect to the variable u and
lt,u) <r(t,u) everywhere.
13.10. Theorem . Letf be a ()-finite, atomless measure, and 1.'1 \{7fP
be locally integrable and satisfy the conditions to) and (00) of 13.1,
and let \f'" \p (see Definition 8.16). T hen there is in L'-f a homogeneous
nQrm equivalent to the F-norm I I generated by the mod.ular lX) =
85
S \f (t, Ix (t)Qdf- 0
..>2.. P f S' --
...l:9..9....o nce If' is locally integrable and \f(t,u) I.(J\t,u)J so tp is
also locally integrable. Moreover, if is convex. Applying B.18(a) to
't' = W' 10 = 1, we obtain existence of a homogeneous norm in Lit eq uivalent
to I I .
We shall introduce now the Orlicz norm II " IO generated by the mo-
dular (xJ= f (t, Ix ltll)djlA-in the Orlicz space LIf', where LffN. The con-
nection bet en norms II I and II /1/ defined in 2.6 will be examined in
,1 0
13.20.
13 .11. Theorem . Let the measure fL be -fini te, lffN and If locally
inte Grable (see Definition 7.5 ) , 1ft complementary to tf in the sense of
Young, and let Lr =[y : S t, Iy It)Udr- 1, y measurable 1. Then
.A
II xII :;:; 10 up 5 x (t I y It) dl\,(.
IO y Lf .a.
is a norm in Llf (called the Orlicz ) and
IIxlI lIxl IQ 2UXI for all x fLip 0
Proof . Let x'" LIf, then
( x ) _ f to (t Ix ltH ) d M. 1 0
I:x!lt -.Q.l \ ' IIxl + t I for every E> .
Taking YfL 1Ir and applyimg the Young inequality, we obtain
S IX(t)y(t )ld <r\1/t IXltH ) dM. + f"f ( t I I t)I'dtlA.<'.2
ItxlL+ f. I \; J 1 \ ' "xll+ '(, I J,\,' Y 1-1'" 0
.
Hence
Ix It) Y It)ldj''t- 2 xll + 2t
Conseq ue:Al y ,
Ifot
for every e>O, y€ L 1 0
II xlL < 211Xfl/D < 00 .
tO -
Hence II xlI (o is finite. I t is easily observed that II II IQ satisfies the
axioms of the norm. Thus, it remains only to sttow that IIxl Rxl IQ 0 Sin-
cellxlL =lIlxlIL , so we may suppose x(t\20 i . By homogenity of the
O (O y
norm, we may restrict ourselves also to the casenxl IO 10
First we show that if ZEL't' ,lIzlI 1 and olY)""Sf();,IYltll)d/,,<->1,
, (O'"<..a.
-then there holds the ineq uali ty
(+) Szlt)y( t1d l\,(. <?o(Y)o
Of course, we may assume y -:;-Lr. By convexity of 1(",
(' Ujlt{t Iy It/l 'dU < ...L S ,.#t, l ) dl\.(. "" 1 ,
J 1 \.. ' olY)/ I -... AlOfy) 'f \: 1J0C,y) r-
..n.. It"'.n.. I
and so y/ 0(y) L1. Hence we get
86
i.e. we have C+) .
S 'y(t)1
.dZ(t \/ 't:y) d"" IIZ'/ /O 1 ,
Let .o..1 C {).2 C ' ..
00
!2..= £1 ' and ji't(,Q,n) < 00 for n = 1,2,... Let
x n (t):: fxtt)if t E ..Q.n and x(t} n
lo at remaining points of..Q,
xott)= xtt) for tE:- '2. . Since x'" f!, so Xn€ L'f for n:z 0,1,2,... lVlore-
over, denoting y It):: p(t,x It})for n =" 0,1,2,..., we obtain y It) y It)
n nit'" n 0
a..e. inj)., and by Fatou lemma, Yo e- Lr if only Yn E L for sufficiently
large n. Now, let Us suppose that Yn 4 L'f, Le. e'tyd > 1 for some n 1.
By 13.6(d), we have u ptt,u):a<.p(t,u)+tpA-(t,p(t,u)) for u)O, tf.S2.. Subs-
tituting here u = x It) and integrating over..!l. we obtain
n
l. xn (t) Yn (t)dfA-= <f(t,x n It))df+ l..(t'Yn It)Jdt = (Xn) + olYn).
Applying the inequality (+) to z = x and y :z y , we obtain
n n
f'x (:t)y It)d j \.t. < ()ofJ) fornb-n.
1.n n _"i n '0
be a sequence of measurable subsets of..Q such tha.t
Hence
o 0
(Xn)+ LYn) (Yn)'
In case when 1< 1t"tYn}<OO , this yields e(X n ) = 0, Le. xnlt)= 0 a.e. in
..Q 0 Hence Yn It)= 0 a.e. in..D., a contradiction to the inequality olYn)
>1. Now, if <?0lY,J= 00, there would be .txnlt)Ynlt)d '" 00. However,
since x It)y Lt) n p(t,nJ, so taking into account the inequaLity
n n 2n
n p(,t,n) S plt,!) d't 'P (t,2n),
n
we get, by local integrability of If '
lt)Ynlt)d .( n Plt,n)dr- 4tftt,2n)d <ca..
This is again a contradiction. us we proved Yn € L,\It. for n == 1,2,...,
and so Yofi: Lf .
Now, we apply the above quoted inequality 13.6ld) to u = xlt) ,obtai-
ning
xlt)Yo<..t)= 4>(t, x(t\) + 'f t'Yolt)) a.e. inJl.
Integrating, we get 0
(t)Yolt)¥= lX)+ (yO>.
However, since Yo fi: L1 ' we have
S xlt) y (t) d IIxll ll = 1.
.J2. 0 )10
Hence
Lx) lX) + (J.0 (yo):a ix It) Yo It) d t"t 1 .
87
This gives II xll 1.
13.12. Remark . Let us note that fini teness of ,.... and local integra-
bili ty ff in Theorem 13.11 were needed only in order to prove the in-
equaJ,.i tY\lxUQ l1xll .
) IO
n.13. Theorem . Let 'f€N, <f compleI;Bntary to \f in the sense of Young,
x&L'f, yE"L . Then there hold the following H81der inequalities:
J SX(t)y(t) d I lIxllf.JIYll o , ISxlt) ylt} d I IIXU II YII ,
whe re.:n. l = &, Ix It)Qd p.-, o lyl::: r '! y (tnd fA-. f'/O
. We show the first inequality - the second one follows by
changing the roles of x,y and 'P, 1t. Taking any 'C»O, we have
S . ( t Iy l t) ) dk < 1
'Y ' JlyH + J- .... ,
and so ...a. o
I S l y (t )l I
.o.x(.tll Y11 + f: dr-- lIxl \o'
and since c.7 0 is arbi t ra f thi s give 10
S xLtl ylt) d[u. ttxII., f} ItYIL .
...'1- 11 10
13.14. Co rollary . If 6 N and If*is complementary to If in the sense
of Young, y L then f(X)=&,xLt\ylt) d is a linear, continuous func-
tional over Llf with normU:rIf=A yt o .
.. ,0
This follows from the second of the HBlder inequalities 13.13 and
from the fact 1/ fll= :o I 'x C:tl y Lt) dAA.f ::: -S jJ 11 x (t\ Y ltl df'tl due to left-
II £.1..n.. ( L4 4 I
continuity of and 0 1.13.
We shall see that Corollary 13.14 may be strenghtened under some
additional assumption on'P in this sense that LX).Sxlt)y(t)dfwith
- £
yfi: L'" is not only continuous with respect to the norm, but also -con-
tinuous (see Definition 5.9), Le. xi O in L<f' implies f(x{-?,O.
13.15.Theorem. Let r- be (J-finite and let the function lffN satisfy
the following condition: for every uJDthere exists a c'70 such that
tilt. U\ ::>. 0 for u"Z u and e.ll t 4i:Jl . Let the function l( complementary
u Y 0
to \(I be locally intec;rable. Then f Lx) = lx It) Y (t) d is a -continu-
ous linear functional over Lip for every yE: L<f
. First, let y be an integrable, simple function on..Q.. wi th
lylt)l l for teJl. , ylt)= 0 for tf.n.'A, wheref"/(:4.)<co. Let axi) O
for some a> 0 and let £...,. 0 be given. Let Ai = te-Jl: IX i It)I 2 A) 1 .
By the assumption, there exists a c '> 0 such that \f(t,alx i It l)f1ajxilt»)
for t E: st' Ai. Hence
88
IflX i )I IX i <.t)ldf + 1 S CD(t,alx, (tjUdf f. + 1,o(ax.).
Ai c t A'Ai 1 2 C e ')
, . !ci,
Taking so large that (axi1<2 ' we obtainlf(x_JI<E .
pff' J:
Now, let US take y (i L arbitrary. By 7.6, the set S of inte grable,
simple functions on il is o"'danse in LIf'II'. Let y -" y, Y E: S and let b:> 0
o n n
be such that (blYn-Y)) O' Moreover, let (ax '-" 0 for an a> 0, xi'" LCf.
Taking an arbitrar<J f70 one may choose an index n for which o(b (Yn- y )
( ab 0 Writing f n (x);:: S x(t) y (t) d , we have then
..Q.. n
If (xi>l If n ( x 1 1+ I f(x i ) - f n lX1! I f n lx1 1 + ....L b f (j(t,alx. lt11)d(-t +
a ...1"2.
a r\f :\t,b Iylt) - Yn (t) df = IfnCx 1 + b (ax + a qO (b(Y-Yn))<
1
<. I fnlx i , + ab la.x + '3 .
N < 1 1
ow, we take io so large that laxi) '3t: ab andrfnlx I<'3£ for i>io'
Thenlf l x,)J< t for i> i .
J! 0
We are going now to prove he converse statement to 13.15. First,
the following Lemma will be shown
13.16. Lemma . If f"I.- finite and Cj)E-p is locally integrable and .?(A
is the characteristic function of a set Af: , then ft(AI-.;>O implies
"\i{ O.
Proof .
Let A.f ,AJ(A.) O, and let
locally integrable, we have (a A:J =J Ai
this shows that " Aill -O.
13.17. Theorem . Let f'1-' be ()-finite and let \fE-N be such that for
every u .., 0 there exists a c:> 0 for vihich "> C for uz u and tEJ!.
o uP, 0
Moreover, let both the functions If and tp (complera.entary to If) be lo-
cally integrable. Then for every linear, q -continuous functional f over
Llf there exists a function yE: L'flt- such that
f(x)= SXlt)y(t)df for every xELtf.
Proof . 1 0 First, let us4iuppose that f«J2.) < 00. Let f be a linear,
-continuous functional over Ltf; then f is also continuous with res-
pect to the norm and so there exists a K> 0 such that If(x)\ K\lXI for
all XE-Llf. Let 1A be the characteristic function of the set AE-% f
then \A E L , because is locally integrable. Hence we may write veAl =
ffx.,J . We shall show that IJ is a count ably additive set function in
a,. 0 be arbitrary. Since <p is
tp(t ,a) dr-<-"" 0 as i..,. <D. But
89
!", absolutely continuous with respect to the measure r-. Indeed, if A ""
A 1 VA 2 U... with Ai pairwise disjoint, AiE: , then writing Bn = A 1 V...
VAn we have Bnlt) A(t)and If(t,I BJt)- Alt)l) <Kt' A(t)) for tEJl .
By Lebesgue dominated convergence theorem, (t, I Bn{j;) - A lt j -:;. 0,
Le.. VB Y A ' ByO-continuity of f, we thus obtain v(J:i1= Z "LA.) 0
1 n' ) =1
In order to prove r--absol ute continuity of V, let us estimate the to-
tal variation of a set AE-.£. : IVI(A)= v+(A)+ V-(A) 2 supD,,(nt: BcA ,
B I 0 But lV.lB)I=lf(xB)I KI\XBII KII i and so Ivl (A) $ 2KIIXAI . Now, 13.16
shows that y is lA-absolutely continuous. Applying Radon-Nikodym theo-
rem we deduce that there exists an integrable function y on..n. such
that ffxJJ = S XA Lt) Y It) dfA'. Consequently,
(1).n. flxJ=)xlt)y(t)djIA-
for every simple function x. ..n.
Let us suppose y(t)10 inll. First, let US observe that (1) holds
for every measurable, bounded, nonnegative function x. Indeed, taking
o x It)f x (:J;\ in..D., x simple, we deduce easily thatS If(t, a (x (t) - xntlJ»d!'t
n n Jl
"";'0 as n-»oofor sufficiently small a)O. Hence fLxd-tf(x), Le.
ixt ) y It) df'A-"i' f(x). On the other hand, 0 x n It) y ltl t x It) Y It) , and so
jx It)yC.t1d S x(t)ylt) dr-<-. This showsl1). In particular, we obtain
!L n ..f!.,.
(2) I S x (t) Y (t) dtA-1 = j f lx)l Kllx II
...t'2- .
for every measurable, bounded, nonnegative funct on x.
Assuming y It\ O in.!t, W8 shall prove now that y LIf'. Let us take
{ yet) if Ylt) n *f. Yn lt » )
Yn lt )== n if yLt»n' xn(tl:: p \.t, '
where p* is defined by 13.3. Then, by 13.6lc),
,, f. Yn It) ) Yn It ) * ( Yn (t )\ lOb. k ( Yn (t) ))
\t'\.t, K == K P t, K J-'f P t, K t
i.e.
x ltl Y (t) ...IJ X l..t J )
n K n = (t, xnlt)) + \jilt, 0
Integrating, we get
( y l t) )
(3) S xnttlY n(t}dl'l-""t 'P ,Xn(t))dfA+ lft< t, -7 ar- .
..n.. ,....
Moreover x G L(f because:x: is bounded. Let us first suppose that (xJ>1.
, n' n
Then we have (xrl qf...xJ) 1, and so 1\ xnl q(x n ). Hence, by inequali ty(2),
we have
90
whence
Itxnlt)Yn(t)djlA) xnlt)Ylt)djlA- K/lx 1/,, K(JlX).
w n) } n
Substituting this to (3), we obtain
IrM Yn lt )
q(xn>+ $ r,t, l{ ) dll.(., ftxJ '
.sz.
S ( Y l t» )
,_51..0/ t, -T dll.l.= 0 ,
Leo ynlt)= 0 a.e. Hence xn(t)= 0 a.e., i.e. (xnJ= 0, a contradiction
to the assumption (xJ ;;.1. 'rhus, we have qlxnl 1. Hence also IIxd' 1,
and so
xnlt)Ynlt)df fxnlt)Ylt)df KI/xnl/ K,
.$'..
bY(2) . Applying(3! , we thus obtain
r .,t( Y n l t) 1 Y l t» )
J \.t, K I dp.. S 'X n (t))d}A- + S ((t, + dj.(. =1}xn It) Ynltl df- 1.
..n. !2.. f K .a.
But Y n It ! O and Y n rt\t yLt) , whence
olil= Sf(t, :if'}dr-= lim 5 (t, Yn t) df"'- 10
...Q n 00 :re
Consequ.ently, YEL4' Now, let us write g(X)=S x(t)y(tl ¥for all XELif.
Then f(x)= g(xl in the set S of simple funct ns, -dense in L If , accor-
ding to 7.6. Since y E L'I1 , so g is a linear, -continuous functional over
L'P, by Theorem 13.15. Consequently, f(x)= glx)for every nonnegative x"L
If xf::LIf> is arbitrary, we get f(x)= g(x) applying this to the positive
and negative part of x, separately.
Now, we resign of the assumption y(t! O in.!2.. . Let y(t) 0 :\nO, se'
A c;,..r a.'1.d y ltl <. 0 in a set B" 2 . Applying the above proved case to each
of the sets A,B, separately, we obtain
f{x1A)= S xlt)y(t)d,rvtand f(x W ) = r xlt)y(t)d ,
A B
and so fCx) == fCx + x = f\x A) + f(x B)= SX(t)yltJdr-<-' which finishes
. ..rt.
the proof 1n the case ft(.ll,) <. 00 .
2 0 Now, let f-«..Q.) == 00 and il=..S2.. 1 vJl 2 V... , where..a n are pairwise
disjoint and f(..sld< 00 for n = 1,2,... Let Ll.f[..Qn) be the generalized Or-
licz space generated by the modular nlX) == \ttlt, Ix ltX)dll.<. and l t f n
be the restriction of the functional f to the subspace of functJ.ons
xE; L L..Q.I equal to zero uotside . First, let 'US observe that f n is a
linear, (j -continuous functional over LIf(.9-. n ); indeed, if :xs. e:- LIf(.R. n ),
n ,... Sl'J1. we
x 0 and we put x_ ltl == x. Ltl for t (:-...Q. , x. It)= 0 for t n '
i J. J. n J.
91
get xiJ.o in LIf(.Q.), and so fnlxi):::f(xi)--,>O. Applying Part 1 0 of the
proof of our theorem we conclude that there exists y E: L ) such that
S n n
fnlX):: .o.nxlt)Ynlt)djI.l. for every Xf/i:L<f(A.. n ). Let yet)::: Yntt) for tGJ2 n ,
n == 1,2,... We shall prove that f (x)= i x It) Y tt) for every Xc L .st)and
ye LIft.Q.). Let xeL'f(..Q.) be given arbitrarily; it may be written in the
form x(t)= x1lt)+ x2ltJ ... for tE-..Q. , where x n = x e.Ltf(...sl.n)' Let
k 70 be so small that . S tf(t, k lx, (:t;)'dllL.. a( hi < cx), then
n ..1 ,- ")
CL xi - x))= . 1:: S. If (t ,k IX i ttJI) df-+ 0 as n-;ooo ,
fi=1 =n+1
and so t 1 x. )t. Hence
=
n n
f{x) == lim f(l: x. \ = lim L S x (tj Yi It) dj'k== S x(t) y It) ft .
n CX) i=1 'iJ n....oo i=1 ..Q.
It remains to prove that yf/i: Ltf'(..Q.). Let An = fJ.. 1 V .. .V..Q..n and let de-
note the :estnction of f to the subspace of functions x €LIf(..n,) e'lual to
zero outside An' then lX)=t x(t)znLt)dt-<-, where Zn(t) = Yl(t)+ ...
+ Yn(t) for tG..!L . But the serres1;. 1 S xlt)y(t)dlA... fx<..t)y(t) du..-is
- I"V r -
convergent. Hence .Q.
CX)
f (x)- lx)=. ,L 4.:. x It) ylt) df--'P 0 as n-'D.
Thus, (X)....., f (}c) for every xn+J tlf(..sz). Since are G-cQntinuous, so
they are also continuous with respect to the norm in L'P(...Q.). Consequen-
tly, there is an M>O such that IIg II IiI for n = 1,2,..., where
n
I\ K= SUP{SA X{t)zn{t)d : i If(t,jxlt)/)djlA- 1I=nznll f),o.
n n
Hence, by 13.11, nZnl B1Zn!I o .M. Thus, </\zjr-4) 1 for n == 1,2,... But
Y (t I = 11m Z (t) for 8.l1 t eJl. . Consequently,
n,"7CO n
S((t, IY,it)l )d",. lim f((t, Iz t)) )ol1'l 1,
..Q n 00 ...'l.
and we obtain y €LIfc .
Applying Theorems 8.13 and 8.14, we obtain from Theorem 13.17 the
following result, iIrJlD3diately :
13.1B. Theorem . Let the function \fE-N and its complementa:ry be
locally integrable. Let us suppose that for every u o ::> 0 there is a c>O
for which ) ');, c for u u and tt-...n.. . Moreover, let us suppose that
U 0
one of the follovnng two conditions is satisfied:
a) pA- is 6"'-finite and atomless and cp satisfies the oondition 8.13 (1).2) ,
b) J1.={1,2,...} 'f-({ )= 1 for n = 1,2,... and 8.13 (02)holds for If.
92
Then the general form of' a linear functional over:& continuous with
respect to the norm is
f(x) == S xtt/ yet) dr- for x E- L tf
with Y €: L'f"" , and Iff" == II ylL . .Jl.
o
Theorem 13.18 enables us to turn back to the problem of conjugate
modulars f:* in the space X of all linear, continuous functionals over
the normed space<X ,II ), where q is convex, inve.stigated in 2. Ta-
king as the generalized Orlicz space generated by the function
<r € N we $all show the following
13.19. Theorem . Let us suppose that the function <pc, N, its comple-
mentary \fit and the measure tN satisfy all assumptions of 13.18, Let
J.lyf<J = sup [x - qt..x>: X€L'f} for every x,lt"e X;=(L.qt. Then .lx = f°(y)
= St(tt, ly(tIVdf"\' ' where y L!ft is the function from 13.18.
..tt
Proof . By Young inequality, 13.6, we have
x"'x = x(t) Y It) df-'<- (x) + t, !y ttllidl ,
-'l ..a,.
and so
'I< (x*> f1(t, iy ttJVd\",,- '
.n;
In order to get the converse LDequality, we apply the equality 13.6lc)
with v '"'!YLtJI and u =lxoLt)\= p (t, ly(t)Q, obtaining Ixolt) ylt)l=
q{t, IxoltH)+ qflt, IYlt)!) . Arguing tiS in the proof of 13.11 (with Yo re-
placed by x ) we may deduce easily that x E LI(1. Hence
o 0
r!lxf<1 x1fxo - lxd '"' h Xo It) Y (t) I.t - llftt, Jx o It)Vdr-: ' IY (t)If:1/1A.. ,
ana tlo.B desired result follows.
From Theorem 13.19, the following corollary follows, irrur,ediately
13.20. Corollary . Let Ip€:N, (" and f- satisfy all assumptions of 13.18
and let II II be the norm in Lip, defined in 2,6 with l..x)=r (t, IXlt\I)df"<- ,.
2
Then II xII =Ilx/f! for every x Eo L .
itO
We shall now investigate cases when there are no linear, oontinuous
functionals over L , with the exception of the zero functionsl.
13.21. , Let the measurer"- be finite and atomless and letlf€-
be integrable over..Q. for every u?- O. Let J:: 1.'\I(t, ly It)IJdf for every
;[-measurable y. Then for every nonnegative, simple function x over..o.
and for every s.et AE£ there exist pairwise disjoint sets A 1 ,A 2 ,A3'" .
-n 1
,A2nE such hat A = A 1 v ... u A 2n and lx;U{ = 2 (x A! for i '" , ..
,2 n , where \B means the characteristic function of the set B.
93
Obviously, it is sufficient to show the lerr a in case n = 1 ; the
easy proof is left to the reader.
13 .22 0 Theorem o Let I'A- be a finite aad atomless measure and let the
function tfe <P be integrable over..sl. for every u O. If Em ) = 0
u:roo u
for almost all t ..Q. , then there are no linear, C?-continuous functio-
nals over the space Lf¥ besides the zero functional.
Proof . It is sufficient to show nonexistence of different from 0
linear functionals continuous with respect to the F-norm. Let x be a
Then
nonneg&tive, simple function, max x(t) = IiI/; 0 and let A O be arbitrary.
tEi'J2.
i (Ax) S If (t, AM) dt'1- = I'll S tf(t M) df'l-+O as A _ 0+.
n..
Let W be a convex and symmetric set in Llf with the property that there
exists an E>O such that if yEL'f and (y} t, then yEW. As we have
seen above,
-n (, n )
Hence 2 q\2 x f,
,." n.
.'" and 2 x n place of A and x, respectively. 'Chere are pairwise dis-
j oint sets A 1 ,... ,A 2n E: such that -0.= A 1 V.. .V A 2n and ({ L2 n x x'A{ ==
-n (. n ) . n n
2 \2 x e, for == 1,2,...,2 . Hence 2 x;v...G W for sufficiently
large n, and so
-n 0 ( n )
lim 2 "j 2 x = O.
n eo
for sufficiently large n. Now, we apply 13.21 to
by convexity of W.
Let f be a linear, continuous functional over L<f' and let us take
-n #-- n
x '" 2 L 2 x . E W ,
i",l
W = y(, Lit: If(y)! 11. We shall show that there is an E'7 0 such that
if yrc. L and (llY\ t ,then yeW. Indeed, conversely, there would exi-
st Ynf:LIf , n == 1,2,..., with (Yd and If (YJI> 1. but this contra-
dicts the e-continuity of f. Hence, as we have proved above, every non-
negative, simple function x belongs to W, i.e.I:f<.x)1 1. Splitting arbi-
trary simple function in positive and negative part. we obtain If (x)1 2
for an arbitrary simole function. But, by 7.6, the set of simple fun-
ctions is -dense in If . Hence Iflx)/ 2 for every x E-LIf, but this impli-
es f = O.
13.23. Remark . If q> satisfies also the condition 8.13l6 then, by
8.13, there exist also no linear functionals over L continuous with
respect to the F-norm I I 'besides the zero functional. In particular,
94
there are no nontrivial linear oontinuous functionals over LI.\' for
p(t) .
(t,u)= u Wtl!thmeasurable O<Pltl<'l a.e. inll,
14. Interpolation of operators in generalized Orlicz spaces
14..1.. Let (.!t,I:,tt) and (.Q..o'%:o'v) be two measure spaces with <r'-fini-
te, complete measures fA- and v and let Ml.!2.,"E. ,,..) and M (...(2.0,2:. 0 ,17) be the
spaces of -measurable, finite r--a.e. Vio-measurable, finite v-a.e.)
unctions on.Jl.lresp."'O"J with equality f-a.e. lv-a.e.) . Moreover, let
'1" ' <:" ) tr" )
E"= E C..Q,'-,f'I- ' E "::: E't'ft(J1.,Z.,1) , i '" 0,1, be the respective genera-
o 0
lized Orlicz spaces of finite elements (see Definition 7.2), where
i'o/i are N-functions (see Definition 13,1). The aim of this Section
is to give an interpolation theorem of Riesz-Thorin type for operators
between the above spaces. Let us recall that the relation Cfl y> 0 for
such functions means that there are a constant K) 0 and an inte grable J
nonnegative function h onA such that 1f1 1 Lt, u) o (t,Ku)+ hlt) for all
u) 0 and almost all t E:!l. l8.1 (2)}; supposing fl to be locally inte-
grable (see Definition 7.5) , 'fo <-Pl is a necessary and sufficient
condition in order that L f(..Q, I )C.Llfo(.Q..,i.,/,,-), this embedding being conti-
nuous (Theorem 8.5). Saying that lfJ o and <f 1 are equivalent, <Po"'" lpl'
if Ifo 1f1 and lfl' If 0 and supposing both Cfo 1 to be locally integra-
ble we thus obtain Lllol..Q) I )::: L<1>f(.!2,'£., f"') topologically, if and only if,
0'V'f l' Let us still reoall that is said to satisfy the condition
C£l2)' if lt,2u) K {!;'9+ h(t) for u O and almost every tE-...a , with
a constant K)O and a nonnegative, integrable function h (see 8.13) .(/J 2 )
is sUfficient in order that E ::: L'P .
-1
14.2. Definition . Let o,fl (see Definition 7.1), and let o '
., 1 be the respective i;'lVerse functions (for every "i J2., separately).
Let
-1 r. -1 \\ 1-'t' [ -1\'1:'
Ifc (t,U)=L'P o (t,ul) 'P 1 (t,u),.i
Then the function <rt" inverse to i.f
ftnction between <Po and 1'
14.3 .Theorem. If 'f o ,flE; lare convex, are N_functions), then
I.f-r €- <P l is convex, is &'1 N-function) for every 0 'C" 1 .
Proof . It is easE:{ seen that if \{Io,<f'1&<t' , then \f (f' for 0 1.
for t E:Jl , u 0, 0 r 1.
is called the 't'- intennediate
95
For the remaining part of the proof we may limlt ourselves to the case
of functions independent of the parameter t. Suppose 0' 1 to be con-
vex. Applying the H81der's inequality
l-t" 1:" 1-'r" 't'" tI
a o b o + a 1 b 1 (a;; - + a O -'C")P ) 1/P l b1:<l + b'tq ) l/ q
... -1 -1 1 0 1 J
a i '''i1 0 , with p ::(1-1:) , q =: 'T:" , 0 < 't"< 1, we obtain
f. 1-1:" ( ) T 1-1:" 1:' 1-'t'" 'V
\.a 0 + b 0 ) a 1 + b 1 a 0 a 1 + bob 1
for ai)bi O, O 't" 1. Applying concavity of inverse functions cp l,
1 and the above inequality with a i =: 'P:1lu1), b i = q>:1 l u 2 ), u 1 ,u O,
i = 0,1, we obtain
2 1 ( Ul: ) = 2[ 1 ( U 1 :U 2 )] 1-1:" [ 1 C 1 : U2 }] t'
[tp \ul)+ 'f 1 (U 2 ,]1--r [f \u11 + 1 (U 2 J]'l4 [ 1( >J 1-'r[ 'P 1 ( )r+
1 -'t'
+[1j' 1 (u21J [1f 1 LU 2 I r= } (u 1 ' + f ( ).
-1
Hence f is concave; consequently, is conv,ex. Now, let 0' 1 be
N-functions, Le. ilu)/u O as u4rO, ilul/ufoo as ufoo, i = 0,1.
Then If: 1 lu)/ul'oo as u O and'f:llu)/uIltO as ufoo, i::: 0,1. Hence
tf (uJ/u too as uJ... 0 and lf \ul/uJ... 0 as u 100 , and this proves that
\f U/ /u fo as uJ,- 0 , fU) /u too as u 100, 0 r 1. This proves lf r to be
an N-function.
14.4. Lemma . Let (be complementary to the N-functionl(> (see Defini-
tion 1 1-), and let if-l, «f1-r 1 be the inverse functions to <p , <e*' ' res-
pectively. Then
-1 -1
u \(J l t, U)(\fk) (t, u) 2u :for every u 0, t Sl .
Proof . Evidently, we may limit oursel es to a function independent
ofttle parameter t. Taking in the Young inequality u.v tp(u}+ qYtv? (see
13.6) \f-1 lu )in place of u and *)-llu)in place of v, we obtain the ine-
quali ty
tf-l (u}( )\U) 2u for u O.
Let p (u)be the right derivative of \.flu), then \{tu);:;:S plIO} ds for u 0
(see 13, formula ( I), and since pis nonincreasing, we get \(1(u) p lUI u.
This inequality may be replaced by a stronger one) applying 13.6(d):
\flu)= u P(U)- ftPluJ) < u p(u) for u>O.
From this inequality it follows that Pls) 'Plul/u implies s u, and
consequently, if O w lflu)/u, then
96
p w) p"(*I/u)= sup f s O : p(s) ({1(u)/U J u ,
by Definition 13.3. Hence, applying 13.4, we get
tflU)/u
If*['f(U)/uJ= S r/tw) dw \.fLu).
o
Taking in this inequality tplu}= v, we obtain vl tfl tvi) v for v> O.
-1 -1
Consequently, v 'f Lv)(<f"'} lv) for v O.
14.5 .Lemma . Let If,,/, be two N-functions, and tplt",'t" the respective
complementary functions.
la) If If(au)$\f'lbu) for some a,b>O and all u O, then
t*t au ) "\bU) for all u O.
tb} If tp*(aul 't''\buj for some a,b,?O and all u O, then
"fI(au) <p(bu) for all u O.
. La} follows from the inequalities
((bU)= sup [tbu)(av) - If> (av») sup [tau}(bv) - 'f'lbV)]= 0/* (au) ,
v O v 0
where we applied Theorem 13.6(b). Part (blof the Lemma follows obser-
ving that \f',o/ are complementary to tp 't'; respectively, and applying
Part La) .
14.6. Theorem . Let <fi, i = 0,1, be N-functions with intermediate
function t.p'[ , 0 '(" 1, and let 1tJ-t! be the respective complementary
,... {
functions. Let tf''t'denote the complementary N-function to the 't-inter-
mediate function 4>'(" between and 'f , where 0 't' 1. Then the N-fun-
ctions 'f-c and 't" are equivalent, i. e. If'!:''''' 't" ' and
(1) q>t' ' ) ,v) ,Jt,2V) for every v O.
Proof . Without loss of generali 1;y we JfJ.8:y assume that Ifi'tp{, i = 0,1,
do not depend on the parrorreter t. By 14.4, we have
(2) u \f lu )Llf r\ u) 2u for u,? 0,
'"
where q> is complementary to \f't" From the definitions of 'f and If't' fol-
lows J-
\P-J lU)C" )\U)=(t.p \Ui(W (uU ['f \u) \U)J
and applying 14.4 to each of the factors on the right-hand side of the
last equality we obtain
-1 -1
G) u 41'(' (u)(if't"') (u) 2u for u O.
Hence and from <..2) we obtain
1 -1 -1 /,-1
'1'; lU}{\f ) lU) 2u 2 q> luklff) lU) I
and thus
97
( ) (- ) -1 -1
4 f' (u) $ 2(lft) lU) for u O.
Now, applying the ri ght-hand side part of the ineq uali ty (2 J and the
left-hand side part of the inequality t.3J, we obtain
C'-\U)( Vl(U) u p \U}(ft.)\U),
i.e.
(5) ¥ :,llU).$(ft..)-l (u) for u O.
Inequalities (41 and(5) yield together
1,. j!\-1 ) (. { - ) -1 1-1
(6) "2yf l lu '" <fit" (u) 2 <p;) (u) for u O.
Taking (If;) -l lu )= v in inequalities l6) we get
( r lL\f;tv)J 2v for v O,
i.e.
(7) If't' ( k r((..V) (fJ2v) for v,? O.
However, cpr is complementary to <it and f is complementary to {f't'" when-
ce (7) may be written in the form
(8; (If.rf( ) V) S(lp'ftzV) .
Now, the inequalities (1) are obtained from (8) easily, applying 14.5.
(j J imply in turn t{t- and te't' to be equivalent C see 14.1).
14.7. Theorem . If the functions tpo'lfl E <p satsify the condition l.b. 2 )
(sea 14.1) , then their 't'-intermediate function t and the function lft.
from Lemma 14.6 satisfy (..6 2 ) for every 0 1:'" 1 with constant K» 0
and function h independent oft" .
Proof. From 14.6, inequalities (1), follows that it is enough to
prove (6 2 )forlf't" Let Ifo,lf/l satisfy(t.i with positive constants Ko,Kl
and nonnegative, integrable functions h o ,h 1 , respectively. Let K
may(K o ,K 1 ) , h(tl = max(holt) ,h 1 tt»)for tf.l2 , then there is a set.Ql cJl
of measure zero such that
i (t,2u) K 'Pi \..t,W+ hlt) for u O, tE,Q' l' i '" 0,1,
or eq ui valently ,
llt,u) \f l(t, (U-h(t») for tESl,.Ql u h[t), i = 0,1.
Consequently, l-T T
;1 (t,u):s[i lf l (t,u)] . 'P 1 tt,u)]
[<p 1 (t, (U-hltl»)] :[I.p llt, (U-h(t»)J""= \f 1 (t, lU-h(tJ1)
for u hlt), tE..a......Ql' O 't" 1. But this is equivalent to the inequality
<rt-(t,2U) KIfJt,u)+ hLt) for u O, t(;.Q.....n. 1t O 't:" 1.
14.8. Theorem . If lfo'CP1 CP and <fo l' 0 Ll T2(lt then
q'o l\' 1 <{'1'
98
Proof- It is sufficient to prove that lfo fl a..'1.d 0 't'1 r-2 1 im-
ply !f'-r -5 l ' Le. that
1 (t,u) \fL-(t,KU)+ ii(t) for I.qo 8J.'ld t ..Q....Jll'
_ 1 A.. _
where K> 0, }.{(.JZ,1);:: 0 and h is a nonnegative, integrable function. But
this is equivalent to the condition
lf (t,u)
(*) KfortE-.Q,-ill' u O.
rp ) (t,u+hlt)
By the assumption o l' we have
I.{i \t,u)
11 K for tE 1"'..Q.l' u O
tp {t, u+hltl)
with some constant K>O and a noanegative, intet;rable hunction h, where
p.tlJ2. 1 ) = O. Thus, we 0 bt ain
tp;;' (t,u) [ ;1 tt,u l_Ti'f l (t,u)]'ii
;::
1f lt,U+hLtl) ('f 1 (t,U+hlt)81-"G[f 1 It,u+hCt>)]t'1
[q> 1 Cot, u+h ("t))) 1-t" lO _ 1 1 (t, uil L ,2, [ {D _ 1 1 (t, u) j -r;.-r 1
1 1 < K't'.z.-t'1.
'" [\p; 1 Ct,u+h l t))] ' -1i [1f - ' ,uIT': l(t,U+h(t)) -..:
Hence l"'l follows with K = K l,t-'t"1 and h = h.
14.9. Theorem . Let If'0,fl,%,t1E<P and let 'Po-Yo' <fl.-v"'t"l. Deno-
ting by 'f...resp.y"C" the 'L-intermediate function between 'fo,Cf 1 resp.
't "Y1' 0 't'" 1, there holds CPt""'; "Y'C' .
Proof . By the assumptions, there exist constants Ko,Kl 0, nonne-
gative, integrable functions ho ,h 1 on A , and a set ..a 1 c:.Sl- of measu-
TI':: zero sucb that
ti(t,Kou)- h1lt) lfilt,u) "Yi(t,K 1 U)+ h 1 (t)
for u:fO, tf.sl....jll' i = 0,1. These inequalities are equivalent to the
following ones :
0'\f' 1 (t,U+hi(t))1'f 1(t,u) and l'f \t,U) lf l(t,U+h(t))
for u O, tE..Q....il 1 , i = 0,1. Hence, by 14.2,
o"t';l(t,U+hi(t» .;l(t,u) and l ;l(t,ul:S ll',;l(t,U+h(tJ)
for u....O tC'..(l.....Q O<::"t"L 1. But the last inequalities are equivalent
9 ,... "t' - -
to
't'.Jt ,K o u) - ho (t) 't'(t, u) 't''C (t ,K, u) + h 1 (t)
99
for u;"O, t€.C!..\.J2. 1 , 0 "C 1, Le. to the relation91'\:""" '
14.10. . If 'fo,cple- and t"is the't"-intermediate function bet-
ween 91 0 and tf 1 , 0 f: 1, then
min[\folt,u), <fl(t,utJ <ft'(t,u) max[lfo\t,uj, 'f 1 l t ,W]
for u 0, t€.,Q. .
Proof .From the definition of \f>.;..1 we have
r, -1 ( ) -1 ( ) 11 -1 f, -1 -1 ]
minL<ro \t,v , \f'1 t,v:J tp'l (t,v) maxL<fo (t,v)t 1 (t,v)
for v 0, t E-..Q. . The right-hand one of the above ineq,ualities means
that
-1 (, ) -1 -1 -1
(*") tp'("l...t,v \{10 (t,v) or 'et" (t,vl If'l(t,v) forY O, t€Jl.,
and the left-hand one means that
-1 ( J -1 -1 -1
(ffl 1/'0 t,v 't" (t,V) or \fJ1 (t,v) tpt" (t,v) for v 0, tE'...Q. .
Let us choose an arti trary u» 0 and let us apply 1.*') with 'f.(t, u) in
place of v. Then we obtain that u = lp;l(t, 41.(t,u),S 'f 1 (t, cp'C'(t,u)}
or U:: 'f;.l(t'\VL(t,u}) q> l(t,<P'C'(:t,U), Le. \.f>o(t,uIS!f'(; ,u)or Ifl(t,u)
t,u). This mea:a.s that min(cpo(t,u), £fl (t,u)J <{lr-(t,u). Applying (ff)
with \f t,u) in place of v, we obtain the inequality 't'tt,u) m8X[fo(t,U),
1\>1 It, uJ].
Co roll a;rx . If the functions "'0 '1f1 E: ip are locally integrable, then
their<r-intermediate function i.f'C" is also locally integrable for every
O "t" 1.
14.11. Theorem . Let !fkc for k ::: 1,2,... and let us suppose that
there exists a set 1 c.....Q. t (.Q.1)::: 0, such t hat the sequence (lfk It, u)
is nondecreasing and such that
\0 (t, u)::: lim \D k (t , u) < OJ
TO k-KJO T
for every u 0 and every t ..ct' l' Then o may be extended to the whole
.Q.X R as an element of the class <l? and we have n Y/I k tU Y II for every
+ 0
2:: -measurable, }A--a.e. finite function y defined on...a, Where Il yU k :::
inf{u"70 :S kQy(t)l/u)d,.. 11 for k =: 0,1,2,... if y/fL\fi-.,l[YU k == 00 if
Y ;, L I<.. .A.
. The first part of the theorem being evident, we limit our-
selves to the proof thatUYllk1nYllo supposing .J'l.1 ::: p. By the assumptions,
we have
k (t, ly ltll) 110 l t, lyltJl) as k 1 00 for t e-...!l.
and so
100
trktyj=rlfk(t,IYltll)d t 54' (t,[y(tIOdM.= 0 (J)as ktoo.
o I 10
Since the sequence ( kty» is nondecreasing, we have
t\'k(;) k+l(;) qoL for u>O, k = 1,2,...
Hence, supposing y <: LIVe, we have
UYl/k II Yllk+l II Y/10 for k ::: 1,2,...
Thus, the se<qluence (itYIl k ! is nondecreasing and bounded, whence conver-
gent. Denoting g ::::k /lYl/k ' we have g lIyl/ o' Supposing g</I Y110 and
taking a number v such that g(V<IIYll o ' we have Ily/lk< v for k = 1,2,...,
and so k( ) l fork= 1,2,... Takingktoo, we obtain 0{ ) 1,
whence IIYllo:!; v, a contradiction. Thus, g =lIyl/o' Now, let us suppose
Yi- Lifo, La .UYll o = 00. If Y4 L k" for some k , then Y' L If " for k k and
o 0
so imIIYl\k ==lIy/I o.Suppose y Lifo but y L K for k == 1,2,... Applying the
cond: ionUE-)with u- 1 y in place of y, u>O, we see that kt;J100as ktoo.
Hence for every u O there exists an index ku such that i'kl;»' for
k'9 k u ' Consequently, uS.IIYllk for k ku' This shows thatUYllk-ro.
14.12. Corollary . Let \fk'lf'k <P for k:=: 0,1,2,... and let klt,u)'1
lfolt,u), 't'klt,u/1'lfo It,uJ as kf 00 for all u 0 and tE'-\'l with the excep-
tion of a set ...Q., of t of measu:re f-{..S2. 1 ) = O. Then, denoting by t{Jk,t" t:\'re
't"-intermediate function between I{)k and f k , 0 1:" 1, k = 0,1,2,..., we
have 41k,t' It,u}t O,1:'(t,U) as ktoo for u O, tE-J2.,.lll' Consequently,
IIYll k ,T1'IIY/l o ,'l" as kt oo , O 't' 1 and l:-measurable,t"-a.e. f:iJnite fun-
ct-Lon Y on...o., where IIYh k 't is the Luxemburg norm of Y generated by
the function <fk,'t' ' i.e. ';Yllk,'t" == inf{u) 0 : Stfk,'t (t, IY II) "' lL k == 0,
...!l.
1,2",.
Now, we are going to formulate the interpolation theorem for opera...
tors in spaces L<P and Elp . This cannot be done in the whole class :, we
must limit ourselves to some of its subelasses.
14.13 . DeJJtlb.i tion . By 'YT(.C.Q.) we denote the class of all functions
If&<F of the fOIr"ll n
tp(tc,u)= lfilu)I(.Q,lt),
==1
where 'Pi are convex <f-functions without parameter (see Example 1.9),
Jl==12 1 V oo.uJ2 n ,Qi pairwise disjoint and is the characteristic
function of the set .Q i. By'Yfl 1 (Q) we denote the class of all functions
for which there exists a sequence of functions CPl'iil.(J2J ,k==1,2,...,
101
such that 'fkLt,u) t£f{!;,u) as k 1 00 for all u 0 and tE-...S2.. besides a set
.!1 1 of t of measure zero. Finally, by €1rCz(Jl) we deno'toe the dass of all
convex functions <pc for whcih ,Iwre exists a function € 'rn 1 (.Q.) such
"""
that If',.,; .
Let us remark that superpositions of functions from the class 'Yf((.Jl)
with simple functions are again simple functions. Non trivial examples
of functions from the above classes will be shown at the and of this
Section (14.18 - 14021).
14.14. Definition . Let P be an operator defined on a nonempt y i-
near subspace L of the space M(S2,!", ) with values in M (..Q. , ,\') (see
I 0 0
lij.1). The operator P is called sublinear , if the following conditions
are satisfied :
la) Ip (xl+x ts)1 (px 1 )(sJI+ K!' x.}lsJI for v-a.e. 10 €- ..Q.o' where xl ,x 2 E L,
(b) IPlc:x}lsJI;jclltpx)(sJI for v-a.e. se-...n.o' where xEL and c is an arbit-
rary complex number.
The following Hadamard's three line theorem for subharmonic func-
tions will be need in the following :
14.15. Theorem . Let f be a nonnegative, bounded function on the strip
15 =={z : 0 Re z 1 I of the complex plane such that log f(z} is subhar-
monic in the open strip D ={z : O<Re z < 11 and continuous in 15. If
theoCe are positive constants K .,K 1 such that f(OI-iy} K and f(1+iyl"K 1
o 1-t t 0
for every real y, then f (t+iy) -6: Ko K 1 for every te-[O,l] and any real y.
The auxiliary notions concerning subharmonio functions may ba found
Q og. in T . Rado (1).
In the following, /lxI/if' will mean the Luxemburg norm of x in the gene-
ralized Orlicz space L'f, generated by the functionlp (see e.g. Corol-
larY 14.12).
14.16. lnterpolation theorem . Let 0' 1 be locally integrable N-fun-
ations on .5t.X R belonging to the class rrrr (..Q.) and let 't. ' \11 1 be locally
+ 1 0 ,.
integrable N-functions on };LoX R+ belonging to the cla<;;s rm, lA-o)' Moreo-
ver, let P be a sublinear operator defined on E lfo (5l.,'i. ,r) V EII (.n., 1:,1") with
values in Iv1(..a. ,J:. ,17) such that
o 0
t+} IIPxllu.. K /lX/l for xE:- Elfo(.Q, ,}d
10 0 TO
and
(j-+)
II EX/Iii'" K 1 /1 x1,<& for x € EIf'((Jl ,1 ,f-) .
102
Then:
1) For eve ry simple function x e I:il(Sl.,7:. ,f-) equal to 0 outside a set of
finite measure/l'" and for every O 't"' 1 there holds the inequality
l) l-t" t"
1 U Px Il't"4 Ko K1_xllyt"
Moreover, supposing P to be a linear operator, there exists a li-
near extension P of the operator P to the whole space E ( ,Z, ) pre-
serving the inequality (l}with P instead of P.
3) If, in addition. there holds fo 4lf 1 or ,- 'f 0' then the subli-
near operator P is defined on the whole space ECYT'(.!)., , ) and satisfies
the inequality (1) for every Xf:Ei¥r("Q,'I., ).
. We define the following auxiliary functions
(2) Kz (t, u)::llp llt, u»)1-zl't 1 (t, u : (3) Qz (t ,u)=[C\ft,l (1;, u»)l-zLt:ff) 1 (t, un z
for O Re z$1, where If1 is complellientary to f 1 in the sense of Young,
i = 0,1. Let S($L,z.., ) and SeA , ,\?) be 'the subspaces of all simple fun-
o 0
ctions, equal to 0 ou:tside a set of finite measure, of the spaces M{J2,
!,IL) and IvI[Jl , 'v ) , respectively. :Wet xG S(.Q.,I:.,u ) , yrc SLJl ,1: ,J) ,
r- 0 0 'r' 0 0
y ls) O for 10 e- J2. , /lxI/ ::lIyll :: 1. Thus x and y are of the form
p 0 1ft' q
x(t}::a .L a i 'VA. (t), y(s)=- L b, Y B - (10) for t 1 , se-...Q.o'
i=l 1\ j=l J '\ J
where a 1 are real, b'hO, A.e:l: ,u(A,l<.oo, B_e ,'1(B,)< 00, the
J' {" J 0 J
sets A"A 2 ,...,A p are pairwise disjoint and the sets B"B 2 ,...,B q are
also pairwise disjoint.
First, we slAppose that <Pi t<m (.Q) and ii €qf{(..a.J for i = 0,1. We may
wri te these functions in the form
m n
tf,lt,u)= '(1). h lu)X.a. (j;), '\Ii.ls,u)= L. "\IIi klu)y..Q. lS), i:: 0.',
1 , "\-h 1 k;=l f:, '\ ok
where tD, h ' lV. are if-functions without parameter,..Q =11 1 V ... l.,)J2 m ,
1 , T J ,k
,a. P airwise disjoint ..Q. ",n.. V ...v.Q , . pairwise disjoint.
-1. ' 0 01 on oJ
. Now, we define the functions
gz{t);= Kz(t,cpjt, !XltH)] I i, for xlt)F 0,
h z (10);= Qz [10, 'I'.2c s , y (10)8.
= a.1 I a. t and taking into
J:
gz t)= 0 for xlt)= 0,
Denoting oC i := lai!' u i
,o)= 0, we obtain
g It):: K fi; ,((L-t t ,cU]u. \A It) ,
z z \'t'" .
i=l
account that Kz It, 0) =
h It):: Q rs,,, ls,b.)1\B,tS).
z j=1 zL J J
103
'faking 0 Re z 1, we have
111 p
gZ(t)= L ?:. KhZ[lfhrlDli']UiX'A ..a (tJ,
h=1 l=1 '\ i 1'1
n q
hzl S )= L 2:: IoLkZ['t':'t'lb)]XB (10),
b1 j=l J l'-%h
where \fh'l is 'L-intermediate between lD &Yld f "ft is t'-interme-
loh 1 h' ik't'
diate between k and rl ' and
Khz (U)=[tp will-z [ip LU z , QkzCu)== l('t' k)1 { l-z [Ct kf\ uiJz .
Now, we define the function
fezl == ll(pgJ(s lhz ls\/ dv == S /P(h z (10\ gz (sl)/dv
o -1Lo
for 0 Re z 1. I t is sufficient to show that f is bounded and conti-
nUOus in the strip O Re z 1 and that log fez) is subharmonic in the
open strip 0 <. Re z <. 1, i. e. 211"
1 W
log f( zl 2'!r Slog f(z+ e ) d(1
o
for every z such that O<Re Z( 1 and for every > 0 such that the
ball K(z, ) with centre at z and with radius q is contained in the
strip 0 <.Re z < 1. Then one may apply 14.15. We start with proving
log f(z) to be subharmonic for 0 (Re z < 1. We have
Hence, denoting
;k == QkzL't' 'tlbj)J, klz)::: B}..a !p((;k)(sJ/dll,
J ok
q n
fLz)", 2.'. 2: r.kl Z \.
j=l k=l J
In order to prove that the function log f(z)is subharmonic it. is suffi-
cient to show that 10g klZ)iS a subharmonic function for j '" 1,2,...,Q
and k =: 1,2,... ,n. This follows from the fact that log flz) and 10grjIJ-Z)
are subharmonic functions in the strip 0 <. Re z < 1, if for every harmo-
- l
nic function ii in this strip, the functions e hlz ) flz) and en z) rjk(Z)
are subharmonic in the strip 0 (Re z <. 1. Let h be an arbitrary harmo-
nic function in the strip 0 <.Re z <. 1 and let H be an analytic function
in this striP, 10 uch that Re H LZ) '" ii (z). Let us denote
O:jk "" e H (z) f;k, g ::: [cph't"(o(. 1.
n q
fez) == L 2:. S I P(QkZ[ 't t" (b .}Jgz)ls)/dv .
k= 1 J= lB .(1..0. J
J ok
we obtain
We have
104
'k ill
0; = Qkz['t' t' lbjJ],L
Further, denoting h=1
jk )!- II hi *jk H(z) jk
Ahi (Z) '" Qkz[",, lbjljgz ' Ahi ) '" e It hi (Z) ,
we have
jk '" i J.. Aj ('7) v*jk ill P "" jk
Z hi "- AA 1'I..!1.. ' 0 Z ::: L ):. hi lz) ui"l. A (1..\1.
h=1 i=1 i- h h=1 i=l '\ i h
r;k lz) '" e hlz1 k (,z) = SIp (r: jk ) lS)ldV .
. B j(1,Stok
First, we shall prove that Ip (i Jk}1 is a subharmonic function. Obviou-
. \ jk k . . ,
sly, the funct ons {lhi and)thi are analyt c n the stnp 0 <.. Re z< 1 .
Thus, we have
*jk 1 2rr *"k'e 1 r If 'k
(4) Ahi (z)= 2tr s Xtl (z+ e ) de = lim r L: >.rl (zl),
, 0 r oo 1=1
h IJ e e -1
were zl'" z+'i e '1 = 2Tilr ,1= 1,2,...,r. Indeed, we have
1 2tr jk 'e 1 r It' '$ 2fr 1 r . 'Iv 'a
2T- S Am (16+ e ) de = 2!f" li m L Jk ( e L) '" lim - L \.Z+ e I.).
o r «I) 1=1 r r oo r l ::: 1
Now, we shall prove that the functionl(P jk)@)lis continuous with res-
pect to z for every 10 e...o.. . We have for every z from the strip 0 < Re z
o
< 1, by subaddi tivi ty of P,
J ICp/zi:z} tsll-I o:-jk)(s)j 1 Ip:C{J:z - :jkills)1 =! ( i i/ tlk(z z)-
'k 1 ill P ) .*-'k 'k IP
>. lZ)J u iXA.(1SL) ls)l L. tl (Z+6..Z)-A (Z fUiXA,I'\.o)(S)I-'?o
- h 11=1 :::1 -"11
as AZ-;>O, since P is a positive-homogeneous operator, the functions
iJ.klz)are continuous, the valuesl(P Ail'1.QJ(s)1 are finite and the sums
are also fini te. We have
P hi
t-l gz ui Ai .
and
(5) 1 jk)ls)I 11 (P jk)@)I-I Glt1 t k)JlS)il [p(-;l l k)}sJlo
Further, we have r
HCP jk)Ls)r-I (-; £ k)] lS) 1/ I[p eo;jk - -;2: : kfl (10) I
(6) 1,,:.1 1=1
f il klZ) - -;f A:ik(z011[p(ui A.C\.!lh B(s) 1-;>0
11=1 i=1 1=1
as r......oo, byl4}. By(5) and (61 we have for every sE-...Qo
[(PO:jk)&3)I lim -;£\(pt; )ls)l= ;yr 2sj(pd{l e i9 )lS}lde .
r....oo 1=1 0 '7
105
The last \!'CJ.uality follows from continuity ofl(r jkKs)lwith respect to
z for e':,ery fixed 10 €-...D..o d i t s eq Yalent to
eh(Z)KP jkj( I ...l2 S it eh(z+ e e) / fn/ . ) LS) / dt3
Z 211" IJz+ e 11 '
, hf.3,)o 'k
which m.ea."lS that e I(P KSJl is a subharmonic function of z for every
z ,
10 " .!lo' Hence it follows that log If}? ,k)(,s)liS a subharmonic function of
z for every fixed 10 Ii:...a , in the strip 0< Re z < 1. Hence, we have also
o
for every z with 0< Re z <: 1 and for every such that the ball K(z }
wi th centre at z and with radius is contained in the strip 0< Re z -<. 1,
S lZ):: B J Jtl k ILPr;jk)(S)ldV J:j fl- (2 1 rr 2Sbd':1 eiJ(s)\d\))d -
-"0 J Ok a
1 2tr ( S Ok ) 1 21i .
'" _ 2 J B 1'\ yl\"J - -\Is)jdv de::: _ 2 S r,f k (z+ e e) de ,
IT 0 j"" ok vz+ q e IT 0 J
Le. the furlctions r1l: k (z\ are also subharmonic in our strip for j = 1,2,
J-
...,q and k = 1,2,... ,n. This means that log rjk(Z) is a subharmonic
function in our strip end, by the previous argument, the function
lOB flz) is also subharmonic in the strip 0 < Re z < 1.
l'OW, we are going to prove f to be continuous for 0 'Re z':( 1. It
is sufficient to prove the functions r to be continuous in this str-
Jk
ip. Applying first sublinearity of P and then H31der's inequality, we
have
I Jjk Lz + t:.z}-
Sk(Z) I B S )1 (P :6z)(s)J-ltp ;k)Cs !dV
j".a. Ok
S I [P(K L\z - r; ls)ldV 21/ jp ( ;:L\Z - r; IIIu, II B ,(\$l "'riA-
B ,(\ll k 10 J ok"
J 0
2. K 0 1/ (; AZ - o kl o II X B ,,,-'2. I"*-t
'1r 'k J ok
by hypothesis, shce d';:AZ - 0; t S(;Q.,y.,f4') c:. LIf ,2:,f'-)' But
II Q'jk _ ojkl/ i. ..f: I;>" lz+ A Z) - (Z)IIIU i A.o1L U 0
z+.6z Z % h=l i=l --titrb
as .1z...0, since:;>' are continuous in Ow:' strip end the double sum
i finite. Hence, applying the previous estimate, we obtain continuity
of r. 1 in Ow:' strip.
J ':
We prove now f to be bounded in the strip 15 :::{Z : O Re z l}. It
is 10 uf'ficient to P rove r to be bounded in :I5. We have
jk
106
k\Z}= B.k I(PO;kKs)/dV 2l1p kllllJ II B f'\ II 2K n jklL II'Y An Uu,Jt-
J ok z TO j I)"' k %*' 0 z '% \B j''''-ok "10.
m p 'k
2Xo L ) tZ Ui1A,I\-'l.lIll 'V B .A lilY*' .
h=1 =1 - -h"\fo '\ J .l\ ok 10
'k
Hence it follows that rjklZ) are bounded in , since A are bounded in
15 and the sums are fini te .
Now, we shall show that 1 0 flZ} 2Ko for Re z '" 0 and 2 0 fLzl 2K1
for Re z ::: 1-
1 0 B H I! ,. .
y older 10 nequal ty and by the assumption(+), we have
(7) flz)=S ICPgils)llb.z ls>!dv 2UPgJLUh II (- 2K IIg IL h U .
.JZ... to z 'fio 0 z If. z 1't
Next, taking Be z :s 0, we obtain 0
lOCgZ) '" S tpJt, Ig z It)l) dr- :: S t, If:' (t, <f't"Lt, ix LtlID] df'l- '"
1(1 A ...n.
= llft'(t, Ixlt}l) df '" elft"'lx, '" 1,
and thus I!g "'llx. 1 for Re z "" O. Next, applying Re z :::: 0, we have
-, ( 0 It
h z lS) =(tf>:J s,lf (s,y(s»). Hence we get
'd(hz) = 'tlty)"" 1 and II hzU't'tf"'IIYI := 1.
Applying this and the previous observation in the ineq uali tYL 7) , we ob-
tain fez) 2Ko if Re z '" O.
o
2 Let Re z '" 1, then
Ig z (tll= (<e,frJ- 1 (t, t (t, Ix It)l)), h z l 10 1 ::: ttfr\s, 'f:ts,y ls 1)1
Hence, we obtain f/If/gz}=ff/X) '" 1 and o//hzJ ::: 'f'tly) "" 1. This implies
R gzV,,:1I hz'\; 1. Thus, applying H81de r' 10 inequality and the ass umption( ++),
we obtain for Re z
flz) IIPg Hh 1t 2K J JtJt'= 2K1"xl yn-r:'" 2K, ,
hi 0 1 "' l.f "
W ch proves 2 .
Now, we may apply Hadamard's theorem 14.15 to the function f, obtai-
ning
f "2 K l-Re z r2: e z
(Z) 0 1
for every complex z such that 0 Re z 1. lvloreover, putting z =T in
the definitions of functions g and h , we obtain
z z
g-r{t)= x It) and h't'lJ3) = Y ls) for every 0 't" 1.
Hence and from the last inequality for flZ) , we get
If(")I 2 K -'l xt for every 0 r 1.
Hence, taking into account that II XII '" 1 and applying 14.6, we obtain
(
/lPx!l 5: sU P { S !Q>xl@)!YLS) dv : IIYII 1, y lS)9 0 } 2 Sup1S l()?x)ls)IYlS) dV'
't't' "...Q. 't 1:' ..0.
107
II " II " 1 (', J ., 0 '" 4 Ko l-'t"" K1 'f:" . T hi (
./ 'l't , y s 9' ) s shows that the ineq uali ty 1) holds
for every function xe S(.n,I:,I-<-) withIlXII/f; 1, Taking an arbitrary func-
tion Xli! SC.a.,i:,f/-) , X f.: 0, we obtain
IIPX.'t';llp (II HCf r)llft'I/xIIIft' 4 K -'l"' K; IIx"lfr'
Le. part 1) of the theorem holds under the restriction\fiemeJl),'i'ie-111lJ2,,)
for i ;::: 0,1.
Part 2) of the theorem wi..,ll be proved in a standart way. Let Xt'
E C(.st,t,f') be arbitrary. By 7.6, there exists a sequence of functions
xnc S(jt,"f..,f'-) such that IIX-XJI<jt- _0 as n_co. Thus, (PXn) is a Cauchy se-
quence in the space LlfC'(.,Q ,I: ,,,) ; indeed, we have for arbitrary m,n,
o Q
I/Px m = pxnqljlt: =!lP Gem - xn)IJ't't- 4 K -'t" K lIxm - d 7 0 as m,n_oo.
Let us denote Px = lim Px . Obviously, the operator P is linear. Now,
n-tOO n
we prove that Px is independent of the choice of the sequencelXn)' For,
let (Yn)be another sequence of flli1ctions from S(.!2.' 'f-) such that IIX:-Ynll
-'0 as n_oo. Since (pYn) is again a Cauchy sequence in 1'/1:"(:.0.. 0 , o ,oJ) , so
Px1,PY1,Px2,PY2"" is also a Cauchy sequence in 1'i'1:"(.!l.0,z.0,\1) , and
so it is convergent to an element of L (..a. ,L ,\7 ) . This shows that
o 0
Px for every Xt Sl-'2. ,-L,f"<'}. W.
lim Px = lim Py . In particular,Px ;:::
:.,.p0::; n ...;> 011 n
have also for every XE:E<i''t''(.sl,I:.,ft):
I - 1- -r 1-"- '1::
I pd = lim IIPx IIL.( 4 K K 1 lim'IX 11i.. = 4 K K 1 IIx/lIJ)'
''/''I::' n-oo n ''I:" 0 n,"*oo r!n: 0 I't"
We are going now to prove part 3) of the theorem. Let X E E<Pc- (.a ,£ ,f-)
and let e. g. fo f l' By 14,.8, we have then fo tf 1 and so, by
8 . 5 l a) ,
EI.(I4 (...Q.,"i. ,r) c: EIft-(12 ,I. , ) C E'PD(..n,or. ,r)
and
iIXI Kollxlllf't' K1 II xa'f, with some Ko,Kl> 0
for functions x belonging to the respective spaces. By 7.6, there exi-
sts a sequence of functions x n S(.sl;r ,,....) such that Nx-xJ C1- 0 as n...... 00 .
Hence liP (x-x }/lu,-'O as n_oo . 'rhus, the sequence of functions Ip lX-x n ) I
n TO
converges to zero in measure v on every set of finite measure y in..Qo
(see e.g. the proof of 7.7, part lV. Hence there exists a subsequence
(xnk) of the se<iuence(xd such that)Plx-xnJi (s)/.....O as k_oov.. a . e . in...a. o '
Since the operator P is sublinear, we conclude thatlQ?XniJ(S)I'"*'I(Px) )las
k_oo v-a.e. in.!1 o . From the continuity of 'tJ.-s,U) with respect to u
we obtain
108
t I (PXnlc)(S) I 1 [ I (Pxj(S)/ ]
lim s, - 'ti s
k_ro 4 K -'("" K llxnklllfr - t:' ' 4 K -T K llxlllfr
for \1-a.e. sc..n.o' Applying Fatou lemma and part V of the theorem, we
obtain
S t llpx)(s) I r t J lPxnk!(s)[
ttt' S, 1-'t" 1;' d" lim rl1 s, 1- 't" 'r d \7 < 1 ft
.0,0 4 Ko K 1 fix II k_co () 't' 4 K K 1 IlxnkU ....
. 1_'£':'t".....0 0
Hence we get npxh. 4 K K 1 "xli .
I/'c- 0 Ifr
Finally, let us remark that if IfIO' l{imrl(.a.} and %''rlei)Ttl ( ),
we obtain the proof of our theorem applying 14.11, 14.12 and 14.16 in
case of functions %k''fl kf;11l( , Yok'i'lkf (4) satisfying the condition
l{iiklt,u)t ilt,u) and tilc(s,uJtl-f\ls,u) as ktro, i = 0,1, defined accor-
ding to the Definition 14.13.
14.17 . Remarks . (8) Theorem 14.16 holds also for Cf 0 ,If ,e-'tfl 2 L ..12.} and
'l'0' 1 2l ), however, the constant 4 Kl-rKJ in inequality 14.16(1)
o 1-t: 1;
has to be replaced then by the constant 4 K Ko K 1 ' where K is an ab-
solute positive constant.
lb) If the functions fo' l ' o'o/l in 14.16 and in 14.17(a) satisfy the
con ition (Ll 2 ) (see 14.1), then E%..(t1.,r'f) = Llfo.{Sl.,Z,t') and E'+'«>t...sz.o'Z.o'v)
:= L ...Q. ,2: ,v) for a = 0;t",1, and consequently, 14.16 becomes an in-
o 0
terpolation theorem for operators in generalized Orlicz spaces.
At the end of this Section we shall provide some examples of fun-
ctions belonging to classes mCJ2.} , 1Tl. 1 (,52,) and 1Jn2l.Q.) .
14. 18. Examples . (a) Let =[a,b], -oo<.a<.b<ro, and let f'" be the
Lebesgue measure defined on the (t'-algebra:I' of all Lebesgue measurable
subsets of.A . Let cp(t, u) be a continuous and nonde cre asing (or non-
inareasing ) function of t .Sl for every u 0 (however, the type of mo-
notoni ci ty of 1ft., u) has to be for every u 0 the same), 4> . We
shall show that l{IEttrt l t..!2.). Indeed, let us denote.5l' k =[a+ ;k Lb-a),
,
a+ b-a . Supposing 'Ptt, u) to be nondecreasing in (a, bJ for every u 0
we de fine
'fk(t'U)=\f + i (b-a), u) if tf'J(ik' i '" 1,2,...,2 k ,
and Ifklb,u)::::tp(b,u). Supposing ,u) to be nonincreasing in ,b1for
every up 0, we define
tpklt,u)=tf(a + k(b-a), u) if tf...Q.ik' i:::: 1,2,...,2 k
109
and <fIk(b,u)=\f(b,u). Obviously, we have lfkE' )' the sequence (tc (t,u»)
is nondecreasing and tpktt,u)f (t,u)as k1' 00 for every teJ2 , u O. Thus,
tpe.'Wr, (.A) .
(b) Let us remark that the above example la)may be modified to the case
whereQ. is a finite union of pairwise disjoint, bounded, open or clo-
sed or half-open intervals .(1.1, ,...,.Q..n on the real line, Z and are
as above. Let If'tt, u) be a continuous and monotone function 0:]:' t on
every.SL. i , i = ,,2,...,n, e :, here, the direction of monotonicity of'f
does not need to be the same on every .Q.i' Then again G (..a). Namely,
the previous argument shows that there exist functions <fik&'rn(...Q.i)'
i = 1,2,...,n, k", 1,2,..., such that \fiklt,u (t,u)for every tE ..sli'
u O. Taking lpklt,u)= 1 iklt'U) i (t)for every t€.Jl and u O, we
obtain kE-4Jt(..Q.) and \fklt,t:j}"'ft,u) as k1\oofor all te-..Q and u O. Thus,
\f' lfI(, (..2.) .
(0) Let -oo<c<d<oo ,..a. 1 =(-oo,c] 0r -'1 1 =(-OO,c»),-'ln =@.,ro)(or "'-n ==
(d,oo)), and let ..a.2' ""' _' be a collection of pairwise disjoint
intervals, disjoint both with..!ll and..A n . Let <.pet, u) be defined forte
Q=.Q1v..s22V ...vJl n and u O, continuous and nondecreasing on.i'll' con-
tinuous and nonde creasing (or nonincreasing) on each .!l.2' 'l3' . . . , ..Q.n-l '
continuous and nonincreasing on ":>Ln' ljIe <f . Moreover, let us suppose
that there exist 'f-functions 1 and o/rI. without parameter such that
lim (t, u)== if,lu) and lim \f ' u)= in lu}
t -00 t oo
for every u O. Then lpc rr1l 1 (...Q.) . Indeed, let us choose a sequence
A k. k
jk' 00 such that jk> d and -jk < c , and let us take Ao == (-Jk,C), n+'=
(d,jk) for k == 1,2,... Now, we may apply Bxample {P)to the set..n: =
01 .Q., v A k vA k in place of -'l. , which leads to a sequence of functio-
l==' l 0 n+l
ns k (!;, u) of the form
I:fklt,u)== 1fk(!;,u) .Q.Y t)+ Cf 1 (u} ,-iJ)+ <Pnlu)Xy OC1?)
such that fk tt, u}'t-<f(t, u) as k1bo for t Eo..n., u O. Since tpk dlYfU....!1}, we
ge t tf tYrt 1 ("Q,1.
14.19. Example . Let A= R+ == [c,oo) with c eR and let 'If; R+ ....R+ be
a \f-function without parameter (see Example 1.9) and let Pl tt P2Lt) be
continuous functions on.1L such that 1 Pl It) P 2 lt)< 00 for tE-J'l . J\lore-
over, let p (t)be nonincreasing and let P (t)be nondecreasing.Let us
1 2
de fine
110
) l'l'Clljf1 (t) for tlu, 1
lt,u)", t )
[ »)P2l for "{Ilu),;;>1.
Then lfG-<t and moreover, If is a continuous and nondecreasing function
of t for every u O. Thus, cp satisfies the assumptions from Example
14 .19laJ in every compact intervel [a, b)CR+ and so tpG-M 1 ([a, b]for 0 a
<b< 00 .
14.20o Remarks .taJLet tpECYlLl...Q,) , Le. G is of the for.m tptt,u)=
n
1CPi Lu) It) with CPi - convex\f-functions wit hout parameter, ..Q.=..Q1 V
... v ,.Q.i pairwise dinjoint. 'l'hen tf is equivalent to a CP-function""
without parameter, i.e. \fJ'""'I" , if and only if, \fi""'""¥ on i for
i = 1,2,...,n. Indeed, it is obvious that lfi""l\jI onJ2 i for i = 1,2,...
,n implies 'f'V"f . Conversely, let us suppose that 'f1V'f ' Le. there
exist positive constants K 1 ,K 2 and nonnegative functions h 1 ,h 2 :..Q..."'-'J> R+
integrable on..n. such that besides a set ...!l1 c:...S'l of measure t«-.sL 1 ):s 0
there hold the inequalities
"f'lK 2 u) - h 2 Lt J 1ft!; , \).) +(K 1 u)+ h 1 It)
for all u O and t E: ..n."JL 1 . We may assum.e without loss of generality
that ...a. 1 = fJ 0 'J: aking t (:0 Ai' we have for every u 0
i' (K 2 U) - h2lt) <.pi It, u).s: 'f(K 1 u)+ h1lt).
rlence, because tfi is independent of t, we get for every u 0 and t .J2i
l'lK 2 U) - inf h2lt) Y'i lu) 't(Kl t9 + inf h 1 (j;) .
t ..Q.i t" i
:Let us denote
",", = inf h1lt) , d, :: inf lt) for i '" 1,2,...,n.
t t.d'l..l..
11' rtl,j},.:J = 00, then c i :: a = 0, sinoa b,1 and a.:re. in"egrable i n .J2. i .
Hence we have
'\f lK2 u)- <R i lu) 't'lK1 u)
for every u O, i.e. 'tv'fi on..!li' Let !"tl-o"i)<oo 0 Obviously, c i < 00
and d i < 00 . Thus, there exist positive numbers u 1 ,u 2 such that c i :f
I.\'lK1 u 1 ) and d i 'Y lK 2 u 2 ). Hence we have for every u max l , u 2 )
'Pi lu) 1.fi{cl uJ+ c i 2 "fl K l u)
and
\.Pi ( u1 1 "rl K 2 u)- d i ),.. i' (K 2 U) - t't'l K 2 u 2 l1' t t(K 2 u)'?I.f(t K 2 U).
'l'hus, we obtain
111
'fa K2u) Lfi lu) 21{'(K 1 u)
for every u1' m.a.x (u 1 ' u 2 ). This means that 'f"""'1.f i on ...Cl. i .
lb) If -'L)<OO and 4'&&J'Tr(..Q.} is defined as in@), where 'fi is not equi-
valen t to % for i f: j , then the generalized Orlicz space LCf( ,z.'f'v)
is not of symmetric type, i.e. there exists no norm 1\ \I on LIt(JZ.)r,f1
eq uivalent to the norm 111' , under which Llf(..Q. ;:i:..,N is a symmetric space
(rearrangement invariant space).
14.21 .Examples . Ca) Let..Q.= [a,bl, 'rand /A.- as in Example 14.1B(a),
and, let p : -'2.....,> [] ,OO)be a continuous and nondecreasing function onll .
Let lft be a If-function without parameter and let
{*) If (!; , u) = [lfC;t)jP (t 1
for u 0, t £:...0.. . Obvipusly, we have '1G . We define a new function
1€ by the formula
J tl\.t) if \f(:l) 1
lD 1 tt, u)= (,
1 ['ftl)]p Lt) if \f(u} 1 0
By EXQillple 14.19 with Pl(t)= 1, P2ltJ"" pLt), we have <flectr[1C.s2.). More-
over, there hold the inequalities
,0 1 ' ' +
for every u:f 0 and t GSl. . Since ftt-St) < 00 , this implies <P 1 "" Cf
Thus, <VE: qn 2C.A.) .
lbl Under the same assumptions on p and "'I' as in (a) let us take
(t, u)= 'Y(p tt )u) .
Then, by Example 14.18(a), we have 4',l)1t1(..Q..}; the same holds if we sup-
pose the function p to be nontncreasing in place of nondecreasing in..Q..
CriAP.rER III
COUNTABLY hiODutARED SF ACES
15. Count ably modulared spaces Xt: and X o
15.1 Q Defini tion . Let a sequence of pseudomoduJ.ars ( n) f n = 1,2,
..., be defined in a real or complex vector space X , and let the con-
di tion n(). x) = 0 for all ).. > 0 and n = 1,2,... imply x = O. Let
00 1 n(X.)
lX)= L 1 + f. <;x: ) , ot x ) = IO n up n(X};
n=l n
here, n lX)/ (1 + e x »= 1 in oase when n (X) "" 00. Then
X ={x,"X : ( x)..,.O as ...,(}j. and X o::&{x x: oL)..x)..,.o as A (}j.
are cailed the countably modulared space and the uniformly countabl
modulared space generated by the sequence ( n)'
It is evident that and fo are semimodulars in X. If n are s-con-
vex, 0 <s 1, then o is also s-convex.
15.2. Theorem . (ap.Jet X i ={XGX: qi(>.x)->o as ,.\ Orj, then
00
X ::; (\ X i .
(b) Let x X. There holds x e ' Il and only if, i l >. x)-;.O as ;.. -9 (}j.
for ever! i, separately.
lO) :Wet x f- X. 'fhere holds X€ o ' if and only if, q i l>- x)-:j)0 as :\...., (}j.
uniformly with respect to i.
Cd) Let xn€,X. There hOlds\Xnt -+O as n oo, if and only if, fiP.xJ-'O
as n -'too for every >. "7 0 and every i, separately.
(e) Let x n (; X. There holds IX n l ,: 0 as n oo, if and only if, f i[A xJ- 0
as n oo uniformly with respect to i for every ).. '::0 O.
(f] There holds X ;- Xi?' this embedding being continuous both with res-
pe ct to modular conve-rgence and norm convergence.
Q (?)Let x€, l' then U' xJ O as ),4 (}j., whence i(}. x)-? as
-",(}j. for every i, and we obtain xG:i lXft. Conversely, let xGi 1 X ,
then i (). x)-'? 0 as );....;.(}j. for every i. Let 0.( f. < 2 be given and let io
be so large that , 2- i < _ 2 1 t. . Let ). > 0 be such that i LA x)<V l2- )
= o 0
for 0<>-<" and i = 1,2,...,i . 'fhen we obtain
o 0
113
io 1 iA x)
((;x) 2 i :1 + ,&\x) + E < E + E ::: E
=1
for 0<'><>'0' which proves that XfX . Lb)is another formulat'ion of (a},
and tc) is obvious. ld) and te) follow from 1.6 applied to semmodulars <f
and o' respectively, and (f) follows from the preceding part of our
theorem.
In the following we shall deal in this Chapter with a more detai-
led analysis of connections between both types of countably modulared
spaces. In the general case, we have oc. . We shall ask the question
whether both countably modular spaces may be equal and under which con-
di tions this may happen. This analysis may be performed only on special
examples of pseudomodulars. It will be done below in Sections 16, 1V
and 18 in tl e cases: spaces of infinitely different able functions
whose all derivatives belong to an Orlicz space, the Orlicz space 1.9.1
and the space 1.9.II with a sequence l<fn) of tp-functions.
1 o Spaces of infinitely differentiable functions
16.1. Notation . Let X be the space of all real-valued (or complex-
al ) R n ,
v ued functions x over the n-dimensional real euclidean space
possessing in R n continuous partial derivatives of all orders
'i}tc(J x
:rfvx oLt ' where 0(= (ot l ""' ), 10\1= 0(. 1 +.. .+d. .
ut ""' t OlJl n n
1... v n f(;
Let Ij7 be a convex N-function (without parameter), and 'P - its comple-
mentary in the sense of Young (see Definitions 13.1 and 13.4). Let us
suppose that If is differentiable at every point. Let
oe (x)= {n ClDO\..x: ltJl)dt
in the following, writing integrals over the whole R n , we shall omit
the symbol R n under the sign of integral.
In order to obtain the modular , defined in 15.1, we may arrange
the mul iindipes 0<.. in a sequence (0«(;i1) arbitrarily and take lX} =
; 1 2- lx) (1 + (j.(<.t!) -1 we shall do this in such a manner that
always d,W::: (0, .. .,0) and that i < j implies I o\.li l (jJI. The se cond of
the modulars in 15.1 is equal to folx)= s F tj..Lx). The respective
countably modulared spaces X and X \1 will be denoted DIf (Rr!j = Dtp =
114
0 1 , n , 0
and DC\' \.R :: DIf = x'o .
16.2. Lemma . Let be a difLerentiable -function (without parMoote
and let be its complementary in the sense of Young. If f is a diffe-
rentiable, real-valued function in R 1 , then
d (lflt)I) a d fltllr cp(2lf ltJI) for t €R 1 .
. Let us extend r.p to the whole of R 1 by the equality 1.jI(-u)=
lfLuJfor u70. Then, applying the Young insqualay uv If(u)+ -ft.'? for
u. v 'l 0 ae 13.6), we get
d Llf (tIP= d (flt»)= 4J '(f ltQf'(t) (\f' L IJ+ (( '(jf ltJj».
Now, it is sufficient to show that t((\f'(li'lt}I)) (2If(t)O, i.e. that
f(tf"M) (2U) for u O. However, we have lplt9=S p(t)dt' and tp*0.)=
o d6"" for u,>o, where p is the derivative of if and p'Yo1= sup{t':
fCC-l ()S ee 13.3, 13.4). Hence
f( 1 (u>)=\(fp (u))= Pf\U) P1G")du p (u) PlP lul)= u P l u) 2t pll') dt'
o u
2u
S p(T)dt" = lfl 2u ).
o
Now, let us adopt the following notation. Let A be the set of all
multiindices d-=l9l 1 ,... ,dI. ) such that r:t:. assU11le only the values ° and
n
1. The ineClualitylt slfor t =(t 1 ,...,t n ), 10 ={sl'....'sn)' will mean
that It.I'ff Is,l for i = 1,2,...,n. If r>O is given, we shall writs
J k =ft k : sk tk sk+ r if sk"?-O, sk-r tks..sk if sk< OJ,
C(s,r)= J 1 XJ 2 X ...XJ n .
It is obvious that C CS,r)c::.{t : Itl lsl for every r»O.
16.3. Theorem . Let.Q. be an open set in R n and let 1i be the closure
of . Let us suppose that there exists an r;;:>O such that for every
s<eli at least one of the S8tS Cls,r)is contained in12.. If the f\)llc-
tLon x is defined and possesses in.a.. continuous parUal derivatives
d...
D x for d...c A, then n-trx.l
Of-) 1f()x:(sJl) 2:: (1 + ) S (2n-l«t/D lt)l)it
_ eA JL
for every 10 C ..S1.. .
Proof .Let us denote
IkLx] = $ ... S \fClx(sl....,sk_l't k ...'tdl)1t k ...dt n ,
J k J n
then
1 1 LX]= j If(\K(yiVdt .
C (10, r)
Let
115
Ak ={ A : o(':::(9l.1""''\_1'O,...,O)}, k::: 1,2,...,n+l.
We prove by induction that k-l-1e<1
(1:*J IklX) + ) 1 1 L2k-l-lttlJt<.x], k::: 1,2,...,n.
k
Clea Ly, this inequality holds for k::: 1. Let us suppose it is true
for some k, where 1 k n. Then, by the mean value theorem, there is a
\9 k ::: f k lSl"" ,sk_l ,t k + 1 ,... ,tn)tJ k such that
S 'P(lx(sl""'Sk_l,t k ,...,\ll)dt k ::: (lx(sl,...,sk_l' 9 k ,
Jk )
t k + 1 ' . . . , \}\ ·
Hence
I k + 1 Lx]:::
J k + 1
+ .
J k + 1
;{ (lX(Sl"" ,Sk_l ,sk,t k + 1 ,... ,t }-
n
tp (j X(sl ' . . . , sk-l ' k' t k + l' . . . , tJI)} dt k + 1 . . .dt n +
} Clx(sl"" ,sk_l ,tY k ,t k + 1 ,... ,t n )i)dt k + 1 .. .dt n :::
n
::: ... S lk .:5t-<f(jx(s1,...,sk_l,tk,tk+l,...,tJl)dtkdtk.f.1.dtn
J k+ 1 J n ek k
+ ) } ] (jx(sl,...,sk_l,tk,tk+l,...,tdDdtkdtk+l...dtn .
k+ 1 n k
By lJemma 16.2 and by monotoni ty of tp , we thus obtain
I k + 1 [x1 S (J u x(sl,...,sk_l,tk,...,t )dtk...dtn +
J k I n k
+
j
J k
1 (2Ix(Sl,...,Sk_l,tk,...,tn)l)dtk...dtn +
n
+ 1 S (lx(sl,...,sk_l,tk,...,\)j)dtk...dtn
r J J
k n
Ik [ : 1 + 0 + ) I k [2x].
l\,alcing use of the induction hypothesis, we apply (if*-> to the right-hand
side of the a.bove ineq uali ty, obtainine; k ioil
(, l ) k-l-p;q Lk-l-I( 'Jf-f<)x 1 .£: [. 1 ) - [ 2k- '\ J o
I k+ 1 ) d.4A, V + r I 1 L 2 out + ot. \..1 + r I 1
K k eA/:, A . A such that
Since to every O\f:A there corresponds exactly one !- k+l
k
116
H)I"4 I+l and D c:t:: ::II Dr$x, and exactly one It'€, A. such that IA/I =1£(1 and
d AI k ' -"k:+ 1 't-
D-x = D\'x, so the above inequality gives
I [x] < L ( 1 + 1 ) k-I(?111 ( 2k-I@IB J
k+l .... eAk+l r 1 .
This proves the equality (k*) for k = 1,2,... ,n. Finally, from the in-
eq uali ty
If>Oxt.sj)= ,: "'O n (lx(sl,...,sn_l,tmdtn + j Dx(s1,...,sn_l,trl dtn
' t n
l follows that
aXlS)n In[ tn J + (1 + ) In[2xJ.
Applying ( , this gives the required ineq uali ty C*) .
The following- theorem is an immediate consequence of 16.3 :
16.4. Theorem . Let the function x be continuous and possess conti-
nuous derivatives D otlf orders d.€ A in R n . 'rhen for every s€R n
there holds the inequality
(lX(?)\) A S (2n-l«l}l:fx It)!}1t .
IS/ ttl n " n
In the following, up to the end of 16, we write LI./(R;= Li@. ,2m'rry,
where ill is the Lebesgue measure and is the 6"'-algebra of Lebesgue
m
measurable subsets of R n ; is a convex N-function independent of the
parameter. The norm /11/ generated by the modular 9bcl= J (\)(lx(t )it
Rn T
will be denoted by II 'l .
16.5. Corolla . Let x be a continuous function n R n with continu-
ous derivatives D of orders d..€:A, and let rJX-x L<fUtn)for ole A. Then
the function x is bounded in R n and x It)-+O as l t l-+co .
!!:QQ!.By the assumption, there is a ),. '>' 0 such that
K =Sq>( I:r:f-xtt dt ..( 00 for o{E- A .
From 15.4 it fmllows that
& -I1+ I} Ix Ls)\) S cp(>' iDQ,x (j;)I)lt 0( A Kot.. < <D.
IS' 1
Since lffu.)too as utoo , this implies x to be bounded in R n . Moreover,
for an arbitrary >O there is an r>O such that S ()ID ttJVdt< :n
l r
for <f...f/::A. Hence we have for\sl r
l2-n+Ic:'tI'\lx )D A S [>dDc(xlt)'f1t <1. A :n f,
lsl 1
Since tplu) = 0 if and only if u = 0 and f is continuous at 0, we may
concl ude that x LS) -;>CO as tsl-'Pco .
16.6. Corollary . Let xi be continuous and have continUDus deriva-
117
tives J1Xx in R n for G(E- A and i '"
i '" 1,2,... andUD xiU o as i......oo
. V n
are urnformly bounded in R and
pect to i.
1,2,... lVloreover, let D .cLi{(RnJfor
for all cf eA. Then the functions x,
Xi (.t}-+O as itl-oo uniformly with res-
This Corollary is proved in a similar wa.y as 16.5, applying 16.4.
16.7. Remark . It is obvious that DyC.DIf. From Corollary 16.5 it
follows tha.t if X€:D" then the function x and its derivatives D<1\.x
of arbitrary orders are bounded in R n , where the constants 1\1 :: sup
I 0 t Rn
.D x (t)/ < 00 de pend on 0(.. We shall show now that if x €' DIf' then
sup Mot. < 00 .
(j.. 0
16.B. Lemma . Ii' x (,'DCf' then there exists a constant M>O such that
lDQ(x It)( M for all ot and all teRn.
Proof . Let xeD . By 15.2lc), there exists a A> 0 such that
S")'IJf.Xlt )dt 1 for all<t. Denotingd-.+ 1+ 1'''.' + n) for '"
(ci 1 ,...,ot. n ),(.1=Q;1",',fn) and applying 16.4 to 2- n DO(x in place of
x we obtain
(2-n)'tDc(xlS))) 11.2A Jq(;z.-I :rf+ xlt\l)dt A (AIDo:.+f> xLt)!}it
fiE ttJ lS/ 'Y (>(;;
irA 1 = 2 n
.Let K = supfu >0 : 'fQ<.) 2nJ , then 2- n ). \Iflxls)\ K, and we conclude that
[:rfxlS)! 2 ..\-1 = Ii! for allot-and 10 € R n .
o . n
16.9. Theorem . Functions xeD.p are expansible in power ser:Les in R .
Each of the functions X€D is either equal to zero or the set of points
at which x tt) /: 0 is dense in R n .
Proof . .Let x eD . From 16.8 we conclude that x 8..Yld all derivatives
D«X are uniformly bounded in fin :, hence x is expansible in a power ser-
n
ies in the neighbourhood of every point to €' R . Hence, if x \.t) == 0 in a
. n I n
ball n R , then x u;J = 0 for eveT'J t E R .
o
16.1O. Corollary . DlfF Dtf for every tf .
Now, we are going to find some necessary and sufficient conditione
in order that a function x D belong to Df' limiting ourselves to the
case R 1 . First, we shall recall some au.x:i.liary notions and tffi oreEls,
which may be found by the reader e. g. in the monograph by A.Zygr.uund [2] .
16.11 . Defini tion . A..YJ. entire function F of a corilplex variable is
called of exponential , if there exist numbers a> 0 and K>O such
118
tha t IF f z)1 Ke a1z ! for
\.. sufficiently large Izj . The greatest lower bound
of such constants a> 0 is called the of the function F.
:Let us remark that an entire function F is of type ('J''7 0, if for
every '7 0 there exist numbers K)o 0 and R;> 0 such that IF (z))(K e(q-+ Jjzl
for every IZI R. For example, algebraic polynomials are of type () and
trigonometric polynomials of degree n are of type n.
16. 12. Paley-Wiener Theorem Ql..Zygmund[?], Chapter XVI, Tho702). Let
Flt)be a function defin:d in (-00,00). There exists a fUl1.ction f L2l-f,5")
for which F (t) = l ls] e st ds for -m'- t < m , if and only if, the follo-
wing two conditions are satisfied:
1 0 F e- L 2 (R 1 ) ,
2 0 F may be extended on the whole complex plane to an entire function
of type () .
16.13. Theorem ( Bernstein Inequality , A.Zygmund[z), Chapter XVI, Th.
7.32). Let the entire function F be of type for some 6",. 0 and let
F be bounded on the real line. If <f is a convex t-function without po.ra-
meter, then
00 CD
f f(<flIF'(t)l)it S T(lFlt)I}it .
-CD -CD
16.14. Theorem . Let XE:-D (?l). There holds x€-D (R1J , if and only if,
function f may be extended on the complex place to an entire func-
the
Uon of type 1.
. Let us suppose that xG:I{p and x may be extended to an entire
function of type 1. By 16.7, the derivatives xlkJare bOli.1'lded on the
real axis for every k = 0,1,2,..., separately. Hence, by 16.13, we have
t IpU..lx' (j;)/)it L q>(.:\ Ix It)l):it for every A 7 O. If we prove that x'
may be also extended to an entire function of type 1, then this ine-
quality may be continuad,.thus obtaining lO) )hf(:I)Qx);'fc:::)(AX) ...,
where o.) (j\ = t <f( >-Ix ( I(i)[)dt , for every >-.> O. Since x f', 100
)l>'x o as f",...... 0+ and consequently, we may conclude that f(j,)(j\x) 0
as ;\ """$) 0+ uniformly with respect to i, which yields XfD , by 15.2 (c).
Thus, it remains to show that x' may be extended to an entire function
of type 1. Let
{ 4€ctzl- x(O»
y (z)=
x' (p) for z =
for z ;. 0
o ,
119
then y is an entire function of type 1 and \y tt)l 2IVIltl- 1 for t real,
where M::: sup Ixlt)). Hence y.e;.L 2 (R 1 ). By 16.12, there exists a func-
t. - <OO) P 1 ist
on y 1 6- L"'ij.-1 ,"0, such that ;)' Lt) ::: J -1 Y 1 t 10) e ds for -oo( t < co 0 Hence
xlt)". x(O)+ tS 1 Ylls)eist dt for -a:(t < 00 .
Differentiating this equality, we obiain for all real t
pl ' t pl .
x'lt)", Ylts)e f! ds + it J sYlls)e st ds ::: ylt)+ itY2lt.»)
-1 -1
) pl ist 2((
where Y2(:t =!lsYlls)e ds, ;)r2"'L L[-l,1]). Extending the last equality
to the whole complex plane we obtain x' (z):: ylz) + iz Y2lz)for all com-
p;ex z. Eut y is an integer fl ction of type 1. Since tYlltlbelongs to
L (l-l ,1)) so, by 16.12, Y2 may be extended to an entire function of ty-
pe 1 on the complex plane. Hence x' may be extended to an entire func-
tion of type 1 on the complex plane.
o
As regards the problem of completeness of spaces Dcp and Dip' there
holds the following
16.15. Theorem . Spaces ¥n) and D&(Rn)are complete with regard to
norms /I ' and II I o' respectively.
Proof . Let (xi) be a Cauchy sequence in D n). Then (p J is a Cauchy
sequence in L\flRn)with respect to the norm j/ "\f=III1 for every 0<. . App-
lying 16.4 we deduce, similar:Ly as in the proof of 16.8, that for every
mul tiindex I1\. the re holds the :meq uali ty
\f(2- n A /1)'\:i (10)- D , ls)1) A Aflp(.AtD x, (!;J- DrA.+t> x, d;J])it.
gC .) ... n J
Thus, (!> xi Lsjis a Cauchy seqwmce, uniformly in R . Hence, the sequence
(Dd... xi (.») is uniformly convergent to a function xO«(.) belonging again
to X, and x.... ls) '" D"'x ls), wheI'S! x ls):: X (, ) (10). On the other hand,
V\. ,0, 0 . . ,0
from completeness of Llf(:a. n ) (see 7.7) it foll ws that\p i) is conver-
gent in L\.f(R n ) to a function f.L<f(Rn). Thus, xo(.= xIX. '" D""x, which pro-
o
ves the completeness of . In order to prove completeness of D ' let
us take a Cauchy sequence LXi) in D9' It is also a Cauchy sequence in Dty
and so, by completeness of , there is a function XE-D f such1J;lat xi-;>x
in Df' By the assumption, for every 7 ° and A;>O there exists an in-
dex io such that S fC >-1 DolX i (t) - D"'-X j (j;JI)at .( E, for i, j io and allot,
wi th i independent of OC . Applying Fatou inequality, we obtain
o
S d.
(>'IDOC:xi tt)- D xlt)l)db t. for i io' i.e. O\.C>,(xi-x)) tfer i io
120
and every . Consequently, xi-x in Df
it is evident that x e Dr
17. Spaces of cp-integrable functions with supremum modulars
17.1. We shall deal here with sequences of modulars defined as in
xample 1.9.11. Let L be a tJ-algebra of subsets of a nonempty setA
and let If'\t')'6-! be a nonempty fa.'Ilily of nonnegative, complete measures
/At- on 2. not vanishing identically, where r is a set of indices. Let
X be the space of extended real-valued -measurable functions on..a. ,
a.e. finite for every 't"12[ , with equality f"t--a.e. for all'tef .
Finally, let l n) l be a sequence of -functions independent of the
parameter. 'i/e consider in X a sequence of l:lodulars
10 PC);;;: :su S f (Ix (t)/) dt-<-t" .
\: n 'CE-l.. .9- n
In case when all measures fLt" are identically equal to a measure r"" tin
are lllodulars defining Orlicz spaces L'f for n = 1,2,... We shall inve-
stigate when the count ably modulared space X and the uniformly counta-
bly modulared space X, defiaed in 15.1 are identical. Since always
X t> <:::: X , this is reduced to the problem, under which conditions on (ftt')
and there holds the converse inclusion X"jC .o' First, we shall in-
troduce some auxiliary notions .concerning the family of measures ( ) .
17.2, DefinJ,tion . The family of measures )'t"6-t will be called
la) uniforJlll r bounded , if sup ft-r(S/.} < co ,
C"f:-t'
\9) e'.'uisiJlittable , if there e:x;ists a number '1.. 0 such that for
1
eveT'J sequ8::1Ce of numbers Ek to for which k "1 and +/ e.k 2 for
k ;;;: 1,2,.., there exist consta..YJ.ts ill, m and a sequence of pairwise liis-
joiYJ.t sets Ak:E:- , k;;;: 1,2,.." suchthat M m70 end
m k s. 't" klAkl IiI k for k;;;: 1,2,...
Obviously, if )t'I2'C consists of one measure only and this
measure is ato;nless, then ( C\'G-L is el;lu.isplittable.
Vie shall now illustrate the notion of equ.ispli ttability by some
nontrivial exa.mples. This will be done in two cases: a fa.mily of mea-
sures (r<d defined b;)r means of Lebesgue integra..ls with a kernel k It,r) ,
and a cauntable f8111ily of pUJ.'81y atomic measures <f«-n1 defined by means
of a matrix ( i).
17 .3 ;J.'heorem . :Let ..Q;;;: [t ,00) and let!: be the <r'-algebra. of Lebes-
'0
121
gue measurable with respect to the first variable for every t:e-7:' and
let
(+) f'tt-(A}= 1 k(t,<r)dt for Ae-2: .
.Let us suppose, there exist numbers 'V}'r0, JIiI m O such that for eve-
ry £ with ° < 'I'l and for ever<J T to there exist tA, , (?, .,T, for
whi ch
m S < 10 up f{. (t ;t" d t ME...
, ..... t'6-'r i:1\; I
Then the fam ly of measures fU_ ) is equisplittable.
"1""'11 t'"frt"
Proof . We define a sequence of sets Ak satisfying the conditions
of Definition 17. 2lb) by induction. We take A 1 = 1 '(>1)' where we take
E..= 1' T = t o ,d...=oll'(z=(\l. Now, let us suppose A 1 ,A 2 ,...,A k _ 1 are
defined in such a. manner that A. =(ol....,A. ) for j = 1,2,...,k-l, "'. . 1
J J L 'J V' J J +
for j = 1,2,...,k-2 and the conditionr from 17.2(b)is satisfied. Taking
in the assumptions of 17.3 E= £-k' T = A-l' o/..;: , = and denoting
Ak = 'fk), we observe easily that Ak satisfies also the condition.
17 . 2 (b ).
17.4. Example . Let 1=['i:"' oo), where 't'1\,,>O and let us define klt,t')
as the Cera.ro kernel of orler I with r 1, i.e.
k ,t ;rl:: { (1 _ ) I'-1
o for t;;>'t"
'Then the family of measures If...r;) defined by formula 17.38-1 is uniformly
bouaded and equisplittable. It is sufficient to verify the assumptions
of 17.3, with t = O. First, we have ll.tc( )=S: kU;,t"")dt = 1 for C',;> O.
Taking J:i1:: m = in 17.3 and choosing 70 and T>O arbitrarily, we
have for f.. ?<X..4- T
{ o for 0 < 't ()(.
tk(t,r;1dt(1- )r for cL<'t"<
0<. d.. r ( ) r
(1 -:c-) - 1 - t- for 'f: .
Hence, the above integral assumes its largest value fort"= and 1Jhis
for 0 < t <;
.... -
value is equal to
{/:> (It r
10 UD f k (t ;t") d t = ( 1 - ) .
'C 6 0- .
17 .;S will be satisfied if we choose a... and so that
A ) 1/ r _ '
= 1 - (:2. e . Thus, the family (f'tclt' s eq1.n-
The conditions from
d<. r
(1 - ) :: £ i,e.
122
spli ttable.
17.5. Theorem . :Letl2..=={1,2,...; and let be the 6"'-algebra of all
subsets of the set..(2.. Let fa ,) oe an infinite matrix of nonnegative
n
numbers with no kolumn consisting of zeros only. We take
whence f'n are
exi;3t numbers
and for every
for which
f-n (A)== !:.
i A
measures on I;forn= 1,2,... Let us suppose there
.,,"0 and Ivl m70 such that for every £ with 0 <. "t
positive integer N there exist indices p,q, where q) N,
a,
n
for A G- z:. , fn. (j5 1= 0,
q
mt sup" a. I.iE'.
n L. n
i
Then the family of JJleasures (I\i ) is equislittablc.
'fll n== 1
Proof . Let us denote [p ,q) = i.P, p+ 1, p+2, .. . ,q and let a sequence
k) such-/;tlat J,O, t.k " +/ for k == 1,2,... be given.
Let us take in the assumptions of our theorem A 1 == [Pl , q 1 1, where £== E 1 ,
Pl = p, ql = 1, N = 1. Now, let us suppose that Al,A2""' _1 are
already defined in such a manner that A, == l p , ,q . J ) j == 1,2, > . . , k-l ,
J J J
<o..(P, 1 for j :::: 1,2,...,k-2 and that the conditions of 17.2(b)are
J J+
satisfied for j < k. Let Us taKe in our assumptions £.= k' hI = qk-l+ 1,
Pk = p, qk = q and let us put Ak == [Pk,qk)' I t is easily observed that
the sequence of sets Ak satisfies the condition 17.2lb).
17. 6. Example . :Let lanJ be the C9 ,r) -Cesa.ro matrix, where :r=1 ,2,...)
i.e.
( n+r-i-l )
r-l
(n:r)
for i,n :: 0,1,2,... We sAall show that the family (1,\):'0 of measures
defined as in 17.5 is equi plittable uniform bOllildedness of this fa-
mily follows froHl the fact that z:OO a, = 1. In order to p"t'ove the
=o n
equis p littabilit y of IAA-)OO we shall verify the assumptions of 17.5.
lorn !J=O
For this purpose we denote
Sln\ =
p,q
Co}
a , =
n
for i n, a , == 0 for i» n,
n
We shall prove
(1) st n )
p,q
t2) g.rq )
p,q
q
,2:, ani for 0 p q .
=p
the follo'ifing inetl uali ties
r-l
<. r ( 1 _ ..L ) (1 _ ) for q?-p and all n,
.... q+1 l r q+l 1
r-
r (1 - ) (1 - )
for q p, p+q even.
123
First let us observe that
SID) < SN)
p,q" p,q
q
== 2:
i==p
q;l == (q-p+l) q11 == 1
--2....
q+1
which gives both inequalities (1) andl2). So we may suvpose in the fol-
lowing that r> 1. First, we prove the inequality (1). In order to esti-
m ate S ln) " hall ' d -f' ./ <
p,q ; we conSl er our cases: n<p, p n""q, q<n rq,
n;;. rq. The case n<.p is trivial, since then S<.n) == O.
p,q
lJet n p and let us denote
d:n)_ (n+r-i-1) 1
i - en-i) !
for i == 0,1,...,n.
We have
en)
-L
c{n)
i-l
n-i+l
. < 1, i.e.
n-l-rr
c n) <: 6''1)
l i-l
for i
1,2,. . . ,n.
l.loreover, n
fDJ r 2: C';fl) for p n q
== (n+1){,n+2) .. . (n+r)
p,q i==p l
and
SLn) r q C n}
(3 ) == Ln+1)(n+2) ... (n+r) 2: for n> q.
p,q l
i=p
Hence we have for p { n q,
In)
S
p,q
r en )C(n)_ r ( l _--2.... ) . ( 1 _ P+ 1 J .:fl _.2:!:l )
In+1J(n+2)...(J1+r) -p+l p- n+l n+2 \. n+r
r-l
r (1 - q;, )'0 - q 2 ), . .. ,( 1 - q r ) r (1 - ) (1 - ) ,
because q+i r(q+l) for i r, and we obtained (1) .
Now, let n>q, then
S(!l) r lq _ p+ 1) C(p.),,, r ( 1 _ --2....).(1 - J2:t.1 ) . .' (1
p,q..... In+l)(n+2) ... (n+r) P q+l \.. n+2 \..
:< r ( 1 _ .....IL ) ,. ( l _ ..L ) <..:{1 _ --2.... ) ' .
x \ q+l q+2 l: q+r
Supposingq<n rq, the above inequality gives
S :q r (1 - q;,),(1 - }(1 - 3 ):o:(1 - T1;;;)
r-l
r .....IL ( P )
r (I - q+1 ) 1 - r q+l
hall show f ir s t that Sln)
Le. the inequality(l}. Supposing n>rq, we 10 p,q
+1). Applying the identity
p,q
..i2:tl )
-
n+ r "
. :.. ( r- r_ 1+ 1 j ) - - (S+ r r )
l for 10 == 0,1,2,...
J=O
124
it is easily ve rified that the condition S(h) firi+ 1) is eq ui valen t to
p ,q p,q
the condition
r
A = Il (1 - = ; ) - O.
But) according to the assumption n> rq, we have
A » ( 1 _ q-])+l r - 1 _ ..JL
rq-p+2 q+l
The right-hand side of the last inequality is easily shown to be al-
ways nonnegative. This proves the inequality S(n) ;jP-+l) for n :>rq.
p,q p)q
Consequently,
r-l
S<n} g:rqJ r ( 1 _ ..JL ) ( 1 _ ---Ii..- )
p,q p,q q+l rq+l fOl'n/rq,
by the former case 0 Thus, (1) is proved for every n.
Now, we procede to the proof of inequality (2) , taking p+q even
and q p. By equality (3), we have
FlJ
p,q
S , r
p,q (rq+1)(rq+2)... (rq+r)
applying the ineq uali ty cf ' < C )
[ 1 ] -1
r '2tp+q) - \p-l) C(;rq) _ --L rq-r-p+2r
(rq+l)(rq+2) ... (rq+r) .I!::t9. - 2r rq+l
2.
1
2J:q) d '
,
=p
Hence,
for i
1,2,... ,n,
we obtain
r-l
IT (2r-l) q--p+2j
rq+j+l
j=l
r-l ) 1
....L ( 1 _ J.1L) IT ( 1 _ P-j l ) ¥ -L ( 1 _....DL.)( 1 _...E- -
2r rq+1) , 1 rq+J+l 2 r rq+l rq+2
J=
But 1 - 1 1 (1 - ) , and consequently we obtain (2) .
1
Applying ineq ua1i ties (1) 8"."ld (2) we show that for every 0 <. € < 4
and for each positive integer N there exist positive integers p,q with
q ':> p N such that
t.
(41
< sup
r2r n
By (1) and (2), it is sufficient
q
L. ani < 2rE .
i=p
to show that there exist positive inte-
gers p,q such that q,/ P9 N, p+q is even and
r-l
(5' t«l - )(1 - ) (1 - )(1
r-l
-1L- ) <:
- r q+l
2 E .
Clearly, the inequality in l5) is satisfied. Now, let
r-l
g(X) =:: (l-xJ(1- ;) for 0 x 1.
Evidently, gll)= o< g(1_2- 1 / r ). By continuity of g, there exists a
nU11lber x such that 1/2 (x <. and 3 S /2 < g (x ) < 2 E . l\.oreo"\e:', one may
000
125
assume that x
o
are arbitrarily
given N one may
is a rational number of the fo rm X :o.....P-.. where
o q+1' p,q
large positive integers and p+q is even. whence for a
assume q>p>N. Hence
2 r-l
2 £ {(1 - q l )(1 - ) <
in particular, the right one of the inequalities(5J is
2 e. ;
satisfied. Now,
we have
o
..... rq+2
_ -.!:L.
r{q+l)
-' -1...
rq+2
for r 2
and so, taking sufficiently large, we get
(1 - (-1 _ (1 _ r1 <:
It
2
for r 1.
7'
This inequality remains valid after multiplying the left-hand side by
- . Hence
r-1 r-1 1 2 1
( 1 ...lL )( 1 ) ( ...lL )( ---E- ) - 2: E '7 2€.- 2 at,
, - q+1 - rq+2 1 - q+1 . 1 - r [q+1)
l.e. the left one of ine ualities (5/.
Now, applying inequalities (4) we observe that the assumptions of 17.5
t ' f ' d . th 1 1 - 1 2 - r ' 1
are sa lS le WJ. "1.= '4 ' \1 == 2r, m = -; . Consequently, the fa.m:t Y
of measures W'n):o defined by the ,natrix 17.6 (0) is equispl'ittable.
17.7. Remark . Not all families of measures are equisplittable. For
example, defining hA. ) co by means of the matrix (a ,) with a . :z 1 for
'f'"n !'J;::: 1 nl Ill.
n::: i, ani::: 0 for n f. i, it is easily seen that fn(A)= 1 if nEA,
iA LA)= 0 if n 4 A, and the family itA ) 00 is not equispli ttable because
In 'I"'n Th=1
sup u LA)::: 1 for every nonempty set A. Similarly, considering the case
n ""n
= (to' 00), '2:= the famil r of Lebesgue measurable subsets of , 'l= tro'oo),
'C'= &r-;::: the Dirac measure concentrated at the point "(' we observe easily
that lfl't')'t?,... is not eq uispli ttable.
Now, we tlUTI back to the considerations of 17.1, taking n' X and
X q as defined there. We shall form sufficient conditions and necessary
condi t ions for eq uali ty X 5 ::: X fo ' se parately.
17.S .Theorem . Let cttd't,,'l be a uniformly bounded family of measu-
res and let the sequence ll.D)OO of lD-functions be equicontinuous at O.
'Tn Th=1 1
If there holds the following condition:
(+) there exist positive numbers k,c,u o and an irdex io such1hat
\f,lcu) k .(ul for every u u and i i o '
l lO 0
126
Then x = X o .
ProoL First, we shall show that the condition (+) is equivalent to
the following one
(++) there exist an index io and a number c ')0 such that for every u'>O
there is a k'>O for which "}'luJ k'\f)i (u) for eve ry u'7.u'and i'7. i .
, 1 0 "7 0
Obviously, l++) implies (+\. Conversely, let us suppose that (+\ holds. Ta-
king u' cu , we have .9- u for u'b u' and so (J), (u)= (D. l c .9- ) ..::. k ( _ ( .9- )
o c 10 I T '1. C - 1 o c
:; k 'Pio lc'u), where c'== . Now, let us suppose that 0.( u'< cu o and
u u'. If u cuo' we obtain ifi(U) :iCCPio(c'U) as previously. If
u' u< cU o ' we get i (uJ <Pi (cUd k io (c' cU o ) == k I.fi )==
i LU o )
k ,o (u') ioLu' ) k' io (u),
o
, ,I -1
where k =: k (h6!tclt.lf1 o lU1) . Thus, we proved(++).
How, we prove the theorem. :Get xf , then <fiCAxj-l"O as A-?>O+
for ea.ch i separately_ Let us choose an a.rbitrary £>0. Since 'Pi are
equicontinuous at 0, there exists a u'>O such that Cf'i (u'j< f Pftc(Jl)r1
for i == 1,2,... Assuming L+), we ba ve also C++) . Taking u' in l++/ as abo-
ve, we can find a k'>O such that fi(u) k'lfio(c'u) for u?u' and i io'
Let ,\ '> 0 be so small that io (c 'Ax I -< 2 for 0 >- 6:>"0' Let us write
A == {tc..Q. : .\ Ix (t)1 u'l, B == {tE-.Jl: >-Ix Lt)1 < u'}.
Then we have for i i and O '>- >- ,
, 0 0
K),f.t\:i) sup S If' (;\\Ix(tll) dl4 + sup S 'P' (>. Ix <.t)ll df4.,
') n-T: A 1'1:" n.'t B
S sup k'S<ri (c' lx(t\l)df{ + sup S tP, lu/d k' i (c'.\x)+ e<e .
C'e"C; A 0 t" ,tit" B 1 0
Taking eventually >'0>0 still smaller, we obtain <¥i(>'x)<E for O o
and for all i. Consequently, x f. Xro .
17 .9. Theorem . Let <ftd r « be a uniformly bounded and equisplittable fa.-
mily of measures. l\ioreover, let the -r-functions q>n satisfy the following
two conditions :
(a) for every index i there exist num1:a's i ' i ' vi;> 0 such that
tpi (Ai 1.1) i CfkluJ for every u v i and k i,
(b) for every 0 and for every index n there exist numbers u£ ,eL E ;> 0
such that for eve ry u u t In d every ct satisfying the ineq uaJ.i ties
o .,( Q/.t there holds the inequality \f n E <f n (u) .
127
If X ::: XeO ' then n} satisfies the condition 17.8 (+) .
Proof . Let US suppose that 17.8(+) does not hold. ':C.hen for arbitra-
ry indices n,m,k there exist an index i k . n and a number u k m
n,m, n,m,
such 1hat
CD. (; -ku k J '? 2 k tJ) (u k ).
h k n,m, 1n n,m,
n,m,
We define now an increasing sequence( )of indices and a sequence( )
of posi ti ve numbers as follows. First, we choose m 1 so that m 1 v 1 '
1 (m,) Inin 0, 4J, where 0'} '? 0 is the number appearing in the Defini-
tion 17.2 of eq uispli ttabili ty of the family (f'tr), and we take u 1 = ul ,l'
Let us suppose that the indices ml,m2""' _1 and the positive numbers
,u2""' _1 are already chosen in such a manner that ml<m2<'''< _1'
11'J.;(m')b 1, m.2-v, fori = 1,2,...,k-1 and,n.(m,»lo. 1 (u. l )fori = 2,3,
't... ., ., T '1" _ -
... ,k-l, where U 1 = u, ,for i = 1,2,... ,k-l. Then we choose m.. m. 1
,I!lj_, K K-
so large that klmJ 1, :?vk' 'fklmJ:;>l.fk_1C'-\:-lJ, and = '-\:,IllJ:c,k'
Since )' , we have
tfk( ) <Pk{ P k-ll'-\:-ll for k = 2,3,...;
hence the sequence (<fk( )is increasing. Let us choose
Ek = 2 k k(uJ for k = 1,2,...
T t> L 1 !!t d
hen !Ok v 0, Ek £1 2lfl (ml)'" 2 <, , an moreove r,
f k + 1 \fk l I <. .1
for k = 1,2,...
Ek 2 (fk+ll +lj 2
From eq uispli ttabilfty of the family [ftd we conclude that there exi-
st numbers m and l\.L, lVi m/ 0, and a sequence (Ak) of pairwise disjoint
sets Ak r , k= 1,2,... such that
1Y\. ;' JAk) 1i1 Ek for k = 1,2,...
Let us take
{ ''\: fo r t c; A k , k = 1,2,...,
xttl"" 0 for t c.Q..... &, Ai
Evidently, xE: X. Denoting K = sup j'1.'((Jl.) , we have
CD t'E- 't'
IDn( nx}= sup Ll('nC?-nuk)kc<. ) l n(.1.n'-\:)I4 ) +
''7 -ce'(' k=1 <-\O f6;1 n
00 1 00
+ sup L: n(An )ft&-k)" K sup L: \fin tXnl)} + rn f <e tbf:CAk) .
a-'t" k=n tee' k= 1 K-n
But
128
00 00 00
L \f k ('\! sup. ftclA. k ) M 2: lfkll)J E. k := 1il L +
k=n Cff"1: k=n k=n 2
Ivl
2 n - 1
Hence
n-l l'iJ.
en( nx) K sup 2: IfnC'>'n ) + 2n- < 00.
rr't k= 1
We prove now that n ( x)...., 0 as ...., Of-. Taking an a.rbi trary "7 0 let
us choose E/2 rP'nX) in place of €. in the assumption(bt. The respec-
tive values ot.ft we shall denote byol', u'. Let us take '>1'>0 so small
-1 )< E.
that A'<' 'An and CPn (? 'u' >. n 2K ' and let us denote A =ft Jl: u:'
nxlt) , B =...!l' A. Then we have for 0<>-<>" :
n(.\X) i tfn' n AnX{tJ) dkv:- + :i l n ( nX(tl) dft
5 2 L x} sup J<fnl'>ln X (\;)) d + fn ( u') sup p.t (..S2)
n n c-t't" A
E + Klpn ( u')< E + K 2i =f.
Hence n(>.X)"'" 0 as >.,...... Of- for each n, separately. Thus, X€ . However,
on the other hand we have
i (2-kxJ 1- sup S tpi (2- k xltl) dlfr.= i (2-k,\:)sUPM )
k, ,k -wr Ak k, ,k k, ,k M'
k k
2 tfkll)J ( ) 1 2 tt'k('\:) m f k = m 70,
whence nfJ..x) does not tend to zero uniformly wIhth respect to n as >.. Of-,
and so x t X .o' Conseq uently, x o F X .
CHAPTER I V
FAlVI[LIES OF MODULARS :DEPENDING ON A PARAlV.1ETER
18. Various families of modulars depending on a parameter
The notion of countably modulared spaces may be extende.,d replacing
the sequence (/{n) of pseudomodulars by a family of pseudomodulars ( }.)
depending on a parameter f which takes values in an abstract set 8o.
18.1. Definition . Let X be a real or complex vector space and let
8. be a nonempty set (called in the sequel the eet of parameters ).
Let H be a G"-algebra of subsets of the set !Z and let m be a cr-fini te,
complete measure in H not identically equal to O. A function e: :
E x X -R+ '" [O,co] will be called a family of .modulars depending Q!! a
parameter , if tJ;J.e following conditions are satisfied
1 ( ,x) is a pseudomodular in X for every € E ,
2) if t;(f,x)= 0 for m-almost every E-E' , then x'" 0,
3) ( ,x) is an H-measurable function of J for every x X.
Let us remark that from <f-fini tene 1010 of follows existence of an
H-measurable function p in E such that 0 < P ) < 00 and 3 P( / dm '" 1.
:r
18.2. Definition . The following functionals will be defined by means
of a family of modulars depending on a paraLileter :
flJ;.}= S p( ) 1 + I(f:X)' dm , where p :; _(O,co)is H-measurable in, E
B
and J pij)dm '" 1 ) we take here 1 € 7 ,x >'''' 1 if q,x) = co,
o(x) = sup ess (i ,x) "
. lX)= S q,x) dm .
16 E
18.3. Proposition . "l?o and s are modulars
a convex modular in X for m-almost every G E
in X :, if ((,.)
, then o' fs are
is
also convex.
18.4. 2.l . Let us remark that taking as :; the set of positive
integers, E = {1 ,2,.. .1, H - the o'-algebra of all subsets of E , and
defining m(A) as tbe number of elements of the set A if A is finite,
mtAI= 00 if A is inf:LYlite, p(n)= 2- n , the modulars f, Po' and fs
130
become equal to the respective modulars defined in 15.1 and the Com-
ments to 17.
18.5. Theorem . The following inclusions hold:
La) X<;'oc X ,(.)X s c X,o , Ce) if mea) <co, then X" ex...,
. _ 00 I >0 -..
@) f .!:= £An' where An E-H are pairwise disjoint and iii f mf;.Jto,
and if B CAn' B H imply B = fJ or B = A , then X.. c.X_ .
n ..s - o
.la) follows from the inequality f0)'-('0(X), andlc)-from the
inequality s Lx) m(!n o lx). In orde.r to get (b) let us denote Bn ={ff-i:
p( ) nl, then En == J p( )dm....,.O. Hence
n
Dc) pt )dm -I: EJ B p(€J ( ,:X:)dm en + n <?sCx/.
n n
Now, let x , i.e. s (t\x) O as A...OJ.. Let €:> 0 be arbitrary. We
choose first n so large tl'.at f n < and thm, keeping n fixed, a
>-0> 0 such that sO\ox)<fn. Then we have Q.xJ < £ + n sf.t.x)<£ for
0< it. "o' Conseqrnntly, XEX . Finally, in order to obtain{;J.) let us
rero.ark that an H-measurable function must be constant on each of the
s<4ts An' Hence if x X, then
,x)= c n for € E An. Thus, we obtain
CD CD
s(X)= 1 cnm(!l.n i ':? r1 c n iff' ml ) sjiP c n ' iQf m( ) = fo Qc)i f mV\:).
Co sequently, X s c: c .
18.6. Theorem . An element x <: X belongs to X , if
q, Ax)_o as ).-0+ m-almost everywhere in E.
Proof . Let \: 0 and let
and only if,
( , >. k X )
\: ( ) '"' p( ) 1 + Lf, AkX) .
Let us suppose that q,>'xJ o as >. O+ m-a.e. in 11 . Thal O (f)
p ), (f, kx) 0 as k co m-a.e. in , where p is m-integra.ble in
!Z. Hence S (t1dm....,.0 as k.....,.co, i.e. xEX... Conversely, let'xe-XS',
E )
i.e. ! )dm_ 0 as k.....oo . Then .......o in measure m, and so, by F.
Riesz -theorem, hk.( 1-" 0 m-a.e. in E for an increasing sequence of in-
diceslki) . Since ( )7 0 in E. , we deduce hence that (J,J.kix)_o
m-a.e. in as k,_oo . Since to( , AX) is a nondecreasing function of
-
the variable >o, this implies (j,)"x)....,.O as A-O+ m-a.e. in r:!. ·
From the above theorem we obtain the following corollary, immedia-
tely :
131
18.7. Corollary If (f' x}-;r.O as '>'-;.0 in me ure m, then xe- .
The converse implication is not true in general, as shows Example
18.4, since then (€, ).x)= n()lx)..., 0 as A-> 0 in measure m is equiva-
lent to the condi tion n{A xl... 0 as A 0 uniformly with respect to n,
and this in tur.a is equivalent to XE-1o. However, this converse impl"i-
cation may be proved un.der some additional assumption on the measure
m. Namely, there holds
18.8. Theorem . Let us suppose that the measure m is absolu.tely con-
tinuous with respect to the measure n(A);:: 1 p(fJdm, i.e. for every f.,.. 0
there exists a r>O such that if AEI: and n(A)< , then m(A) < t.
Iv'ioreover, let xfX. Then :here holds xe: , if and only if,<:>ct,)..x)
as A 0 in measure m in .
Proof . According to Coroilary 18.7; it is sufficient to prove the
"only if" part of the eql;dvalency. Supposing xeX we have (f,;>"x) O
as ;'\ O m-a.e. in B , by 18.6. Since n is absolu.tely aontinuous with
respect to m, this implies ({,.Axl-O as .\-..0 n-a.e. in E . But
n(2 J < 00, and so we may concl u.de that (f, ), x) _0 as ,\ _ 0 in m6asure
ninE. Since m is absolutely continuous with respect to n, this imp-
plies (j, >- x) O as >.,..0 in measure m in E .
18.9. Example . In order to give an example of a family of modulars
generalizing the case of countably modulared spaces with modulars gene-
rated by a sequence of f-.functions C see 17.1 with f'4.r;= r- independent
ofT), we shall take a measure space (.1l..,L.,f') with complete, nontrivial
measure fv and we shall define X as the space of extcended real-valued,
I'-measurable and -almost eVerr'ywhere finite fu..YJ.ctions on ..a., with
equality -a.e. £et (3,H,m) be a measure space with a complete, non-
trivial measure m. Let tp: E X [O,oo) P,oo) be a fu.."lction satisfying t'he
following conditions:
1 0
q ,u)is a continuous,
for every f: E ,
2 0 if \fJ(i , u)= 0 m-a.e. in!:! , then u = 0,
3 0 if If\j,u/-1loo as u..,oo for m-a.e. E:;
4 0 ,u)is an H-measurable function of for every u O.
Then it is easily ohserved that
ij,x)= S <p( , Ix Lt)l) d
increasing function of the variable u 0
132
is a family of modulars depending on the parameter f ' according to
18.1. In the fo loYang, we shall investigate the modular spaces X
o
and s ' generated by the modulars
eft-J )ol.XI= SUPisS l'PC , IXlt)lld and slX)=SStp( , Ix (tJl)dr- dm .
First, we shall prove the following B'Jl
18.1O. Theorem . Let s be defined as in forw.ula 18.9(1f) and let x EX.
Then x e-X fi ' if and onl.y if, fso.xk 00 for some A/O.
Proof . One has only to prove the sufficiency of the above condi-
tion. Let s (l-OX) < co for a >'0> 0 8,(ld let >n wO, An >-0' Then there
exists a set A€H of measure m(A)= 0 such tllat 1<p( 1}.0IX(tJUdIK<co
for '2 'A. Applying the Lebesgue dominated convergence theorem to
the sequence of integrals M( },An Ix Lt)/J df--, we obtain <f ( I >'nX) O
for all i€ 3' 'A. Applying the same theorem once more to the sequence
of integrals l ( ,Anx)dm t we get s().nxJ-l,O, i.e. xG-X es '
=. 00 00
18.11. Theorem . Let L = L (.st, ,H with the normtfx«a,= su esslx(t)1
and let 0 be defined by 18.9 (If 1 . Then
Call supp,osing (J2.I<co, there holds LOO<::. o' if and only if,
supess \D( , u) 0 as u-;>O+,
, J 00
supess (D( , u ):: 00 for some u '? 0, then X 41 c L ,
i 1 0 0 )0.
SU? GSS r(f,l1)-O+ A4 IA- 0+ }
there exists a sequence (Bn) of pairwise disjoint sets Bne-I: ,
such that 0 < ft (Bd""O as n .oo :,
imply that from X DcLOO it follows SUP{SS CP( ,ud= co for some
U o /' O.
Proof . La,) If SUprBS
t{(.o) supess \P(f, 'xllxl) ) -;. 0 as
00 , 00
L c. X o, then 1 cX o and we get
),. O+.
(a 2 ) if
( ) 1 0
2 0
If( ' u)-.:>O as U-4O+ and x L co , then <folAX)
(1'70+ , and so x XcJ . Conversely, let
)0 1
suP ss rcg,>-.)= o (A.l) /1.(-1.) O as
2)Supposing sup,ess f( 'uo)= 00 for some uo>O and Xf q and writing
An == t€..Q: n lx(t)l<n+q for n = 0,1,2,...,
we have co
00 '/ o(}.. x) sUjJess L: AS lp( , An) dr- (An' supess cr( ' An)
n=o n
for every n and for ).. / 0 sufficiently sillall. Hence tf-(An) = 0 for suf-
co
ficienly large n. Consequently, xfL .
133
t 'Let supess If(f,U)<ocl for every u,>O and let the conditions(a.o.)
o 0 . )
1,2 be aat sfied. Let an == supess 'PC! ,h). We choose a subsequence
(BnIJ of the sequence (BrJ such that f MlBnJ $ ruin (2- k a;1 ,2- k J, and a fun-
ction x(tl == k for t E: Bnk ' x It} == 0 if t does not belong to any of the
sets1Jnk' Then XfLOO. Now, let an :;:>O be given and let US choose N so
large that +,2-k < t: . Let 0 < X< 1. 'rhen
J;"'JQ == supess rlBnk) 'f U, Ak) supess
f k:=1
co
f n,?i N) 2:. fi(BnJ +
k==l
00 1
of. 2: ft(B nk \ supess ( ,A N) +"2t .
k=N+1
However, by assumPtion( jll°, the first term on the right-hand side
tends to zero as ""_ 0+ and so is less than f for sufficiently small
),70. This shows that o(AX)"'O as >.....0+, whence x ,"X O' ':rhus, (a j )
holds.
18.12. Theorem .Let L oo : L C1J (.n.,Y,r) with norm /lxI/CD == su ess Ix (1;)1
and let s be defined by 18.9 ). Then
lb,) supposing ftl.fU< 00, there holds LOOc:. s. if and only if,
S lD(g,u) dm < 00 for some u ';> 0,
;;1 0 0
" - 00
(b 2 ' if f( ' u o ) dm == 00 for some u o '> 0, then s<::'L ,
0:>3) if the condition 2 0 of 18.11 ( ) is satisfied and if x. sc L CO , then
j (J,u) dm: C1J for some u >0.
E 0 0
. (bl) Let f f(f, 1..1. 0 ) rim < 00 for some u o > 0 and let x € L Then
flOCk x} =l( 'P(f, )"l:J \t)l) dm) dfl t«..Q) J tp ( , >.l\xlfcJ dm fi(JU lfij, u o ) dm < 00
for 0 < ,.>, ulllXlIoo . From 18.10 we conclude that X .s . Conversely,
let LOOC , then 1 E- and so t«.n.}ltp( ,A)dm: fs(.\.1) <co for suf'fici-
antly small :> O.
(b 2 )Supposing t C!,uo)dm: C1J for some Uo)O and x E $ , let us define
sets as in the proof of 18.11la2). Then
00 > s (?x) £ C o n \f (f, An) dr) dm: ! rl )lf{tJ,n)dm -1
for sufficiently small >. :;:> O. Since) If( ,\'\) dm == 00 for n uoA ,we
-1 E C1J
have f'A.lAn} == 0 for n+ uo and so x EeL .
(by Let us suppose that i.f> , u.) dm < 00 for every u:> 0 8..1'J.d let t"Ae. condi-
tion 2 0 of 18.11t )be satisfied. Let us take == f( ,n)dm and let
US choose the subsequence lB nk ) and function x as in the proof of 18.11 (af
134
00 00
Then x.L , but s(X)= 1 f'ilBnk) < and Xf- , by 18.10. Consequ-
ently, (b 3 )is proved.
18.13. Example . Now, we shall provide specis example of a family of
modulars depending on a parameter, Generalizing further t:ru:. eneralized
Orlicz space L<f= Llf(p.,"t.,/-t) with a ...;function <f depending on a parameter.
Namely, let r-- be a (f'-fini te measure in a o--alge bra of subse ts of a non-
void set.Q. and let X be the sJ)ace of all extended real-valued, .[-measu-
rable and finite /A-almost everywhere functions On..o. , with equali ty
-a.e. We consider a faLilily of modulars depending on a parameter, ta-
king.E =..fi, i e. a family : ..rr x)(_ R+ ' satisfying the following
condi tioniq :
1) lt,X)is a pseudomodular on X for every t"..Q.
2) if lt,x)= 0 for fA-a.e. t&-Jl , then x = 0,
3) (:t ,x)is a t-measurable function of t for every xeo;:(
A t some places, we shall ass urne additionally that
4) if x,ye'l and Ix It)j Iy It)/ A- -a.e in...Q., then (t, x) q(t, y) fL-a.e .in J2. .
We denote bY' X the vector space
X =-{X€..t : (t, >. x)..., 0 as A Of- for f--a.e. t ..Q t
and we restrict now the family to the product ..a X X ; of course,
: ...a. X X...",. R+ is also a family of modulars dep:eding on a parameter.
We shall investigate the modular
<;>s(x)= 1 (t ,x)d JA-
in the vector space X ; the respective F-pseudonorm in X S is
Ixl s =- inf{u> 0 : s( ) u} (see 1.5).
Evidently, if the pseudomodulars (t,.) are convexforr-a.e. tE-J'l.,
then & is a convex modular.
Let XC;!o be the modular space generated by the modular 910' The follo-
wing analogy of Theorem 18.10 holds
18.14. Theorem . If X is defined as in 18.13, then an element X€ X be-
longs to X !., if and only if, there exists a '>"o 0 such that <?sC>.ox) <00 .
Proof. Let x/i:X and s{. ox)<oofor a 40 0, then the Lebesgue do-
minated convergence theorem yields s ();nx}_O for every sequence An O.
Hence x 6- !>.
18.15. Definition . An element x !o is called finite , if ( JtI<.QClJpr""'At'O.
The set of all finite elellBnts of is denoted by E(!, (see also 7. 2 ).
135
18 0 16. Theo:rew . IfX sis defined as in 18.13, thenE $is a vector
sUbspace of X closed in X > with respect to the F-pseudonorm I I s . If
satisfies additionally the condition 4) of 18.13, then the condi-
tions y&E s. ' x measurable and Ix It)l IY It)1 It-a.e. imply XE\ .
Proof . Since s (>'(ax+by) 10 (2).. ax) + .s (2 by) for very x,Y€
X@ , )O and a,b real, so E", is a vector space. Let x eEp , x }[
)S )s n >$ --H
and let rXh-:x:J s""O' i.e. s(2)o.(xn-X))-f'0 for every >.,.0. Now, let us
fix an arbitrary "'/ O. Let n be so large that to (2)..(x -xJ)<.co . Then
'110 n
s(;..x) s(2)-txn-X)) + s C2Ax n ) <00 . Hence xe-"s . Finally, let us
supvos e the condition 4) of 18.13 to be satisfied and let y E s ' x
measurable, \X(t)I \y(tMf\-a.e. Then \,,(t, ).x) (t,>.y) y.,-a.e. for every
;.. > O. Hence x€-X and s().x) (UsfAY) for all ;\>0. But fsf}..y,,,,,,O as
0 and s()..y} < 00 for every A;;- o. Thus, the inequality 10 (Ax)
fs(,\..y) implies X€E S
18.17. Example . We are going to specify now the modulars <?(t,x)from
18.13. Let k : ..fl.X..Q..xR _R be a measurable function inJ:lX-Q.xR ,
+ + +
k(t,s,O)= 0 in...ax.S1, and let k(t, ,u)be a continuous and increasing to
00 function of variable ufR for every (j:;,s)e-..Rx..Q. . Then it is easily
+
seen that
{1t-1 (t,x)= 1. k {t,s, IX (,;>11) dfA-
satisfies the conditions 1) - 4) of 18.13 and X = ; moreover, fs (x) = 0
implies x = O. If, additionally, k(t,s,U) is a convex function of uER+
for every (t, s) f..Q..xJl , then (t, .) are convex for r-a. e. te..52 and
consequently, s is convex. Let us remark that if t.f:..fLXR+ R+tlff.
( see 7.1) is an increasing function of the variable u ER+, then
k(t,s,u)=I.f(s,u) satisfies the above assumptions and the generalized
Orlicz s ace L ( ,Z, I is a special case of the space s generated by
the f81llily of mOdulars lt,x) defineo. above.
18.18. Theorem . If Ct,x) is defined by the formula 18.17 C*) withf
<J-finite and if for(!r'ery U)O there holds the i.aequality
.h.k (t, 10, u) dft(t) > 0
for a.e. 10 e-..Q. , then the space with F-norm I s is complete.
. The proof of this theorem is obtained by an extension of
the arguments performed when proving Theorem 7.7 on completeness of a
generalized OI icz space. We shall show only that supposing <oo
136
and wri ting
viA): S fts)dtHs) for AE- , where fls]=fk It,s,uldfttt),
A ..$2
the measuref- is absolutely continuous with respec.t to the measure v:,
the remaining- part of the proof is obtained repeating the arguments
used in the proof of 7.7, replacing single integrals S.. .df"- by dou-
ble ones R. .dJotls, dt'-lt) . Supposing fA- to be not absolu't}ely continuous
wi th respect to V , there would exist an £ >0 and a sequence of sets
Anf t. such that v(A n )(2- n and (An) f, for n == 1,2,... Supposing
Bn A n + 1 vA n + 2 v , B == B 1 f\B 2 "..., we have B 1 ='B 2 :> '" and
00 00
vU3d L v(A k ) 2" 2- k =: 2- n .
k==n+ 1 k=m+ 1
Hence v(B) V(Bn) 2- n for n = 1,2,..., and consequently, £ f(s]dft =
\lCB)=: O. Henc€ (j,i)=: 0, because f(sl> 0 r-a.e. in-'l . But ftl.Jl./ <.00,
and so 0 =: ftC») == p..t Bn)= n l fl03 n l f , because (Bd f(An+l )?-e,
a contradiction.
19. Families of Orlicz classes
19.1. Notation . Let (...Q.,Z,/'t), (E,H,m) and E*,H ,m ) be measure spa-
ces, where f4-(.ft.1 < 00. During this whole Section, the following non-
atomici ty condition for f' will be suvposed : for every sequence (en)
of positive numbers satisfying e. 1 + €2+'" <. f<-(..st.) there exist a se-
quence <Pn)of pairwise disjoint sets in L such that f!C-Sl. n )= En for
n = 1,2,... We shall consider two fixed nonempty families cHand
kc. H such that O<m(p)<ooand O<n(Z )<oofor all ZeZ and Z*'E *
192 . Defini tion . The faruily fZ will be called v- absorbed , if there
exists an increasing sequence lZd of sets Zn 6. su:::h that for every
Z e-'Z there is an index k for which Z CZ k . The family!(,*will be called
6"'- absorbing , if there exists a sequence (z ) of sets Z ($ !Zit such that
for every Z t<: tz* there is an index k for which z c. Z;IIc for all k., ko'
o 0 0
19.3. Definition . Let If: i::x[O,oo)-t19,oo) satisfy the conditions 1 -4
of 18.9 and let t:E It x.[O,oo)-(9,oo) satisfy analogous conditions, repla
ci.ng E and ill. by E"" and m*, respectively. Iiio:r;eover, let 'fe, u) be m-integr-
able over Z for every Z(; , u ).0, and let t<:, u) be m*-integrable
over Z for every Z Ii: €-Z* , u O. Let denote the quadruplet
137
==(Z,Z*,C,U O ) with ZfZ' , Z Z , c:,?O, U070. The function't'wili
be called DC- weaker than f ' if
) tH*,u)dm K c S If (j,uJ dm for all u u o '
Z
Let X be the set of all r-measurable functions x :..!t_ (-oo,oo) with
eq uality f'-a. e., such that the functions ( , Ix (tll) and f ( , IX ltlJ) are
H)( L -measurable in EXJ2 and H J(.[ -measurable in .E xJ2. , respectively.
Then, evidently, there holds:
19.4. Lemma . Let (u i ) be a sequence of positive numbers and letl
be a sequence of pairwise disjoint sets from I . Then the function
x(tl == u, for t It..cl., xlt\= 0 for tcdl....,ra ..a., belongs to X.
==1
According to the notation of 16.9, the following fawilies of modu-
lars depending on a parameter will be used on X :
(h x ) == S If> ( , Ix ttH) d ' ( 11 ;X) == S t (g , IX ttH) dr-
for SE:e: ; *If;EIt, XE-X. In t;he sequel,4ve shall write briefly in.
place of t',:x;) and *in place of f{\.,xl, if x x is fixed. Moreover,
we shall denote by L 1 (Z ,H,m) and L 1 (Z* ,Hit" ,m"") the spaces of functions
integrable over the set Z Z or m*'-integrable over the set Z1tt<: * ,
respectively.
We shall be concerned with the connection between integrability
of over a set Z E: aad m"'-inte{jrability of over a set zlfe!Z
In all the theorems, the assumptions of 19.1 and 19.3 will be preassu-
med.
19.5. Theorem . Let Z be G'"-absorbed and let 'Jvlt be a'-absorbing. Then
the following two conditions are equ.ivalen- :
1 0 if t L 1 (Z ,H,m) for all Z €- , then thE re exists a Zi'tE 1t such that
" Ll(z1t,H ,m*) ,
2 0 there exist sets Z f:-Z , Z1tE-!Z- and numbers c,uo O such that '0/ is
c;('-weaker than with c:t=(Z,Z ,c,uo}.
Proof . Writi.ag T =={t€..Q.: rxlt)l uot for an arbitrary x X, we have
S 1I:( jt",x) dm ) (S 't'lr, Ix ltll) dm*') df + f(J2.) S j\-yl%i'r, u o } dm1'-
T Z
c S ( s lfi( ,Ix It)l) dm)dM. + ft(.Q) S :4'l uo) dm ,
TZ I
i.e.
(H) ( x)dm c S ( ,x)dm + M(-4) jlt uo) dmit' .
Z ,- Z
138
Supposing the condition 2 0 , this inelluality implies the co.adition 1 0 .
Now, let us suppose that 2 0 does not hold. Taking Z*= Z , Z -= Z.,c -= 2 i ,
l l
U o suf"ficiently large in 19.3 (11:-) , we Llay define an increasing sequence
(u i ) tending to 00 such that
* t( u. ) drrt 2 i S ( , u.) diD. for i =: 1,2,...,
i l Z. l
l
where zt and Zi are taken from the Definition 19.2. Since I.flS' u i ) 100
form-a.e. t(:-E, we obtainzt <flj,ujdm1'oo. Hence we may choose(ui)so
that ! \.f'( , u.) dm > 1 for all i. Let us ta.v.:e
l ,tt ( ...Q.)
E- = - for i -= 1,2,...;
l 2 l ; If( , u i ) dm
l
then i l E. i <. f<-($l..). Hence there are pairwise disjoint sets €- 2::
i =: 1,2,..., such that u.(..lt..) -= E, for all i. Defining x as in 19.4 and
r- l l
taking arbitrary Ze:2 , we find an index k such that Z CZ k . Since ZkC
Zi for i k, we have
k-l 00
S .o( ,x)dm 2: A./..(.fL_) S tD('-,u_) dm + 2. CS2..1 S W(f,u,)dm
Z ') - _ 1 ( - J: Z 1 5 l - k ' l Z 1 ') l
l- k - k
k-l
1 /,,-C.t:ti) Q \fl ' u i ) dm + 2 -lc+ 1 fl(.S1..) < 00,
l- 1k
and so G L \Z ,H,meJ for every Z ;Z . On t other hand,
S olt-( ,x)dm h 2: E- S /Mf u,)dIJ1J(-:? I.t. S \JJ( )t,u_)dm*
Z .\- ') , , - k l Z )/: T 1. '- k l Z * I l
l- 0 l- 0 i
00 .
? .L E i 2 l S <f l ' u i ) dm = 00,
i =:ko Zi
ti l l *' *" .... ) 0
whence I!f i L Z ,H ,m" . Thus, 1 does not hold.
19.6. Theorem . The following two conditions are equivalent
1 0 if there exists a et Z Go Z such that EL 1 (Z ,H,m), then
GL1(p*,H ,m*J for every Z !Z*,
2 0 for every Ze-:l and Z E- jf there exist constants c,u o ,>0 such that
"f is -weaker than \f with 0/.= (Z ,Z:f-, c, u o ) .
. Supposing 2 0 and applying the iaeCj,uality 19.50H") with ar-
bitrary Ze-Z , Z , we observe that 'Ll ,H,m) implies
Ll(Z.,H ,m ), i.e. 1 0 . Now, supposing 2 0 to be not true, one may take
sets Z e-% , Z*"e- It such t at
1' lu,)dm*' > 2 l j ( ,ui}dm for i 1,2,...
Zit l Z
139
wi th a sequence u i 1- 00, where it may be supposed that CP{l, u i ) dm ::> 1
for all i. Choosing
(..Q)
( ' ui dm
for i => 1,2,...,
E. == .
2 S
0t0 Z
we have i l f i < 1t and so there are pairwise disjoint sets .....(liE-I: ,
i = 1,2,..., such that ... (..(2.> == t, for all i. Taking x as in Lemma
r
19.4, we obtain
00
S , x) dm =: 1: (..a....) S If( , u.) dm =: fA..l..tt < 00,
Z i== 1 Z
i.e. €Ll(z,H,m), but
00 00
S l(f7:,x)dm!t == 2:. E.. J j'G u.) drrr L: £, 2 i S z ('" u i ) dm == 00 ,
ZY< i==l Z y i=l
i.e. "'4 L l \Z*',HII:-,mil-). Hence 1 0 is not satisfied.
19.7. Theorem . Let the familyZ be O-absorbed. 'l'hen the following
two conditions are equivalent
1 0 if E L l (Z,H,m} for all Z6-'Zt , then (€L l (z*",HI't",m-'") for all z E3Z,f,
2 0 for every Z!!' there exist a set Z c-'Z and numbers c, u ';> 0 such
o
that 'Y is -weaker than <p withot=(Z,Z ,c,uo J.
Proof . Supposing 2 0 , the inequality 19.5(H) implies 1 0 , iltJlilediate-
ly. Let us suppose, 2 0 does not hold. Then there exist a set Zft& ./I: and
a sequence u.1' 00 such that
,
! i' u.)dml '7 2 S f(%,u,)dm for i == 1,2,...,
Z Z .
where(Z.)is defined as in 19.2:, moreover, we may suppose that
}, I.fl , u i ) dm > 1 for all i. Let us choose the sequences (r i ) and i)
1.
and the function x as in the proof of 19.5. Then we have for every
ZE- with ZCZ k :
k-l -k+l . nl
S ct ' x) dm I;i S \{I ( ' u) dm + 2 ftt <.;f < 00 ,
Z i=l Zk
00 .
3 ..f(I.= x)dm;ll- "# L €. 2 S Cf( 'u. )dm == 00 ,
,,-05'-5 .
z ==1 Zi
i.e. 'Lllz,H,m), but "" L1CEjf-,HJt-,m , a contradiction with 1 0 ,
19.8. Theorem . Let the family !tit be (j-absorbing. Then the following
two conditions are equivalent:
1 0 t:'1 h L L l (, Z H ) t hen there exists a
if there exis ts a ZQ.A. for whic "'j '" \!,.,m,
Z!f;C;S:: such that lt'eLl(zjt'-,H ,m""),
140
2 0 for every Z & !<I there exis t a zlf:e!zil" and numbers c, u o > 0 such that Y"
is -weaker thanl.p WithOC=(z,Z.*-,c,u o ).
. 2 0 implis 1 0 , by inequality 19.5 . Supposing 2° to be not
true, there exist a z and a sequence u i 100 With} q'{f,ui)dm '/ 1
such hat
S 'f ( '\ u i ) dm tt > 2 i S I.fl$' u.} dm
Z Z
l.J",t US chooselthe seY.uences (E i \ and <:fl i ) and the
proof of 19.6. 'fhen
for i ::: 1,2,...
func tion x as in the
00
S (g,X)dm::: 1: f(JZ.,) S 'P( ,u_)dm = t«.!"/,) < 00,
Z _ 1 . i= 1 Z
i.e. { .w (Z,H,m). Now, let Z e-%it" be arbitrary and let ko be ehosen
in 8uch a lil8l.l.:.ler that Z';C.Z,f; for all k')k . Then
K ? 0
00 ()() ,
*( "',x)dm'1t?:- L: E. S'r(r,u.)dm ?L e, 2 9<p ,u,)dm::: 00,
Z"t i=k l Z i=k Z
.):1 0 0
whence <) L (z!t-,H*",m'1. Consequently, 1 0 does not hold.
19.9. Example . Let B =9* = N = the set of positive integers, H = H)f-:::
the O"'"-algebra of all subsets of N, m = m*'= the measure assigning to each
one-point set the number 1. In place of 9( ,u)and "fCt"",u)we write now
itu)and't'ilu), where i = 1,2,... We have (f,x)= (tX(j;)l)d ,( X)=
l til (t)I)df-' a.."'ld the conditions ...Ll(z,H,m)and llZJf-,HJf-,mit-Jmean.
.sz, L If- Ux It)l) dAL <. ao and 2: S \V, (Ix It)1) d K < 00,
i'ZJ'2,. /.. i Z..!2..1 1-
resQ8ctively.
Let t=2l1- be tIle fwnilJ of sets 11f, 1,21,{1,2,3 ,... Then Z is cr...
absorbed with Zi ={1,2,...,i}, an.d Z*"is (}-absorbing with z = {ll for
i = 1,2,... ",oreover, sLlce the S0tS Z Z and ZIl-&.2 it are finite, so
G 1, \z ,H,m ) tlefulS til3.t x c-L ' = L '-l.J2.' 'fI--) for all i (:.Z and 'EL l(z* ,H*",m )
means that x L\fc:= 1 ';(.Jt,2,1A), ,',here 1 tf . and L4' - are the Orlicz classes
. 0 0 f' 0 0
generated by -functions i and If'i' resjJectively. Taking Z ={1,2,...,k
and Zjf=i1,2,...,lJ, c,uo>O we observe thatlfi)is -weai(er than (lf i } with
d... = (z ,Z 'c' u o ), if and 0:::11y if
1 k
L't'. lu) c 2:<fi 0'-) for u uo'
j=l J i=l
Conseyuently, 'rheorel!ls 19.5 - 19.8 imply the following
19.10. Corollary . Let us consider the following conditoons
A.l° () L \;C U n L i, B.l° U A L 'C: n L:':,
i=l i=l j=l i=l j=l i=l
t+)
141
00 00 (I) 1 00 i
C.l° (\ L '; C. n L \fi,,' D.l° U n L '¥i' c U n LIi( '
i=l o i=l 0 0 0
i=, j=l 1=1 j=l
A.2 °
there exist indices ktl and numbers c,uo"O such that(+) holds,
B.2 °
for any two indices k,l there exist numbers c, u 0 such that
o
(+1 holds,
C.2 °
for every index 1 there exists an index k and numbers c, u o > 0
such ib.at l+) holds,
D.2° for every index k there exists an index 1 and numbers c,uo>O
suchtb.at <.+) holds.
Then the following equivalencies hold :
A., o 0
is equivalent to A.2 ,
B., o . 0
s equivalent to B.2 ,
C.' o , . al t 0.2 0 ,
s eqw.v en to
D.l° is equivalent to D.2°.
Now, let us take as = * the family of all finite nonempty sets
of positive integers. Then Z is again ()-absorbed, but tzk is not !f"-ab-
sorbing. Thus, one may apply Theorem 19.6, obtaining the following
19.11. Corollary . There holds .f) , L ' c: , L't'';, if and only if, for
= 0 =1 0
any two finite sets Z,Z* of positive integers there exist numbexs c,uo">O
such that
2:. 'i',(u) c 2. !.f-lu) for u u .
j c Zk J i <; Z 0
20. Spaces of analytic functions
20.1. Let X be the space of all analytic functions F in the unit
disc D =1.z Izl < lJ on the complex plane, and let be a <p-function.
Then
21\ 't
(r,F)= S (lF (re )1) dt , 0 r < 1,
o
is a family of modulars on X depending on the parameter r, in the sen-
se of the efinition 18.1 ; indeed, (r,F) is a finite modular on X for
every 0 <.r, 1 separately, due to the well-known properties of analytic
functions. It will be essential in the development f the theory that
<r,F) be a nondecreasing function of r CO,,); this will need an addi-
tional assumption of logarithmic convexity of .
142
20.2. Definition . A -function (without parameter) is called lo -
gan thmicallv convex , if the fuaction lV\ == \P v) is convex for all real
v and such that lim v-l lv'= <D.
V-;OCO
20.3. Example . Every convex function is evidently logarithmically
convex, but there exist non-convex -functions which are logarithillical-
ly convex, as \(>lu) = t u I p for 0 -<. p < 1.
20.4. LeliIDla . If is a nondecreasing and convex function on R 1 and
F is analytic in the unit disc D =iz : IZJ < 1}, then the intec:ral
1 2 IT 11\' ( . it )
2i\ S 's: InIF I) dt
I 0
is a nondecreasing funcbon of r [O, 1) .
The proof of tnis lemma may be found in A.Zygmund [4], Chapter VII,
2h.7 .12', p. 273 .
20.5. Theorem . If tp is a logarithmically con1.lex l(-function and F EX,
then lr,F) defined in 20.1 is a nondecreasing function of r E[O, 1) .
. Taking fEe v} =\fLe v) we observe that,£ sat.isfies the assumptio-
ns of 20.4 and <Q (In u) = Q;t Ifor u,. O. Hence
1 1 2 . t 1 2rr ' t
2fCr,F)= 2"- S\D(lFLre )I)dt = 2 5 (lnIF(re )\)dt
q 1\ 0 1 q 0
is a nondecreasing function of r E- 1.0,1) .
20.6. Definition . Let m(r)be a nondecreasing function in[O,1}satis-
fying the conditions mlO\= 0, m(l)O. We define for F X
olF} = sup (1';F) = lim (r,F),
O r(j r 1-
1-
S (r,F)dmlrl if o(F)== 00 and m ll-}== ml1}
o
10 IF )
1
f ,F)dm('f") if /F)< CD
o
00 if oLF)== 00 and m(1-) < m(l),
where
1- a
S (r,F)dmlr, == lim S (r,F)dmlr).
o a 1- 0
Obviously, o and s are modulars in X.
20.7. Theorem .(,.a) There holds s(F'£ olF)m(l) for F<=X ; moreover,
X eX, this embedding being continucus wi til respect to modula' conver-
o s
gence as well as to nOTIJ convergence.
(b) A function FEX belongs to s' if and only if: JaJi'}<oo for some
143
a )0.
(c) If m(u»O for some u (O,l), then
s(F)::: m(J-)suP{ m ' S Q:',F)dm(r):
for all F EX. 0
Proof . (al is obvious and (bJis obtained C'1Pplying the Lebesgue domina-
m lu) >0, 0 <. u < 1 !
ted convergence theorem to the double integral
1 2/T . it
s F) = f (j.IF (re )I)dt dm(r)
o 0
for >. -+ 0+, In order to show tc) let us take f (U)::: .-L
m(u)
u (O,1)such that m[ul>O. Taking u<,v<l, we get
i'(v) - f (U):;; m ) S (;t';F)dm(r)- ( m u, - m v) )S (r,F) drJ.(r)
u 0
(m ul - v» ) [m(u) <RCu,FJ- I (r'F dm(r)] 0
and the desired result follows from illonotoaicity of f, rillaediately.
u
S (r,F)drJ.Lr)for
o
20.S. Examples . tal Let us suppose that mer) is constant ia an inter-
val [R,l), where O(R<l', and that m(R-J(m(R+). Vie shall show that then
X = X and there hold the ineq uali ties :
lm(R+) - m@- )] (R,F) 10 (F) mlR+) <;'(R,F) for Fe X.
Of course, it is sufficient to prove the iaequali ties. {in ting C == mlR+J
- m (R-) , we have
R-
slFl== <!(r,F)dm(r) + (R,F)C ? C ,F),
R-
s ) S (R,F)dm(r) + f(R,F)C
o
lim (.R,F)[mlr\- m(O))+ ,o(R,F)C = (R,F)mlR+).
r R- )
lb) Let us suppose that m(l-l<m(J). We shall show that then X = X o,
the identity map being an isomorphism between X >and . Indeed, let
us write c = mll)- m(J-). Supposing F EX, olF}<oo, to every E:> 0
one may choose an R such that 0 <:R (1 and (r,F) > o(F)-e. for R(r(l.
Then 1
s(F) S (r,F)dmlr) ( olF)-t)[rn(1)- m(R)1 c(foG')-f.) .
R
Since t O is arbitrary, we conclude that s(F' c 0Qi". By the defi-
nition of s' the same holds if oQr)== <D. This together with 20,7 (a)
gives the double inequality
c 0(F) s(F) m(1) olF) for F X,
and the result follows.
144
Let us still remark that Examples Ca) and (b) provide .the largest and
the smallest possible space S'
20.9. Definition . The modular space X o with o given by 20.6 is cal-
led the Hardy-Orlicz space and is denoted by ; the corresponding class
of elements FE X such that o(;Fk 00 is called the Hardy - Orlicz class
and is denoted by H . By Kq we shall denote the space of all finite ele-
o
ments of H'P, i.e. of functions Fe X such that o(AF)<oo for every ":>0.
20.1O. Remark . From 20.7 (b) and Example 20.8lb) follows that a func-
tion F €oX belongs to the Hardy-Orlicz space HCf, if and only if, o(a]j<oo
for some a :>0.
We shall study now the connection between the space H and the Or-
licz space L = L( , )). For this purpose, we need two following auxil-
iary classes N and N' of functions F E'X.
20.11. Defini tion . (al A function Fe: X is said to belong to the Qlass
N, if 2 IT - t
sup S In+ IF (re H dt < 00 ,
0 <1 0 +
where In+ u = 0 for 0 <..u 1, In u = In u for u >1.
lb} A function F f:X is said to belong to the ylass 1'1', if the integrals
x , t
g In+IFlre )1 dt , 0 r<1,
are uniformly (with regpect to r) absolutely continuous functions of
the variable x, 0 x 2'!1".
Obviously, we have N'CN.
The fo lowing may be proved
20.12. Theorem . If F€N, then the non-taagential limit Fle it )=
lim , F 0;)exis ts for almost every t E-(O,2n]; here A is a domain bet-
A z-te t it _"
ween two chords of the disc D, starting at the point e . liloreover.
In IF (eit)Jis integrable in (0 , 2!T] and F(eit);i 0 almost everywhere on
(0,211).
The proof of this theorem, ai>plying the tecJ:mique of Blaschke
products, may be found in A.Zyglllund fj), Chapter VII, Theorem 7.25 ,
p.276.
20. 13. Theorem . la\ The function
1
F(z): exp 21'(
2 f T it
S e + z
it
o e - z
d '>'(t)
belongs to N for every- real-valued fllilCtion >. of b01.4"1.d.9o. variation
145
in (0,2nJ .
(b\ Let be a nonnegative, nondecreasing and convex function for u?- O.
Then
2 IT ' 2f1
S (ln+'F(re 11) dt ! (In+IF(eit)l)dt for Os..r<l.
o 0
function F € H beloilgs to the class N I, if and only if,
2ff , 2 ' - 1
+ t 't
lim SIn IFlre )I dt = S In+IF{e )1 dt .
r..l- 0 0
The proofs of parts (aJ - (c) of this theorem may be found in A.Zyg-
mundL4), Chapter VII, Tileorems 7.30 ,p.277, 7.50 ,p.282 and 7.53 ,p.
283.
(t*)
(c\ A
.
It is easily seen that both classes N aad N' are linear sub paces
of the vecto? space X.
20.14. Theorem . If 'f is a 10garitbluicallY convex If-function, then
al.9cN' .
.Proof . Due to tile linearity of H', it is sufficie.at to show that
olF)<oo , F X, imply FEN'. Let F X, olF)<co, and let B be an ar-
bi trary Lebesgue-measurable subset of the interval [0, 2 It) of Lebesgue
L18asurefBI, O r< 1. Applying Jencen's inequality to the fU.Clction<Elvl::o
\ft ) t;r r I. Ire i tll dt ) I I (YP(lnIF lre it)1 )dt + 20 lO»)
I ' ( otF)+ 21T <P(1»)< co .
Det us now suppose that F does not belong to N ' . Then there exist a
sequencelBn) o¥ illeasu.rable subsets of(0,2rr] and a sequence (rn) of num-
bers and a nWJlber M > 0 such that 0<113 1-> 0, 0 r < 1 and
L . t n n
t In+IF(rne )! dt ';:}1 for n = 1,2,...
n
deuce, applying the above i.aequality(+), we obtain
co > olF)+ 2ITi.f(1) '? lJ3ni CI n \ i In+j F(rneitll d1 l13nl ( ).
However, since v-1 {y)-I00 as v.-yeJ:, the right-hand side of the last in-
e'luali ty tends to 00 as n......oo, a contradiction.
20.15. Theorem . If If is logari tLOO c ally convex 'f-fun.ctio n aad Fe- N ' ,
then
21t ,
o(F) = S (IFle t)l)dt .
o
146
where F{eit)is the non-tangential limit of Ftz) as z_e it .
Proof . By Theorem 20.13lb), we have the inequality 20.13( ' for
every r [O, 1) . Applying Fatou lemma to the integrals on the left-hand
side of this inequality as r-.l- , we conclude easily that
2 If , 2tT ,
Em (ln+IF(rel"tjl)dt = jPlln+IF(eltU)dt ,
r.... 1- 0 0
i.e.
2it ,2 f T '
(1} r S max L (j} , 'f Q F (re 1 t}lEd t = S max[ 6) , (fOF le l I)]dt .
o 0
Moreover, by 20.12, we have
lim [min \f(11 , UF (rei I)J = mir{\fb) , (I F ei tJl))
r-+1-
for almost every t E[0,2!f) t whence
21T 't 2ii '
(2) lim S min[q;Ll), OFlrel )I)]dt = J minl ), \f(lF leltWl dt .
r l- 0 0
Adding both equation.s (1} and l2} and taking into account the fact that
max la, b) + min (a, b)= a + b for every re al a, b, we get
2ti 't 21f' 't
lim ! [ (11+ aFlrel )I)]dt .. S[ (J)+ (lF(el )I)]o.t ,
1'_1- 0 0
which yields the desired result.
20.16 . Examples .I. Let us consider the function
l+z
Fe z )= exp l-z for I Z I<l .
We shall show that F EoN and F 4 H for every logari tbmically convex If-
2iT 't
function , but £ IF (e l >I) dt <. 00. This shows that the assumption
F e-N' in 20.15 cannot be replaced by F N. Indeed, we have
2
1 - r
't
IF (rel )/ = exp
2
1 - 2r cos t + r
for 0 r < 1
and so
2!T - t 2ft" l _ r 2
S In+ IF lre l JI dt = S 2 dt = 21\" for O'r< 1,
o 0 1 - 2r cos t - r
which shows that F £ N. Now, let 't' be a logari tbmically convex 'f-function
and let us suppose that FE: HI/'. Then there is a ),.,. 0 such that
lim 1 tf '\jI (' iF (rei t)n dt < 00. The If-function Lu1= 'fi u)is logari thmi-
r 1- 21T ( . it.
cally convex and such that to IF)= lim t If IF(re )I)dt < co. Apply:mg
) 0 r 1- 0
the easily verified inequality
1 - r 2
2 1 r for Itl l-r, O r< 1,
- 2r cos t + r
147
and writing v)= (eV), we obtain
2ri' it 2ft , 2 IT' ( 2 J
(r,F)= S<pQF (re )Odt:s S (lnIFlreltJl)dt = S 1 - r 2 dt
o 0 0 1- 2r cos t + r
S pr 1 - r2: 2 ) dt '?- 2 ) (1 _ r).
1 l-r II - 2r cos t + r
But since v -l (v} o as v -+co, the right-hand side of the last inequa-
lity tends to 00 as r l-. Hence IF)= lim tr,F)= 00, a contradiction.
o 1'...pl-
1 - r 2
Finally, we have
it
IF (e )/= lim exp 2 =
r...l- 1 - 2r cos t + r
for 0<: t < 2tr ,
and so
2tr . t
9 OF(el )1) dt = 21T ll) < 00
o
for every logarithmically convex I{J-function If .
Let us still remark that F belongs to some spaces Xi> J X. l'iamely,
t5, vil1+ )
let us take m(r)=:: rand '1.lY\=[ln(l + e a with 0.(a<:1 in Definition
20.6 of {ls. Then we have
'] 2ti"f- ( 2 l+a
lF,F)= S Lln 1 + exp - r J dt
o - 2r cos t + r
2{ (1 +
o \ 1-
1
slF)= S (r,F)dr:s 22+a rr (1 + l=a ) <00.
o
Hence F € X<; .
II. There will be shown that for every function FfN' there is a
logari thmically convex If-function lD such that F t H q1 . l'i ly, let
Tit 0
An ={t f[0,21T): n-l IF (e 11< n .
Since In l1+u) In 2 + In+ u for u 0, we have
00 2lT. 2rr + it
L:{Anlln n ) In(l + IFlel)l) dt 2 In 2 + f In IF (e )1 dt < co ,
2 0 0
by 20.13 {c/o Now, let us take a seCJ.uence(Sn)of numbers such that
o (a l' 00 and SnIA Iln n < co, and a 2 < a... In 2. Let us define
n n=2 n :>
{ a 2 t for 0 t < 2
plt) =
an for n-1 S t <. n , n = 3,4,...
Then lu)= r t- l plt)dt is easily proveQ to be logarithmically convex
o
2 r 2 ) 1+a dt
- r 22+a [ 1 + )aJ,
2r cos t +
and so
148
f-function. Ho\ ver, we have
n
lfln} = + [a. (In k - In (k-1J) 5- a r In
k=3 11: " n L
for n = 3,4,... Hence, by 20.15,
2rr it 00
tFI = S;WCe )I) dt \f(2X!A l ' + 1A21J+ I" «{n)lAnl 2 'IT \f(2) +
o n=3
00
+ [. a In n'IA I <
n=3 n n
n
2 + 2:.. (In k - In (k-1Jlt= anln n
k=3
co .
Thus, Fe Hlf .
o
The following important theorem gives the cOllilection between HIf,
H% ' Kif and L = L If (fo, 211)), Lt-, Elf, respectively.
20.17. Theorem .A function Fe x belongs to HIf(to Ht ' Kif, respecti-
vely), if and only if
1 0 FEN'
,
2 0 F( i") €L<¥ (to L ' ECf, respectively),
where F(e ') means the function Flei of the variable t defined in 20.12.
Proof. For convenience , we limit ourselves to the case . Let
- 0
F H ' then 1 0 follows from Theorem 20.14. Conversely, let us suppose
o 0 ( i. ) If
that 1 and 2 hold, then FeE to implies £:0 IF)(co, by Theorem
20.15. Consequently, FEHer .
o
20.18. Remark . Taking into account the fact that the correspondence
between functions F € N and limit functions F(e i .) is a linear isomor-
phism, so neglecting the difference between ispmorphic spaces, Theorem
20.17 may be written by
Htt= N '"Lip ,
means of the following eq uali ties
HI{) = N'"LIf, Ktf= N Etf.
o 0
20.19. As an example of applications of the above result, we shall
prove necessary and sufficient conditions for inclusion of Hardy-Orlicz
classes Hlf C. H'i" and Hardy-Orlicz spaces Ift'. For that purpose let us
o 0
recall that if If and \f' are If-functions (without parameter) , then
a' :Dr C.L't', if and only if, the re exis t constants K, u .,. 0 such that
100 0
W) K'fU-!) for every u uo'
1:j) BCL't, if and only if, there exist constants K l ,K 2 , u o > 0 such that
'flu) K l <flK2 u) for every u U o
see Theorems 8.4, 8.5 and 8.2 . Now, we may prove the following
20.20. Theorem . Let q> and't' be logarithmically "Convex 'f-functions.
Then
149
a} H;;fc. H ' if and chnly if, there exist constants K, U o ';> 0 such that
't'(p.J S:K \jI(u) for eve ry u u o '
b) HIfc.H'+', if and only if, there exist constants 1,K2'uo-::>0 such that
'I'(u) Kl \f(K2 u) for every u uo'
Proof .We limit ourselves to the proof of a}. If the condition I\'(u)
K (U) for u 7. '/ u holds, then by 20.17, F € N' and F (e it) E L<f ; ..conseq uen-
. 0 0
tly, F(e ') L't' and applying 20.17 again we conclude that F H't'. Con-
o 0
versely, let us suppose that the condition q{.u) K <fLU) for u U o does
not hold. Then there exists a function geLIf, gfL'f. Let
o 0
{Ig (t)[ if Igltll 1,
f It) =t 1 elsewhere in [0,2[(") .
Then
2 If 2«
S I{>(f (t)j dt SIf(1g It)1) dt + 2rr <Pll) ,
o 0
Conseq uen tly, f e L'¥ and f 4 LIf'. We shall prove
o 0
is integrable in [0,2rr] . Let f<..v) =\fl e v}. Since
-1;C\
re exists aV o "4 0 such that v lv) '11 for all
for all real v, and so In u v 0 + IF(u) for all
all t rO,21T), we obtaiI'
21\
S Iln f It)\dt =
o
Now, we apply
21T 2ft
J 'i' (f (tl) dt Sift\€; ltJl) dt.
o 0
that the function In f It)
v -1 <2Cv}.." 00 as v _ co, the-
V.....v . Hence v v + V)
?' 0 0
u'70' Since f(t) 1 for
because f L%.
integral of In
211'
Slnflt)dt
o
20.23 la)with
2tf
2ft V o + S (f <:tl)dt < co,
o
Alt) equal to the indefinite
f (t). Thus,
1 2 IT 1x
S e + a
F (z ) = exp _ 2 '
IT x
o e - Z
In f (x)dx
belongs to N. Moreover, we have
1 2tt ix it 1 2 It l-l
IF (rei 11= exp _ 2 S Re e + rei t In f(X) dx = exp-:::- S 2
tr 0 e ix re 21t 0 1-2rcos (t- r
.In flx)d:x: .
However,
2n- 1 _ l
-L S 2 hlx\ dx-,? hlt)
2n- 0 1 - 2r cos(t-x)+ r
a.i;.most everywhere as r....l- for every function h integrable in {0,2rr}
(see e.g. A.Zygmund[4], Chapter III, Theorem 7.9 , p.l0l). Hence
IF(reiil-yexp In flt) = fltlas r-1- for almost every tE{0,2tr).Conse-
150
. t
quently,IF<e I= f(t)almost everywhere, i.e. F(ei') L ,F(ei.).LIf
o 0
We are still going to show that FeN'. We have
+ it r, 1 2cr 2
In IF(re ;1= ma:x:LO, 2-rr S 1 - r 2 In f(X)dX]__
o 1 - 2r costt-x!+ r
- ma:x:(6, In f(tJ] = In+ f (t} as r_l-,
and besause
and
2
l-r .l=.E
2 .... 1+r 1,
1 - 2r costt-xl+ r
2 IT'
r Inl f txlldx <. 00 ,
o
so
2!f . 2ft" 2/T. t
Sln+IFlre tll dt - Sln+ flt)dt = Sln+IF(e )\ dt .
000
liIaldng use of 20.13lc,we conclude FtN'. Now, applying 20.17, we obtain
FE'Htf and F.tHIf'.
o 't 0
The following Corollary is obtained from Theorem 20.20, immediately:
20.21. Corollary . Let If be a logarithmically convex If-function. The-
re holds H:!= HIf, if and only if, (.1 2 ) there enst constants K,u o > 0 such
that Cf"l2u) K \f'(1.1) for every u u o '
CHAPTER V
SOME: .APPLICATIONS OF MODULAH SPACES
21. Application to integral equations
21.1. Notation . Let ,t:,IA-) be a measure space)l- the space of all
extended real-valued, L-measurable, fini te almost everywhere functi-
ons on with equality f-a.e., and let I : .n...X X....... (-co,oo] be such
that "tt,X) = II It ,x)/ is a family of convex Ulodulars on £ satisfying
the assumptions 1}- 3} of Example 18.13. Moreover, et X, fs and X r ,
be defined as in Example 18.13. We shall be c,oncerned with solving
the modular equations
x(t):::; aI (t,x) and
in the modular space Xf '
For this purpose we shall
x(t). aI(t,x)+ :JCol t) -a.et in...Q.
where x f X.. and the number a are fixed.
o ,,'
consider the following operators A and B
A\X/lt)= aI(t,X) and B[xKt)= aILt,x)+ xo(t) for xE- , a" O.
Solutions of the above modular equations are fixed points of the ope-
rators A and B, respect vely. We shall seak after conditions under whi-
ch A and B map X (or a ball lr) ={x" X<:,s: I(X( $ rj) into itself and
are contraction operators. Suvposing s to be complete with respect to
the norm 1/ U s ' this will enable us to apply Banach fix-point princi-
ple to the solution of the above equations. An application to Urysohn's
integral equation will be provided.
21.2. Theorem . I f there holds the condition
la) for every x '" XE:s and A l ,. 0 the re exis t ;:} ').2 ';7{) such that
[t, J- 2 (: ,x)J C \t, "1x) a.e. in..!l ,
then A maps X. in L. If for some R,r>O
- ., - s I 1
lbl for every x e- x. and >.e lO ,R-) there holds
- $ R
[t, a (<, ,x)] (t, Ar x) a.e. in..o. ,
then A maps Kp Lx? in Kt:J ill) .
.,.1 "j!>
Proof . Supposing la) t we get by integration
s ( a -1 A(x»)= s( A 2 I l ,x») :::; s (>'2 <> (., ,x:») C s(>'l x)
and applying 18.14 we see that xE-X .. implies A(x) s. Now, supposing(b'
and taking x e-K s(y)' we observe that there holds also La) and so A(Xlf s'
152
liloreover, integrating the inequality in (b) we obtain f(R- l A(x»).s Q:--lx),
whence A(x) K (R) .
21 .3 . Theorem . I f there holds 21.2 (a), then B maps s into itself.
If kxon s .9T for some itE(O,l) and if'21.2lb)holds with R '" (l-,9)r,
then B maps K lr) into itself.
Proof . The first part follows from 21.2, immediately. The second
one follows also from 21.2, because
lIB lx)/ '.A(x)lI +Uxoll {l-J)r +.$r = r.
21.4. Theorem . Let us suppose that for every E>O
o 0 sueD. that for every u:;.O and all x,yt ..(;r:) there
quality
S [ t, I ("X1e: I<'-'Y) Jdf"- S (t, Xa /) dr---.
A .P-
there exists a
holds the ine-
Then
o
1 if satisfies 21.2lb)with R '" r, then A maps )into itself, con-
tinuously,
2 0 if I\Xou !. .a.r and satisfies 21.2lb)with R = ll- )r for some 0<.$<1,
then B maps V) into itself., contimLOusly.
Both statements remain valid for r = co with K <;(co)= \. 0
Proof . Evidently, we may limit ourselves to the case 1 . From 21.2
it follows that A maps K r}into itself ({)<r 00). We have
II X't !.= lal inf{U> 0 : 1. (t, :6 )dr- 1 L
I/ A(XI-EA(Y) 'U=ta\inf{u>o: ) [t, I(..)\:H"Y) ]d ll,
' ...a..
whence
j j ll l
6 S
21.5. Theorem . Let
. 1 . II A (x) - A LvJ I I ,. 1
mp es t: .
co s
us suppose that there e::d1:ts a number
(t> 0 such
that
f [t, H ';X:)-uI(.,y1 Jdr'L:$ J (t, l-Y ) d
A ...n. C\. au
for every U)O and all x,y X1 . Tten
1 0 if satisfies 21.2lb)with R = r, then A : lr') and
IIA lxl- A(y) U s C1..ux-YI s for x,Y€ rj ,
2 0 ifIPCJt;.. -8-r and satisfies 21.2 Lb)with R =(1- )r for some 0($<1,
then B : :ri"'" ,[1-') and
153
liB (XI- B(.Y)IL c::(lIx-Yllp for x,yGKtr').
s ") <g
Both statements remain valid for r = 00 with K t co )=: X.
o <;s f
Proof . 1 is proved applying formulae for norms from the proof of
21.4 with f= 1 and Q= -1 , obtaining
IIA(x}- A (y)/ = lal inf {u >0 : S [t, I(',x} I(..y) ] dp.t 1 f
'> ...Q.
lalinf{u >0 : S ( t, 1- Y ) d 1 r = at IIx-y/l. .
....0. oC. au S
2 0 is proved analogously.
21.6 . E.xamp : Let f«..fl.} <. 00. We are going now to apply the above to
the special case Of defined in Exa:mple 18.17. This means that we de-
fine
I (t,x)=fLt,x)= lk(t,s, Ix (sllflfl(s/ ,
where k :..!l. )<....Q.x R _ R is a measurable function in ...Qx...sl.x R ,
+ + +
k(t,s,O)= 0 in ..o..x...Q. and k\.t,s,u)is continuous, convex and increasing
to co function of variable u ER for every (t,s)e.Jl x . As we have seen
+
in 18.17, I and sat.i.sfy then all the assumptions of 21.1 and moreover,
X = and if x,y Xt IxltJ/ lylt)1 }A.-a.e. in...Q., then S'lt,XI (t'Y)f-a.e.
inA. By 18.18, if for every u>O there holds Sk(t,s,u)dfLt/> 0 for
...Q
-a.e. 10 e,.!2 , then the space s is complete with respect to the norm!l s'
The modular eql,lations X\t)= a I (t,X) and x(tl= aI(,t,X)+ xoltlbecome
now Urysohn's integral equations
O -, x<.t)= aSk(t,s,/X(SJl)dtt Cs ),
J1.
( ) x (t)= aSk (t,s, Ix (1011) dMLs) + Xo It) ,
...Q.
and the operators A and B are now of the form
Alx)(.tl = as k(t,s, Ix(s)/) dt«ts) ,
BLxlltJ '" aj'k(t,.c;, Ixls)/) d{"tLs) + xott).
.£t.
Let us denpte for every .\>0,
k (t,u,v)=S k[t,s,Ak(S,u,v)] df-Ls) , (t,x) = Sk l (t,s, IxtS)I)dfttsJ.
...a. ..Q
By the above notation and assumptions we shall apply now Theorems
21.2 - 21.5 to the case of Urysobn's integral equations.
o -1
21.7. Theorem . 1 Let 0 < r< 00 , 0 <. R'( co , t a I (ft(..s2.I) . If for eve-
ry x EK..[r) and each 6(0,R-l] there holds the inequality
$ ). ( R'.
1 (t,x) f«.n.' t, "r Xi a.e. wSl,
154
then A maps K lr) in I\,W}.
o , 1
2 Let 0 <r< 00', 0<.9-< 1 ,!Ix IL ..i7r, ta I (j(l..ct/f . If for every x K 0:')
-1 -1 0 '; f5
and each A € (0, (1-)j) r ) there holds the inequality
(t,x) l.!l.) (t, >'(l-S)x) a.e. in..rl.. ,
then B maps K trJ into i ts-elf.
Proof . We limit ourselves to 1 0 . If xc , O<A R-l and lalfi(...a.' l
then, by ensen's inequality,
tt, }..a (' ,x») f/.:.a-» ) Hk[t,s, Xk(s,u, Ix (WV]dttlU)} ) =
.!ts}..
= ) It,x) f ' ). x)
a.e. in.Q , and la)follows from Theorem 21.2lb) .
21.8. Theorem . Let us sUl?pose that for every (?:,?O there exists a
¥",. 0 such that for every x,y € K (;t')there holds the inequality
s
1 { fot .a.) 1 k [t, u, Ik (u, v, Ix {V)I} - k (u, v ,I y (v) !) I] d t«V) 1 d ft(U)
}k[t,u, I X(U I - r X(V) dtt(J:t)
..t7. -1
for a.e. t€..Q. . Let lal (jfl4)) . Then
1 0 if satisfies the assumptions of 21.7 (a) with R = r, then A maps
,}r) into itself, continuously,
2 0 if (J satisfies the assumptions of 21. 7lb), then maps Klr) into
.. q\
itself, continuously.
. I t suffices to prove the continu.i ty of A. Let E,;:. 0 and '1;10
-1 -1 -1 .
be arbitrary and let us take 0 == e'lJf«-!l/) ,0= 6[al iVt . Then we obtaJ.n,
applying Jensen's inequality:
S [t, I(..Xl t '1. H '. Y ) ) dftlt) :f
f S1,) Uk (t, u, I u,v, (xLv)!) - k (u, v, ly(v)11 I)dfllV)] d ( ull d(1(lt)
!l:st:.J2. . L
$ S (t, 9)d (t).
..Q. at)
Thust 21.4,1 0 yields the desired result.
21.9. Theorem . Let us sUj)pose that there exists an r;t,)I> 0 such that
for every > 0 and all x,y€:- (r)there holds the inequality
S f f't A) Sk[t,u, ft;} lklu,'l,IX V'I) - k(u,v, ly(v""l dfX(V)j d(l«U)
+.a. Jl
Sk [ t,u, l lxlul- y(U)I] dft(ul
.A. Ia
155
:for a.e. tE . Let lal $ (rtC.$t))-l . Then
o
1 if satisfies the assumptions of 21.7 (a) with R = r, then A : r)
l ¥)and IIA(x}- A(y)/'- d.JJx-ynii for x,Y6K(TJ,
o 'c;S)i s
2 if satisfies the assumptions of 21.7 (b), then B: K 1 V 1 -, (r) and
liB lxl- B (y)I s d' IIx-y!l(,> for x,y € tr) .
Proof. Vie may limit ourselves to the case 1 0 . Arguing as in the
proof of 21.8 wi th = 1, we obtain
S [t, I(.,X) I(-,y) ]dr(J;) S (t, x_7 Y )d ),
..it 4'_ cl. a
and the result is obtained apJlying 21.5.1 0 .
Applying 'J.'l1eorems 21.9 and 18.18 (see also 21.6), Banach fix-point
principle leads to the following theorem on integral equations 21.6 ( J
and (Irk) , in1l11ediately :
21.10. Theorem . Let us suppose that the fUnction k is defined as in
Example 21.6 and satisi'ie 10 the following condition : for eve ry u> 0,
the il1.eq us_i ty 5 k It, 10, u) dl"tlt I '> 0 holds for (k.-a.e. 10 eJ2. . Let O r<oo,
-1 . 4l.
0<lal«t«4\) . l.ioreover, let us sUQfJose that there is a number OCf- (0,1)
suchtj;lat :for ever>J 'VI> ° arld for all x,y€KLr)there holds the inequa-
. $
li ty 21. 9 ) for a.e. te:.l2. . Then
10_if_l tlt,X) tt( lt, >.x) a.e. in...Q for every xe'f-r)and each
A lb, r J, then the integral equation 21. 6 ) has in the ball K lr) only
s
the solution x = 0,
2 0 ifllxoll. $r and (t,x) ftl.aj (t, (l- lx) a.e. inJl for some !tf(),1),
.,s (, -1 -1 J
every xe: l s(I) and each 7<.f\.O, (1-,(}) r ,then the integral equation
21.6t 1t1 has in the baa K ) exactly one solution x. This solution is
obtained as the linn t in X p of the seuuence x defined by X n lt):z
.,s - n
aIlt,x 1 )+ x (t) forn= 2,3,..., where X l cKlrJ is arbitrary.
0
21.11.A special . Let us consider the case when k(t,s,u):z
k (t,s)tN'u) , where k is weasurable and positive in ..a..xSl.. and is a
o 'f\: 0
convex c.p-fun.ction. Then, by 18.18, the space X '> is complete. IiIoreover,
we have
(t, x)= 1 ko (t, 10) CP(IX lS)1! dtt( 10) ,
LX):;: S wls) If(l x (10)1) dft<'s) , where VI' lSI=Sk o It ,10) d lt) > 0.
S..n.. I ..n. .
Thus, X is the Orlicz space L (J1.' 'fI-) with weight w an.d II /( s s the
w
Luxemburg norm in L l.n. ''-'}k}. Specializing further (flu) = I ul, we obtain
156
Le t US
(t.x)= lA .lko It,U)ko(u,s)Jx(s)ldfLu) ls).
consider e.g. the equation
t
x(t)= a S tslxls)/ds + x Lt), 0 t 1,
o 0
(++ )
then
{ ts for O s t
koLt,s)= 0 fort<s l
lvloreover, we have
t 1 1 2
(t,x)= S tslx(snds , (i) (:xJ= - S s(t - 10 J/x(s)/ds
o ",\10 2
1 3 3 0
.>. { '3/>'ltu(t - u)lvl for O u t
k"' 1 It, u, v) =
o for t (' u 1 ,
; (t,x)= 1)"11 ts(t 3 - 103) Ix(sHds .
o
Hence the inequality ;(t,x) C<l.st) (t, Hl-S>X) in the assumption
21.10.2 0 ia certainly satisfied if O<S< . Moreover, the inequality
)
21.9(.+) is equivalent to t
1 t 3 3
'3 r vlt - v )jIXLV)/-ly(vJ//dv$ lal J vlx(v)- ylvJ{dv for o t 1
o 0
and this is certainly satisfied iflal 3((. Thus, if 1Ix/ Sr, r'?O,
o < 9- < f ,I at -<. ruin (1, 3ct), 0 <:C«1j , then the equation (++) has exactly
one sol ution in the ball K (r) .
...
ce(tV =1 ul ,
WlS)
1 10 (1 _ s2)
2
22. Application to approximation theory
22.1. Notation . et lJl, , ) be a measure apace with a finite measure
fA- and let f be a space of extended real-valued, l:'-measurable functions
on.a, with equality yt-a. e. (compare EXaLlple 18.13). ':t will stand for
the subspace of f consisting of functions finite ft-a.e. Let n: .a)(' R+
where n = 1,2,..., satisfy the following conditions
1) nlt,X)iS a pseudomodular in r for every tE:.Q and n = 1,2,...,
2) if nlt,x)= 0 fOI'f-a.e. tE-J2. and n = 1,2,..., then x = 0,
37 n (:t,x) is a X-measux80ble function of t for every x£-X and n=1 ,2.,..
In order to get a modular in , we may ap£Jly two procedures : either
first inteGr8-te with respect to t and then sum with respect to n , or
10
conversely. This leads to two following modulars and s
00
lt,x}=f r n n(t,x)(l + en(t,x)-l/
n=l
ns Uc)= S nlt,x)d,M-'
.J'l
157
00
sl..x)= L 2- n n s tx)6 + Lx)fl, tD s Lx}= S A) ,x)d .
n=1 ns ') Jl '\
Evidently, s and s are modulars on . We shall denote the respective
modular spaces by '$ and ..' First, we shall see that under a sui table
assumption on n' both spaces x and !o are equal.
22.2. Theorem . Let US suppose that xE':l and that there exists a num-
ber >'0-> 0 (dependent on x) such that
l n(t').ox)d <CO for n = 1,2,...
Then the folloVdng conditions La) , lb) , lc) are pairt1ise eq ui val en t
(a)xE-X s ,
tel there exists a set A E- s: wi th....tA) = 0
for every t &.st \. A and each n.
. (cl==>ta\. If (c) holds, then applying the assumption on existen-
ce of >. 0 > 0 such that r f (t, A x) du< 00 we obtain S 10 (t," x) dLt - 0 as
n 0 ,- ..."1.... n (V
.i\ 0, by Lebesgue dominated convergence theorem. Hence lim O cr' (,>.x)= 0,
-1 >I n
where <t (}.x) S Lt,)I. x) <¥A: * [1 + S. It,). X)d",, ] . Given an E.? 0 let
n ..n.. n 00 -n A l n
us choose N so large tha.t 2 <'2£. There e s a ). 1 such that
N -n n= +1 1 s 1 1
0< 1 <: o and n l 2 O"'nL>'x) <'2€. for 0< <;>'1' Hence <; (.\xl<'2 E +'2£.=E
for O<A< E. ' Le.la).
tcl lb). Let us writ -1
f,,(t) 2: 2-n n(t,.xx)[1 + n\t,)..x)) .
n=l
Since n (t, x) is for fixed n,t,x a nondecreasing function of the var-
iable >- for).> 0, so f). Lt) is also nondecreasing with respect to ;.. > O.
Moreover, 0 f"lt) l and,so if,>. tt)df't < 00 for every .>. O. In p ti-
cular, S f' l tt) dl"'- < co. Supposing (c), we have
...A -1
(+) tt'n(t, ).x)= ntt,>,x)'[l + n(t,Ax)] -" 0 as >...,0 for tE-JbA.
- -n <: t B
Given an E.,. 0 let US choose N so large that 2: 2 3 ',sL) ' y
n=N+ 1 fI.(
Egoroff's theorem, ther€ exists a set B6- with AC..a' B and r-ti'!z"'B\<
_ 3 1 e. such that (+1 holds unifor-.JJly in B. 'rhus, there exists aA 2 >0 such
N £
that n 12-n n (1;, ;. x) <. 3/{{;u for t Band 0 < <A2' Hence f(t, A xk=
g:.2- n q;- (t,?\x)< 3 2e for tEB and O<A<>-e . HenceJ t'Ax)df.t.'3e
rf;; 1 n ft(.Q.) 2 B
for 0 <?« A 2 . Consequently, s(Ax) <'3E. + B dfA- <€ for 0< >'<)2.
Thus, x e .
(al lc). Supposing xe X. , we have Em S' It, ). x) df- = 0 for every n.
s 0.rt n
(b \ x E- $ ,
such that lim IJ n tt,,>,xl=O
O )
158
Hence n It,k -lx)...,. 0 as k_oo in measw:'e inn., and so 4'n (t,k lx)...,. 0
as ll,"-oo/4.-a-e.in..a , wherelk ) is ful increasing sequence of i.a6.ices. Sill-
I Hi
ce n It, }>. x) is a no.ndecreasing function of A> 0, this implies n (t, A xl
-to as )...... 0 N..-a.e. in...Q. . Taking as A the set of t G.J2 for which
{ n 00
(.I n It, f.-x) does not tend to 0 as A 0 and writing A = V A , we
\ n=l n
obtain (c) .
(b) lc). Supposing xeX , we have lim Sf, (t) du. = O. Since, as we
! _Oft'" ,-
have already shovm, fX (t} is nondecreasing with respect to A:;' 0 and
J f l lt)df< 00, Lebesgue dominated. convergence theorem gives lim f;\.Lt/=O
.It >....,0
-a.e. Consequently, there is a set Ae2;: with ft(AI= 0 such that
nlt, (\x}..,..O as '>"...,.0 for t6-.Q.'A and all n.
22.3. Remark . Let us note that the assU11lption in Theorem 22.2 con-
ceming integrability of () l t, ), x) for some " ';> 0 and n = 1,2,... was
'In 0 0
needed only in the partlc)=!l(8.) of the proof. Thus, the equivalency(b/.:,(P)
and the inclusion X sc: s do not require this assU11lption.
22.4 /eheorem . The following relations hold for any sequence (X of
elements of 1:
(a') IX0es""" 0 as k_oo, if and only if,
1 <?n (t, ). df -J 0 as lc co
o and each n,
for every >. :;>
(b'}1"1c1es-0 as
k_oo, if and only if,
n(t, A "1c)_0 as k_ooin measure fA..in-'l.
for every ). > 0 and each n,
(c ') if E: X e , and 1 I<:s..... 0, then )"1c1 !. 0 as k.-.oo .
Proof . La') The condition Jxklf!o O as k oo means that rS(A )..;)O as
k.....cofor every 1\>0 (see 15.2(dJ), i.e. that ns( as k.....cofor eve-
ry A? 0 and each n. This proves la').
lb') Arguing as above we see that !xk!" _0 as k_co is equivalent to
'>s OO- n
l tt, )..x dlA-- 0 as k- i.e. tOkl 1 2 O"n(,),.x1<)= 0 for l O
where we apply the notation fromfue proof of 22.2. Thus, 1 1... 0 lmpll-
\$ ,
es a-nO _O as k-foOO :t'or 1\:;>0 and n = 1,2,..., and so we obtaln
n It, ).. J 1 ,) _0 as k....,.oo in measure in..Q., for every A> 0 and each n.
Conversely, assuming this condit on to be satisfi d, we obtain
()n )-O as k_cofor ).>0, n =: 1,2,...
Let E.,. 0 be given and let 14 be chosen in such a mall.i1er that
159
oo -n <::
l + 1 2 2 ttC-SZ. )' Taking >- >0 fixed we choose k 1 so large that
-nh 1 00 n
_12 V' )<'2E for k k1' Then £, 2- G"nl>-Xk)<f for k k1' but
this proves IX !I_ 0 as k-Poo.
tc'/follows from(a')andCb'), iJIlli1ediately.
22.5. Approximation J(roblemQ The problem of approxiraation of func-
tions by nonlinear singular integrals may be formulated in the 10 etting
of modular spaces as follows : under what assumptions on a seq uence m )
of modulars depending on a par8.1Jleter and on an element xE3( , the re ho-
lds c; 'I,X) X(') as m-,>ooin the sense of convergence in X or x... ?
m s
Wi th this form.ulation it is clear that we have to 1 imi t oursel ve 10 to
the case of normegati ve functions x; an arbitrary x ?( should be fir-
st splitted into positive and negative part, approximating each of the-
se two separa ely. We shall limit ourselves here to the case of the spa-
ce s .
In the f'6110wing we shall assume all the time hat n) satisfies, be-
sides 22.1, - , also the following assumption:
4} if x,ye-l and lx(t)I lyltl/ -a.e. in..st, then nlt,x) nlt,y),...-a.e.
in for n == 1,2,... (compare Example 18.13).
22.6. Definition . We shall say that the sequence(SOn)is constant pre -
serving , if
a) constant functions belong to X s ,
b) nlt,c)= c for every tG-.il , c O and n == 1,2,...,
c) n It,x - xlt) is a l:"measurable function of t for every xte){; , x? O.
22.7. Theorem . If (fn)iS constant preserving, then for every Xf-X s,
x O and arbitrary 1\'70, 0..,[»0 with""+ == 1 there holds the inequali-
ty
f n {t, ).[ m Co ,x)- xL.)] 1 dM. S n {t., 2 m[" X-;(I):]i df4- + S n {t, 2,\ xl'l} df
.!l- ..!1 .tl
for m,n == 1,2,...
. Let XE'X s ' x O. Since (fn) is constant preserving, we have
m(t, x (tl)= xlt' for tf.12. .
Hence
m(t,x) emlt, x tl] + m ' X- lt) J= x(t) + m ' X- (t) ).
x It)= m (t, Ol xlt») 11l tt; ,x)+ m[t, lX-Xltl)]. m (t,x) + m [t, X-% LtJ j ,
and we obtain
160
mlt,x)- xlt) m[t, X- (t' J, otxlt)- mLt,X) fm(t, X- (t} J.
This gives
mlt,x)- x(t}= [ m (t,x)- xtt>] + xlt} m[t, X- {t) J + xttJ,
x(t)- mlt,x)= f-xlt)+ [d.xlt)- m(t,x)] f.>xlt)+ m ' x- (tJ J.
Consequently, denoting
Ymlt):: I m(t,x)- xlt)/
we get
O YmttJ m[t, x-;{t) ] + x(t).
Hence
n(t,>'Ym) n{t''' mt' X-;(" j + A xLt)l
n f t ,2 '\ m[" X-;{I) J1 + n{t,2). xt')f .
Integrating this inequality over..o., we obtain the required result.
22.8. Corollary . If{fJis constant preserving, t.I:Bn for every A"'O,
€ :> 0 and xE Xt: S wi th x+-O there exists a constant > 0 such that for
every m = 1,2,... there holds the inequality
St>.[ m ( ,xl- xl' )]f s {2 A m [" X- {. > jf+ f .
Proof .From the inequality given in 22.7 follows that
S{"[ ml',X) - Xl')]} S{2.x m[" x-r') Jl+ S{2 X("f.
However, from X€X ,it follows S{2 £ Xl"} O as -I"0, which proves
the Corollary.
22.9. Definition . The sequence( n)will be called singular the
point X e , if for arbitrary a, b :> 0 and n 1,2, .. ., the seq U3nce
J:mlX)= 1 n{t,a ml.,b(x-X(")JJdjIA-
tends to zero as m...p(X). The sequence ( n)will be called singular , if it
is singular at eve ry point x? 0, x e .
22.10. Theorem . Let the sequence (f n' be constant preserving and sin-
gular at a point :it E. X , X? O. Then
m (. ,x) _ x(.) as m....,.oo
wi th respect to the F-norm convergence in X .
Proof . Since ( n)iS singular at x, we have
ns{a m[.,b(x-X('»)]f 0 as m-.oo
for n = 1,2,... Conseqwntly,
S{a m[.,b(X-)(('»Jt O as m_oo
161
for arbitrary a, b > O. Let us take any e. '> 0 and A > 0 and let US cho-
ose p.> 0 satsifying the inequality from 22.8, a = 2 and b = . Then
there is an index M such that
s.f2 >"<>m(" )t- (') Jl< E for m M.
Hence .S{>.[ m("x>- X('»)! < 2& for m Nl, i.e.
S{).( m("X)- x(,'H 0 as m...oo.
Consequently, ,1 m("X)- X(.) s - 0 as m oo, where I I s is the F-norm in
x generated by the modular <f.
22.11 . E)r:ample . We give now a concre te example of a seq uence (enl of
modulars depending on a parameter, satisfying the above required condi-
tions. Let ..Q= [0,1), and let fA- be the Lebesgue measure on the O"-algebra
of Lebesgue measurable subsets of..n.. As 1 we shall take the family
of all measurable :functi.ons in [0, 1), which will be extended periodical-
ly with period 1 to the whole R l . Let K u)be measurable functions,
n
Kn,u)O a.e. in[O,l), r: KnlU)du = 1 for n = 1,2,...; the functions Kn
will be also extended to R l periodically with period 1. Let fbe a con-
vex cf-function and let -1 be the inverse to tp for u O. We define
( n (t,x)= fY S l Kn(U)<POXlurtJl)dU]-
Let us remark that (f n) sat sfies all the assumptions 22. 1 - 3) and
22.5, 4) . Indeed, if ct,f- O, d..+(b = 1, x,y€:;(, then
n(t,o(x + y) flf d..S l KnluJ (lx,U:!-tltl du + f' S l Kn<.u)<p(ly(urt)l) dU}
o 0
n x) + n( Y) n(xl+ n(Y) 0
T he remaining conditions are satisfied, trivially. lJoreover, we have
n(t,x)= 1{} Knlv-t)le(IXlV)l)dV
and from the assumption ,.1 K ,"-u)du 1 it follows ,t,c)= c for c O,
)0 n In.
o t< 1, immediately. Since 22.6,c) is satisfied, evidently, sO n) s
constant preserving.
22.12. Definition .The sequenceCK n ) defined in 22.11 will be called a
singular kernel , if
1 -a 1 )
lim S K (,u)du = 0 for every OE (0, 2: .
n -xv 6 n
22.13 . Theorem . Let lKn) be a singular kernel and let lp be a convex
tp-function satisfying the £O.Llowing conditions :
(D. \there exist constants K ') 0 and u > 0 such that
2 0
tp(2U) K4'0)for u uo '
162
Ip- l tu )
(w) lim v sup = o.
v v u
Let the sequence'fn)be defined by 22.11(11-). Then
m(e,x)- x(.) as m oo
with respect to the F-norm I j 5 in X(S for every x X rLCf(o, 1) , x ¥ O.
Proof . According to 22.10 it is sufficient to show that the seq
ence (g J given by 22.11 (*) is singular at every 0 x € x s n LtffD, 1). First,
we shall prove that 1 1
C*-) lim S Km(V) fcp(b/X(V+S)- x(sJ/)ds 1 dv = 0
m oo 9 lo >
for eve ry b ) O. By con'ti tion (6 2 ) and by 7. 13 we concl ude that
W't\bX, sup S l.fJ(b Ix(s+v) - x <'S)j) ds _ 0 as f 0
fV1 4 0
for every b > O. Hence
1
r Cf (b Ix (v+sl - x (10)1) ds W't'(bx,6') for )v/< c .
By l-periodi ci t of the fu.'1ction x, the same holds for t1-v J < 6
Hence
& 1 II
S Kmlvl{ <.f(blx(v+s)- xt.sll)dsl dv S KmlV)W bx,SJ dv" wr(?x,o)
000
and 1 { 1
f Km tv) Cf (b Ix(v+s) - XlS)/)dS } dv 41(; (bX, 5) .
1-6 0
Hence
1 1 1-0 1
S Km lv)J l f cp(b Ix(v+s)- X(S)\)dS } dv 2 t"(bX, 5) + S KmlvJ. S!f( lx (v+sJ -
o 0 6 10
1-0 1
Xls ) ds J dv 2wibx,o) + 2 f Km(v)dv r tp( 2b lx l s)l)dS .
o 0
Now, given €.> 0 and b ;::'0, we first choosi;j 0'> 0 so small that
,,1t-(bx,&)<{£ . Since cp satisfies (Ll , so \f(2b/Xls)l)ds -< 00 (see 8.13)
and according to the assulllQtion of singularity of(K n ) we may choose 1V1
so large that
1- 1
Kmlv)dv' i \f(2b/xlS)I)ds < i E for m>M.
Thus, 1 1
., tKmLv>{!<f!(bIX(v+SJ-XlS)/)dSrdV< E+if=c form>IvI,
whicil proves l*).
Applyifig (it") we are going now to show that {j n 1 is singular at x, i.e.
that J lxJ.......o as m_co (see 22.9). We have
m
JmlX)= 1 f l { S KnlU)f(atp-l(S KmlV) (bIX(urt +v)- xlurt)\)dV)dU} dt.
o 0 0 ,a 0
Let us take a, b> 0 and n fixed and let us choose an arbl trary ;;>.
163
By (tl 2 ) , there exists c>o (depending on e.) such that
(auJ c'('(uJ for u f .
We divide the interval [0, 1] in two sets A and B of values u :
1 1 t t
At ={u'P,l]: 'f- (f Km(vJ\f(blx(v+urt)- xCurt)l)d,?<EJ ' Bt =[0,11,A t .
We have 0
Sl Kn l u}q>[a \fY S KmLvHp(b Ix (v+urt) - x (urt)i}4 v ] du = r + f
o 0 At Bt
1 1
' (aE.) S K lu)du + c I K (uJ K (v/lf(b Ix(v+urt)- x (urt)1 )dVSdU .
o n B n m
t
1
\pCa EJ + c S K Lu f K (v) qJ(b I x(v+urt) - x (urtJI)dv duo
o n m -1 >
Let us denote v t = <f(aE.) , 0e. = v f irf t. If u lu) . Since is continuous,
we have vf, ° as O. Hence 6'e. -i' ° as !. -;0 0, by condi tion (w).
Moreover,
-1 u
&' 1{1 lu) If(ae.l l for u? v = If>(ae.).
Writing = lei... we thus have If (ul feu for u (a!) . Hence
1 1I. a ...) 1 1
J m LX) S fl ! (ae.) + c S Kn lU)r Km(V) If'(b IX(v+urt) - x (urt)\) dV] du dt
o 0
1 1 1
o!:. {<f(aE,) + c i KnlU)li Km(v) (blx(v+urt)- X(urt)l)dV) dU dt =
1 ",,1 1
= 5 6 + Q c Kmt'1 (bIX(V+S)- xls)t)dS dv.
1
Now, let '1»0 be arbitrary. We choose E> ° so small that df< 2
and then we take, by condi tion ( ), an index 111 so large that
1 1 ..2L
Km lv1 U \fCblx{v+s)- XlSJl)dsl dv < 2q c for ID»l¥i.
ThenJm(X)< form>l'Ii. Thus JmOCI-O as m (J)and so ( n)is singular
at X. Consequently, by 22.10, we have mL',x)"",x(.) as m cowith res-
pect to the F-norm I I in X i'
c O;,il'iJE NT S
1.
The theory of modular spaces was founded by Hidegoro Nakano, who
developed an extensive theory of such spaces in his two fundamental mo-
nographies H. Nakano [Jj, [2] in 1950 and 1951 (see also H. Nakano[3]).
The first attempt was made in H. Nal\:ano [11, where a modular is defined
in a universally continuous semi - ordered linear space X, i.e. a vector
lattice X such tlat for every system(x,d;\E-/\ of positive elements there
exists the "infimum" 1\ x . Namely, a modular 10: X-(O,co]is sup-
AE-/\ '" 'I
posed to satisfy the following conditions:
a) for every X€ X, x f 0, there exist a, b > 0 such that <f'(ax»)O and
tbx) < 00 ,
b) for all XEX, d..., ;;.0, there hOlds ( d...;8 x) (<plO(:XJ+ ((3x»),
c) if /xlfSlyl, then (X\ (yl ,
d) if x y, then <>(x+y) =: (X' + (y) ,
e) if 0 x). t x, )0.,/\ , then (xJ::; sup (x)o,).
).E-!\
The space X with such a modular was called 1959 by S. Yamamuro [1}, a
Nakano space .. H. Nal\:ano (11 developed a spectral theory and found inte-
gral representation for projectors acting on X ; a short description of
these ideas may be found in S. Yamamuro L11 (see also S. Koshi and T. Shi-
..JlOgal\:i (11). As regards concave modulars, see H. Nakano [4]. The theory of
Nakano spaces was further developed extensively by Japanese mathemati-
cians from the Hokkaido Dni versi ty i,n Sapporo, most ly belonging to the
school of Nakano, as I. Amemiya, T. Ando, J. Ishii, T. I to, S. Koshi,
T. Shimogaki, S. Yamamuro and others. Some further results concerning
modulars in semi-ordered spaces of functions are presented in 12.
In his second bookH. Na..l{ano[21( 78. p.204) introduced the notion
of a modular in a real vector space X as a functional : X-[O,coJsa-
tisfying tllli following conditions
1) lO)::; 0 , 2) l-x)::; (x),
3) for every x E: X .the re is a A '> 0 such that (>. x) < 00 ,
4) if ().x)=: 0 for every ;;.0, then x ::; 0,
5) lo!..x + Y) d. (X)+ y)for every x,yE:X, f1..,g, 1-0'd..+ ::; 1,
165
6) (X) = sup o X} for every x € X.
" 0 A<1
et us remark that such a is according to our terminology, a left-
Jontinuous semimodular (see Definitions 1.1 and 1.7), where means
that X = X . :I:he term "semimodular" is due to R. Lesniewicz [61, 2.1,
p. 245, and the definitions of right-continuity, left-continuity and
continuity (see Definition 1.7) together with Theorem 1.8 with s = 1
are given in J. Musielak[91, Prop.l.l, p.235-236. Let US still remark
that H. Nakano uses the term " normal modular " for what we call a left-
continuous modular (see e.g. H. Nakano[5J). H. Nakano (2] ( 81, p. 213}
defined also the norm II I/ from Theorem 1.5 in the convex case s = 1.
The first attempt to a theory of general modulars without any
reference to convexity was made in case of Orlicz spaces of fuactions
L\f!(O,l) and of sequences 11/1 by S. Mazur and W.Orlicz[l) (pp. 105 and
116), who defined in these spaces the F-norml from Theorem 1.5.
Defini tion 1.1 of a pseudomodular and of a modular in a general,
not necessari.ly convex case and in an arbitrary vector space X was
given by J. I1iusielak and W. Orlicz[3], 1.01, where also the F-norml
o
was generally introduced(1.21). The main idea lies here in the axiom 3
loLx +('Iy) :::: lx)+ (Y1 for x,yfX, rA.,f1,?-0, C1o-+(!>= 1 from Definition 1.1,
which is the proper tool for defining the F-norm i I .
Theorem 1.10 concerning the s-pseudonorm n ft<s t the so-called Amemiya
norm, is given in case s = 1 in the book of H. Nakano 1 (see 81, p.2H).
A similar theorem is true also in the non-convex case, with the same
, , , 0, 1 + (4x) ,
deflnl bon of the Amemiya F-norm : I x I = In; 0 J ' belng a
modular; namely, S. Koshi and T. ShimOgak i[21( 4, pp. 215,216) pro-
o
ved also in this case the inequali ties IXI iX I 2 1 xl e .
As regards the general notions of a modular and amodular space as
presented here, they are presented also in the book of S. Rolewicz [2]
in the convex case and, in the books of J. Iviusielak [13) , (14] in the ge-
neral case.
Further development was done 1968 again by H. Nakanot?l, (6], who
introduced a unified theory of modulars in vector spaces, embracing
both convex and non-convex case. Namely, H. Nakano[5) defines a mod
lar on a real vector space X as a functional : X [O,coJsuch that
1] l-X) = lx), 2) (ax) (bx) for 0 a b , )] /?o..x).... 0 as '). 0+,
166
4] lx+y) (( ('txJ+ (ry») for x,y ex with a p.O
independent of x and y .
The number r calls Nakano the character of the modular f4 ; obviously,
a convex modular is of character 2. Let us note that the condition 3]
states simply that X == X . Now, an element x X calls Nakano the null
element o:f X, i:f (Ax)= ° for all ). >0. If x == 0 is the only null
eleuent of X, Nakano calls a pure modular ; in our terminology, this
is exactly a semim.odular. There is developed by H. Nakano l51, (6Ja gene-
ral theory of such modulars including norms and conjugate modulars -
a notion we shall discuss in 2.
I t should be noted here that some other attempts to get a unified
theory of modulars or modular spaces, namely a theory of F-modular spa-
ces and of modular bases, will be presented in 4 and 6. Rece#!-y, a
theory of modular spaces oven non-archimedean fields was developed by
R. Urbanski [1]
Here, we shall notice still another approach towards a general the-
ory which was done 1978 by Ph. Turpin[5] . He introduced the notion of a
modular scale, generalizing that of a modular. 1Iodular scales are defi-
ned generally in preordered semigroups , that is in additive seilllgroups
X with neutral element ° and with partial order which does not need
to be compatible with the algebraic structure of X. For example, a real
or complex vaator space becomes a preordered semigroup with partial pre-
order x y defihed by x == ay for some scalar a withlal l. Now, the fol-
lowing definition is given:
Definition . (Ph. Turpin[5], p.332).
red semigroup X is a family == ( a) a>O
indexed on (0,00), such tat
A decreasing scale on a preorde-
of functionals qa : X_[O,oo) ,
[al atO)"" 0, (b] b(x) a(X)for xfX, O<a'b,
lc] if x,y E X and x y, then a lX} a lY) for every a >0.
A decreasing scale =CRa)a)O 0.Cl X is called a modular , if
lb'] a+b (x+y) aC.xl+ ( 't(y)for x,y X and a,b>O.
Of course, conditions [a1 and lb'J imply (b] .
In case of a sernigroup X of functions x : .f:l. lo,oo1, a modular
scale = a) a'jJ on X is called a -)!lodular , if
167
00
[b"J a = 2:' a <.cowith
n=l n
00
x 2: x , x ,x X, imply
ll=ln n
an';> 0 and
co
alx, L A_C.xn)'
ll= 1 -u
If X is a preordered semigroup, then Ph. Turpin [?] calls a functio-
nal r : X lO,(;IC)]a sub additive modular functional on X, if the family
f! = ( J a)O of functionals defined by a = r for every a';> 0 is a modular
scale on'X.
Now, if = (fa) a)O
X, then tJ ) 0 defined
a a)
is a decreasing scale on a pre ordered semigroup
by
(x):::: infta')o: a(X) bf,
where we put inf rJ :::: 00 , is again a decreasing sc e on X ; if, More-
over, C'?a)a)o is a modular scale (resp. o--modular scale) then(J 'b>O is
again a modular scale (resp. <t-modular scale) and J is a sub additive
Crespo (f-subadditive) modular functional on X (Ph. T urpin [5] , Prop.
1.2.2, p. 334). Adjoining with every modular scale ::::(fa)a>O the de-
creasing scale :::: ( a) a.>O defined by a(X):::: a- l a(x), the functional
I I defined by ,xl = JfLX)for xEX is a subadditive modular functional
on X called the associated subadditive modular functional on X. Now,
let X be a vector space with a modular scale ' then the associated
subaddi tive modular functional I I defines a group topology rf on X,
where sets fx X : a (.xl br, a, b >' 0 , constitute a basis of neighbour-
hoods of 0 in X. Let
X :::: {x GX: I>-x I O as A 0+ f,.
Definition {Ph. Turpin [5] , Def.2.1, p. 341). A generalized modular
vector space is a vector space with a modular scale = ( al a)O such
that X = X , with the associated liilear topology r
Ph. Turpin[5] develops a theory of such spaces with special empha-
size to generalized Orlicz spaces (see 7).
Among other attempts in the theory of modular spaces let us still
mention here that of ee Peng-Yee D], 1978 , who defines sectionally
modulared spaces as some modular spaces of sequences with a OJ nvex mo-
dular such that the "modularizing" procedure is applied to a certain
"sectional mapping" of the sequence x, involving blocks of components
of x.
168
As regards Examples 1.9, cOIDlllents to I - IV will be given in 7
and to V - in 10.
2.
The notion of a conjugate n dular is due to H. Nakano who introdu-
ced it 1950 in case of a convex modular in a semi-ordered vector spa-
ce (H. Nakano[l], 38, p. 138) and 1951 in case of a convex modular
in a vector space (H. Nakano[2J, 80, p. 209). Nakano calls the con-
jugate modular jt - the associated modular .Theorem 2.1 is version of
a result of H. Nakano. Theorem 2.5 is to be found in H. Hudzik, J. lilu-
sieJk:ak and ii. Urbanski [5]. The norm II II (see Theorem 2.6) is called
by H. Nakano the first , while /IIXfIJ= sup{fxfrXI: x","X;,llxltt It- the
second norm (see H. Nakano [1], 40, pp. 179 and 181). As regards the
general form of the conj ugate modular It' and the norm /I I' in a gene-
ralized Orlicz space, see 1heorem 13.19 and Corollary 13.20. A survrey
and further re sul ts concerning conj ugate modulars may be found in H.
Nakano [5] ; it was also the subject of investigations of many authors
mostly from the Nakano school in Sapporo (see also S. Yamamuro(l1
Some results concerning this problem in connection with modular conver-
gence will be presented in 5.
3.
The not'ion of anQ',s)- modular space was introduced 1974 by J. hIu-
sielak and J. .Peetre»1in case s = 1. In J. blusielakll0)tms notion is
generalized to the case of an(F,lfJ- modular space, where tp ;Ls a strict-
ly increasing, continuous function of ufO such that 1.f(0) 0, (ul-;Q)as
U -')00 and (uvl ')/ (u) tf'Lvl for u, v O. T he only diffe l'enCe from Defini-
tion 3.11 is in replacing the eq uali ty (A,s + (';, 10 = 1 in condition 3 0 by
10
)+ IfL ) 1. The F-norm I 1 k becomes now
/)( I ,k =' inf{u > 0 : ( ) k (<f(u»)f.
Results presented here until 3.13 are essentially taken from J. Ivlusie-
lak and J. Peetre ll). As regards Theorems 3.14 - 3.16, they may be found
in case 10 = 1 in T. ill. J dryka and J. Musielak [2].
4.
All the results of 4 are due to T.lVl. J dryka and J. I\lusielak(J],
169
1971. Some further development may be found in J. 1,lusielak[9].
5.
The notion of modular convergence in the sense, of Definition 5.1
was introduce.a first 1959 byJ. Ivlusielak andW. orliczL31(1.04, p. 50),
where also the condition B and SOlile elementary properties are presen-
ted. It should be noted here that the term "modular convergence" was
used already by H. Nakano [110 47, p. 205), where it meant exactly what
we call noI'lil convergence, i.e. convergence (A ex -x»_O as k-",oo for
. k
every ).. '> O. Let us still remark that 1967 H. H. Herda (1) conside-
red nonsy etric modulars, i.e. modulars without condition t-x): (x))
which lead to two kinds oj modular co,lvergence xk x if (>'(JSc-xJ)....o
and xk x if (Atx-X1l)...,O(for some ).. '?' OJ.
As regards examples of modular convergence, one should mention he-
re also the - convergence defined by A. Alexiewicz [11 as follows : let
11', and n II' be two norms in a vector space X (/I II 'may be also an F-norm),
then a sequence (xni of elements of X is called (- convergent to an ele-
ment xeX, if /lxn-xl/'...,o as n....,oo and (IIXn/l) is a bounded sequence. Now,
assuming ""xiI to be a monotone function of ;.. 0 for every x,,"X, R.
Lesniewicz observed that -convergence in X is equivalent to modular
convergence in X with a modular defined by
x ) II xII' if!lxll1t 1
<.:) 1. 00 if !lx/l::;;. 1 .
It should be still remarked, that applying Definition 5.4.1) of
boundedness to a sequence of ele.uents (x1J of X one can say that l is
- bounded , if k ..!.O for every sequ.ence of numbers Ek O. Thus, if
two modulars and e,r are defined in a vector space X, a modular
l- coflvergence of a seQuence (X k ) to an ele/"ent x may be defined by the
requirement that lX k ' be -bounded and x. The triple(X, , () may be
termed a two - modular space . Such two-modular spaces are introduced by
J. h'iusielak and A. Waszalc1.12].
The notion of -boundedness of a se'l uence of elemel1 ts of the modu-
lar space X and the equivalency of conditions 5.4(aland(blin case of
sequences in place of sets is given in J. l,iusielak and A. Waszak [12].
All the results of 5.4 _ 5.12 with s = 1 are obtained by H. Huddk ,
170
J. Iilusielak and R. Urbanski[5].
The contents of 5.13 - 5.17 are due to J. IJusielak (18) and were
further generalized to the case of two-modular convergence in J. Mu-
sielak (?O] and to non-linear operators Tv in J. liiusielak [19].
Categories of modular spaces are investigated re cently by T. Ku-
biak [1].
6.
The concept of a modular base and all the results presented in
6 are due to R. Lesniewicz [5:), [6], 1975 (see also R. Lesniewicz and
VI. Orlicz [1].
7.
First results concerning Orlicz spaces in case of a convex func-
tion f independent of the parai!!eter t were obtained in early 1930 th by
W. Orlicz (11, [21 and by Z. Birnba1.llil and W. Orlicz [2], in connection
with problel1lS concerning orthogoL1al expansions. The problems of these
spaces were later developed in 1950 th by three schools: that of H.
NMano in Sapporo, Japan, of 1,1. A. KrasnoselslQr and Ya. B. Rutickit'
in Voronez, S.S.S.R., and of A. C. Zaanen and VI.A.I. Luxemburg in
Leyden, The Netherlands. As regards the Sapporo school, its direction
was to develop a general theo:cy of I,lodular spaces. Tlri.s direc.tion, pre-
sented 1950 by H. Nakano (1) in his book !ll iodulared seuri.-ordered linear
spaces", was scatched in our COlfll!!ents to 1. The Voronez school deve-
loped mostly applications to integral equations ; he influence of this
group on the future development of the theory of Orlicz spaces owned
also a lot to the fact of the first monographic exposition of this
tlleory by Ill. A. Krasnoselsktt and Ya. B. Ruticki [111958 in the book
"Convex lUl1ctions and Orlicz spaces". As regards references from the
period before 1958, the reader is sucgested to consult this monography.
The results of the Leyden school went in a more geL1eral direction of
Banach function spaces. A theory of such spaces was developed 1955 by
Vi. A. I. Luxemburg (lJ in lri.s Doctor Thesis "Banach Function Spaces" and
followed by a series of papers of W. A. I. Luxemburg and A. C. Zaanen
[11-[15] and of W. A. I. Lu.'(emburgl21-[41in the decade 1956 - 1966.
171
As regards a systematic treatment of Orlicz spaces, it was done 1960
by A. C. Zaanen[l} in his book "Linear Analysis". A further develop-
ment in the direction of Banach function spaces was done 1974 by p. P.
Zabreiko[l]. Much later, 1977 and 1979, Orlicz space's of sequences and
of functions were presented from the point, of view of bases, isomorphi-
sms and Boyd indices in the books of J. Lindenstrauss and L. Tzafriri
[11, (2), "Classical Banach Spaces I,ll".
The case of a function independent of the parameter t but not
necessarily convex, together with the F-norm I I in this case, was con-
sidered 1958 by S. Mazur and Vi. Orlicz (:1] in both cases : spaces L'1 of
Lebesgue integrable fu.o.ctions in [0, lJ and spaces 1<1 of sequences. A ge-
neral theory in case of L was developed 1960 - 1961 by W. Matuszewska
and Vi. Orlicz (2) and by Vi. 11latuszewska [11 , (2]; an exposition may be
found in W. Orlicz 13]. In direct connection with the paper of S. l{lazur
and W. Orlicz [1] stand the results of So Rolewicz [11, concerning mostly
vvith local boundedness, existence of an s-homogeneous norm and existen-
ce of nontrivial linear continuous functionals in these spaces. S. Ro-
lewicz uses for Orlicz spaces the notation (LI or tpll) in place of Llf
or l . The same notation is used later. in early 1970 th , by P.L. Uli-
anov [11- [6], where the author investigates the Orlicz class L of Le-
besguef-inte;:;rable functions in (0, 1J, denoted by him / 0 Supposing
: R R+ to be even, nonnegative, finite, nondecreasing in(O'c9and such
th-lt (u)"""oo as t.....co, P.L. Ulianov investigates the influence of the
conditions lim lPlt)::: <f(0)::: 0 and lli t )1) < 00 , and also of the con-
t=illt. 1fl2 u j t . 00 "1 .nll \
di tion f.A 2) (u) 0( 00 on the behavicur of the classes ,. ,; let us
rem k that the second and third condition were introduced and investi-
gated already 1930 by Z. Birnbaum and W. Orlicz l1]. Introducing a pseu-
dometric d(x,y)=j fOX(tl- yttJ/}dt in CftL), P.L. Ulianov considers the
problems of density in CP(;L' of some subclasses of 'f(;L) and the problem of
basic property of the Faber-Schauder system inqg,} . He investigates also
various topologies and the problem of existence of a countable base of
neighbourhoods of zero in (L) 0 It should be also noted that P. L. Ul-
ianov [51 introduces a class (;LJ which he calls conj ugate to i.p(;L), de-
fined as the collection of all measurable functions y such that
l\(>(lx(t)+ Yl:t)l)dt < 00 for every XE- c..LJ. As regards further biblio-
o
172
graphy in this direction, it may be found in the above quoted papers of
P.L. Ulianov (see also D. Stachowiak - U-nilka (1) -(31 in case of. an abs-
tract measure space, and S. Gnilka and D. StachoV{ ak - Gnilka(l) as re-
gards an abstract formulation of the conj ugate class lp lL)). Let us still
remark some results connected with appro:x:iJmation theoT![ in Orlicz spa-
ces, as a Bernstein-type inequality for fractional derivatives in Or-
licz spac& L -oo,oo) or Orlicz space L of 21f-periodic functions with
an s-convex function 'f ' or :Bernstein and Privalov-type inequalities
for trigonometric polynomials in Orlicz spaces, obtained by H. Musie-
lak (1) (Theorem 3, p. 49 - Bernstein inequality for fractional deri-
vatives ), [21 (for someotre r results concerning estimation by If-moduli
of smoGthness, see H. Musielak[3 . As regards Bernstein inequality for
trigonometric polynomials in generalized Orlicz spaces, see H. lliusie-
lak [4].
A very important tool in dealing with Orlicz spaces, particularly
in connection vnth problems of interpolation of operators (see 14)
are the indices of Orlicz spaces. They were introduced by W. Matuszew-
ska and W. Orlicz [21, (3) (see also Vi. hiatuszewska [)1) in the following
manner. Let for a -function ,
, !u.vJ - ---:--
(ul= hm (v) ¥U)= hm br)'
v oo v--,oo
Then the Hatuszewska - Odicz indices and <5"1f of the function If are
d efiue d as
lOt» ::: lim
T 1 (u-tOQ
log lU\
log u
) _ log J;plu/ = inf
= sup log u ' (J, = 1 < l m 10g u U)O
u)1 u
log lU ) .
log u '
they exist for every -function without parameter, although may be
infinite (see also PoL. Butzer and F. Feh r (11). Besides these indices,
-functions If satisfying the condition (Ai ;for all u (see 8) , the
function
q>(uv)
(U):: sup [I)(y:'
11"70 "t'
is also of importance; it is always nondecreasing, left-continuous
for u,v O and submultiplicative and in case of q> convex, 't is also
p
convex (see H. £iiU:Sielak[21, LemmA.l,pp. 57 - 58). In case CPlu)=!UI,
- p
p>O, we have obviously hplU)::: lU)= 't(u)=lul
dices 10 and (j are invariants of the relation
, u?-O,
and 10 = (J" = p. The in-
1 f 'f
....... between -functions,
173
defined byW. MatuszevvskaQ]as follows: fl 2 means that a l lf1(a 2 1..1)
'f2(U) bl'llb2u) for all 1..1,+0, with some constants a l ,a 2 ,b l ,b 2 >0
(see also 8, condition 8.2l2l)) .
The notions of Natuszewska-Orlicz indices are generalized to the
case of symmetric function spaces by D.W. Boyd (1) , (2) and by 1,1. Zippin
[1] in connection with problellls of interpolation of operators (for a
detailed study and further references, see L. I,laligranda(2JJ.
One should mention also investigation of upper mean values in lat-
tices of measurable functions, originated 1966 by Wo Orlicz [7J.
A further development of the theory of Orlicz spaces consisting in
application of a function depending on a par81Ileter t, l.ftt, 1..1), in pla-
ce of a function \flul , was originated 1950 by H. Nakano [11 (APpendix I,
p. 281) and developed 1959 by J. Musielak and W. Orlicz[3] as an example
of modular spaces; in fact, under some restriction on a modular senn-
ordered vector space, it is always a generalized Orlicz space (see 1 .
Generalized Orlicz spaces of functions 1d of sequences are also called
modular function spaces (see J. Ishii [1)) and modular sequence space s
(see J. Lindenstrauss and L. Tzafriri (11), or p1usielak - Orlicz spaces (see
e.g. Ph. Turpinl51). There are many possible generalizations of the class
't from Defini tion 7.1, which furnish more general spaces Lip. These
generalizations consist in replacing the continuity assumption on (t,uJ
with respect to 1..1 by a weaker assumption of one-sided continuity, and
in allowing the function If to take infinite values. I shall describe he-
..,
re briefly two of such possibilities, by I.V. Sragin and by Ph. Turpin.
V'
I.P. Sragin(1), 1970, int-:roduces the notion of a pre-genfunction and of
- -
a genfunction if . Namely, a pre - genf'unction is a function : '1x R+ R+,
where R+ = [0,00.], if it is measurable with respect to t E;.,Sl. for every
u?O and nondecreasing and left-continuous with respect to u,"R+ in the
sense of topology in R for a.e. t ..5l , and if (f(0, t) is integrable
on .!l. . A pre-genfunction is called a genfunction , if qJ(O,t)= 0 and
,¥(OO,t)7 0 for a.e. tE-..D. , and \P(+O, t)= 0 a.e. on the set of t ose
t e.st which satisfy the inequality sup{u: I.f(u,t) < CD > O. loP. Sragin[11
investigates properties of generalized Orlicz spaces L , classes L and
spaces Elf in case when tp is a genfunction. Ph. Turpin [5], 1978, consi-
ders generalized modular spaces as an example of his theory of modular
174
scales, which were described in the comments to 1. He considers fun-
ctions qJ:..Q. )( R+ R+ measurable with respect to t .S2. for all u 0
and increasing, left-continuous as a function of u O , equal to 0 and
continuous at u = 0 for all t J'2. , which he calls Musielak-Orlicz fun-
ctions (or Orlicz functions in case when , t) does not depend on t).
Now, taking
rex):: S !I(t, Ix G;!I) dM.
x "SlT r
and alxl:: r( ) , :: Cfa)a)O becomes a (J-.modular scale on X and the
associated subaddi ti ve modular functional j., is equal to the F-norm
'.11" generated by the modular r on X :: X .
In the present exposition in 7 and 8 we limit ourselves to the
case of lplt, u) - a 1-function for every t e..Q , in place of a. e. t e-J2.
this is indeed no restriction, because we may exclude an exceptional
subset of -'>- of measure 0 without changing the modular . Also the
assumption of continuity in place of left-continuity of If with respect
to u does not affect the generality much. Finally, supposing tp to ta-
ke only finite values We exclude the space LOO which in cese of r- finite
may be considered as Llf wi th cu)= 00 for sufficiently large u. How-
ever, restricting ourselves to functions lpe- we avoid a number of
technical difficulties which do not contribute much to the general dea
of spaces L'P.
The same reason of simplifying the course is responsible for restri-
cting ourselves to the case of scalar-valued functions, mentioning the
case of fQ ctions x with values in a Banach space at this place only.
The theory of such spaces was developed first in case of a function
wi thout parameter t, and the modular over the space X of measurable
functions defined in...Q.. and with values in a Bana:ch space E is defined
as tX)::SI/1C1lxC. U)d . Next, there were considered functions 4': E _ R+
..tl. - - . th-
in place of {o: R -+:Ii satisfying sui table assumptions, however, Wl
1 + +
out supposing q> to be a superposition of a function on R+ withthe norm
II lIon E only. The respective spaces are called by B. Turett[2], 1980,
Fenchel - Orlicz spaces . The paper of B. T.urett l2J is an expository one
and the reader may find there the solutions of problems of completeness,
linear continuous functionals, strict convexity rotundity and F chet
differentials, together with a bibliography concerning such spaces.
175
Fenchel-Orlicz spaces over non-archimedean fields were intraduced 1981
by F. Fuentes and F.L. Hern&1dez [1] .
Let us still remark that instead of "strong" Orlicz spaces of vec-
tor-valued functions one can also consider "weak" Orlicz spaces of fun-
ctions x wi th values in a Banach space E such that x""x E L for every
linear continuous functional x1\o over E (see J. f,Iusielak and W. Orlicz
[6], where the connection between such -Qpaces and some spaces of set
functions V lAI = fAx (t] dtA- with finite generalized Rie sz variation is
established) .
The most general case consists in considering the function : ..Q.X E
--+R+ ' where E is a Banach space. A systematic study of this case may
be found in two papers of A. Kozek [11, l21 (1977 and 1979), where one may
find also historical remarks, applications to the theory of subdiffe-
rentials and an extensive bibliography (see also N. Dinculeanu[1J-[4],
V.R. Portnov ll] , 11.S. Skaaf [1], [21, A.D. Ioffe [11 , [21, J.Y.T. Woo [1],
Vi. Sch8rner ll1, F.L. HernMdez (1). Let us outline now the way of con-
structing a modular defining the generalized Orlicz space by A. Ko-
zek [11 .
Let(J1.,I,r-J be a measure apace with a nonnegative, 6-finite, comple-
te measure r- ' E - a Banach space, and J'?J - the (j-algebra of Borel sub-
sets of E. There are considered N II - functions , i. e. functions \of : .ilx E
R = [0, 00) satisfying the following conditions :
+
(i) there is a set oE!: with fA-(.!Lo) = 0 such that
1. q>(t,O)= 0, 2. \flt,-x/= If(t,x) for every xe- E ,
3. I.f(t,x) is a convex function of x E ,
4. tt, is a lower sellJicontinuous function of x '-E for every
1; f...tt ,..a. 0
lii) If is a 2,:xP.a -measurable function.
If, moreover, there exist functions rA,'>'-: ..a..'..o.o""'(O,ro) such that
l{1(t ,x) 0{ It) for x € E, "XIf "0 It) , then the UN-function tp is called
an H'- function . Finally, if E is a separable, reflexive Banach space,
'P is anU'-function and there are functions S' : ..n.......no-.(Q,oo)such
that cp{t, x) 6 '7(t) for x €;E ,lIxll 'll t) , then tp is called an N- function .
Supposing \p to be an H" -function, denoting by X .the class of all
L-measurable functions Jet<): ...$\.-. E with equality -a.e. and by L!-
176
the class of all x t' X for which
Orlicz space Ltf
contain:Lng Lip .
o
Generalized Orlicz spaces of vector-valued functions in case when
lX):= S 'f(t, xlt»)dj4- < CD , the genera-
::a.
is defined as the smallest linear subspace of X
E is a topological vector space were defined and investigated by M. Wi-
s;ta[l) ,(2].
The theory of generalized Orlicz spaces with values in a Banach
space is in close connection with convex analysis (see R.T. Rockafel-
lar [11, l21), the function <p being a specialization of a convex normal
integrand.
Let us remark still that in case of usual Orlicz spaces, there was
proved for a sequence of convex -functions l<fnl and a convex l{>-function
that Ux :lIxl for the respective norms in spaces :d'.. and Lf41 in an
interval ta,b1 for every measurable bounded function in [a, b), if and
only if I.\In Cu) \f ) for u") 0 ( W. Orlicz [6J, Th.3 .1).
Products of Orlicz spaces were considered by G. Dankert III and
later by C. Bylka and W. Orlicz[21.
Orlicz spaces based on a finitely additive measure are considered
in W. Orlicz [81.
Orl':icz spaces Llf were gene-ralized also, replacing one measure fA by
a family of measures M-t" with "C'e-f , as show Examples 1.9,II-IV. Exam-
ple'1.9.II1 with a . = 1 for i n, a. := 0 for i»n, was investigated
n n 'n
by J. :Musiel ak and W. Or1icz D1 , L51, and in the general case by J. lIlu-
sielak lJ1 and by A. Waszak t!l-[5]. Example 1.9.III in case when the mea-
sures f-'lt:" are absolutely continuous with respect to one measure j\'L, i.e.
LJq = sup S alt, )crOXltH) dt'«- ' was considered by A. Waszak(2], and the
't"oA,
general case by A. Waszak(5) , L. Drevmowski and A. Kaminskall1 and A.
KaminskaL21 (the last two for functions with values in a Banach space).
Example 1.9.IV in case of a pseudomodular \X}= TL..; o S 1(\x ltJI> dt
was investigated by J. Albrycht [11, [2], J. Iilusielak and W. Orlicz [3]
and by H. Musielak ll); for the general case, see J. lVlusielak (8) and
J. JYlusielak and A. Waszak [41 (see also 17) .
Coming over to the approximation problems considered in 7.11 -7.24,
let US first remark that the notion of -boundedness from Definition
7.11 was considered first by A. Kaminska [l1l see also 9, Theorem 9. 6 ) .
177
Theorems 7.15 and 7.16 were first proved by H. Hudzik, J. Musielak and
R. Urbanski (11 in case when V = N '" 1,2,... and 1l() is the filter of
complements of finite subsets of N. The results of 7.11 - 7.24 are to
be found in J. Musielak [18]. Let us still remark that Theorems 7.13 -
7.16 are generalized to the case of two-modular spaces (see the com-
ments to 5) by J. 11usielak [20J, and to the case of non-linear trans-
forms Tw by J. l'ilusLelak [19].
' 8
In case of convex 'f-functions without paramete r t, some relations
of the form presented in Theorem 8.2 we re considered alre ady 1931 by
Z. Birnbaum and W. Orlicz [2] and developed in the book of I'il.A. Kras-
noselskiC and Ya.B. Rutickir 11]. If 'f is a f-function, not necessarily
convex, indepen.dent of the parameter t, a study of these relations and
applications to embedding theorems for Orlicz classes and Orlicz spaces
being special cases of Theorems 8.4 and 8.5, was done 1961 by W. Matu-
szewska ll], l l (see also the comments to 19).
The first result concerning convex functions lop tp is due to J.
Ishii [1] in 1959, who obtained Theorems 8.5 and 8.11 as regards inclu-
sion of spaces, even if lflt, u) may be equal to 0 for some u::> 0 and may
assume infinite vaJ.ues. S. Koshi and T. Shimogaki(ll in 1961 remark
that the assumption of convexity of the functions is not essential here;
they also conclude Theorem 8.13 with condition (6 2 \ for functions de-
pending on a parameter, in case of Lip with Lebesgue measure in [0,1} and
in case of 1'Y .
For l sults concerning superpositions in generalized Orlicz spaces
and in particular the condition 8.12( ), see also A.D. Ioffe(31.
A generalization of the above results to the case of functions with
values in a Banach space, supposing the measure r- to be atomless, was
done 1979 by A. Kozek [2]. Supposing If and I./' be N" -functions (see com-
ments to 7), A. Kozek [ 1' Theorem 1.7 states first that the condi-
tion 8.1 (1) implies L CL'b and secondly, that if fA is atomless, «(t,.)
is continuous at 0 for a.e. t ..n.. and E is separable, then L C::L im-
plies 8.1 (1). The proof of Theorem 8.4 together with Lemma 8.3 we have
adapted here just from A. Kozek l21. Restriction to scalar-valued func-
178
tions does not change the idea of proof and replacing the space of reals
by a separable Banach space one obtains easily the vector-valued case.
As regards the case of generalized Orlicz spaces ltj) of qequences,
we adopt here a general approach given 1976 by LV. Sragin [2] . In fact,
the simple and elegant method developed by IoV. Sragin may be applied
to various problems concerning spaces of sequences.
The importance of the conditions (.6 2 ) and (6"2) from Theorem 8.13
Will be seen also later in this book; the original condition (D 2 ) for
large u>O without parameter (see Remark 8.15(21) was given alre dy by
W. Orlicz [1J .
Theorems 8.17 and 8.18 in case of f indepcndent of the parameter t
are due to W. Matuszewska and W. Orlicz[l1in 1961:, the conditions in
Theorem 8.18. are then also necessary.
There is still another approach to the problem of embedding of ge-
neralized Orlicz spaces given by Ph. Turpin [11, [3] , [4) in connection
with his galb theor".f. The notion of a galb GlE) of a linear topologi-
cal space E defines Ph. Turpin (11, 1973, as the vector space of seq uen-
ces )..== (>yJ of scalars n such that for every neighbourhood U of 0 in E
there is a neighbourhood V of 0 in E such that 2 A V C U. He defines
n>.o n
convergence of a filter in G(E, and applies this notion to investiga-
tion of products of Orlicz spaces and local boundedness of Orlicz spa-
ces in case of a concave function cp . In 1976 Ph. Turpin(3) applies
galbs in order to obtain embedding theorems for generalized Orlicz spa-
ces of scalar-valued functions, proving the following result:
Theorem. (Ph. Turpin,L3J). La) Let fA' be a:tollless and let >.,0{ ';> O. Let
us denote
ltl= sup {1l 0: \jI(t,?\.u)oltp(t,u)l,
, ).,,, 1-
where we take sup /J ::: O. Then L'PC L't', if and only if, there exist ,\,;> 0,
ot.< (X) and 10 '> 0 such that
S 'f'(t,s oO (t») dt < 00 .
.A 1",<11-
lb' Let fA- be purely atomic on...Q. and let ).,£,cl. > O. Let us write
"t;f t)= sup\.u O: 4'£t,u) E. and 't(t, A u)? o( (t,un,
,I',
where we take sup /J = O. Then l<.fc IIV, if and only if, there exist
E, X,;> 0, 01. < (X) and 10';> 0 such that
:L 'ttt ,s't"E Q\. (t») <00.
tE /'100.,
179
A survey of the galb theory with applications to Orlicz spaces, both
atomless and purely atomic, with emphasis put on embedding problems and
products and unions of such spaces, may be found in Ph. Turpin [4J, where
also a wide bibliography is given.
Let us still remark that following the method of Ph. Turpin, :M. Wi-
sla[3) extends embedding theorems to the case of generalized Orlicz
spaces of functions with values in a topological vector space E.
Finally, it should be noted that embedding theorems in vector-val-
ued case with a Banach space E, both in the atomless as '1'1811 as in the
puxely atomic case, are proved 1981 by A. Kaminska(21 in the more gene-
ral case of a family of measures (ftd , 'L'E-- r , where the modular is gi-
ven by the formula
(x)== s p 1 'P(t, (t» d
(see Example 109.II and comments to 7). The proof of A. Kaminska uses
the argwnents of A. Kozek [2] in case of an atomless submeasure vtA) ==
v
sup 't(!l.), and those of I. V. Sragin [.21 in case when...Q consists of a
countable number of atoms.
9
A systematic treatment of criteria for conditional compactness of
sets in spaces E\fI in case of an N-function tp without parameter and a
compact set ...Q. Co R n , may be found in the book of M.A. Krasnoselskil:
and Ya.B. Huticki! L1J, Chapter II, 11. In fact, our Theorems 9.3,
9.12, 9.14 and 9.11 correspond to Theorems 11.3, 11.2, 11.4 and Lemma
11.1 from the above book indeed, Theorem 9.12 for an N-function
wi thout parameter satisfying the condition (f:::.2) may be traced back as
far as to T. Takahashi L1], 1934 . Crt teria for conditional compact-
ness in Orlicz spaces may be found also in the book of A. Kufner, O.
John and S. Fucik [1], 3.14, pp. 172 - 178.
Investigation of criteria for conditional compactness in subspaces
E({I of generalized Orlicz spaces were originated by R. Pluciennik(21 in
1979 and then continued by A. Kaminska and R. Pluciennik [11 in 1980
and by A. Kaminska(1] in 1981 . The results presented in 9 are to be
found in the above papers. Namely, Lemma 9.2 and Theorems 9.3, 9.6 in
the more general case of vector-valued functions x with values in a
180
real Banach space, but for convex , are to be found in A. Kaminska (11
as Lemma 1.1, Theorem 1.2 and Theorem 1.6. Theorems 9.10, 9.11 and suf-
ficiency part of Theorem 9.12 are in A. Kaminska and R. Plucie nnik[1]
as Lemmas 2.3,2.2,2.4 and Theorem 2.9. Necessity part of Theorem 9.12
may be found in A. Kaminska (1), Theorem 3.1. Theorem 9.14 is from A.
Kaminska and R. Pluciennik[l), Theorem 2.10.
Let us still remark that some results concerning the translation.
operator in Elf may be found also in T .1,1. Jliidryla and J. lYiusielak.[61.
As regards applications to problems of compactness of operators
in generalized Orlicz spaces and to linear integral equations and in-
tegral equations of the type of Ha.IIUllerstein and. of Uryson in these
sp ces, one should consult R. Plucieanik [31.
10
A systematic-study of generalized Orlicz-Sobolev spaces i due to
H. Hudzik, who developed a theory of these spaces in a series of pa-
pers H. Hudzikll1-17] written 1 76 - 1979. Definition 10.1 and Theorem
10.2 is to be found in H. Hudzik [11 ..J:heorem 10.3 is proved in H. Hu.-
dzik (5). Applying a method of A. Kozek (2], H. Hudzik proved also a con-
verse to Theorem 10.3, namely:
Theorem (H. :audzik [7J). Let If,'r be N-functions (see Definition
13.1) satisfying both the condition )of 8.13 and the following con-
dition : for every compact set B C..Q., there is a set A.scA of Lebesgue
measure zero, a positive constant C B and a positive function inte-
grable in..Q.. such that u C:(3 min[\flt,u), 'ttt,u)J+ (t)for u4 0 ,
t B'A.s' Finally, let W (..a.)CW {..o.), where W l..Q.) is dense in (...o..)
with respect to the normll'Qmk . Then there holds the condition 10.3l2J.
"'f'(..st)
To most important problems in generalized Orlicz-Sobolev spaces
W l } belongs the problem under what conditions the set C t. ) of infi-
ni tely differentiable fllilctions with compact sU[.JQort in-'l is dense in
W (A) . In the case of lu) = lu ,p, p;;> 1, this was answered by V.I .Bu.d-
renkov lll. 'rhe answer in case of W lRn) was given by H. Hudzik f.?1 and,
applying Budrenlwv's method, was extended by H. Hudzik l4]to the case
of Wq(.Q.). Let i(t)= K el/0t -1)forltl 1, ilt,= 0 for!tl;;>l, where\tl=
!(t l ,...,tJI =lt + ... + t )1/2 and K';;>O is chosen in such a man-
181
ner that SRni It) dt ::: 1. Let
-!La = t&...a: inf' It-sl at
sEb.Q
for any open set ...Q.. c:: Rn , where' a> 0 and 1)J2. is the boundary of...Q. .
Moreover, let
X ttl'<ll a- n S i ( t-S ) x(s/ds
a ..a al
for any function x locally integrable in.n.. H. Hudzik considers (4]
the following property
( ) Iff(t,lxalt)l)dt CS<P(t,lxltl/)dt forO<a 1,
arb t:t'ary Lebesgue measurable set A c....Q. and any measurable function x,
wi th an absolute constant C> O. For example, any convex If -function <r
wi thout parameter satisfies P . The following stateroont holds :
Theorem (H. Hudzik [4] la)Let !.pC- cf be convex, locally integrable,
and let it satisfy the conditions inf \p(t, 1) :;:. 0 and sup 1/;; 2 1 < 00
te:.n. U)O,u ..,k
a.e. in . Moreover, let <p satisfy the property P . Then if x'Mtf{...Q)
o -k 0
and 1 a 1\ xR:rt(.Q.." ) is bounded as a _ 0+, then 2 there erlsts a
00
ssquence (yJ of ystCo(,.Q.) such that 1(X - YsIJw ) O as s.....oo.
0» Let'f be an N-function(see Definition 13.11 such that tf(2u1 KifQtl
for all u O with a constant K >0. Let K lu1= Ull and et the in-
-1 ..?"l tJ1 u/ -1 n
verse K of the function K sat sfy the cond t on J d K (.U J < 00.
o
If :X:EW e,.n.) satisfies the condition 10.6(b), then it satisfies also
the condition 10.6 (al .
For further properties of the spaces W l....a) as separability, general
form of linear continuous functionals and reflexivity, the reader is
referred to H. HudzikL7].
As regards properties and applications of Sobolev spaces and other
spAces of differentiable functions, see the monography by A. Kufner,
O. John and S. Fuci k l1].
The origin of considerations concerning generalized variation lies
in the paper of N. Wiener [1], 1924, con-cerning quadratic variation. In
the late 1930-th this paper was followed by a series of papers of L.C.
Young [1], (21, and E.R. Love and L.C. Young (1], (21, finally 1951 by
E .R. Love l11, where generalized variation with power P 1, the respec-
tive generalized absolute continuity E.R. Love (1) and application to
182
Riemann-Stieltjes integrals and to Fourier series are given. Limit pro-
perties of these families of spaces were investigated by J. Musielak
and Z. Semadeni (1], where one can find also Example 10.10.1 in case of
p
lfC-u)=iul , p 1. In case of a function 'P<..u) without parameter, varia-
tion V\p and the respective modular space '\I\f were defined and investi-
gated by J. Musielak and W. Orlicz (1), 1957,l2], 1959, as well as the
notion of If -abs(}lutely continuous functions. Let us quote the last no-
tion here. A function x defined in la, b) is called - absolutely continu.-
m
..Ql!2, if for every E. '/ 0 there exists a ;- > 0 such that £1 CP (Ix ({\>
- x L ) I) < E for all finite sets of non-overlapping intervals (ft_,A-)
m
C [a, b) , i '" 1,2,...,m, such that i 1q>(t'i- oli> < 6 . Denoting by ACcp
the space of all functions x in la, bl such that x (a)'" 0 and kx is
-absolutely continuous in (a, b) for some k >0, AC'f is a closed subspace
.4- -1
of VIf . If q> is convex and u <fLul...., 00 as u oo, then AC((I is a sepa-
rable with respect to Vq>-convergence subspace of the nonseparable space
1J , and there is easily possible to establish approximation processes
for functions from AC . Also, general form of linear, continuous func-
tionals over ACIp may be obtained in the form of a Riemann-Stiel tjes in-
tegral, assuming to be a convex N-function without parameter satisfy-
ing the condition (t.1. 2 ) for small u,,>O , i.e.(52) (see J. liiusielak and W.
Orlicz l2]). Examples 10.10.11 and III for If without parameter may be
found in J. Musielak and W. Orlicz [3]. Linear continuous functionels
over these spaces were investigated further by R. Lesniewicz and W. Or-
licz [2] . Results for spaces VIf were obtained also by H.H. Herda[2].
Spaces of sequences of a fini te -variation were considered by J. jylu-
sielak(2) .The parameter t in lt,u) in connection with spaces 1J1{I and
ACtp was introduced and investigated in a series of papers in years 1976-
11178 by S. GnB:ka (1)-[6); in particular, Property 10.7(e)was proved in
case of without parameter by J. r.iusielak and W. Orlicz [.21, and in the
general case by S. Gnilkal1J.
When speaking about generalized variation let us still mention the
interesting notion of q-integral -variation introduce.d by A.p. Tere-
hin [1) in 1972 in case of lu)::: I ul p , p 1, and developed in the gene-
ral case by A. Borucka-Cieslewicz [11 , (2]. If x is a function defined in
la,b1 and a c<d b, 1 q<ro, then we write
183
'"'q (c,d;x)= sup (dS-h I x(t+hl- X(t)jq dt )1/ q
O<h<d-c c
and we define the ct.- integral tf- variation of x by the formula
m
V tl'J q (X)= 10 up r {f.I ( u ( t " l ' t , ; x») ,
T ' 1'( i 1 1 . '1 -
where 1T: a = to < t 1 < ... <: t m = b runs over all partitions of the in-
terval [a, b] .
11
Uniform conve:x:i ty of an Orlicz space LI{> wi th Luxemburg norm II I
(see 1.5), with,!, without parameter, was proved by W.A.I. Luxemburg (1),
P. 65, under the as sumption (1::. 2 ) as in Theorem 11.6, with h l tl == 0, 10 uppo-
sing If to be uniformly convex (in the terminology used by Luxemburg,
strictly convex). The method of proof applied for T"heorem 11.6 is just
that of Luxemburg, replacing Orlicz space by a generalized Orlicz spa-
ce; let us here remark in connection with Definition 11.5 that a notion
of uniformly convex modular was considered already 1950 by H. Nakano
[11( 50, p. 217)and[2J( 87, p. 226). Uniform convexity of the space
LIf' with \V<:t, u) = f ul P (t), with 1..( K2 P l t) K1 < CD was proved by H. Nakano
(2). Necessary and sufficient conditions for uniform convexity of if
wi th Orlicz norm U II (see 13 .11) with CO without parameter, were gi-
,o T
ven by H.W. Milnes [11 in case of an atomleBs measure. V.A. Akimovic[11
has shown that uniform convexity of (wi thout parameter) and (Ll 2) wi th
hlt):: 0 are sufficient for uniform convexity of Ltf' both wi th resJ?e ct
to the norm /I /1411 as well as the norm II II . In case of the norm 1/ II.
'I , 0 '<';
and without parameter necessary and sufficient conditions are gi-
ven by A. Kaminska 1)1 :
Theorem CA. Kaminska[3]). Let f- be atomless and let cp be a convex
-function wi hout parameter.
(a) Let ttl...Q.):: co, then LI.fI with Luxemburg norm II I is uniformly
convex, if and only if, If' is uniformly convex on R+ and satisfies the
condition
( , 6 2 ) Cf(2U) K \flu) for arbitrary u 0, with a K> O.
tb) Let flAl.J:2.) <. 00, then L'f with Luxemburg norm 1\ II is uniformly con-
vex, if and only if, <p is strictly convex on R+, uniformly convex for
u u with some u '> 0 and satisfies the condition
o 0
184
(A 2 ) 'f(2u) K (u) for u u, with some K70 aw;lu '70.
, 0 0
In case of f'\(..a.)= 00 , the sufficiency of this theorem was proved
by W .A.I. Luxemburg [1], and in f{l.!l) < 00, properties of strictly con-
vex and uniformly convex functions are applied. In the necessity part
of the proof, the conditions (° 2 , A) , (6 2 ) and strict convexity of t.p
are obtained, applying B. Turett [1]; uniform convexi ty of tp for u uo
is proved indirectly.
In case of a purely atomic measure, there holds the following
Theorem (A. Kamiri.ska[3]). Let lp be a convex f-functionoowi thout para-
meter. The Orlicz sequence space Ilf with modular (x)= t;1 cpntnl), x =
ltn) l<P, is uniformly convex, if and only if, cp is strictly convex on
the segment [0, uol, where If'(u o ):=: ' uniformly convex for sufficiently
small u 0 and satisfies the condition
(6 2 ) lpt2u) 'K (u) for o u uo' with some K>O and uo>O.
H. Hudzik L9] obtained sufficient conditions and necessary conditions
in order that a generalized Orlicz space of vector-valued functions be
uniformly convex. In fact, these results are formulated for product of
two generalized Orlicz spaces with Luxemburg norms. Let 1 ,1: 1 'fl) and
ta2, ,r-2) be two measure spaces withe-finite, atomless and complete
measures 1"-1'/1-2 and let tf l ,lf2 E be convex functions of u O for every
t E-..S'2.. 1 resp. se:..s1 2 . Let X and Y be Banach spaces with norms II II andll 01,
respectively, and let
l(xj= lfl(t,l(x.--lt)!l) d]\..1. 1 , 2ly)=..t..CP2(S'II(YlS)lIIldjlA..2'
H. Hudzik cons ders the Carte sian product L = L 't(J21;X) X L '-(..Q2;Y) of
Orlicz spaces of vector-valued strongly measurable functions from ..J1. 1
to X and from 2 to Y , with Luxemburg norm IIl' generated by the convex
modular e((x,y) = 1 tx)+ 2(:y)' Under the above notation there is proved
the following result:
Theorem (H. Hudzik {9J). tal Le,t (fl '9'2 <P be uniformly convex and let
lf1' 2 satisfy both the condition (4 2 ) from 11.6. Moreover, let the spa.-
ces X and Y be uniformly convex. Then the space L is uniformly convex.
(b) Let L be uniformly convex. Then the functions 'P l ,tp2 satisfy the con-
di tion (Ll 2 )from 11.6 and the spaces X and Y are uniformly convex.
'rhe proof of (a)is reduced to showing to be uniformly convex:, CPI
is obtained applying the fact that (£1 2 ' implies etxl= 1 to be equiva-
185
lent to IIX/I = 1.
Concerning Remark 11.4 let us note, that V.A. Akimovi (l)obtained
necessary and sufficient conditions in order that a function 4>€- N
(see Definition 13.1) without parameter be uniformly convex:
Theorem (V.A. Akimovi6, [lJ). Let 4' : R -R be a convex, continu.o.......
+ +
function such that \flO} = 0 and 1lul»0 for u >0, '-9 0 as u O+ and
-1coas u-.co. Let p(ul be the rigi:J.t derivative of at u. The fun-
ction \f is uniformly convex on R , if and only if, for every 0 <. e< 1
th . + p((l+t ill 4
ere eXJ..sts a K" "7 1 such that for every u O there holds I K .
, p tu
There are strong connections between uniform convexity atld strict
convexi ty of an Orli cZ space. A normed space X with norm II II is called
strictly convex or rotund ,if the condition Ux+yK=llxII +)Iyll with some
x,Ye X , Y f. 0, implies the equality y "" cX with some constant c>O. Ne-
cessary and sufficient conditions for strict convexity of an Orlicz
space L If with function If' wi thout parameter and atomless measure p.t.. were
given by H.W. Milnes [1Jin case of Orlicz norm and by 13. '.l'urett (11 in
case of Luxemburg norm. The results of 13. Turett [11 were extended by H.
Hudzik [81 and by A. Kaminska [41 to the case of function If with a para-
meter. The same paper by H. HudzH: L81 contains a detailed study concer-
ning uniformly convex and strictly convex functions 'P <.f and spaces.
Theorem (Ei. Hudzik, [8]). Let the measure 1J.- be finite and atomless
and let 4'f . (a) If 1f1 is strictly convex in R+ for -almost every
t ..n. and if it satisfies the conditionl1l 2 ) from 11.6, then the space Llf
is strictly convex. (b) Let us SUljpose that either 4' does not satisfy l42 \
from 11.6 or there exists a set A€'2: of positive measure and I-mea.-
surable functions u, v such that u It) .( v Lt) and <.p( t, A u(t) + (1->.) vtt):::
\f(t, u ltJ)+ <:I-A) I.p(t, v (t}) for every tEA and a M lo, 1); then L is not
strictly convex.
Theorem CA. Kaminska [41). Let \{J == ( n) be a sequence of convex functi-
ons iO: R -.,.R , HI lol: 0, n::: 1,2,... The Orlicz sequence space l'f is
ln + + "In
strictly convex, if and only if, the following con itions are satisfied:
1° there exist numbers &'/0, K,.O and a sequencelan)of nonnegative nu.m.-
bers wi th a <. oofor which ,n 01<6 implies l2U) K l!;l)+ an for
I1=1 n "In n
u 0 and n == 1,2,... (see also 8.10),
2 0 if An == {u O: tt'JU) J' then the functionsTn are strictly convex on
186
sets A n ' besides at most one function (/)
1 nl '
3 0 if a == inf {lfnllU): there exists a v I: u such that
1 1 l
'2\fh l lul+ "2 4In1 (v)s, then tpn are also strictly convex on
n lul l-a t for every n I: n 1 .
Let us still remark that the above theorems are proved under some
Ifnl d (Uf-VI) ==
sets Bn ::.;: u O
more general assumptions on lfCt,.) or Cfn' which do not need to be fi-
ni te on the whole R .
+
A theorem of Clarkson says that a uniformly convex Banach space is
always reflexive. There is no such connection between strict convexity
and reflexivity, as show the following examples of Orlicz spaces given
by H. Hudzik (81, where ItA.. is a -fini te and atomless measure. If <flu! ==
lU+1) In(Ui-1), then L is strictly convex but not reflexive. If
{ u for 0 <: u <: 1 { - u for 0 { u <' 2
(uJ" 2:-' for < -:' 2, then '1'(* L: for 2: U:4 .
212
u - 1 for u '> 2 "4 u + 1 for u> 4
and both Ltf and L are reflexive, but are not strictly convex.
There are considered also other properties related to uniform and
strict convexity. Among them let us still mention the results of M.
Denker and R. Kombrink (1) concerning B-convexity and uniform convexi-
fiability of Orlicz sequence spaces.
The abo.ve presented results were re cently further extended by H.
Hudzik and A. Kaminska.
12
The notion of a modular defined as in 12.1 (b) was first introduced
by H. Nakano tt1in the case of an abstract vector lattice. Theorem 12.4
is a result of L. Drewnowski and Yi. Orlicz [1], 2.1; the actual exposi-
tion was kindly communicated to me by L. J)rewnowski. This result ( see
also W.A. Woyczynski [1), (2) and K. Sundaresan[l)) shows that modulared
vector lattrbces of measurable functions are nothing but some generali-
zed Orlicz spaces. This was further generalized and extended. L. Drew-
nowski and W. Orli cz [2] - [.5) investigated the same problem for orthogo-
nally additive functionals over,vector lattices of measurable extended
real-valued functions, obtaining the general form F (X)"" S f (t,x It))d1'L of
.Sl. .
orthogonally additive, continuous functional F, the function f be ng of
187
a class Cw,a (see L. Drewnowski and W. Orlicz(5), 'l'heorem 3.2)0 As re-
gards applications and extensions of heorem 12.4, it was applied (in
the form of 12 0 7) by N.J. Kalton [1}, Th. 2.2, in order to prove that
if T : E\f_X, where X is a topological vector space and Elf is defined
by 7.2 with independent of the parameter, is a continuous, linear
operator such tha"t' TfIx : ItxlI 1n is relatively compact and if .J:i!g
\ 00 oo
u == 0, then T == 0 (see also Theorem 13.22 ; compare D. PallaschIte(l)
and Ph. Turpin(11). Lately L. Drewnowski (1) extended the result of N.
J. Kal ton applying Theorem 12.4 for X =: E'f and Kal ton's method of Ra-
demacher functions, to functions depending on a parameter. The result
of L. Drewnowski, kindly communicated to me by the Author, is the fol-
lowing one :
Theorem (L. Drewnowski l11). Let tfE- and let f- be a O""-fini te,
atomless measure on a O"-algebra 2: of subsets of a set.Jl , LI{1== L41(.!t,I,/-1,) .
. l{ttM, )
Let Y be an arb trary Hausdorff topologocal vector space. If lim == 0
u.......oo u
Ior -a.e. t(j...s1.. and if T: Etf-+y is a continuous, linear operator such
that T(Nx)is precompact for every x E\f, where N x ={Y€ EP: \y/!IXI , then
T = O.
Theorem 12.4 was also extended to Orlicz lattices. The notion of
an Orlicz lattice, whose study was initiated and developed by W.J.Claas
and A.C. Zaanenl1J, is defined as follows. Let L be a vector lattice
(Riesz space in the terminology of W.J. Claas and A.C. Zaanen ; see al-
so W .A.J. Luxemburg and A.C. Zaanen [16]) and let be a finite. disjoin-
tly additive modular on L such that (>.x)......O as A--;O+' If L is co
plete with re spe ct to the F-norm I I generated by ;the modular ' then
L =(L,I I ) is called an Orlicz lattice ; if is convex, we call Li!.=
(L,tlll ) with I( II defined as in 1.5, a convex Orlicz lattice . iiiore ge-
neral, if I (I is any F-norm on L such that x,y(;L, IXI "Ylimply\:kI/ Iyl/,
and L is complete with respect to the F-noI'lD.!l f , then\L,! If) is cal-
led an F- lattice . A linear isomorphism P : Ll L2' where L 1 and L 2 are
F-lattices, is called a Riesz isomorphism, _ if P(XA y)== P lX1I\P(y) for all
x,ye,; L l ; the lattices L l ,L 2 are then calJ_ed Riesz isomorphic . An ele-
ment x of a vector lattice L is called a ]reak unit , if fl\e == 0 and
f ;> 0 imply f == O. With this notation, the folloV'fing result is proved
Theorem (W. Wnuk , 111). Let L be a convex O:clicz lattice with a
188
weak unit and let (2x) K (;x;)for all xc;L with a constant K)O. Then
there exists a measure space l..!l,'I,r) with a finite measure /A- and a
function <Fe wi th the following properties :
1 0 Cflt,u)')o for U>O, tE-
2 0 <e(t,u} is right-continuous at u = 0 for t ..SL
3 0 f:!;,ul is convex with respect to u O for t €-
4 0
cp satisfies the condition 8.13 ( 2)'
such that LE; ahd LIfL..Q., !, ) are isometrically Riesz
morphism P : L L lJl,t'r)satisfies the condition
x L 0
isomorphic. This iso-
(x) = I.f (Fx} for all
This result W8J3 next extended to the case of cr not convex and not
satisfying the (Ll ;[-condi tion :
Theorem (F. Kranz andW. Wnuk[ll). IfL is an Orlicz lattice, then
there exist a measure space ( ,f-) and a function <p € satisfying
the conditions 1 0 and 2 0 above such that is isometrically Ries-z. iso-
morphic to ElfbQ;r.,ftJ. If, moreover, L has a weak unit, then..Q. may be
taken compact and,.,.. finit'e.
In case of Lf!, with a weak unit,...c. is defined as the Stone space of
the algebra ( ):t< (d)'" xA(e-x) = 01, lLf)+ being the positive cone of L ,
and Theorem 12.4 is applied to a modular lVI, defined on the algebra of
open-closed subsets of..n by means of . In the general case, another
Orlicz space with weak unit is obtained using Kuratowski-Zorn lemma
and the prenous result is applied.
13
TDe theory of functions complementary in the sense of Young and
the definition of the Orloicz norm II lit:) as in Theorem 13.11 in case
"\,0
of convex functions without parameter is due to W. Orlicz [11 ,l21.
In fact, W. Orlicz defined the space L as the set of measurable fun-
ctions x such that Ilxl/ p < 00 . A detailed exposition may be found in
the books of W .A.J. Luxemburg [11 and of M.A.Krasnoselski:( and Ya.B.
RuticJd.l.ll] (see also J. i'liusielak [14]). H. Hudzik and A. Kaminska [21 ob-
served that the same argumentation may be applied to the case of fun-
ctions depending on a parameter. The complementary function may
be defined also in case of a non-convex function ; then one may de-
189
fine also the associated function . This definition together with
1'heorem 13.10 in case of independent of the parameter are due to W.
Matuszewska and W. Orlicz111. Theorem 13.18 was proved in case ofcp in-
dependent of the parameter by J. l,iusielak and W. Orlicz[4] for finite
measu:rep... and by J. fiiusielak and A. Waszakl1:5J foro-finite measu:re .
In case of 91 independent of the parameter and .!L= [a, b) , the general
form of a -continuous, linear functional over L If was obtained by W.
Orlicz L5) without assumption of convexity of If, only if tf u ,....., 00 as
u oo. As regards some problems of convergence of -continuous, line-
ar operators over Orlicz spaces with values in a nOYilled space or an F-
norIJ.ed space, see W. Orlicz[41. Theorem n.19, i.e. the general form
of a conjugate modularG in a generalized Orlicz space, was given al-
ready 1950 by H. Nakano [1], Apuendix I, p.283. Theorem 13.22 was first
given in case of norm convergence in the space Ltf over[0,11 with Le-
besgue measure f. by S. Rolewicz [1], and extended to q-convergence in an
atomless modular lattice by J. l"usielak and W. 0:clicZ(4). Theorem 13.18
may be applied to derive results on reflexivity of generalized Orlicz
spaces. The necessary tool is here the following observation:
,..
ilxlI = SUPt X[t)YLt\dfA: YE:L , lIyll o,o 1!,
done in case of tp indepehdent of" the pararueter by W .A.J. Luxemburg111
in Chapter II, p.50, and easily extendable to depending on t. Apply-
ing this observation and 13.18, one may easily deduce the following sta-
tement :
Theorem . Let the following assumptions be satisfied
1 ) If ,f € liT are locally integrable,
2) for every u '7 0 there exists a c>O such that
0
):>; and ) for u U o ' t Jl ,
v c U ? C
u
3) there holds one of the following assUIiiptions :
a) is O'-fini te &"ld atomless and both fmctions tf ,f satisfy the
condition 8.13 (A 2 \ ,
b) SL= {1 ,2, . . . , ft({n J=: 1 for n == 1,2,... and both functions ,tp*
s a.ti sfy the condi tion 8.13 (0 2 ) .
Then the space <L ,IIII > is reflexive.
(& if <0 satisfies
Obviously, if {l'f , II II ) is uniforwly convex 1 e. g. T
the assumptions of 'rheorem 11.6 , then it is also reflexive.
190
14
The inte rpolation theorem of strong type for operators in L P spaces
was proved first by ill. Riesz bJin 1926 and then extended, applying co
plex method, by G.O. Thorin l1J in 1939 (see also G.O. Thorin[2J). I t can
be found under the name of the theorem of Iii. Riesz and Thorin in the
book of A. Zygmund [2] , 1.11, p. 95 (see also J. Bergh and J. L8fstr8m
(1), theorem 1.1.1 ; for historical remarks, see A. Zygmund (21, Notes,
p. 332).
Besides this theorem, there is also known the interpolation theorem
of wea..l.c type due to J. Marcinkiewiczl1), 1939 (see A. Zygmund(2), Theo-
rem 1.3.1). In contrary to the Ill. Riesz - Thorin theorem proved by means
of a complex method, the proof of Marcinkiewicz thDrem is based on real
methods. Here, we deal only with extensions of the M. Riesz - Thorin
theorem.
First extensions of the M. Riesz - Thorin theorem to the case of
Orlicz spaces were given 1963 independently by I.B. Simonenko and by Ya.
B. RutickiL I.B. Simonenko[1'J interpolated between powers, but the in-
termediate function needs or y to be a so-called uasipower function.
Ya.B. Ruticki:L (1) interpolates between subspaces E of Orlicz spaces
generated by N-functions w;i. thout parameter, where..Q. is a compact subset
of R n ; he gives in this case the definition of -intermediate functions,
and proves a theorem of the type of ou.r Theorem 13 .16 adopting the com-
plex method of proof in case of linear operators and Orlicz norms (Ya.
B. Ruticki!: DJ, Theorem 11 on p. 125) ; in Ya.B. Rutickit[21, 1964, the
same is done in the case of Lt,q:emburg norms.
The Swfie method is applied by M.M. Rao[l], 1966, who proves theorems
on interpolation of linear operators between Orlicz spaces (Th.l on p.
549 and 'rh.2 on p. 552). The role of symmetric spaces in interpolation
was pointed out by B.II1. Semenov (1), 1968.
The interpolation theorem in the form of our Theorem 14.16, Le.
for sublinear operators, but only in case of Orlicz spaces without pa-
ra.w.eter which are symmetric spaces was given first by W.T. Kraynek(ll,
1972. He proved it for s-convex f-functions and also in case of vector-
valued functions with values in a Banach space.
191
The present version of the interpolation theorem 14016 wag expanded
also in a more general case of functions with values in Banach spaces,
by H. Hudzik, J. 1iusielak and R. Urbanski[2J-[41, 1980. In[21, there
are treated linear operators, and(.3]conte.ins results for sublinear ope-
rators. Most results which are presented here were given in these two
papers (e.g. 'rheorems 14.7,14.8 and 14.11 are from [2] and 14.9,14.10
and 14.16 are from(31).Auxiliary theorems on't'-intermediate functions
are mostly simple modifications of those provided by Ya.B. Rutickil IJJ
in case of N-functions without parameter. Let us still remark that the
results presented in the papers by H. Hudzik, J. Musielak and R. Urbari.-
ski l21, [3J concern vec-wr-valued functions with values in Banach spa-
ces and were given here in the scalar case for the sak of simplicity_
In order to obtain the vector case one needs only to apply the methods
developed by A. Kozek (1] , l2], H. Hudzik (JO) and by H. Hudzik and A. Ka-
minska (11, L2J. In H. Hudzik, J. IvIusielak and R. Urbanski [41 there are
obtained results on compac ness of the interpolating operators under
assumption on compactness of the given operators ; this paper concerns
sublinear operators in spaces Elf for functions with val ues in Banach
spaces. Results concerning interpolation theorem for generalized Orlicz
seq uence spaces are to be found in H. Hudzik, Vi. Orlicz and R. Urbanski l1].
Interpolation theorems were also the starting p0int of a theory of
intermediate spaces and interpolation spaces for pairs of Banach spaces.
A wide exposition of these ideas, originated by J. Peetre [11 and by J .L.,
Lions and J. Peetre 81 in 1964 and further developed in a series of pa-
pers by J. Peetre and other mathematicians may be found in the books of
P.L. Butzer and H. Behrens 01, 1967 , J. Bergh and J. L8fstrBm [11, 1976,
S.G. Krein, Yu.I. Petunin and E.M. SemenovUl, 1978, and H. Triebel[l],
1978, where also wide bibliographies may be found (see also A. Kufner,
O. John and S. Fucik(l), 1977, p. 207}. The theory of intermediate and
interpolation spaces in case of Orlicz spaces was developed by J. Peetre
[21 in 1970, and by J. Gustavsson and J. Peetre[lJin 1977 (see also C.
Bermett [1]); in case of general modular spaces with application to Or-
licz spaces it as developed by A. Godziewski(l], 1978, and lately by
M. Krbe c [11 .
As regards intermediate and interpolation spaces for various types
192
of spaces (e.g. for Lipschitz classes etc.) and applications to weak
interpolation, it has to be referred to the papers of A. Calderon (11 ,
1964, G.G. Lorentz and T. Shimogaki (1), 1968, T. Shimogaki Ul, 1968,
F. FeMr111, 1980, L. Maligranda (1], 1981. One should also mention the
role of indices in Orlicz spaces and more general function spaces in
the theory of interpolation. The respective papers were quoted in the
Commenst to 7 ; for a wider bibliography, see L. Maligrandal2).
15
The idea developed in 15 together vdth Theorem 15.2 is due to
o. ilbrycht and J. I\i.usielak [1), 1968.
16
The theory of spaces D LP ' i.e. Dip with tp(u)==IU\P, P?l, was given
by L. Schwartz [1]in 1951, who applied it also for defining the space Diq
of distributions as the dual of D._ P with 1 + 1 = 1. The respective the...
.lJ p q
ory of spaces D'f and their duals D wi th a convex N-function indepen-
dent of the parameter t was originated 1961 and developed by J. Musielak
(3)-(7]; in particular, L4'C:D;CD' for any convex N-functionr' where D'
is the space of all distributions, the above embedding being continuous.
Theorems 16.2 - 16.6 were proved by R. Bojanic and J. I.iusielak(l]in 1964,
Theorem 16.9 is due to J. Albrycht and J. i'lusielak[11in 1968, and Theo-
rem 16.14 was proved by Z. Ciesielski and J. I,iusiel alr in 1969 (see J.
I\1usielak [8]). Spaces Dtp were extended to some more general case inclu-
ding function If> with parameter by J. l\lagdziarz (1), t21. Let us quote the
important Theorem 16.4 in case of \f with a paramete r :
Theorem (J. lYiagdziarz (21, p .170). Let y> : R n x: R R+ and let the de-
(I> '\)1 l tp + .
rivatives '9 'flt,u)= t(!,1... vt ... (t l ,t 2 ,...,t n ,U) ex st for all t =
f. ) n 1 n
\..t l ".', tn,E;R , u o, 0= Cf'1'" . ,l"n) A. Let us denote for every func-
tion x : Rn-"R by ol!tp{t,xlt») the derivative i)!\.p(t,u> with u = x(t)
and let us suppose that 'Q 1f6't? , '() tp(.,u)is convex with respect to
u O and 'i) lflt,u)/u.....o as u O+, o(!>lj>lt,u)/u....oo as u oofor every
n n -
t ER and € A. Finally, let x : R -..R be. continuous and posses con-
n n
tinuous derivatives DeXx of prders (}..f:A in R . '.J:hen for every s R
the re holds the ineq uali ty
193
If ( .,)) I x (s) p
L S 0 £f[t,2n-jCi+ I]Do!.xltJ!Jdt.
d.+ Eo A Isl It'
17
The problem of sufficient aL1.d necessary conditions foX' equali ty
i. o == Xe for modulars n(x)== r 'PnCIX (j;lIId was solved 1968 by J. Albrycht
1 ...11.
and J. V1usielak [Jlin the following special cases of measures: 1) fini-
te, atomless measure fA-' 2)...Q consists of' a countable set of atoms t,
, J
va th measure t4 {t J .n == Ci), '70, 0 <, lim tV, < OJ, 3)..Q and fL as in 2) ,
. OOJ J-+oo J
but l).m W J ' == 0 and 1; 1 c.v, == 00. The notion of eq uispli tt abili ty of a
J-.OO J- J
familY(f-i'(") of measures was introduced 1970 by J. hIusielak and A. Vla-
szakl31. where also Theorems 17,3.17.5,17.8,17.9 and Example 17.4
are given; Example 17.6 may be found in J. i,lusielal: and A. Waszaktl10
The purely atomic case for a countable family of measures was treated
1970 by J. Musielak and A. WaszakL41. Sufficient conditions for X == X..,
'7Q "
in case of Ifnlt,u) depending on a pararneter t are given by M. Toure [11.
An analogous problem to that of 17 in case of a sequence of modulars
qn lx)::z lim J 4'n(lx It))) df't.'t" ' where r is a topological space (see Exam-
(-"11: 0
pIe 1.9.IV )was treated 1969 by J. lVIusielak and A. Waszak(;2].
Besides spaces X and X , J.o 1Iusielak and A. Waszak 1 introduced
also other countably modulared spaces, namely corresponding to the modu-
00 1 n
lars () (x/== 1:. (i) eX) and to (x ) == sup - ,:s:;. I). (x). Evidently, we have
,\s n=1 "'In )4"'- n n )."'1 ,,).
X sc. oc c Xf:" The following statement was proved :
Theorem (J. 1'Ilusielak and A. Waszak, (5)). Let fA- be a finite and atom-
less measure in..a.. and let the If-functions lfn' n == 1,2,..., be equicon-
tinuous at 0 and satisfy the assurnptionlb)in 17.8. Moreover, let us sup-
pose that there exist constants , ,v 0 such that for every index i
there holds the inequality liPi(>,U){ lfk(u)for k i and u=7v. Finally,
let tJ lx) == S cp, (Ix It}f) d i ll\,. Then X == X,.o, if and only if, Xl) == X.(7'
An ) 1b ,
M. Dawidowski [1] has shown that in case of modulars defined in
16.1 with n '" 1, there holds always X ; o'
Taking qn LX)== 1. tpnCIx (t)IJ d with finite measure f'l ' one may compare
the countably modulared spaces under consideration with the space L 00 '"
1.. CD(..!l).. The following condition will be of importance : Of) there exists a
seq,uence (B) of pairwise disjoint, measurable subsets of...Q. with (lS!3ri""O
n
194
and fALBd O for all n. Now, the;re hold:
Theorem (J. Iifusielak and A. Was ak[5J). The following implications
hold :
(a, LcoC ' if and only if there is au» 0 such that n 1 t.fn luol < co ,
(l So 00 0 00
b) if there is a u o '" 0 such that &,<fn (uol= ro, then X <C.L ,
O f 00, 00
(C) l C*J hold'S and Xt.I c. L , then there s au,,> 0 such that £ cp (u I
') So 0 n== 1 n 0
=-00.
Theorem (J. Musielak and A. Waszak(5]). The following implications
hold :
co
(a')L c X o' if and only if
lb 1 if there is a u o > 0 such
(c ') if (if) holds and X<;o C L oc;
Cfn are equ.icontinuous at 0,
that sup 10 (u I:: OJ, then X... C L co ,
n Tn 0 )0
then there exists au;> 0 such that
o
s p tfJn lU o )== OJ.
Generalizations of the above two theorems to the case when n is re-
placed by an arbi tr ry (possibly also uncountable) parameter are gi-
ven in Theorems 18.11 and 18.12.
18
The notion of a faLOily of modulars depending on a parameter was
introduced 1971 by J. lIlusielak and A. Waszak [6],where also Theorems
18.5 - 18.8 are proved. Theorems 18.10 - 18.12 may be found in the pa-
per (7] of the same authors. The space Xq-s introduced in Example 18.13 was
considered 1976 by T.lvl. J dryka and J. hlusielakf2]; Theorem 18.18 on
completeness was also proved there. The space of finite elements E s to-
gether with Theorem 18.16 is to be found in T .1,1. J dryka and J. Musie-
lak [6]. The following result from this paper extends 7.6 and 7.10 :
Theorem (T.lVl. J dryka and J. 1\lUsielak(6)}. Let the following assump-
tions be satisfied :
1.At:£ and ,t-tlA)<oo implY A€ X, where i\A is the characteristic fun-
ction of the set A,
2. for every X" s there is a set ..Q.1E- £ of measure C.Ql):: 0 such that
fQr every "70 there exists a 0.,. 0 such that for any tE..Q......n. l
and B"r , the inequality j\t(B)<b implies ,Jt,x B)<t
3.for every XEX there exists a set ..a2 L with f"--CA 2 )== 0 such
that for every E> 0 there is a set A , tA\<.oosuch that for
195
an arbitrary tE'..n.'.Sl.2 there holds (t,xX!l\A)<f.
Then
(a)if ze- .. and ynE'X, 0 Ynlt) zlt) a.e. in..,Qfor n = 1,2,... and if
Yn (t) O r--a.e. in..Q., then (t'Yn) 0 f""a.e. in.a,
(PI the set S, of simple functions in E is dense in E witn respect
to the F-norm I I ,> .
If, moreover, the measure r-- is se.l:'arable, then
lC) if for any Ae t the condition 'XA is equivalent to t'tlA.)(ro,
then is a separable sUbspace of with respect to the F-norm
I L .
Let us still notice two natural faIDilies of modulars depending on
a parameter. One is defined on the space of analytic functions on the
unit disc and is closely connected with Hardy-Orlicz spaces; we sh8J.l
investigate this family in 19. Another one is the family of modulars
( ,x)== : <pC tD It>1) dt + Ix(o}J, where Dol is the derivative of fractio-
nal order c( E (0,1] and is a convex -function ; this family was con-
sidered by H. liiusielak and J. Musielak111, 1980.
19
First results concerning inclusions of Orlicz classes in case when
6A- is the Lebesgue measure in an interval.ft.= la, b I were obtained by W.
I.Iatuszewska 111in 1961. She proved among others the following
Theorem lW. Matuszewska [.11). Let i' ,9>1 .lf 2 ,... be tp-functions.
1) The inclusion i l L '; la,bl C L't@,b) holds, if and only if, there
exist an index m and a number d:;' 0 such that 't(U) d1 m qi (1.11
La) for u U o with some u o > 0, if , b) is bounded,
(pI for u-ZO, if la, b) is unbounded.
;>' 00.
2) The inclusion LI.fCa, b}C h!1 L\¥ la, b) holds, if and only if, there
exist an index m and a number c m > 0 such that tVm lu) c m \flu)
(a, for u ;;u o with some U070, if la,b) is bounded,
(PI for U'70, if (s;!.,bl is unbounded.
20
The notion of Hardy spaces is due to G.H. Hardy (1),1914 ; for other
historical remarks, see A. Zygmund [1], Notes to Chapter IV, 7, p.381.
196
A theory of Hardy spaces H P with p>O (i,e. with U1)==luIP, p>O)as well
as of the classes N and N' may be found in A. Zygmund (:1), Chapter IV,
7, pp. 211-285 . Definition 20.11lal, lb} is to be found in A. Zygmund tn,
p.271 and 7.51 , p.283 ; for convenience we take the necessary and suf-
ficient condition 7.51 as the definition of the class N'.
Hardy-Orlicz spaces were introduced and investigated by R. Lesnie-
wicz[1],1971 ; among others, the idea of the role of logarithmic con-
veri ty in the theory of Hardy-Orlicz spaces is due to R. Lesniewicz l11
(for further investigation of logarithmically convex functions, see Cz.
Bylka and W. Orlicz (11). The paper of R. Leaniewi.cz tl1contains among oth-
ers also Theorems 20.5, 20.14, 20.16 - 20.21 ; Theorem 20.20 is proved
in a more general setting of a sequence of tp-functions (like the theo-
rem of W. Matuszewska [11, see comments to 19).
General spaces s with modular s as in Definition 20.6 were intro-
duced and investigated in the papers of T .M. J dryka and J. Musielak l4)
and of J. Musielak [14], [17]; Theorem 20.7 and Examples 20.8 are to be
found there l171 .
Hardy-Orlicz spaces Hf are complete wi. th re spect to the F-norm gene-
rated by "the function \.f (R. Leaniewicz [11, Chapter III, Theorem 1.3.2).
'l!his is not true any more for general spaces ; indeed, the space X s
defined in Example 20.8 La) is not complete (J. Musielak [17J, pp.22-23).
The Hardy-Orlicz space Hip is separable, if and only if, If sastisfies
the condition ( 2) from Corollary 20.21 CR. Leaniewi.cz (1), Chapter III,
Theorem 2.2.2); on the contrary, the space !. from Example 20.8 (a) is
always separable (J. Musielak(17), p.23). It is not known under what
conditions on the function m in Definition 20.6, the respective space
X )is complete. Also it is not known under what conditions on m, J
from 20.21 is a necessary and sufficient condition for separability
of X !o .
Let us still remark that the theory of Hardy-Orlicz spaces was fur-
ther developed by R. Lesniewi.c z [21- [4], in the direction of problems of
duali ty. An important property of spaces Iff is that there are nontri-
vial continuous linear functionals over HCf even in case when f is not
convex.
197
21
The idea of this Section consists in making the space of solutions
of an integral eqUation dependent on the kernel of this equation, and
not a fixed space like C(..!L) or LP(.Q..), as in the classical theory of
integral equations. This idea was developed first by T .M. J dryka and
J. Musielak [31,1976, where however in place of a ball K'scr)in X s
there ...as considered a subset ={XE ,..: Ix It} )[ a.e. in.A..} , with fi-
xed M. The same was done by R. Pluciennik [1] , who applied more general
Edelstein's fix-point theorem in place of Banach fix-point theorem. The
present treatment was given by T .M. J dryka and J. Musielak(51 (see also
J. MusielakCt4). Applying the modular o(x)= It.£. (t,x)in place of t;s'
the theory of modular equations of t he above considered type in the spa.-
ce ce X was g:i ven in J. Musielak 05], where one can also find the spe-
cialization to the integral equation xttl:: a p(t) rlsN{Jx;(s)l) ds +
o
x (,t) with p,r positive, measurable in [t ,(0), t real, sup p (t)(oo,
o 0 0 t t.o
I:o rlt)dt < 00 and with a convex tp-function Cf satisfying the <... :z)-condi-
tion for small u (i.e. the (lJ condition).
22
The approximation problem 22.5 was first put by J. MusielakLl1),
1979, where also the proof of Theorem 22.7 may be fbund. A larger expo-
sition of the results of 22 is to be found in J. Musielak \j6]. These
results were further expanded and generalized by S. Stoinski [11,\2), whe-
re among o"thers the assumption on the integral of lu)in 22.11 is re-
placed by lim t 1 K lu)du::::: 1 and approximation of not necessarily non-
n....oo 0 n
negative functions x is considered.
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base
modular - , 27
non-weaker modular -, 27
- - , 26
sequential modular -, 27
sequential - - in , 36
topological , 31
base 10
equivalent modular - , 27
bimodular, 15
bipseudomodular, 15
class
- ' 33
- Iffi (..Q.) of functions, 100
- (S1..) of functions, 100
- (.Q..) of functions, 101
- N, 144
- N', 144
conjugate -, 171
generalized Orlicz
LY' , 33
o
generalized Orlicz sequence
Hardy-Orlicz -, 144
Orlicz -, 34
-closure, 19
condi tion
- (LJ 2 ), 52, 54
l6 2 ), 52
¥- -, 61
contraction
( IO")- - , 21
restricted ( ,u') - - , 21
convergence
modular - , 18
IN1EX
lip , 34
o
218
modular ¥- -, 169
- -, 18
with respect to a filter, 23
element
finite
null
of L4' , 33
of X , 166
eq uation
modular -, 151
family
of modulars depending on a parameter, 129
equispli ttable of measures, 120
-absorbed -, 136
()-absorbing - , 136
uniformly bounded of measures, 120
filter, 23
F-operation, 10
function
of exponential type, 117
associated - , 84
It-weaker , 137
complementary , 83, 84
F-superadditive -, 11
f- , 4
Ip-absolutely continuous
locally integrable
logarithmically convex
Iilusielak-Orlicz -, 174
N- -, 82, 175
N' - -, 175
N" - -, 175
- , 1 82
, 34
, 142
Orlicz
- ,
, 43
174
non-wealcer
Steklov - , 61
s-convex - , 5
type of a - , 118
219
bounded - , 37
-bounded , 41
't" -bounded ,41
+
t-intermediate , 94
uniformly convex - , 75
functional
associated subaddi tive modular - , 167
sub additive modular - , 167
functions
equivalent -, 54
!'-equimeasurable , 81
s-equivalent -, 54
galb, 178
genfunction, 173
pre- -, 173
indices
fuatuszewska-Orlicz -, 172
ineq uali ty
Young , 83
isomorphism
Riesz -, 187
kernel
Cesaro of order r , 121
singular , 38
s-singular - , 42
lattice
convex Orlicz -, 187
F- -, 187
Orlicz
- ,
187
map
( ,(r')-bounded -, 21
\ ,<r)-continuous -, 21
strongly ( ,G')-bounded - , 21
modular, 1
associated -, 168
character of a - , 166
conj ugate , 8
(F,s)- , 12
normal , 165
pure , 166
uniformly convex - , 75
modulus
T- of smoothness, 25
neighbourhood
- of zero, 36
norm, 2
Amemiya - , 7
F- , 2
first , 168
Luxemburg - , 7
Orlicz - , 9, 85
10- -, 2
second -, 168
norms
equicontinuous - , 55
ope rat or
convolution -, 38
subline ar , 101
translation - , 37, 41,58
-bounded -, 23
property
majorant , 77
pseudomodular ,
continuous , 4
convex -,
disjointly additive -, 77
(F , s) - -, 11
left-continuous -,4,11
right-con tin uous -, 4
s-convex - , 1
220
221
pseudonorm
F- , 2
10- , 2
scala
decreasing - , 166
modular , 166
(J"-modular - , 166
se mi group
preordered -, 166
semimodular ,
IF,s)- , 12
sequence
of modulars, singular at a point, 160
B-convergent -, 18
constant preserving
of modulars , 159
J-convergent -, 169
modular convergent -, 18
-convergent
singular
set
- , 18
of mod ulars , 1 60
of pro::ameters , 129
abso:rbing in ' 26
balM-ced. - in X ' 26
bimodular - , 15
-bounded , 19, 169
-closed -, 13,19
-compact -, 19
-dense , 19
relatively -compact - t 19
space
- Elf , 33
of functions of finite generalized -variation, 69
bimod ular -, 1 6
countably modulared t 112
Fenchel-Orlicz
- ,
174
222
F-modular -, 12
generalized modular -, 28
generalized modular vector
generalized Orlicz - L ,
generalized Orlicz sequence
generalized Orlicz-Sobolev
Hardy-Orlicz , 144
intermediate , 191
interpolation -, 191
measure , 4
modular - 2
- , 1 67
33, 176
lW , 34
- w (J2.) , 66
modular function , 173
modular sequence , 173
l'ilusielak-Orlicz - , 173
Nakano - , 164
Orlicz - LIP , 5, 34
Orlicz sequence - , 5
rotund - , 185
-separable -, 19
strictly convex -, 185
two-modular -, 169
uniformly convex Banach - , 74
uniformly countably modulared -, 112
universally continuous semi-ordered linear -, 164
theorem
interpolation -, 101
unit
weak -, 187
variation
- -, 6
generalized If-
q-integral -
- , 69
- , 183