Mathematical
Surveys
and
Monographs
me 125
The Cauchy
Transform
Joseph A. Cima
Alec L. Matheson
William T. Ross
(Hwff ))^ American Mathematical Society
\\ Lull 1U )l~

The Cauchy
Transform

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Mathematical
Surveys
and
Monographs
Volume 125
jj$^&&
The Cauchy
Transform
Joseph A. Cima
Alec L. Matheson
William T. Ross
American Mathematical Society

EDITORIAL COMMITTEE
Jerry L. Bona Peter S. Landweber
Michael G. Eastwood Michael P. Loss
J. T. Stafford, Chair
2000 Mathematics Subject Classification. Primary 30E20, 30E10, 30H05, 32A35, 32A40,
32A37, 32A60, 47B35, 47B37, 46E27.
For additional information and updates on this book, visit
www.ams.org/bookpages/surv-125
Library of Congress Cataloging-in-Publication Data
Cima, Joseph A., 1933-
The Cauchy transform/ Joseph A. Cima, Alec L. Matheson, William T. Ross.
p. cm. - (Mathematical surveys and monographs, ISSN 0076-5376; v. 125)
Includes bibliographical references and index.
ISBN 0-8218-3871-7 (acid-free paper)
1. Cauchy integrals. 2. Cauchy transform. 3. Functions of complex variables. 4. Holomorphic
functions. 5. Operator theory. I. Matheson, Alec L., 1946- II. Ross, William T., 1964- III. Title.
IV. Mathematical surveys and monographs; no. 125.
QA331.7:C56 2006
515/.43-dc22 2005055587
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10 9 8 7 6 5 4 3 2 1 10 09 08 07 06

Contents
Preface
Overview
Chapter 1. Preliminaries
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
1.10.
1.11.
1.12.
Basic notation
Lebesgue spaces
Borel measures
Some elementary functional analysis
Some operator theory
Functional analysis on the space of measures
Non-tangential limits and angular derivatives
Poisson and conjugate Poisson integrals
The classical Hardy spaces
Weak-type spaces
Interpolation and Carleson's theorem
Some integral estimates
Chapter 2. The Cauchy transform as a function
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
General properties of Cauchy integrals
Cauchy integrals and H1
Cauchy A-integrals
Fatou's jump theorem
Plemelj's formula
Tangential boundary behavior
Cauchy-Stieltjes integrals
Chapter 3. The Cauchy transform as an operator
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
An early theorem of Privalov
Riesz's theorem
Bounded and vanishing mean oscillation
Kolmogorov's theorem
Weighted spaces
The Cauchy transform and duality
Best constants
The Hilbert transform
Chapter 4. Topologies on the space of Cauchy transforms
4.1.
4.2.
4.3.
The norm topology
The weak-* topology
The weak topology
ix
1
11
11
11
14
17
20
22
25
30
32
35
36
39
41
41
46
48
54
56
58
59
61
62
64
69
73
76
77
79
81
83
83
91
94
V

vi CONTENTS
4.4. Schauder bases 95
Chapter 5. Which functions are Cauchy integrals? 99
5.1. General remarks 99
5.2. A theorem of Havin 99
5.3. A theorem of Tumarkin 100
5.4. Aleksandrov's characterization 102
5.5. Other representation theorems 109
5.6. Some geometric conditions 110
Chapter 6. Multipliers and divisors 115
6.1. Multipliers and Toeplitz operators 115
6.2. Some necessary conditions 118
6.3. A theorem of Goluzina 120
6.4. Some sufficient conditions 122
6.5. The ^-property 127
6.6. Multipliers and inner functions 129
Chapter 7. The distribution function for Cauchy transforms 163
7.1. The Hilbert transform of a measure 163
7.2. Boole's theorem and its generalizations 164
7.3. A refinement of Boole's theorem 169
7.4. Measures on the circle 170
7.5. A theorem of Stein and Weiss 176
Chapter 8. The backward shift on H2 179
8.1. Beurling's theorem 179
8.2. A theorem of Douglas, Shapiro, and Shields 180
8.3. Spectral properties 184
8.4. Kernel functions 185
8.5. A density theorem 186
8.6. A theorem of Ahern and Clark 192
8.7. A basis for backward shift invariant subspaces 192
8.8. The compression of the shift 194
8.9. Rank-one unitary perturbations 196
Chapter 9. Clark measures 201
9.1. Some basic facts about Clark measures 201
9.2. Angular derivatives and point masses 208
9.3. Aleksandrov's disintegration theorem 211
9.4. Extensions of the disintegration theorem 212
9.5. Clark's theorem on perturbations 218
9.6. Some remarks on pure point spectra 221
9.7. Poltoratski's distribution theorem 222
Chapter 10. The normalized Cauchy transform 227
10.1. Basic definition 227
10.2. Mapping properties of the normalized Cauchy transform 227
10.3. Function properties of the normalized Cauchy transform 230
10.4. A few remarks about the Borel transform 241

CONTENTS vii
10.5. A closer look at the J-property 243
Chapter 11. Other operators on the Cauchy transforms 249
11.1. Some classical operators 249
11.2. The forward shift 250
11.3. The backward shift 252
11.4. Toeplitz operators 252
11.5. Composition operators 253
11.6. The Cesaro operator 253
List of Symbols 255
Bibliography 257
Index
267

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Preface
This book is a survey of Cauchy transforms of measures on the unit circle.
The study of such functions is quite old and quite vast: quite old in that it dates
back to the mid 1800s with the classical Cauchy integral formula; quite vast in that
even though we restrict our study to Cauchy transforms of measures supported
on the circle and not in the plane, the subject still makes deep connections to
complex analysis, functional analysis, distribution theory, perturbation theory, and
mathematical physics. We present an overview of these connections in the next
chapter.
Though we hope that experienced researchers will appreciate our presentation
of the subject, this book is written for a knowledgable graduate student and as
such, the main results are presented with complete proofs. This level of detail
might seem a bit pedantic for the more experienced researcher. However, our
aim in writing this book is to make this material on Cauchy transforms not only
available but accessible. To this end, we include a chapter reminding the reader of
some basic facts from measure theory, functional analysis, operator theory, Fourier
analysis, and Hardy space theory. Certainly a graduate student with a solid course
in measure theory, perhaps out of [182], and a course in functional analysis, perhaps
out of [49] or [183], should be adequately prepared. We will develop everything
else.
Unfortunately, this book is not self-contained. We present a review of the basic
background material but leave the proofs to the references. The material on Cauchy
transforms is self-contained and the results are presented with complete proofs.
Although we certainly worked hard to write an error-free book, our experience
tells us that some errors might have slipped through. Corrections and updates will
be posted at the web address found on the copyright page.
We welcome your comments.
J. A. Cima - Chapel Hill A. L. Matheson - Beaumont W. T. Ross - Richmond
cimaOemail.unc.edu matheson@math.lamar.edu wrossOrichmond.edu

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Overview
Let % denote the collection of analytic functions on the open unit disk D =
{z 6 C : \z\ < 1} that take the form
Jt 1 - Qz
where \i belongs to M, the space of finite, complex, Borel measures on the unit
circle T = 9D. In the classical setting, as studied by Cauchy, Sokhotski, Plemelj,
Morera, and Privalov, the Cauchy transform took the form of a Cauchy-Stieltjes
integral
where F is a function of bounded variation on [0, 2n].
In this monograph, we plan to study many aspects of the Cauchy transform: its
function-theoretic properties (growth estimates, boundary behavior); the properties
of the map /i i—> K/i; the functional analysis on the Banach space X (norm, dual,
predual, basis); the representation of analytic functions as Cauchy transforms; the
multipliers (functions (j) such that <p% C 3C); the classical operators on % (shift
operators, composition operators); and the distribution function y i—> m(\K/j,\ > y)
(where m is Lebesgue measure on T). We will also examine more modern work,
beginning with a seminal paper of D. Clark and later taken up by A. B. Aleksandrov
and A. Poltoratski, that uncovers the important role Cauchy transforms play in
perturbations of certain linear operators. To set the stage for what follows, we
begin with an overview.
We start off in Chapter 1 with a quick review of measure theory, integration,
functional analysis, harmonic analysis, and the classical Hardy spaces. This review
will provide a solid foundation and clarify the notation.
The heart of the subject begins in Chapter 2 with the basic function properties
of Cauchy transforms with special emphasis on how these properties are encoded
in the representing measure \i. For example, a Cauchy transform / = K/i satisfies
the growth estimate
(||/i|| is the total variation norm of /i) as well as the identity
lim(l-r)/(rC) = /*({<}), (€T.
r—*1~
This last identity says that Cauchy transforms behave poorly at places on the unit
circle where the representing measure /i has a point mass. Despite this seemingly
l

2
OVERVIEW
poor boundary behavior, Smirnov's theorem says that Cauchy transforms do have
some regularity near the circle in that they belong to certain Hardy spaces Hp.
More precisely, whenever / = K/i and 0 < p < 1,
sup /'|/(rC)|pdm(C)<oo,
0<r<lJT
where dm = d6/2n is normalized Lebesgue measure on the unit circle. Let Hp be
the space of analytic functions / for which the above inequality holds and let
II/IIhp := f sup f\f(rCWdm(C)) P-
By standard Hardy space theory, Cauchy transforms have radial boundary values
/(C) := lim /«)
r—*1~
for m-almost every £ G T. In fact, the formulas of Fatou and Plemelj say that the
analytic function / on C \ T (where C = C U {oo}) denned by
c
satisfies
f(z)= /"—L-d/x(C), zeC\T,
J 1 - Cz
lim (/(rC)-/(C/r)) = ^-(C)
r-+i- am
lim (f(rC) + f(CM) = 2P.V. [^
r-+l~ J 1
dM(0
^C
for m-a.e. ^ G T.
In this chapter we also discuss when / = K/i can be recovered from its boundary
function ( h-> /(£) via the Cauchy integral formula
f(z)= /-^iUm(C), zeB.
Jt 1 - (z
For a general / = if/x, the boundary function £ h-> /(C), although belonging to Lp
for 0 < p < 1, need not be integrable and so the above Cauchy integral
representation may not make sense. A result of Riesz says that the Cauchy integral formula
holds if and only if / belongs to the Hardy space H1, that is,
sup / |/(rC)|dm(C) < oo.
0<r<lJT
Interestingly enough, there is a substitute Cauchy 'A-integral formula' due to
Ul'yanov which says that if//Cm and / = K\±, then
f(z)= lim / MLdm(0, zeB.
This Cauchy .A-integral formula has been recently used by Sarason and Garcia to
further study the structure of certain Hp functions.
In Chapter 3 we treat the Cauchy transform not merely as an analytic function,
but as a linear mapping \i h-> K\i from the space of measures on the circle to the
space of analytic functions on the disk. From Smirnov's theorem, we know that
K(M) £ p| Hp.
0<P<1

OVERVIEW
3
In fact,
11^11^ = °(r^)' p^1_-
We first cover the well-studied problem: if / belongs to a certain subclass of L1,
what type of analytic function is
/+ := AT(/dm)?
Probably the earliest theorems here were those of Privalov (if / is a Lipschitz
function on the circle, then /+ is Lipschitz on D~), and of Riesz (if 1 < p < oo
and / G Lp, then /+ G Hp). Then there are the more recent theorems of Spanne
and Stein which say that if / G L°°, then /+ G BMOA (the analytic functions of
bounded mean oscillation) while if / is continuous, then /+ G VMOA (the analytic
functions of vanishing mean oscillation). When / G L2 has Fourier series
oo
n= — oo
then
oo
f+(z) = ^f(n)zn, 2€D,
n=0
belongs to the Hardy space H2 and the mapping / h-> /+ is the orthogonal
projection, the 'Riesz projection', of L2 onto H2.
Riesz's theorem says that the Riesz projection operator / h-> /+ and the
associated conjugation operator / h-> / := — 2z/+ + if(0) + i/ are continuous on LP for
1 < p < oo, that is to say,
II/Hlp^pII/IIlp, ||/+||hp^bp||/|Up, feLP,
for some constants Ap and Bp that are independent of /. An old theorem of
Pichorides identifies the best constant Ap as tan(7r/2p) if 1 < p ^ 2 and cot(7r/2p)
if p > 2, while a relatively recent theorem of Hollenbeck and Verbitsky identifies
the best constant Bp as l/sin(7rp).
This chapter also covers the important weak-type theorem of Kolmogorov
m(\Kfi\ >y) = 0(l/y), y -> oo,
that gives an estimate of the distribution function for K\i. It will turn out, quite
amazingly, that one can recover information about the measure from this
distribution function. For example, Tsereteli's theorem says
fi < m <^> m(\Kfi\ > y) = o(l/y), y -> oo.
Other work of Hruscev and Vinogradov, covered in Chapter 7, as well as some
relatively recent work of A. Poltoratski, covered in Chapter 9, shows even more is
true.
In Chapter 4 we treat the Cauchy transforms % = {K/jl : \i G M} as a Banach
space. Since
A>i = A>2 ^ Mi - M2 G H\,
where H\ are the measures {f dm : / G i71,/(0) = 0}, X can be identified in a
natural way with the quotient space M/Hq, by means of the mapping K/i h-> [/j].
Here [/i] is the coset in M/Hq represented by /jl. One defines the norm of K/jl to be

4
OVERVIEW
the norm of the coset [p] in the quotient space topology of M/Hq . Equivalently,
the norm of an / G X is
||/||=inf{||M|| :/ = #/*}•
Equipped with this norm, X becomes a Banach space and furthermore, the previous
growth estimate can be improved to
Thus X becomes a Banach space of analytic functions in that the natural injection i :
X —» Hol(D) (the analytic functions on D with the topology of uniform convergence
on compact sets) is continuous. From here, one can ask some natural questions.
Is X separable? Is it reflexive? What is its dual (predual)? How do the weak
and weak-* topologies act on XI Is X weakly complete? Is X weakly sequentially
complete? Does X have a basis? What type? These questions are thoroughly
addressed in this chapter.
So far, we have discussed the basic properties of a Cauchy transform / =
K/i. An interesting and still open question is to determine whether or not a given
analytic function / on the disk takes the form / = K\i. From what was said
above, certain necessary conditions hold. For example, a Cauchy transform / must
have bounded Taylor coefficients; must satisfy the growth condition \f(z)\ = 0((1 —
l^l)-1); the boundary values of the function / must satisfy the Lp condition ||/||lp =
0((1 — p)_1) for 0 < p < 1; the boundary values for / must also satisfy the weak-
type inequality m(\f\ > y) = 0(l/y). Unfortunately, none of these conditions is
sufficient.
A more tractable question is: if / is not merely analytic on D but instead is
analytic on the larger set C \ T with /(oo) = 0, when is / equal to
/■_L-d/i(c), ^c\t,
J 1 - (z
for some measure \i on the circle? Tumarkin answered this question with the
following theorem: if / is analytic on C \ T with /(oo) = 0, then / is the Cauchy
integral of a measure on the circle if and only if
sup /|/(rC)-/(C/r)|dm(C)<oo.
0<r<lJj
Aleksandrov refined this theorem and identified the type of measure (whether
absolutely continuous or singular with respect to Lebesgue measure) needed to represent
/. These representation theorems are covered in Chapter 5.
At the end of this chapter we examine the question: which Riemann maps
ip : D —-> Q are Cauchy transforms? For example, it is relatively easy to see that
if -0(D) is contained in a half-plane, then ^ is a Cauchy transform. What is more
difficult to see is that ift is a Cauchy transform whenever ip(3) omits two oppositely
pointing rays. What happens when -0(D) is a domain that spirals out towards
infinity?
An important class of functions associated with a function space X are the
'multipliers'. Here we mean the set of functions (ft for which (ftX C X. The
multipliers constitute the complete set of multiplication operators / >—> cftf on X and there
is quite a large literature on the subject. One can show that when X is a space

OVERVIEW
5
of analytic functions, a multiplier of X must be a bounded analytic function. For
the Hardy spaces i7p, the multipliers are precisely the bounded analytic functions.
However, for other function spaces, such as the classical Dirichlet space or the
analytic functions of bounded mean oscillation, not every bounded analytic function is
a multiplier. Furthermore, even when a complete characterization of the multipliers
is known, it is often difficult to apply to any particular circumstance.
Chapter 6 deals with the multipliers of X. Despite some interesting results,
these multipliers are still not thoroughly understood. For example, a multiplier of X
must be bounded, must have radial limits everywhere (not just almost everywhere),
and the partial sums of its Taylor series must be uniformly bounded. However, these
conditions do not characterize the multipliers.
In this chapter we also cover the ^-property for X. A space of functions X
contained in the union of the Hp classes, as the Cauchy transforms are, satisfies
the ^-property if whenever / G X and d is inner with f /d G Hp for some p > 0,
then f /^ G X. By the classical Nevanlinna factorization theorem, the Hardy spaces
have the ^-property. It turns out that X, as well as the multipliers of X, enjoy the
^-property.
For the Hardy space, every inner function is a multiplier. On the other hand,
there is the deep result of Hruscev and Vinogradov which says that an inner function
is a multiplier of X if and only if it is a Blaschke product
zm TT \an\ Q>n- Z
CLn 1 CLn,Z
n=l
whose zeros (an)n^i satisfy the uniform Frostman condition
E°° 1 - \an\
The proof of this is quite complicated but still worthwhile to present since it involves
many earlier results about Cauchy transforms as well as the well-known Carleson
interpolation theorem.
There is also an interesting connection between multipliers and co-analytic
Toeplitz operators, namely, a bounded analytic function 0 on D is a multiplier of
X if and only if the co-analytic Toeplitz operator
(Tff)(z) := / Ml^l dm(C) = Wh(z)
* Jt 1 - (z
is a bounded operator from the space of bounded analytic functions to itself.
Kolmogorov's weak-type estimate m(|K\i\ > y) = 0(1/y) has been re-examined
recently yielding some fascinating results on how this distribution function y h->
m(\K/i\ > y) can be used to recover the singular part of the measure fi. Chapter 7
is devoted to these ideas. For example, it is relatively easy to show that when
\i <^m, the Kolmogorov estimate can be improved from
ro(|A>l >V) = 0(l/y)
to
m(\Kfi\ >y) = o(l/y).
Tsereteli proved the converse, namely,
m(\Kii\ > y) = o(l/y) <^> /x < m.

6
OVERVIEW
The relationship between the distribution function and the singular part of the
measure goes well beyond the improved Kolmogorov estimate. The first of two
important theorems here is one of Hruscev and Vinogradov which says that
lim 7rym(\Kfi\ > y) = ||/xs||,
where fis is the singular part of /i with respect to Lebesgue measure. Notice that
when \i <C m, or equivalently \is = 0, we obtain Tsereteli's theorem. The other
more striking, and more recent, theorem of Poltoratski says that
lim -Kymi\K\i\ > y) • m = /is, weak-*,
y-^oo
thus recovering the actual singular part of the measure and not merely its total
variation norm.
These distributional results are closely related to the distribution functions
y i-> m(\Qii\ > y) and y i-> mi(|!K/i| > y)
of the conjugate function
(Qn)(e*9) = P.V.J cot (^) d/x(e")
and the Hilbert transform
(Xn)(x) = P.V. [ —dn{t),
JR X — I
where \i is a finite measure on R. Some of these distribution theorems are quite
classical. For instance, an 1857 theorem of Boole says that if
n
£(x):=^-^—, OjGR, Cj>0,
3 = 1 J
which is just the Hilbert transform of the positive discrete measure
n
3 = 1
then
1 n
mi({x e R:g(x) > y}) = -^c?,
y 3=1
where mi is Lebesgue measure on R.
Though the material in the first several chapters is certainly both elegant and
important, our real inspiration for writing this monograph is the relatively recent
work beginning with a seminal paper of Clark which relates the Cauchy transform
to perturbation theory. Due to recent advances of Aleksandrov and Poltoratski, this
remains an active area of research rife with many interesting problems. Chapters
8, 9, and 10 cover this connection between Cauchy transforms and perturbation
theory.
Let us take a few moments to describe the basics of Clark's results.
According to Beurling's theorem, the subspaces dH2, where d is an inner function, are

OVERVIEW
7
all of the (non-trivial) invariant subspaces of the shift operator Sf = zf on H2.
Consequently, the invariant subspaces of the backward shift operator
/ - /(o)
s*f
are of the form (rdH2)±.
The functions
kx(z):= V4 W, A,zGD,
1 — Xz
are the reproducing kernels for (dH2)^ in the sense that k\ G (dH2)^ and
/(A) = (/,fcA) v/e(tf#2)x-
Here we are using the usual 'Cauchy' inner product
(/,»)••= [ f(OW)dm(t)
Jj
on H2. Clark's work was inspired by the question as to whether or not a given
sequence of kernel functions (k\n)n^i has dense linear span in (i9H2)±. Clark
showed that for certain £ G T, the kernels fc^ belong to (dH2)^ and are eigenvectors
for an associated unitary operator Ua on (,dH2)±. Using the spectral properties of
Ua, Clark determined when these eigenvectors fc^ form a spanning set for (dH2)^
and then used a Paley-Wiener type theorem to say when the k\n's were 'close
enough' to the fc^'s to form a spanning set.
The unitary operator UQ mentioned above is the following: let S$ be the
compression of the shift S to ($H2)J-; that is,
S* :=PtfS|(i?ff2)x
where P$ is the orthogonal projection of H2 onto (i3H2)±. All possible rank-one
unitary perturbations of S#, under the simplifying assumption that i9(0) = 0, are
given by
Uaf:=S*f + (f,-)a, aeT.
It turns out that Ua is also cyclic and hence the spectral theorem for unitary
operators says that Ua is unitarily equivalent to the operator 'multiplication by
z\ (Zg)(£) h-> C#(C)> on the space L2(aa), where aa is a certain positive singular
measure on T. It is quite remarkable, as we shall discuss in a moment, that <ja can
be computed from the inner function d.
The unitary equivalence of Z on L2(aa) and Ua on (dH2)^ is realized by the
unitary operator
Fa:(i?i?V-+£Va),
which maps the reproducing kernel
1 — Xz
for (dH2)^ to the function
r |—^ __
l-AC

8
OVERVIEW
in L2(aa) and extends by linearity and continuity. Clark uses this unitary
equivalence, as well as the structure of the associated space L2(aa), to further examine
whether or not the kernels (fcAn)n^i form a spanning set for (dH2)^.
This spectral measure aa for Ua arises as follows: for each fixed a G T the
function
is a positive harmonic function on D, which, by Herglotz's theorem, takes the form
where the right-hand side of the above equation is the Poisson integral (Pcra)(z) of
a positive measure aa on T. Without too much difficulty, one can show that the
measure aa is carried by the set {£ G T : $(£) = a} and hence aa _L m.
Furthermore, aa _L ap for a ^ /3. Though many mathematicians, and some physicists, have
used the measures described above, we think it is appropriate to call such measures
'Clark measures' since they are frequently referred to as such in the literature.
This idea extends beyond inner functions d to any (p G ball(i7°°) to create
a family of positive measures {/ia : a G T} associated with (p. It is becoming a
tradition to call this family of measures the ' Aleksandrov measures' associated with
(p. A beautiful theorem of Aleksandrov shows how this family of measures provides
a disintegration of normalized Lebesgue measure m on the circle. Indeed,
/ia dm(a) = m,
where the integral is interpreted in the weak-* sense; that is,
1(1 /(C) <WC)) dm(a) = / /(C) dm(C)
for all continuous functions /onT.
The identity
produces the following formula for
Ta : L\aa) - (tftfV
in terms of the 'normalized' Cauchy transform
_,,_#(/daa)
Poltoratski showed that several interesting things happen here. The first is that for
cra-almost every £ G T, the non-tangential limit of the above normalized Cauchy
transform exists and is equal to /(C)- On the other hand, for g G ($H2)±, the
non-tangential limits certainly exist almost everywhere with respect to Lebesgue
measure on the circle (since (dH2)1- C H2). But in fact, for cra-almost every £, the
non-tangential limit of g exists and is equal to (3ra^f)(C)-
The compression S$ and its rank-one unitary perturbation Ua are covered in
Chapter 8. Clark measures, as well as Clark's theorem and Poltoratski's weak-type
I

OVERVIEW
9
theorem
lim nym(\K/jL\ > y) • m = /is, weak-*,
y-^oo
are covered in Chapter 9. Poltoratski's theorems on the normalized Cauchy
transform
K(fdfJL)
are covered in Chapter 10.
At the end of Chapter 10, we briefly mention an independent and parallel
'Clark-type' theory, starting with some early papers of Aronszajn and Donoghue
and continued in more recent papers of Simon and Wolff, involving the spectral
measures for the rank-one perturbations
A\:= A + \v®v, A G R,
of a self-adjoint operator A with cyclic vector v. Here, the Borel transform
JR t — Z
a close cousin to the Cauchy transform, comes into play.
In Chapter 11 we survey some results about the classical operators on X. These
operators, which have been studied quite extensively on the Hardy spaces Hp,
include the shift, backward shift, composition, Toeplitz, and Cesaro operators. We
also discuss versions of the Hardy space theorems, Beurling's theorem for example,
in the setting of Cauchy transforms.
Conspicuously missing from this book is a discussion of the Cauchy transform
/ dfi(w)
J w - z
of a measure \i compactly supported in the plane. Certainly these Cauchy
transforms are important. However, broadening this book to include these opens up a
vast array of topics from so many other fields of analysis such as potential theory,
partial differential equations, polynomial and rational approximation [212, 213,
214], the Painleve problem, Tolsa's solution to the semi-additivity of analytic
capacity [216, 217], as well as many others, that our original motivation for writing
this monograph would be lost. Focusing on Cauchy transforms of measures on the
circle links the classical function theory with the more modern applications to
perturbation theory. If one is interested in exploring Cauchy transforms of measures
on the plane, the books [27, 50, 73, 78, 146, 154, 169] as well as the survey
papers [32, 33] are a good place to start. There is also a notion of fractional Cauchy
transforms [131].

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CHAPTER 1
Preliminaries
1.1. Basic notation
There is a complete list of symbols towards the end of the book. Here are some
basic symbols and some remarks to help get the reader started.
• C (complex numbers)
• C = CU{oo} (Riemann sphere)
• K (real numbers)
• B = {zeC: \z\ < 1}
• T = <9D
• N = {1,2,---}
• N0 = {0,1,2,...}
• Z = {...-2,-1,0,1,2,...}
• When denning functions, sets, operators, etc., we will often use the
notation A := xxx. By this we mean A 'is defined to be' xxx.
• As is traditional in analysis, the constants c, c', c", • • • c\, C2, • • • can change
from one line to the next without being relabled.
• Numbering is done by chapter and section, and all equations, theorems,
propositions, and such are numbered consecutively.
• If J is a set in some topological vector space,
— V J is the closed linear span of the elements of J.
— J- is the closure of J.
• If A C C, then A = [a : a G C}, the complex conjugate of the elements of
A. From the previous item, note that A~ is the closure of A.
• A linear manifold in some topological vector space is a set which is closed
under the basic vector space operations. A subspace is a closed (topolog-
ically) linear manifold.
1.2. Lebesgue spaces
1 A non-empty family jF of subsets ofC:=CU{oo}is called an algebra if
AuBe? for all A,BgJ
and
C\AGJ for all Ae5.
An algebra jF is called a a-algebra if
oo
[J An E J whenever {An : n G N} C J.
n=l
A complete treatment of this standard real analysis material can be found in many texts.
Several that come to mind are [68, 117, 149, 182, 229].
11

12
1. PRELIMINARIES
Given any collection jF of subsets of C, there is a smallest cr-algebra containing jF.
The Borel algebra, or the Borel sets, is the smallest cr-algebra containing the open
subsets of C.
A function / : T —-> C is a Borel function if f~1(G) is a Borel set whenever
G C C is open. If follows that f~1(B) is a Borel subset of T set whenever B is a
Borel subset of C. If (/n)n^i ls a sequence of real-valued Borel functions, then the
functions
lim /n(C) and lim fn(Q
are Borel functions.
Let m denote standard Lebesgue measure on T, normalized so that m(T) = 1.
This normalization will help us avoid an extra 2n in our formulas. Let L° denote
the Lebesgue measurable functions / : T —> C and, for 0 < p < oo, let Lp denote
the space of / G L° for which
i/p
< oo.
||p:=|jT|/|"dm}
When p = oo, L°° will denote the (essentially) bounded measurable functions with
the (essential) sup norm
(1.2.1) H/IU := inf {y > 0 : m(|/| > y) = 0} .
As is customary, we equate two measurable functions that are equal almost
everywhere.
Holder's inequality
/
\f9\dm^\\f\\p\\g\\q, p>l, - + - = 1,
T P Q
Minkowski's inequality
II/ + sIIp^II/IIp + NU p>i,
as well as the associated inequality
ll/ + sllj^ll/ll? + IMIS, o<P<i,
imply that for 1 ^ p ^ oo, the quantity ||/||p defines a norm on Lp which makes it
a Banach space (complete normed linear space) while for 0 < p < 1, the quantity
d(f,g) := ||/ — g\\p defines a metric on Lp that makes it a complete, translation
invariant, metric space.
A classical representation theorem of F. Riesz says that for p ^ 1, every
continuous linear functional £ : Lp —> C takes the form
*(/)= f fVdm
JJ
for some unique g G Lq (1/p + 1/q = 1). Moreover, the identity
(1-2.2) sup j|^/5dm| :f€lS, \\f\\p ^ l| = ||5||,
implies that the norm of this functional is \\g\\q. Thus when p ^ 1, one equates
(Lp)*, the set of continuous linear functionals on Lp, with Lq. When 0 < p < 1, we
have (LpY = (0) [56].

1.2. LEBESGUE SPACES
13
We will now review distribution functions and rearrangements. Two nice
references for this are [85, 229]. For / G L°, the function
(1.2.3) Xf : [0, oo) - [0,1], Xf(y) := m(\f\ > y),
is called the distribution function for / and certainly plays an important role in
analysis and probability. One can see that Xf is a decreasing right-continuous
function on [0, oo). There are also the following Lp results.
Proposition 1.2.4. Forp>0,
/♦OO
(i) \\fWpP = PJ yp-^f(y)dy.
(2) Xf(y) < y-*\\f\\P.
For / e L°, the function /* : [0,1] -+ [0, oo),
(1.2.5) r(x):=mt{y>0:\f(y)^x}
is called the decreasing rearrangement of /. If Xf is one-to-one, then /* is AT1.
One can check that if
n
3 = 1
where the A/'s are pairwise disjoint measurable subsets of T and
bi > b2 > ••• > bn,
then
n
and
n
/*(X) = H&iX[Bi_i,Bi)(X)'
where
3
B3 -=Ylm^'
i=l
Note that bo := oo, 6n+i := 0, Bo := 0. The first important fact about /* is that
(1.2.6) Xf, = Xf,
where A/*(x) = mi(f* > x) and mi is Lebesgue measure on R. The second is that,
at least for / ^ 0, there is a measure preserving transformation h : T —* [0,1] so
that
(1.2.7) f = f*oh.
See [185] for details2.
In our presentation here, we are really considering the decreasing re-arrangement of |/|. If
one is willing to expand the definition of decreasing re-arrangement, one can prove eq.(1.2.7) for
general real-valued /.

14
1. PRELIMINARIES
1.3. Borel measures
A (finite) Borel measure /x on T is a function which assigns to each Borel set
icTa complex number jjl(A) such that /x(0) = 0 and
/ oo \ oo
5
1 / n=l
whenever (-An)n^>i C T is a sequence of pairwise disjoint Borel sets. Unless we say
otherwise, our measures will be complex-valued. We will denote the linear space
of Borel measures on T by M. A measure /iG Mis positive (denoted /x ^ 0) if
fi(A) ^ 0 for all Borel sets A C T. We set M+ := {/x G M : /x ^ 0}.
Theorem 1.3.1 (Jordan decomposition theorem). ,4n?/ fi e M can be written
uniquely as
(1.3.2) /x = (/ii -/x2) + z(/x3 -/m), Hj£M+.
For fi e M, define the £o£a/ variation of /x to be the number
(1.3.3) ||/i|| :=sup< ]P|/x(Aj)| : {Ai,--- , An] is a Borel partition of T
For a measure /x, define the total variation measure |/x| by
(1.3.4) |/x|CA) := sup < ]P \l*i>(Aj)\ : {Alr •• ,An} is a Borel partition of A > .
Note that
H(T) = |H|
and that if \i is real with \i = \i\ — /x2, Mj; G M+, then
|/x| =/xi + /x2.
For a general /x G M with
/x = (/xi - /x2) + i(/X3 - /m), Mj £ M+,
it follows from the inequality
|a±zfr|^ —-^, a, 6 > 0,
V2
that for all Borel sets AcT,
^l^^^UlAiKAX^Ai^A).
Proposition 1.3.5. T/ie space M, endowed with the total variation norm || • ||,
zs a Banach space.
Let C(T) denote the Banach space of complex-valued continuous functions on
T endowed with the supremum norm
H/IU=sup{|/(C)|:<eT}.
The identification of C(T)* (the dual space of C(T)) with the Borel measures M is
a classical theorem of F. Riesz.

1.3. BOREL MEASURES
15
Theorem 1.3.6 (Riesz representation theorem). Let £ G C(T)*. Then th
i unique /i G M such that £
= £IJj, where
w)-= jW
Moreover,
WA =sup| / /d/x
:/eC(T),||/||ooO| = ||M||.
The Riesz representation theorem implies that the map /^ !—^ ^^ is an isometric
isomorphism between M and C(T)* and one often identifies C(T)* with M.
A measure /iGMis absolutely continuous (with respect to Lebesgue measure
m), written // <C m, if /x(A) = 0 whenever A is a Borel set with m(A) = 0. A
measure fi is singular (with respect to m), written /i _L m, if there are disjoint
Borel sets A and B such that A U B = T and /x(A) = m(B) = 0.
Theorem 1.3.7 (Radon-Nikodym theorem). A Borel measure fi e M is
absolutely continuous with respect to Lebesgue measure m if and only if d/i = / dm for
some f G L1, that is to say,
fi(E)= [ /dm,
Je
for all Borel sets EcT.
The function / in the above theorem is called the Radon-Nikodym derivative
of \i and is often denoted by
Q> .= ,
dm
It is a standard fact that the Radon-Nikodym derivative of \i can be computed as
a symmetric derivative. We spend a little time with this idea since it will become
important in Chapter 9. We follow [68, 182]. For each £ G T and t > 0 (sufficiently
small), let
/((,£) :={Ceis :-t<s<t}
be the arc of the unit circle subtended by the points (elt and Ce~~lt- If M G Af is
real, define, for each £ G T,
and note that £ h-> A*(C) is a Borel function on T. Define
GD/x)(C) := lim At(C)
t—o+
(^)(C):=M+Af(().
When (22/i)(C) = (Dfi)(Q < oo we say that /i is differentiahle at £ and we write
(D/i)(Q := (I2aO(C) = (^aO(C)- For a complex measure \i~ \i\ + z/i2> where /ii,/i2
are real measures, we say that (D/i)(Q exists if both (D/ii)(Q and (_D/z2)(C) exist.
Here is a collection of important properties of D/i.
Proposition 1.3.8 (Lebesgue differentiation theorem). For ^ £ M, (D/j,)(Q
exists for m-a.e. £ G T and
(I>M)(0 = ^(C) m-a.e.

16
1. PRELIMINARIES
Theorem 1.3.9 (Lebesgue decomposition theorem). Any \i G M can be
decomposed uniquely as
where \ia^s G M with \ia <C m and (is _L m.
As a consequence of the Lebesgue decomposition theorem and the definition of
the total variation norm, one has the following.
Corollary 1.3.10. If /i = /ia -\- /is is the Lebesgue decomposition of n, then
wi = ikii + ikii.
Define
Ma := {/x G M : /x < m} Ms := {/iGM:/il m}.
Note from Proposition 1.3.5 that M, when endowed with the total variation norm,
is a Banach space and by the Lebesgue decomposition theorem,
M = Ma®Ms.
In particular,
|H| = ||/Xa|| + ||AlJ, A*a £Ma, Vs eMs
and so Ma and Ms are closed subspaces of M.
For n £ M, consider the union U of all the open subsets U C T for which
fJi(U) = 0. The complement T \ U is called the support of /i. A Borel set H C T
for which fi(H D A) = n(A) for all Borel subsets A c T is called a carrier of /i.
Certainly the support of fi is a carrier but a carrier need not be the support and
need not even be closed. For example, if / is continuous and d/i = / dm, then a
carrier of fi is T \ /_1({0}) (which is open) while the support of /i is the closure of
this set. The following facts are found in [68, 182].
Proposition 1.3.11. If /i G M+ and /i = /ia+/is is the Lebesgue decomposition
of ii, then
(1) Diis = 0 and Dfi = D/ia fia-a.e.
(2) iia is carried by {0 < D_(i < oo}.
(3) iis is carried by {D_H = oo}.
Remark 1.3.12. From time to time, we will be using the following
generalization of the Lebesgue decomposition theorem (see [99] for example): for i/,/iE M,
we say that v is absolutely continuous with respect to /i, written v <C /i, if
|/z|(E) = 0=>i/(£) = 0.
If v = (y\ — ^2)+ ^3 — ^4)5 Vj £ M+, is the Jordan decomposition of z/, the following
are equivalent: (i) v «c M, (ii) ^ <C /i, j = 1,2,3,4, (hi) |z/| <C /i, (iv) |z/| <C \\i\.
The Radon-Nikodym theorem becomes: if v <C /i, then there is an / G L1(|/i|) such
that
i/(A) = [ fdn
J A
for all Borel subsets id
We say ii,v £ M are mutually singular, written /i _L z/, if there are disjoint
Borel sets A and B with i U B = T and HCA) = MC8) = 0. The following are
equivalent: (i) ji _L z/, (ii) |/i| _L |z/|, (hi) /ij _L i/*. for j,/c = 1,2,3,4.

1.4. SOME ELEMENTARY FUNCTIONAL ANALYSIS
17
The Lebesgue decomposition theorem says that for \i, v G M,
where v% <C fi and v% _L /i. Furthermore, this decomposition is unique.
For (1 G M+ and n G N, let Fn := {( G T : m({C}) > Vn} and observe, since \i
is a finite measure, that Fn is a finite set. Also observe that
oo
{C e T : /*({<}) > 0} = (J Fn
n=l
and so the set of atoms of a measure (i.e., those £ G T for which /i({C}) > 0) must
be at most a countable set. A measure fi G M is a discrete measure if it has a
carrier that is at most countable. A measure fi G M is continuous if /i({C}) = 0 for
all £ G T. There is the following refinement of the Lebesgue decomposition theorem
[99, p. 337].
Theorem 1.3.13. If /j,e M, then
li = na + fic + /xd,
where \ia <C m, /ic, \±d _L m; /ic Z5 continuous, and \±d is discrete. Furthermore,
\ia,\ic,\i& are pairwise mutually singular.
1.4. Some elementary functional analysis
We expect the reader to know the basics of functional analysis and so this brief
section is merely to set the notation. For a reader needing a review, we recommend
the books [49, 142, 183, 231].
For a complex Banach space X, with norm || • ||, let X* denote the dual space
of continuous linear functionals £ : X^C. Note that X* is a Banach space when
endowed with the norm
(1.4.1) II^H :=sup{|^)| :xeX, ||x|| < 1}.
We will make several uses of the uniform boundedness principle.
Theorem 1.4.2 (Principle of uniform boundedness). Let J be a family in X*.
If for each x G X,
sup{|^(»| :££?} <oo,
then
sup{\\e\\ :££?} <oo.
We will also make several uses of the Hahn-Banach theorems.
Theorem 1.4.3 (Hahn-Banach extension theorem). Suppose W is a closed
subspace of X and £ G W*. Then there is an L G X* such that L\W = £ and
\\L\\ = Wl
Theorem 1.4.4 (Hahn-Banach separation theorem). Suppose W is a closed
subspace of X and x G X\W. Then there is an £ G X* such that £(W) = {0},
\\£\\ = 1, and£{x) = dist(x,W).

18
1. PRELIMINARIES
For W C X, define the polar of W to be the set
W° := {i£X* : sup \£(x)\ ^ 11 .
-polar of y to be the set
Y := \xe X:sup|^(x)| ^ ll.
For y C X* define the pre-polar of y to be the set
For V C X (or X*) the convex hull of V is the set
n n
]T CjVj : ^ G V, Cj ^ 0, ]T Cj = 1
The convex balanced hull of V is the set
n n I
]T c^7- : ^ G V, Cj G C, ]T |cj| ^ 1 > .
J=i i=i J
Here are some important facts about polars.
Proposition 1.4.5.
(1) IfWx C W C X, *Aen r c W?;
(2) IfY1cY C X\ then °Y c °Yi;
(3) //1^ C X, £/ien °(VF°) is the closure of the convex balanced hull ofW.
For a closed subspace W of a Banach space y, let y/VF be the space of cosets
[y] := y -\-W. When given the usual (pointwise) vector space operations
[yi] + [2/2] := [2/1 + 2/2], c[y] := [cy],
where 2/1,2/2 £ ^ and c G C, and the norm
||[y]|| := dist(y,W) = inf{||y + H| '-weW},
the quotient space ^/W becomes a Banach space. Let W-1, the annihilator of W,
be the subspace
W-L := {^ G y* : ^(W0 = 0}.
Note that W1- is a closed subspace of y*. The following two results follow from the
Hahn-Banach theorems.
Theorem 1.4.6. For a closed subspace W of a Banach space X, the quotient
space X*/W_L is isometrically isomorphic to W*. In fact, for each £ G X*,
sup{KO)| : w G W,\\w\\ ^ 1} = dist(£,W^).
Furthermore, there is a (\> G VF^ so £/m£
|K+ 0|| = dzs^W^).
Theorem 1.4.7. For a closed subspace W of a Banach space X, the Banach
space (X/Wy is isometrically isomorphic to W^. Moreover, for fixed x G X,
sup{K(x)| :£eW±,\\£\\ ^ l} = dist(x,W).
Furthermore, this supremum is achieved.

1.4. SOME ELEMENTARY FUNCTIONAL ANALYSIS
19
We now consider other topologies on X and X*. We say U C X is weakly open
if given any xq £ U, there are t\, • • • , £n G X* and an e > 0 such that
n
p| {xeX: \£k{x-x0)\ <e}cU.
k=i
We mention a few important facts about the weak topology on X. First, X, endowed
with its weak topology, is a locally convex topological vector space. Second, a
weakly closed subset of X is normed closed but the converse is generally not true.
However, as a consequence of Mazur's theorem, a convex subset of X is weakly
closed if and only if is it norm closed. Third, the weak and norm topologies on
X are the same if and only if X is finite dimensional. A sequence (xn)n^i C X
converges to x G X weakly if £(xn) —* £(x) for each £ G X*.
The dual space X* is endowed with the norm given by eq.( 1.4.1) which makes
it a Banach space. There is another important topology on X*. A set U C X* is
weak-* open if for any £o G [/, there are x\, • • • , xn G X and an e > 0 such that
n
f]{£eX*:\(£-£0)(xk)\<e}cU.
k=i
The space (X*, *), X* endowed with this weak-* topology, is a locally convex
topological vector space. A sequence (£n)n^i C X* converges to £ weak-* if and only if
£n(%) —* £{%) for each x G X. An application of the uniform boundedness
principle (Theorem 1.4.2) says that a weak-* convergent sequence (£n)n^i is uniformly
bounded, that it to say, sup{pn|| : n ^ 1} < oo. There is also the important
Banach-Alaoglu theorem.
Theorem 1.4.8 (Banach-Alaoglu). For a Banach space X, the closed unit ball
ball(X*) :={^GX* : ||l|| ^ 1}
is compact in (X*, *).
Remark 1.4.9. If X is also separable (i.e., contains a countable dense set),
then ball(X*) (with the weak-* topology) is metrizable. Thus compactness, in the
weak-* topology, of ball(X*) is equivalent to the fact that if (£n)n^i is a sequence
in ball(X*), then there is an £ G ball(X*) and a subsequence £nk —* £ weak-*.
We will be applying this result to the unit ball in the space of measures many
times. This also says, using an elementary property of the metric topology, that
if E C ball(X*) and £ belongs to the weak-* closure of E then there is a sequence
(£n)n^i C E converging weak-* to £. In several applications, we will have a subset
E of ball(X*) for which we can identify the weak-* closure using the Hahn-Banach
separation theorem. Using only this Hahn-Banach argument, we can say that given
an £ in the weak-* closure of E, there is a net in E converging to £ weak-*. The
above argument using the Banach-Alaoglu theorem says there is a sequence in E
converging to £ weak-*.
Theorem 1.4.10. IfYc X*, then (°Y)° is the weak-* closure of the convex,
balanced hull ofY.
If X is a Banach space, then so is X* and hence one can consider its second
dual X** := (X*)*. For x G X, let Q(x) be the element of X** defined by
(Q(x))(i) = t(x)

20
1. PRELIMINARIES
and observe from the Hahn-Banach theorem that the map x \-+ Q(x) is an isometric
linear map from X into X**, often called the cannonical embedding of X into X**.
The space X is said to be reflexive if this map x >—> Q(x) is onto. One can show
that Lp, for 1 < p < oo, is reflexive while L1 is not. We point out some basic facts
about reflexive spaces.
Theorem 1.4.11. For a Banach space X, the following are equivalent.
(1) X is reflexive.
(2) X* is reflexive.
(3) Every subspace of X is reflexive.
(4) Every quotient space of X is reflexive.
(5) The closed unit ball {x G X : \\x\\ ^ 1} is compact in the weak topology.
The last of the above equivalent conditions is a consequence of Goldstine's
theorem [81].
A Banach space X is separable if it contains a countable dense set. For example,
the Lp, 1 ^ p < oo, spaces are all separable (the trigonometric polynomials are
dense) while L°° is not. A topological vector space y (for example X* endowed
with the weak-* topology), is separable if it contains a countable dense set. The
following proposition is useful in proving a Banach space is not separable.
Proposition 1.4.12. IfX is a Banach space and {xa : a e A} is an uncountable
subset of X satisfying
\\xa ~ Xb\\ ^ 1, a,b e A, a^b,
then X is not separable.
Proof. The hypothesis says that the open balls
A(a, 1/2) := {x £ X : ||x - a\\ < 1/2}
are disjoint. If J were a countable dense subset of X then each ball A (a, 1/2) would
contain at least one element of J, making J uncountable. □
For example, to see that L°° is not separable set
*a(C) :=Xia(0> 0<a<27r,
where Ia := {elt : 0 < t < a} and use the previous proposition.
A few results relating separability and reflexivity are the following.
Proposition 1.4.13.
(1) Let X be a Banach space. IfX* is separable, then X is also separable.
(2) If X is a reflexive Banach space, then X is separable if and only if X* is
separable.
1.5. Some operator theory
Here are a few reminders from operator theory. The sources [49, 173, 183] will
have the details. For Banach spaces X, y, a linear operator A : X —* y is bounded if
(1.5.1) sup{||Ax||y : ||x||x *U} < oo.
The quantity in the previous line is called the operator norm of A and is denoted
by ||i4||. Note that A is continuous if and only if it is bounded.

1.5. SOME OPERATOR THEORY
21
Theorem 1.5.2 (Closed graph theorem). A linear operator A : X —» y is
bounded if and only if its graph
{(x,Ax) : x £ X}
is a dosed subset of X x y. Equivalently, the graph of A is dosed if and only if
given a sequence xn —+ x such that Axn —» y, £/ien Ax = y.
If A : X —» X is a bounded linear operator, we define (J(A), the spectrum of A,
to be the set of complex numbers A such that (XI — A) is not invertible.
Proposition 1.5.3. If A : X —+ X is a bounded linear operator, then
(1) o~(A) is a non-empty compact subset of C.
(2) a(A) C {z : |*| ^ |H|}.
(3)
sup{|A| : A e <t(4)} = lim Pn||1/n.
n—+oo
If £ G y* and A : X —* y is bounded, then £ o A £ X* and this induces a linear
map A* :y* ^X*, by
A*(£) :=£oA.
The map A* is called the adjoint of A.
Proposition 1.5.4. // A : X «-► y is bounded, then so is A* and \\A\\ = \\A*\\.
Furthermore, if the dual pairing between X and X* is written as £(x) = (x,£)%, then
(x,A*£)x = (AxJ)y, xeX, £eT.
Notice that when X, y are Hilbert spaces, then A* is the usual Hilbert space
adjoin in that A* : y —» X and
(Ax, y)y = (x, A*y)x, £ G X, y G y.
In particular, if A is represented by a matrix, then A* is represented by the
conjugate transpose of A.
If !Ki,!K2 are Hilbert spaces, we say a bounded linear operator U : *K\ —* !K2
is isometric if
We say that [/ is unitary if C/CKi = 3^2- Notice that a unitary operator U satisfies
(Ux, Uy)w2 = (x, y)^ii Vx, y G JCi
and [/* = C/_1. Moreover, if C/ : !K —► !K is unitary, then
a(U) C T.
Two operators A : !Ki —* !Ki and jB : 3^2 ~* ^2 are unitarily equivalent if there
is a unitary U : !Ki —* !K2 such that
A = U*BU.
An operator A : IK —» 3i is cyc/zc if there is a vector v G 3i (called the cyclic
vector) such that
\J{Anv : n G N0} = IK.

22
1. PRELIMINARIES
Here V denotes the closed linear span. If a £ M, a theorem of Szego [101, p. 49]
says that
(1.5.5) / log ( -— J dm = —oo
h \dmj
if and only if the operator
Mc : L2(a) - L», (MC/)(C) := C/(C)
has the constant function x = 1 as its cyclic vector. Since M? = M^, this operator is
unitary. As it turns out, this operator is the 'model' for all cyclic unitary operators.
Theorem 1.5.6 (Spectral theorem for unitary operators). If Ji is a separable
Hilbert space and U : 3i —* 3i is unitary and cyclic with cyclic vector v, then there
is a measure a £ M satisfying eq. (1.5.5) and a unitary T : *K —* L2(o~) such that
Tv = 1 and
T*MCT = U.
If A : 'K —» 3i is self-adjoint, that is, A* = A, then it is well-known that
cr(A) cE. If /i is compactly supported measure on K one can consider the operator
Mx : L2(fi) - L2(/i), (Mxf)(x) = xf(x).
Since M* = M^, Mx is self adjoint. Moreover, by the Stone Weierstrass theorem,
the vector 0 = 1 is cyclic for Mx. It turns out that Mx is the 'model' for all cyclic
self adjoint operators.
Theorem 1.5.7 (Spectral theorem for self-adjoint operators). If Ji is a
separable Hilbert space and A : 3i —* 3i is a cyclic self-adjoint operator with cyclic
vector v, then there is a finite compactly supported measure \i on R and a unitary
T : 3i —+ L2(/i) such that Tv = 1 and
T*MXT = A.
Definition 1.5.8. If A : "K —» 'K is either self-adjoint or unitary, we will say
that A has pure point spectrum if the corresponding spectral measure (from the
spectral theorem) is discrete, that is fi = fid (see Theorem 1.3.13).
Notice that fi has a point mass at z if and only if the characteristic function
X{z} is an eigenvector for Mz on L2(fi). Thus fi is discrete if and only if the
characteristic functions on the point masses of fi span L2(fi). Since the eigenvectors
for Mz correspond to the eigenvectors for A (or U) via the intertwining operator,
the operator A (or U) has pure point spectrum if and only if its eigenvectors form a
spanning set. This observation will become important in Chapter 8 and Chapter 9.
1.6. Functional analysis on the space of measures
Recall from Section 1.3 that M denotes the space of finite, complex, Borel
measures on T and C(T) denotes the complex-valued continuous functions on T.
By the Riesz representation theorem (Theorem 1.3.6) the mapping fi i—► t,^ is an
isometric isomorphic mapping from M to C(T)* which, from our remarks in the
previous section, gives rise to the weak-* topology. As before, we write (M, *) to
denote M, endowed with the weak-* topology. A net (/xa)aga converges to fi weak-*
if and only if
/ / dfix -» fdfi

1.6. FUNCTIONAL ANALYSIS ON THE SPACE OF MEASURES
23
for every / G C(T). An equivalent and useful characterization of weak-*
convergence in M comes with the following [156, 210].
Proposition 1.6.1. A net (//a)aga C ball(M) converges weak-* to /i if and
only if
fix(A) -»fi(A)
for each Borel set A c T with n(dA) = 0.
This next lemma is a general fact about weak-* limits and works in a variety
of settings. We state and prove it in the special setting of measures.
Proposition 1.6.2. If (/in)n^i C M converges to /i weak-*, then
sup||/xn|| < oo
n
and
llMll < Um \\fxn\\.
n—+oo
Proof. By the Principle of Uniform Boundedness, we know that
sup||/xn|| < oo.
n
Let
L := lim ||/xn||
n-+oo
and choose a subsequence (n<nk)k^i so that
k—+oo
Given e > 0, there is a K G N so that
Since
lim ||/xnJ| =L.
||/xnJ|^L + € Vk^K.
IMI =sup| / gdfj,\
there is a g G ball(C(T)) such that
w-e<
:5Gball(C(T))J,
/
#d/x
But since \ink —* /i weak-*, we can assume the above K was chosen so that
||/x||-e<
However, since # G ball(C(T)),
/ 0 d/xnfc
/ 0 d/infc
\/k> K.
and so for all /c ^ if,
The result now follows.
D
The Banach-Alaoglu theorem (Theorem 1.4.8) in the setting (M, *) takes the
following form.

24
1. PRELIMINARIES
Theorem 1.6.3 (Banach-Alaoglu theorem). The closed unit ball
ball(M) := {/i G M : ||/x|| ^ 1}
zs compact in (M, *). In particular, if (/in)n^i ^ a sequence from ball(M), there is
a subsequence (/infc)/c^i a^d a^G ball(M) such that for each f G C(T),
/ /d/infc -* / /d/x.
We also make a few remarks about separability and density. For /i G M we let
/z(n):= /rd/i(C), riGZ,
be the sequence of Fourier coefficients of /i. When d/i = /dm, we write
/(«):= //(Ordm(C)
for the Fourier coefficients of an L1 function /. Also define, for iV G No, the
iV-partial sum
N
saKmxo ••= E £(fc)cfc
and the Cesaro sum
(1.6.4) MmXC) := ^3 (So(m)(C) + • •' + 5iv(M)(0) •
When d/i = /dm, we let 0"tv(/) •= 0"Ar(/dm).
Theorem 1.6.5.
(1) (Fejer; /// G C(T), tften ||<M/)lloo < ||/||oo and aN(/) - / uniformly
onT as N —+ oo.
(2) (Lebesgue) If f G Lp, 1 < p < oo, £/&en 0"at(/) —* / almost everywhere and
in Lp-norm as N —* oo.
(3) IffeL00, ihenWcTNifiWoo^ ||/||oo and aN(f) ^ f weak-* as N - oo.
(4) For general \i G M, (Jn(i^) dm —* d/i weak-* as N —* oo.
A computation with the total variation norm shows that the uncountable set
{o~ett : 0 < £ < 27r} satisfies
(1.6.6) ||^-o>||=2, s^*
and so by Proposition 1.4.12, M is not separable in the norm topology. Here, for
C G T, 5^ is the unit point mass, that is to say, the measure on T such that
5^A>-\ 0, ifC^A
However, since every element of Ma (the absolutely continuous measures) is of the
form /dm, f e L1, and
ll/dm|| = 11/11!,
we can apply statement (2) of Theorem 1.6.5 to say that
o"tv(/) dm —* /dm, iV —> oo
in the norm of M and so, since the trigonometric polynomials with complex rational
coefficients are a countable dense subset of L1, Ma is a separable subspace of M.

1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES
25
On the other hand, (M, *), the space of measures endowed with the weak-*
topology, is separable. One can see this in several ways. First, by part (4) of the
above theorem, ctat(/) dm —» d/i weak-*. We can also see this with the following.
Proposition 1.6.7. Both Ms and Ma are dense in (M, *).
PROOF. For / G C(T) and ( G T, we have
/d<Sc = /(0-
/
It follows that the only / G C(T) that annihilates the linear span of the point
masses is the zero function. Thus, by the Hahn-Banach separation theorem, the
linear span of Ms is dense in (M, *).
To see the density of Ma in (M, *), define
d"h := 7^Xih dm, h > 0,
where Ih is the arc of the circle subtended by e~lh and elh and observe that z/^ —» Ji
weak-* (Lebesgue differentiation theorem). Now use the density of Ms in (M, *) as
argued in the first part of the proof. □
We will also make use of the following.
Proposition 1.6.8. The convex balanced hull of
{k ■ C e T}
is weak-* dense in the ball of M.
Proof. If Y = {Sc : C e T}, one can easily show that °Y = ball(C(T)) and so
(°Y)° = ball(M). Now use Theorem 1.4.10. □
Remark 1.6.9. We can combine Proposition 1.6.8 with Remark 1.4.9 to prove
the following: given \i G M, there is a sequence (/in)n^i C M such that each \in is
a finite union of point masses, ||/in|| < ||/i|| for all n, and fin —^ /i weak-*.
There is also the following refinement (see [40, p. 221])
Proposition 1.6.10. Suppose \i G M+ with support on a closed set F c T.
Then there is a sequence /in —» /i weak-* such that for each n, fin G M+, is supported
in F, is a finite linear combination of point masses, and ||/in|| = ||mII-
1.7. Non-tangential limits and angular derivatives
For an analytic function /onD and ( G T, we say that / has a radial limit L
at C, if
lim /«) = L.
For C G T and a > 1, let
(1.7.1) ra(0:={zeB:\z-C\<a(l-\z\)}
be a non-tangential approach region (often called a Stoltz region). Note that Ta(£)
is a triangular shaped region with its vertex at £ (see Fig. 1). We say that / has a
non-tangential limit value A at (, written
Z lim f(z) = A,

26
1. PRELIMINARIES
Figure 1. Non-tangential approach region with vertex at £ G T
if f(z) —■» .A as z —■» C within any non-tangential approach region ra(£). Let us
mention a few well-known results about non-tangential limits. We refer the reader
to [48] for the proofs.
Theorem 1.7.2 (Fatou). If f is a bounded analytic function on D, then the
non-tangential limit of f exists and is finite for almost every £ G T.
For bounded analytic functions, the existence of radial and non-tangential limits
are the same.
Theorem 1.7.3 (Lindelof). If f is a bounded analytic function on D and
f(z)—^Aasz—^C along some arc lying in D and terminating at ( G T, then
Z lim f(z) = A.
Unfortunately, for bounded analytic functions, non-tangential limits is about
the best we can do.
Theorem 1.7.4. Let C be a simple closed Jordan curve internally tangent to T
at the point £ = 1 and having no other points in common with T. For 0 < 6 < 2n,
let C$ be the rotation of C through an angle 6 about the origin. Then there is a
bounded analytic function f on'D which does not approach a limit as z approaches
any point e%e from the right or the left along Ce-
Littlewood [124] proved the 'almost everywhere' version of this theorem while
Lohwater and Piranian [126] proved the stronger 'everywhere' result above.
Theorem 1.7.5 (Privalov's uniqueness theorem [48, 118, 169]). Suppose f is
analytic on D and
Z lim f(z) - 0
for (" in some subset ofT of positive Lebesgue measure. Then / = 0.
Non-tangential limits are important in the statement of Privalov's theorem
since there are non-trivial analytic functions on D which have radial limits equal to
zero almost everywhere on T [25]. There are no non-trivial analytic functions on
D which have radial limits equal to zero everywhere on T [44, p. 12].

1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES
27
We know that bounded analytic functions have non-tangential limits almost
everywhere. To focus on the question as to whether or not a bounded analytic
function has a non-tangential limit at a specific point £ G T, we need the following
factorization theorem [65].
Theorem 1.7.6. /// is a bounded analytic function on D, then
where d is a bounded analytic function that has boundary values of unit modulus
almost everywhere and F is a bounded analytic function that satisfies
log|F(0)|= /log|F(C)|dm(C).
JJ
The function d is called the inner factor of / and the function F is called the
outer factor of /. We can factor d further as
where b is a Blaschke product
b(z) = zmf[
\CLfi\ C^n Z
n=l
whose zeros at z = 0 as well as {an} C ©\{0} (repeated according to multiplicity)
satisfy the Blaschke condition
oo
^(1 - \an\) < oo,
n=l
(which guarantees the convergence of the product) and s^ is the (zero free) singular
inner factor
s^(z) = exp I - / —— d/x(C) J ,
where \i G M+ and is singular. Furthermore, the outer factor F can be written as
F(z) = ji exp [J ^±J log |F(C)|dm(<)
Note that log |F| G L1 (see Theorem 1.9.4 below) and so the above integral makes
sense.
The following theorem of Frostman [48, p. 33] [72], discusses non-tangential
limits of Blaschke products.
Theorem 1.7.7 (Frostman). Let b be a Blaschke product with zeros (an)n^i.
A necessary and sufficient condition that b and all its partial products have non-
tangential limits of modulus one at £ is that
f;i^<oc.
Ahern and Clark [2, 3] refine Frostman's theorem and extend it to general
inner functions.
Theorem 1.7.8 (Ahern and Clark). Suppose that d = bs^ is inner and ( G T
with /i({C}) = 0. The following are equivalent.

28 1. PRELIMINARIES
(1) Every divisor3 of d has a non-tangential limit of modulus equal to one at
c-
(2) Every divisor of d has a finite non-tangential limit at Q.
(3)
^iic-onry k-ci
< 00.
Definition 1.7.9. For an analytic function (ft : D -» D4 and a point ( G T, we
say that (ft has an angular derivative at £ G T if for some r/ G T,
*->C 2 - C
exists and is finite. We denote the above limit, whenever it exists, by (ftf(C)-
The first thing to notice is that the existence of an angular derivative
automatically implies that
Z lim <ft(z) = r\
and that \rj\ = 1. The following result is the key to understanding angular
derivatives. A proof can be found in [6, 51, 195].
Theorem 1.7.10 (Julia-Caratheodory). For an analytic function (ft : D —» D
and £ G T £/ie following statements are equivalent.
(i)
i-i^)i
lim — —— = 5 < oo,
(2)
~C 1
Zlim^WtC)
z-*C z — (,
exists for some r\ G T,
(3)
Z lim 0'(z)
exists and
Z lim 00) = r] G T.
(a) S > 0 zn (%).
(b) JTie points r\ in (2) and (3) is the same.
(c) ^(C) = C^ and
Zlim0/O) = 0/(C).
z-*C
(d) // any of the above conditions hold, then
i-i^)i
<5 = Z lim
♦C l-|*l
We now focus on specific results on the existence of angular derivatives. We
begin with a simplifying proposition which is a corollary of Theorem 1.7.10.
"^We say an inner function ip is a divisor of $ if •d/ij) is also inner.
Such 4> are often called analytic self-maps of ID).

1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES
29
Proposition 1.7.11. If 4>i,4>2 are analytic self maps of 3 and <j> = 0202? then
I0'(C)I = 101(01 +102(01
for every £ G T.
If we focus our attention on inner functions d = bs^, where b is a Blaschke
product with zeros (an)n^i and s^ is the singular inner factor with singular measure
/i, the above proposition says we can consider the Blaschke factor and singular inner
factor separately. Here are two classical theorem that do this.
Theorem 1.7.12 (Frostman [72]). Ifb is a Blaschke product with zeros (cin)n^>i
and ( G T, then b has a finite angular derivative at £ if and only if
v^ l- K|2
Moreover,
00 1 I |2
Theorem 1.7.13 (M. Riesz [175]). The singular inner function s^ has a finite
angular derivative at £ eT if and only if
MO
I
< 00.
ic-ci2
Moreover,
MO
^(01 = 2/
K-CI5
< 00.
If /i({C}) > 0> then the above integral diverges and so s^ will not have an
angular derivative at £. In this case, 1^(701 -^ 0 as r -> T and so s^ cannot
possibly have a finite angular derivative.
Corollary 1.7.14. An inner function d = bs^ has a finite angular derivative
at £ G T if and only if
^ic-anp+2y
^K-an|2 7 |^-C|2
Moreover,
' (01 „^IC-an|2+ J K-CI2'
For conditions on the existence of angular derivatives for general self maps
we need the following factorization theorem.
Proposition 1.7.15. If 4>: D -+ D is analytic, then
(1.7.16) <j>(z) = fc(z) exp ( [ -£±i di/(C)
w/iere b is a Blaschke product with zeros (an)n^i and z/ G M+.
If z/ _L m, then the second factor is a singular inner function.

30 1. PRELIMINARIES
Theorem 1.7.17 (Ahern and Clark [3, 4]). An analytic self map (/> ofB,
factored as in eq.(1.7.16), has a finite angular derivative at £ G T if and only if
i - Kl2 , o f <M0 „
eiC-an|2+2i
-IC-anl2 J k-Cl2
Moreover,
<M0
i'<oi-E^W
l£-CI!
P,(C):=5R(^) = ^^, CeT, 2g
g,(C):=cj(^Ll) = ±^i, (ST, 2GD.
1.8. Poisson and conjugate Poisson integrals
Define the Poisson kernel
'C + z) = l~\z\2
X-z) \C~z[
and conjugate Poisson kernel
X + z\ 2Z((z)
X-zJ -\C-zf
For fixed ( G T, the functions
z*->Pz(C) and z*->Qz(C)
are harmonic on the open unit disk D and so, for \i G M, the Poisson integral
(1.8.1) (Pfi)(z):=jpz(OMC)
and the conjugate Poisson integral
(1.8.2) (Qn){z):=JQz{Odn(Q
are harmonic on D. An obvious closely related kernel is the Herglotz kernel
which is an analytic function of z with 9fti7z(() = Pz(() > 0 and so the Herglotz
integral
/•
(1.8.3) (Hfi)(z):= J Hz(C)dfi(C)
is analytic on D and has positive real part whenever /i G M+.
Observe that for 0 < 5 < 1 and £ G T,
and so
3ft
1 — st 1 — st *-^
s s n=0
n= —oo
where
1, ifn<0;
sgn(n) = ^ 0, if n — 0;
1, ifn>0.

1.8. POISSON AND CONJUGATE POISSON INTEGRALS
31
Thus
oo
(P/x)«) = £ n(n)r^C
n= —oo
(1.8.4)
n= —oo
where, as before,
£(n):= /fdMC)
are the Fourier coefficients of \i.
Here are some standard facts about Poisson integrals [101, p. 32 - 33].
Proposition 1.8.5. For an f e L1 and 0 < r < 1, let
/r(C) := (P/dm)«), C G T.
(1) If f is continuous, then fr—*f uniformly onT as r —* 1~.
(2) If f e Lp, l^p<oc, then fr-*finLvasr-*l-.
(3) If f £ L°°, then fr —+ f weak-* as r -+ 1", that is to say
/ frgdm -* / fgdm, r -* 1~
Jt ./t
/T ./T
/or ever?/ g £ L1.
(4) For a general /i G M, (P/i)(r-) dm —* d/i weak-* as r —» l-.
Here are two important results that will be used many times throughout this
book. The first is Fatou's theorem5.
Theorem 1.8.6 (Fatou). If fi e M, and (D/j,)(C) exists, then
hm (P/i)(rC) = (^)(C).
r—>1~
Remark 1.8.7.
(1) From Proposition 1.3.8, D\i = d/i/dm m-almost everywhere and so the
radial limit of the Poisson integral is equal to the Radon-Nikodym derivative
m-a.e.
(2) If \i _L m, or equivalently D\i = d\±ldm = 0 m-a.e., then the above limit
is zero m-a.e.
(3) The radial limit in Fatou's theorem can be replaced by a non-tangential
limit, that is to say,
Zlim(P/i)(z) = (L>M)(<)
whenever (D/i)(Q exists.
(4) If C G T and fi is a real measure, then [182]
(1-8.8) (Dp)(Q < !im (P/x)«) < En~ (P/x)«) < (S/x)(C).
Fatou's original proof in terms of Poisson-Stieltjes integrals is in [69]. The references [65,
p. 39] or [101, p. 34] have modern proofs.

32
1. PRELIMINARIES
If \i G M+, then certainly P/i > 0 on D. Also note that i7/i is analytic on D
with RH/i = P/i ^ 0. This following theorem of Herglotz 6 is the converse.
Theorem 1.8.9 (Herglotz).
(1) Ifu^O on D and harmonic, then u = Pfi for some /i G M+.
(2) If / is analytic on D, 3£/ ^ 0, and /(0) > 0, then f = i7/i for some
/iG M+.
From Fatou's theorem (Theorem 1.8.6), we know that P\i has finite non-
tangential boundary values m-almost everywhere and we will see in the next chapter
(Lemma 2.1.11) that H\i does as well. Since Hfi = P/i+zQ/i, then Q/i has boundary
values and the m-almost everywhere defined boundary function
(Q/x)(C) := lim (Qn)(rQ
is called the conjugate function. At least formally (replacing z with ez0 and £ with
elt in the eq.(1.8.2)), this boundary function (Qfi)(eze) is equal to
(QM)(e") = fj 5 (f^5) d/i(e«) = fj cot (^) dMe").
Unfortunately, for fixed 0, the function cot(0 — t) may not belong to L1(/i), making
the integral possibly undefined. In terms of principal value integrals, we do have
the following standard fact.
cot(V)d/x(eit)
^cot(^)d/x(e«).
for m-a.e. e%e.
1.9. The classical Hardy spaces
For 0 < p < oo, let Hp, the Hardy space7, denote the space of functions /
analytic on D for which the Lp integral means
(1.9.1) Mp(r;/):=|jT|/(rC)|I,dm(C)}
remain bounded as r | 1~~. This definition can be extended to p = oo by
M^r;/) :=sup{|/K)|:CeT}
and so H°° is the set of bounded analytic functions on D. The function
r k+ Mp(r; /)
is increasing on the interval [0,1), that is,
(1.9.2) A/(ri; /) < M(r2; /), 0 ^ n < r2 < 1,
Theorem 1.8.10. If n
lim
GM,
(Q/i)(re") =
then
/»2tt
P.V. /
Jo
= lim /"
The reference [98] contains the original proof while [101, p. 34] or [65, p. 2] have more
modern proofs.
We refer the reader to several classic texts [65, 79, 101, 118, 234] for the proofs of
everything in this section.

1.9. THE CLASSICAL HARDY SPACES
33
and the quantity
\HP := sup Mp{r-J) = lim Mp(r;f)
0<r<l r|l~
defines a norm on Hp when 1 ^ p < oo. When 0 < p < 1, the quantity
dist(f,g):=\\f-g\\pHP
defines a translation invariant metric on Hp'. The pointwise estimate
(1-9-3) |/(z)K 2^11/11^ (i_^|)i/p, zeB,
can be used to show that Hp (1 ^ p < oo) is a Banach space while Hp (0 < p < 1) is
an F-space (a complete translation invariant metric space). In particular, if fn —-> /
in Hp', then fn—>f uniformly on compact subsets of D.
The following standard facts about functions in Hp spaces will be used many
times throughout this book.
Theorem 1.9.4. For 0 < p < oo and f e Hp,
(1)
/(C):=Zlim/(z),
the non-tangential limit of f at (, exists for almost every £ G T.
(2) This m-a.e. defined boundary function £ h-> /(£) belongs to Lp and when
0 < p < oo,
lim /"|/K)-/(C)|pdm(C) = 0.
r-*l JT
ffence ||/||Hp = ||/||„.
(3) IffeHr\{0}, then
J
Jt
log|/(C)|dm(C)>-oo
/T
and hence the function ( h-> /(£) can no£ vanish on any set of positive
measure in T.
(4) Ifp^l, and f G Hp has Taylor series
oo
f(z) = J2<lnZn,
n=0
£/ien
JT
/(C)C dm(C), neN0.
(5) For 0 < p < oo, £/ie polynomials are dense in Hp. When p = oo, £/ie
polynomials are weak-* dense in H°°.
Every / G i7p has an associated boundary function which belongs to Lp and
has the same norm. We denote this set of boundary functions by
Hp(T) := (/ £LP : /(C) = lim f(r£) a.e. for some / G Hp
Frequently we will not make a distinction between Hp and HP(T). As such, we will
also use the notation
ll/L = 11/11**

34
1. PRELIMINARIES
for the Hp norm of /, or equivalently the Lp norm of the boundary function C h->
/(C)- Throughout this book we will use the following important fact.
Proposition 1.9.5 (Smirnov). If 0 < p < q and f £ Hp has Lq boundary
values, then f G Hq.
We know that HP(T) is a closed subspace of Lp. Turning this problem around,
one can ask: when does a given / G Lp belong to HP(T)? At least for p ^ 1, there
is an answer given by a theorem of F. and M. Riesz.
Theorem 1.9.6. For p ^ I, a function f G Lp belongs to HP(T) if and only if
the Fourier coefficients
f /(OfdmK)
Jt
vanish for all n < 0.
Actually, the following is the most useful version of this theorem.
Theorem 1.9.7 (F. and M. Riesz theorem). Suppose /i G M satisfies
Cn d/i(C) = 0 whenever n G N0.
Then d/x = 0dm, where 0 G H% = {/ G H1 : /(0) = 0}.
/■
Every / G Hp can be factored as
(1.9.8) f = OfIf.
The function Of, the outer factor, is characterized by the property that Of belongs
to Hp and
(1.9.9) log|O/(0)|= [log\Of(C)\dm(Q.
JT
Every Hp outer function F (i.e., F has no inner factor) can be expressed as
(1.9.10) F(z) = e^exp f / f^ log^(C) dm(C)
where 7 is a real number, ip ^ 0, log^ £ Ll, and -0 G Lp. Note that F has no zeros
in the open unit disk and |F(C)| = 0(C) almost everywhere. Moreover, every such
F as in eq.(1.9.10) belongs to Hp and is outer. The inner factor, If, is characterized
by the property that 7/ is a bounded analytic function on D whose boundary values
satisfy |//(C)| = 1 for almost every C- Furthermore, as seen Section 1.7, the inner
factor 7/ can be factored further as the product of two inner functions
(1.9.11) If = bSlM,
where b is a Blaschke product and s^ is a singular inner function.
A meromorphic function / on D is said to be of bounded type if / = hi/h2, where
hi,h2 are bounded analytic functions on D. From Theorem 1.9.4 and eq.(1.9.8), a
function of bounded type must have finite non-tangential limits almost everywhere
on T and can be factored as
, _ IhxOhx
Ih2Oh2
The set N, the Nevanlinna class, will be the functions / of bounded type which
are analytic on D (equivalently 1^ is a singular inner function). The set iV+, the

1.10. WEAK-TYPE SPACES
35
Smirnov class, will be the set of / G N for which Ih2 is a constant. It is a standard
fact that
feN^> lim / log+|/(rC)|dm(C) <oc
and that for / G iV, the boundary function satisfies
/"log+|/(C)|dm(C)<oo.
For / G iV, we have
/GiV+^ lim /log+|/(rC)|dm(C)= / log+ |/(C)| dm(C).
Note also that
(J iF c iV+.
p>0
We also have the following generalization of Proposition 1.9.5.
Theorem 1.9.12 (Smirnov). If f £ N+ with Lp boundary function, then f G
HP.
1.10. Weak-type spaces
We say a function / G L° (the Lebesgue measurable functions on T) belongs
to L1'00, or weak-L1, if
ra(l/l > 2/) = 0 f - J , ?/-> oc.
We say / G Lj'00 if
Define the quasi-norm8
™(\f\ > v) = o[ - ), ?/->oo.
|Li,oo :=svpym(\f\ > y).
y>0
Let Hl'°° be the analytic functions on D for which
H/IIhi.oo := sup UMIllco < oo, /r(C) = /«).
0<r<l
Proposition 1.10.1.
#1,00 c p| Hp
0<p<l
Proof. It follows from the distributional identity
WP=pf yp-1m(\g\>y)dy, gGL°,
«/[0,oo]
This quasi-norm does not satisfy the triangle inequality ||/-f-g|| ^ ||/|| +\\g\\ but does satisfy
||/-h^|| ^ 2(||/|| + \\g\\). See [111] for more on quasi-norms.

36 1. PRELIMINARIES
(Proposition 1.2.4), that for / G Hl'°° and A = \\fr\\L^
/»oo
\\fr\\pP=P yp-1m(\fr\>y)dy
Jo
pA /»oo
= p yp~lm(\fr\ > y)dy+p I yp~2ym(\fr\ >y)dy
JO J A
pA /»oo
^p yP^dy+pA / yp"2dy
JO J A
P
1-P
= AP + -^—Av
- A"
~ 1-p-
a
The following deep result is an equivalent characterization of Hl'°° [9].
Theorem 1.10.2. For an analytic function f onH), the following are equivalent.
(1) fe &•<*>.
(2) The radial maximal function
(M/)(C):= sup |/K)|
0<r<l
belongs to L1,0°.
(3) The non-tangential maximal function
(Naf)(Q:= sup \f(z)\
belongs to L1,0°.
Remark 1.10.3. Compare this theorem to the following equivalent
characterization of Hp by Hardy and Littlewood [87] (1 ^ p < oo) and Burkholder, Gundy,
and Silverstein [35] (0 < p < 1) (see also [79, 118]): if 0 < p ^ oo and / is analytic
on D, then the conditions (i) / G ifp, (ii) Mf G Lp, (iii) Naf G Lp, are equivalent.
Since every / G H1:°° has boundary values, defined almost everywhere by
/(C) = lim /«),
r—>1~
we can define H^00 to be those / G if1'00 for which /|T G Lj'°°. Recall
Theorem 1.9.12 which says that if / G iV+ (the Smirnov class) and /|T (the non-
tangential boundary values of /) belongs to Lp, then / G Hp. Here is the
corresponding result for the analytic weak-type spaces.
Theorem 1.10.4. If f e N+ and /|T G L1'00 (respectively f G Lj'°°), tten
/ G if1'00 (respectively f G i^'00)-
1.11. Interpolation and Carleson's theorem
It will be important for the work in Chapter 6 to gather up some well-known
results about interpolating sequences. We quickly review these ideas and refer the
reader to sources like [21, 65, 79, 191, 193] for the formal proofs. We will write E

1.11. INTERPOLATION AND CARLESON'S THEOREM
37
to indicate a sequence in D. For simplicity we will always assume 0 £ E. Associated
to E is the discrete measure /i# on D given by
HE(A):= Y, (1 - l«D^ ^CD.
aEEHA
The Blaschke condition on E\ that is,
]T(l-|a|)<oc,
a£E
simply asserts that \±e is a finite measure. This condition is equivalent to the
convergence of the Blaschke product
B{z) := J]
\a\ a — z
a 1 — az
aeE
uniformly on compact subsets of D. We write ba for the individual Blaschke factor
\a\ a - z
K{z) =
Ba(z)
a I — az'
and let
B(z)
ba(z)
be the Blaschke product with one of its factors divided out.
We say a sequence E is separated if
(1.11.1) s(E) := inf { p(a, b) : a, b G E and a ^ b } > 0,
where
is the pseudo-hyperbolic distance between a and 6, and uniformly separated if
(1.11.2) 6(E) := inf |Sa(a)| > 0.
aeE
Let 7 be an arc on the unit circle, and define the Carleson square on I to be the
set
(1.11.3) Q= IzeB: ^- e/and 1- \z\ < m(I)\
(see Figure 2). A positive measure /i on D is a Carleson measure if there is a
constant c^ depending only on /i such that
/i(<2) ^ ^771(7)
for each Carleson square Q. We define 7^ to be the infimum of all such constants
c^. We say that E is a Carleson sequence if /i# is a Carleson measure and we set
7(£):=7/zB.
The sequence E is an interpolating sequence if, whenever # G £°°(E), the
bounded functions on the sequence E, there is a function / G i7°° such that
f\E = #. By the open mapping theorem, there is a constant C such that for
each g G l°°(E), a function / G H°° can be chosen so that
(1-11.4) H/IU^CsupflffC*)! :*€£}.

38
1. PRELIMINARIES
Figure 2. A Carleson square Q over the arc I C T
We define C(E) to be the infimum of such constants C above. It is easy to see that
E must be the zero set of a Blaschke sequence, and not too difficult to see that E
is separated. The main theorem here is one of Carleson.
Theorem 1.11.5 (Carleson). Let E be a countable subset of D. Then the
following are equivalent.
(1) E is an interpolating sequence;
(2) E is uniformly separated;
(3) E is separated and \ie is a Carleson measure.
In case any of these conditions hold, we have the following relationships between
the constants s(E),6(E)^(E), and C(E):
(1.11.6) T7^^^(^)^ciT©^c2-^-fl + log X
6{E) ^ v y^ 6(E) ^ 6{E)\ *6{E)
(1.11.7) s{E)>8{E), ^)^-~-y 5(E)>exp(-cs^y
where ci,C2,C3 > 0 are absolute constants.
Interpolation sequences actually exist [101, p. 203].
Theorem 1.11.8 (Hayman-Newman). A sequence (zn)n^i C D such that
sup(i_Z_^ii:nGN\<i
I l-|*n| J
is an interpolating sequence.
Corollary 1.11.9. If (rn)n^i c (0,1) with rn f 1, then (rn)n^i is an
interpolating sequence if and only if
supP^1 :neNJ<l.
Just in case the reader might think that interpolating sequences must approach
the unit circle exponentially, there is this curious result of Naftalevic in [147].

1.12. SOME INTEGRAL ESTIMATES
39
Theorem 1.11.10 (Naftalevic). If (rn)n^i c (0,1) satisfies
oo
n=l
then there is a sequence of angles (0n)n^i C [0,27r) s?xc/i £/m£ (rnet0n)n^i is an
interpolating sequence.
1.12. Some integral estimates
We end this chapter with some trivial but very useful integral estimates that
will be used often throughout the book. The first estimate, through rather easy,
drives everything.
Lemma 1.12.1. There are universal constants c\,c\ > 0 such that
ci((l - r)2 + #2)1/2 ^ |1 - re»\ < c2((l - r)2 + tf2)1/2
/or all r G (|,1) and a// 0 G [0,7r].
Proof. Note that
|1 - rel6\ = (1 - 2rcos6> + r2)1/2 = ((1 - r)2 + 4rsin2((9/2))1/2.
Using the estimate
- ^sin((9/2) ^6> V0G [0,tt],
7T
we get
— ^sm2(6/2)^62 V6e [0,tt].
Hence we obtain constants ci, c2 > 0 so that
ci ((1 - r)2 + (92) ^ 1 - 2rcos(9 + r2 ^ c2 ((1 - rf + (92)
for all r G (|,1) and 0 G [0,?r]. D
Lemma 1.12.2. Given p > 1, there is a positive constant c > 0 so that
r i c
/^ \i-reiO\P ^ (l-r)P-1
for all r G (|,1).
Proof. Observe that
r de _ r do _ r do
J_n \l-reie\P "/_„ (l-2rcos(9 + r2)P/2 ~ 70 (1 - 2r cos(9 + r2)?/2 "
Thus by Lemma 1.12.1,
r do r do / r^ r \
J_n\l-re^\P^CJ0 ((l-r)* + 0*)P/*-C\Jo +J/1_J'
Estimating these two integrals, we get
/,1""r d0 f1'7' d0 _ 1
J0 ((1 - r)2 + 02)p/2 ^ ^ ((1 _ r)2)p/2 - (l_r)p-l
r de r de _ i c
i!_r ((l-r)2 + 02)p/2 ** J^ (02)p/2 "C+ (l-r)P-1 ** (l-r)P"1'
and

40
1. PRELIMINARIES
that
□
Lemma 1.12.3. There are constants ci,c2 > 0 independent of r G (|,1) such
i r i i
ci log ^ / w dO ^ c2 log .
1 — r J_7T\l—re™\ 1 —r
Proof. From Lemma 1.12.1,
r de ^ r i ^ <r r ^
ci / , ^ / - T^r av ^ ci / .
Jo V(l-O2 + 02 7-^ |1 -re^| 70 ^(1 ~ 02 + #2
By integrating, we get
and the estimate follows. □

CHAPTER 2
The Cauchy transform as a function
2.1. General properties of Cauchy integrals
For \i G M, the analytic function
(2.1.1) (*»(*):= /—Vd/i(C)
7 1 - 0*
on D is called the Cauchy transform of /i and the set of functions
X := {Kfi : /JEM}
is called the space of Cauchy transforms. Note that
// oo \ oo „
\n=0 J n=0 ^
and so K\i has the power series expansion
oo
(2.1.2) (ArM)(z) = £/i(n)zn
n=0
where
£(n) :=/rd/i(C), neZ,
are the Fourier coefficients of the measure \i. From the elementary inequality
l£(«)l < IImII,
we can say the following.
Proposition 2.1.3. The Taylor coefficients of a Cauchy transform are bounded.
Having bounded Taylor coefficients does not automatically gain one entrance
into the space of Cauchy transforms. One need only consider the following theorem
of Littlewood [65, p. 228]: If (an)n^i is a sequence of complex numbers such that
oo
lim |an|1//n = 1 and Y^ \an\2 = oo,
n=0
then for dlmost every choice of signs (en)n^o> the analytic function on D defined by
oo
f(z) = ^2 e^anZn
71=0
41

42 2. THE CAUCHY TRANSFORM AS A FUNCTION
does not have radial limits on a set of full measure on T.1 We will see momentarily
(Theorem 2.1.10) that a Cauchy transform must have radial limits almost
everywhere. From here, one can create an analytic function on D with bounded Taylor
coefficients that is not a Cauchy transform.
Definition 2.1.4. For a fixed / £ 3C, let
Rf:={neM:f = K^i]
be the set of measures that represent /.
Observe that Rf is always an infinite set. To see this, notice that if 0 G Hq =
{/ e Hl : /(0) = 0}, then
4>{n) = JCHOdm(C) = 0 Vn G No,
and so
oo
K (d/i + 4>dm) (z) = ^ (/2(n) + £(n)) zn = £ jl(n)zn = {Ky)(z).
n=0 n=0
Thus
/iGfl/=>d/i + 0dm e Rf V0 G #0\
making ify an infinite set.
We leave it to the reader to use the F. and M. Riesz theorem (Theorem 1.9.7)
to prove the following proposition.
Proposition 2.1.5. Let /el
(1) Kfi = 0 if and only if d/i = 0 dm for some (f) G H$.
(2) For /i, v G Rf, d/i — dv = 0 dm for some <fi G Hq.
(3) If n,v G Rf, then fis = vs?
Remark 2.1.6. Using (2) above we have an equivalence relation on the space
of measures M and each element of % corresponds to a coset in M. We will discuss
this further in Chapter 4.
Let us say a few words about the boundary behavior of a Cauchy transform.
A simple estimate shows that K\i satisfies the growth condition
(2-1.7) \(Kn)(z)\ < J^jL.
This follows from the inequalities
'""X"1 S / |rJWd""(<>« /T^Nd,"IK)« rqir
For any £ G T, observe that
(l-r)(tf/i)(rC)= f^—f-d^).
J l-£rC
A routine exercise using the dominated convergence theorem will show that
(2.1.8) lim(l-r)(JrM)«) = /*({<})■
There is a rich history of such types of functions. See [234, p. 380] and [109].
Recall that jjls is the singular part, with respect to m, of \i (see Theorem 1.3.9).

2.1. GENERAL PROPERTIES OF CAUCHY INTEGRALS
43
Thus
lim \(Kfi)(rC)\ = 00
r—*l~
whenever /jl({C}) ¥" 0> which can indeed be a dense subset of T. In fact, Poincare
noticed the poor behavior of Cauchy transforms of certain discrete measures back
in 1883 when he observed that the analytic function defined by the series
00
(2-1.9) /(*) = £**
n=lL ^nZ
where (cn)n^i is an absolutely summable sequence of non-zero numbers and (Cn)n^i
is a sequence of distinct points that are dense in T, does not have an analytic
continuation across any portion of the unit circle. Observe that the above example
of Poincare is the Cauchy transform of the discrete measure
d/i = y^cn£CTi,
where S^n is the unit point mass at Cn-3
Despite the fact that for certain measures /i,
lim \(Kfi)(rQ\ = 00
r—^l-
for ( in some dense subset of T, this pathological set must be of Lebesgue measure
zero. Indeed, there is some regularity in the boundary behavior of the Cauchy
transform. Recall from Chapter 1 the definition and basic properties of the classical
Hardy space Hp (0 < p < 00) of analytic functions / on the unit disk for which
i/p
< 00.
||p:={ sup f\f(rC)\pdm(0)
For example, by Theorem 1.9.4, functions / G Hp have radial boundary values
/(C) := lim /(rC)
r—>-l_
for almost every £ G T and
\\f\\PP= /"|/(C)lPdm(C)= lim / |/K)P>dm(C).
Theorem 2.1.10 (Smirnov). If /i G M, then
0<p<l
and moreover,
\\Kf*\\p ^ Cp||Ai||,
where
Poincare's example in eq.(2.1.9) is more general than what we stated here. He proved,
using a different method, since the Lebesgue theory was not available to him, the same non-
continuability result with the circle replaced by a curve bounding a convex set in the plane [161].
In fact, there is quite a large literature on creating analytic functions on D which have all sorts of
pathological properties near the boundary. Several representative examples are [25, 126, 127].

44
2. THE CAUCHY TRANSFORM AS A FUNCTION
Proof. Using the Jordan decomposition to write fi G M as
M = Oi - M2) + *(^3 - M4), Mj £ M+,
and noting that
7 |i-C*r
the result follows from four applications of the following standard fact [79, p. 114].
□
Lemma 2.1.11. Let F be analytic on D with !RF > 0. Then for all 0 < r < 1
and 0 < p < 1,
/»2tt
Moreover j
f \F(reie)\Pd0^Ap\F(O)\P.
Jo
^ = °irbj' p^1_-
Proof. Since 9?F > 0, then F = |F|e^, where -tt/2 < 0 < tt/2. Since F has
no zeros in the disk, the function Fp (the branch which has positive real part at
the origin) is also analytic on D and
Fp = \F\P (cos(p0) + zsin(p0)).
For 0 < p < 1,
SR(FP) = \F\Pcos(W>) ^ |F|pcos(ptt/2).
We conclude that
/ \F(rel6)\pdO ^ Ap [ " R(Fp(rexe))dO = Ap$l(Fp{0)).
Jo Jo
The last equality follows from the mean-value property of harmonic functions. The
desired inequality follows from the observation that R(Fp(0)) ^ |F(0)|p. Finally
notice that
K = —7^-7^ = o (—^— ), p -> r
p cos(>7r/2) \l-p;' F
D
Corollary 2.1.12. If f G X, then the non-tangential limit of f exists and is
finite for almost every £ G T.
Proof. Since X C #p for all 0 < p < 1 (Theorem 2.1.10), the result follows
from the existence of non-tangential limits of Hp functions (Theorem 1.9.4). □
Observe that the containment
X£ p| Hp
0<p<l
is strict since one can check, by using the estimate in Lemma 1.12.1, that the
function
/(^log^)^

2.1. GENERAL PROPERTIES OF CAUCHY INTEGRALS
45
belongs to Hp for all 0 < p < 1. However, / does not satisfy the necessary growth
condition
1 - \z\
in eq.(2.1.7) to be a Cauchy transform. One can also see that / is not a Cauchy
transform by using Proposition 2.1.3 and the observation that
n=l \fc=l /
and hence has unbounded Taylor coefficients.
Proposition 2.1.13. /// is analytic on D and Rf > 0, then /el
Proof. Without loss of generality, assume that /(0) > 0. If this is not the case,
replace / by g = / —i9/(0). If we can show # = K\±, then / = if(i3/(0)dm + d/z).
With the assumption that /(0) > 0, we can apply Herglotz's theorem
(Theorem 1.8.9) to see that
for some \i G M+. A little algebra shows that
c - ^ i - c^
and so / = if (2/i — m). D
We will see in Theorem 5.6.3 that if / is analytic on D and C \ /(B) contains
two oppositely oriented half-lines, then / £ 3C
Remark 2.1.14. Theorem 2.1.10 is due to Smirnov [200] (see also [65, p. 39]).
In Proposition 3.7.1, we will begin to look at the 'best' constant cp in the inequality
\\Kp\\p^cp\\n\\.
Smirnov's theorem yields the estimate
ii^iip = °(r1^)
For certain measures, we can do a bit better.
Proposition 2.1.15. Iffi^m, then
PROOF. Let d/i = gdm for some g G L1. For e > 0 given, let
-1 N
h(Q= ]T h(n)C + J2kn)C, (GT,
n= — N n=0
be a trigonometric polynomial with \\g — h\\i < e (an appropriate Cesaro polynomial
of g will work - see Theorem 1.6.5). Observe that
N
K(hdm)(z) = Y^Hn)zn.
n=0

46
2. THE CAUCHY TRANSFORM AS A FUNCTION
Thus,
\\K(hdm)\\*
N
Y,Kn)zn
N
^h(n)zn
n=0
/ N ~
= £lM«)ls
\n=0
p/2
<
£
and so certainly
\\K{hdmW = o{
1-p
), P"
Using Smirnov's theorem (Theorem 2.1.10),
\\K(hdm) -K(gdm)\\pp ^ C \\hdm-gdm\\
= C
1
1-p
1
ii^-ffii
Finally,
(l-p)\\K(gdm)\\>^o(l)+Ce*.
Letting e —> 0 yields the result.
□
2.2. Cauchy integrals and i/1
The classical Cauchy integral formula4 says that if / is analytic in a
neighborhood of D , then
Making the observation that
><*> = ^
d(
ICI=i
/(C)
d(.
2ni(
dm(C),
we will write the Cauchy integral formula as
/(C)
/(*) = / y
JJ 1
(z
dm(C).
The question now is: what is the 'largest' class of analytic functions on D that can
be written via the Cauchy integral formula? For / G H1, Theorem 1.9.4 says that
/(C) = lim /(rC)
exists for almost every ( G T and defines an integrable function. Thus for H1
functions, the integral on the right-hand side of the Cauchy integral formula makes
sense. It turns out that the left-hand side is equal to the right-hand side.
See [201] for a historical overview of Cauchy.

2.2. CAUCHY INTEGRALS AND H1 47
Proposition 2.2.1 (Cauchy integral formula). For f £ H1,
Jjl
f(z) = I -/^-dm(C),
that is to say, f = K(fdm).
Proof. For 0 < r < 1, let fr(z) = f(rz) and note that fr is analytic on the
slightly larger disk {\z\ < 1/r}. By the classical Cauchy integral formula,
fr(z)= /-MOdm(C).
J 1 - Cz
Note that fr —>• / in L1 (Theorem 1.9.4) and so for each z G
/awa^
as r —> V
Clearly fr —* f pointwise in D. Combine these two limits to obtain the Cauchy
integral formula. □
By Holder's inequality, Hp C H1 for all p ^ 1, and so we can combine
Theorem 2.1.10 and Proposition 2.2.1 to see that
(2.2.2) [JHP £ X £ p| Hp.
p^l 0<p<l
We have already seen why the second containment is strict. The first containment is
strict since / = (1 — z)-1 = K8\ but does not belong to H1. Indeed, the boundary
function f(eie) is 1/(1 - eie) and
l/(e")l~]Jp ^°'
and hence is not integrable.
From Proposition 2.2.1, every / G H1 can be written as the Cauchy integral
of its boundary function. This next proposition says, in a sense, that H1 functions
are the only ones which can be written in this way.
Proposition 2.2.3. Let f be analytic on D. Then the following two conditions
hold if and only if f G H1:
(1) The function
/(C) := lim /«)
r—>-l_
exists for m-almost every £ G T and is integrable.
(2) For all z£B,
f(z) = J^-zdm(C).
Proof. If / £ if1, then the two conditions hold by Theorem 1.9.4 and
Proposition 2.2.1. Conversely, suppose the two conditions hold. Then / = K(fdm) G Hp
for all 0 < p < 1 (Theorem 2.1.10). But since the boundary function belongs to
L1, then / G H1 (Proposition 1.9.5). □

48 2. THE CAUCHY TRANSFORM AS A FUNCTION
2.3. Cauchy yl-integrals
In this section, we prove a generalization of the Cauchy integral formula
J 1 - C,z
involving the theory of .A-integrals as studied by Denjoy, Titchmarsh [215], Kol-
mogorov, Ul'yanov [225], and Aleksandrov [9].
A Lebesgue measurable function g on T is A-integrable if the following two
conditions hold. The first is that g G L0,oc, that is to say,
(2.3.1) m(\g\>y) = o(l/y), y - oo.
The second is that
(A) [g(Qdm(C):= Urn / g{Q dm(C)
exists. We call the above limit the A-integral of g. One can show, as Titchmarsh
did, that the .A-integral is a linear operation5.
Proposition 2.3.2 (Titchmarsh). Iff and g are A-integrable functions and a
is a constant, then the functions f + g and af are A-integrable and
(A)J(f + g)dm=(A)Jfdm+(A)Jgdm
(A) j{af)dm = a({A) j/dm).
Proof. For y > 0, let
if 1/(01 < V,
and observe that
Jyy^> ' \ 0, otherwise.
(A) //(C)dm(C)= lim /"/(j/,C)dm«).
The functions (/ + g)(y, () and f(y, () + g(y, () equal /(C) + g{Q off the set
(2.3.3) {|/ + <?I>3/}U{|/|>2/}U{M>2/}.
But since
{|/ + 5l>2/}c{|/|>2//2}u{|5|>3//2},
and m(\f\ > y) and m(\g\ > y) are both o(l/y), the measure of the set in eq.(2.3.3)
is 0(1/?/). Also observe that for all £ G T,
\U + 9)(y,0\^y, \f(y,0+g(y,0\<2y.
Therefore,
J f(y, C) dm(C) + Jg(y, C) dm(C)) - /(/ + g){y, C) dm(C)
< / 3ydm«)
•^{|/+9l>y}u{|/|>y}u{|g|>!,}
< 3j,o(l/y)-0
We thank A. Poltoratski for showing us this proof.

2.3. CAUCHY A-INTEGRALS
49
as y —> oo. This proves the first identity of the proposition. The second identity is
obvious. □
The growth condition in eq.(2.3.1) is essential for linearity (see Remark 2.3.18
below).
For \i G M with \i <C m, the Cauchy transform / = K\i has non-tangential
boundary values m-almost everywhere and, as a consequence of Kolmogorov's
theorem (see Theorem 3.4.1 and Proposition 3.4.11 below), / satisfies the growth
condition in eq.(2.3.1). However, this Cauchy transform may not belong to H1 (see
eq. (3.2.6) below) and so cannot be recovered from its boundary function via the
Cauchy integral formula. A theorem of Ul'yanov [225] is the substitute 'Cauchy
A-integral formula'.
Theorem 2.3.4 (Ul'yanov). For \i G M with fi < m, the function f = K/jl is
A-integrable and
(2.3.5) f(z) = (A) [ -^- dm(C), z G D.
Jt 1 - Cz
The version of Ul'yanov's theorem we wish to prove is a generalization due to
Aleksandrov [9] concerning the weak-type class H0,o°. See Chapter 1, especially
Theorem 1.10.4, for a reminder of the definition of H^°°.
Theorem 2.3.6 (Aleksandrov). If f e H^°° then
f{z) = {A) [ JttLdm(t), zeB.
Jj 1 - Qz
Proof. We begin our proof by showing that
(2.3.7) f(0) = (A)Jfdm.
Before computing
(A) j fdm= lim / /dm,
J L-+ooJlfl<L
we make some preliminary observations. Recall from eq.(1.2.3) the distribution
function
A:[0,oc)-+[0,1], X(y) = m(\f\ > y)
for / and, from eq.(1.2.5), the decreasing re-arrangement
0 : [0,1] -+ [0, oo), 0(x) := inf{y > 0 : X(y) ^ x}
of /. Notice that m is normalized Lebesgue measure on the circle and so ra(T) = 1.
We will also make several uses of the inequality
(2.3.8) </>(x) = o(-\ x->0+.
Indeed, a geometric argument shows that when y = </)(x),
xc/)(x) ^ ym((/) > y) = y\(y),
where the last equality follows from eq.(1.2.6) which states that the distribution
functions for 0 and |/| are the same. The estimate in eq.(2.3.8) now follows from
the assumption that yX(y) = o(l) as y —-> oo.

50 2. THE CAUCHY TRANSFORM AS A FUNCTION
From eq.(1.2.7), there exists a measure preserving map h : T —-> [0,1] such that
(2.3.9) 0oh=|/|.
Since h is measure preserving, it must therefore satisfy
(2.3.10) f G(h(C))dm(C)= / G{x)dx
for every G G ^[0,1].
For large L > 0 and our measure preserving map /i, let
(0 if/i(C)^A(L),
5L(0: ll0gX§ if^^A^
[gL(0dm(Q=f log^dm(C)
which by eq.(2.3.10) equals
(2.3.11) jf log-^dx = -A(L).
Thus gL satisfies
—oo < gL ^ 0 and gL £ L1.
Let
9L = PgL, 9L = QgL
be the Poisson6 and conjugate Poisson integrals of g^ and note that <7l(0) = 0. The
function
FL = exp(#L + z#L)
is analytic on D, |Fl| ^ 1 (since ^ < 0), and is outer7. Furthermore, jFl G i/1.
To see this last fact, first observe that JFl G Hp for all 0 < p < 1 (since / G
#0'°° C Hp for all 0 < p < 1 and Fl is bounded) and so, by Proposition 1.9.5,
it suffices to show that JFl has integrable boundary values. Using the identities
eq.(2.3.9) and eq.(2.3.10),
[\fFL\dm= [ |/(C)|^1 dm(C)+ / |/(C)| dm(C)
J Jh^X(L) A\L) Jh>\(L)
= [ <t>(h(C))^dm(C)+ [ 0(MC))dm(C)
Jh^X(L) A\L) Jh>\(L)
rKL) x ri
= / 4>{x)--—dx+ I </>(x)dx.
JO A\L) J\(L)
Now use eq.(2.3.8) to show that the first integral converges. The fact that (j) is
decreasing shows that the second integral also converges. Thus JFl G H1.
We abuse notation a little here and identify gL on the circle with its harmonic extension to
the disk via the Poisson integral. Notice that both functions are the same almost everywhere on
the circle.
7Observe that / log|FL|dm = gLdm = (PgL)(0) = log|FL(0)| (see eq.(1.9.9)).

2.3. CAUCHY A-INTEGRALS 51
Since fFL G H\
9l(0) _ f/nx -A(L)
(2.3.12) J jFL dm = /(0)FL(0) = /(0)e^°> = /(0)e
This last equality follows from the identity
9l(0) = J gLdm = -X(L)
and eq.(2.3.11).
We are now ready to compute
lim / / dm.
-—°o J\f
We have
L-+oojlf^L
[ fdm= [ fFLdm+ [ f(l-FL)dm- [ fFLdm
J\f\^L Jj J\f\<L J\f\>L
= (I) + (II) -(III).
By eq.(2.3.12), the first quantity (I) is equal to
/(0)FL(0) = /(0)e-A<L> - /(0) as L -. oo,
since A(L) = o(l/L) by our assumption that / G H0,o°.
For the second quantity (II),
1/2 / n x 1/2
|l-FL|2dm ,
\^L J
(2.3.13) / \f\\l-FL\dm^ If |/|2 dm) ( [
Afl^L \J\f\^L J \J\f
Let us estimate each of the factors in eq.(2.3.13). To this end,
J |/|2dm = J (j)(h)2dm (from eq.(2.3.9))
= [ (p(x)2dx (fromeq.(2.3.10))
^ / (p(x)2dx.
J\(L)
This last inequality follows from the containment
(2.3.14) {x:(P(x) <L} C (A(L),1]
which follows from the definition of <p. Apply the distributional equality (with
fe^O)
/»1 /»00
/ k(x)2dx = tm1(k>t)dt
Jo Jo
to the function x0, where \ = X[A(L),i]> to get
/»1 /»00
/ c/)(x)2dx= tmi(x</>>t)dt.
J\{L) Jo

52
2. THE CAUCHY TRANSFORM AS A FUNCTION
From the definition of 0, it follows that <p(\(L)) ^ L. But since <j> is decreasing, it
also follows that \(\) ^ L and so
/♦OO pL,
I tmi(x0> i)dt= tmi(x<t>> t)dt
Jo Jo
^ / tmi((/>>i)dt
Jo
= / t\(t)dt (by eq.(1.2.6))
Jo
^ Lsup{£A(£) : £E [0,oo)}
< cL (since A(£) = o(l/£), t —> oo)
Combine all of this to get
(2.3.15) / \f\2 dm ^cL.
We will come back to this inequality in a moment.
Now let us estimate the second factor in eq.(2.3.13). Indeed,
/ |l-FL|2dm= / \l-e9Lel9L\2dm
= [ |l-e^|2dm (^L = 0on{|/|^L},eq.(2.3.14))
J\fUL
'l/l<£
< f \l-^L\2dm
Jt
^ / \gL\2 dm (since |1 - eix\ ^ \x\)
Jt
^ / \9l\2 dm (by Parseval and eq.(1.8.4))
/t
mon2
1<l v A(L)
= J0 V0gA(L)J dX (by e^(2-3-10))
= 2A(L).
Putting this together with eq.(2.3.13) and eq.(2.3.15), we get
/ |/||l-FL|dm^cV/ZA(L)
J\mL
which goes to zero as L —» oo.

2.3. CAUCHY A-INTEGRALS
53
For quantity (III),
/ \f\\FL\dm= [ |/| log-A-dm
J\f\>L J\f\>L X\L)
= [ 0(h(O)^77Tdm(C) (by eq.(2.3.9))
J<t>(x)>L A\L)
/4>(x)>L
fHL) x
Using the fact that (p(x) — o(l/x), one shows that this quantity approaches zero as
L —> oo. Thus
(2.3.16) (A) J fdm= lim / fdm = f(0).
J L->ocJ\f^L
To prove the Cauchy A-integral formula
f(z) = (A) /-^-dm(C), zGD,
Jt 1 - Cz
fix z G D and assume for a moment that f(z) = 0. Apply eq.(2.3.16) to the function
F(W) = ^M-
w — z
to get
(2.3.17) 0 = f(z) = F(0) = (-4) / F(Q dm(C) = (A) f -^L- dm(C).
Jt Jt 1 - C^
When /(;?) is not necessarily equal to zero, apply eq.(2.3.17) to the function
G(w) = f(w) - f{z)
to get
G{z) = {A) / ^§-dm(C)
(A);kwadm(0
Jt 1 - (z
{A) f JULdm(Q-f(z).
Jt 1 - Cz
Cz
The result now follows. □
Remark 2.3.18.
(1) The little-oh condition in eq.(2.3.1) is important in the statement of Alek-
sandrov's theorem (Theorem 2.3.6) and the theorem fails without it. For
example, consider the function
/(z) = z___
and observe (see Theorem 7.4.1) that
2 (\
ra(|/| ^ L) = — arctan —
7T V L

54
2. THE CAUCHY TRANSFORM AS A FUNCTION
which is not o(l/L) as L —> oo. Furthermore, for all £ G T \ {1},
Thus the quantities
/(C)dm(C)
/,
'l/l<£
are real-valued for every L. However, /(0) = z and hence the formula in
eq.(2.3.5) is not valid for this /.
(2) Theorem 2.3.4, and various other results about conjugate functions, were
explored by Ul'yanov [224, 225]. For example, if / G L1, then the non-
tangential boundary function / of the conjugate function (Q/dm),
although defined almost everywhere, need not be integrable, making the
quantity (Qfdm) undefined as a Lebesgue integral. However,
(Qfdm)(z) = (A)JQ,(Qf(C)dm(C),
z g :
A good treatment of this is found in [26, Vol II].
(3) See [20, 184] for other .A-integral theorems.
(4) The paper [77] contains an application of .A-integrals to the study of real
outer functions.
2.4. Fatou's jump theorem
So far, we have been considering the Cauchy transform K/i as a function on
the unit disk. However, the function
(C/i)(s):= /-^d/i(C)
Jt 1 - (z
is analytic on C\T with an analytic continuation across the complement of the
support of ii. Let us use the notation
(C/x)i := Clip, (C/x)c := C/x|Dc,
where De := {z G C : 1 < \z\ ^ oo} is the extended exterior disk. Of course one
notices that {C/i)i = K/jl. One can check, using an analog of Smirnov's theorem -
Theorem 2.1.10, that
(C/i)eG f| ffP(Dc),
0<p<l
where F G Hp(3e) means the analytic function z k-> F(1/z) on D belongs to Hp.
The norm of an F G #p(De) is
\\F\\HPiBe):=\\F(l/z)\\HP.
One can also prove, as before, that
H(CM)e||ff*(De)=0(j4^), P-l-
This means that (C/x)e has (exterior) non-tangential limits almost everywhere on
T. It is also true that
l(CWe(*)|<
\z -1

2.4. FATOU'S JUMP THEOREM 55
and that (C/x)e(oo) = 0. One can even compute, in a similar way as in eq.(2.1.2),
the Taylor series of (C/i)e about the point at infinity as
(C/i)e(*) = -£
Jl(-n)
i z
n=l
Define, for almost every £ G T,
(C/x)l(C) := lim (C/x)i(rC), (C/x)e(C) := lim (CM)e(C/r).
Two important quantities to consider are
(2.4.1) (C/x)t(C) - (CAi)e(C) and (CM)t(C) + (C/i)e(C).
For the first, a computation shows that
(C/x)(rC) - (C/i) (C/r)
[ PrdOMO,
Jt
the Poisson integral of /i (see eq.(1.8.1)). By Fatou's theorem (Theorem 1.8.6) we
obtain the following 'jump theorem'.
Corollary 2.4.2 (Fatou's jump theorem). For \i G M,
(Cfi)i(Q ~ (Cfi)e(Q = ^(0 rn-a.e. C G T.
As we shall see later (see Theorem 5.3.1 and Theorem 5.4.5), this 'jump'
theorem will help us characterize the analytic functions /onC\T which can be written
in the form / = C\i for some \i G M. For now though, we summarize the above
ideas.
Proposition 2.4.3. For fi G M, the following are equivalent.
(i)
lim(AT/x)(rC) = ^(C)
r-+i- dm
m-almost everywhere.
(2) (Cn)e = 0.
(3) The Poisson integral
u(z)= J PZ(Q d^O
is analytic on D.
Proof. (1) => (2): By Fatou's jump theorem (Corollary 2.4.2)
(C/xMC)-(C/x)e(c) = ^(C)
almost everywhere. So statement (1) implies that (C/i)e(C) = 0 almost everywhere.
By Theorem 1.9.4, we get (C/x)e = 0.
(2) => (3): For z G D, one can verify the identify
(2.4.4) u{z) = J Pz (C)dAi(C) = (CM) (z) - (C/x) (1/z).
Thus if (C/i)e = 0, then u = (Cfi)i and thus is analytic.
(3) => (1): Suppose it is analytic on D. By using the identity in eq.(2.4.4), we
know that the function (Cii)e(l/~z) is analytic on De. But since this function is also

56
2. THE CAUCHY TRANSFORM AS A FUNCTION
conjugate analytic, it must be constant (Cauchy-Riemann equations). However,
since (Cfi)e(oo) = 0, this constant must be zero. Thus u = (Cfi)i and so, by
Fatou's jump theorem,
^(0= lim u(rC)= lim (C/x)t(rC)
am r-+i- r-+i-
m-almost everywhere. D
2.5. Plemelj's formula
We now return to the second quantity in eq.(2.4.1), namely
(C/i)i(C) + (CAi)e(O-
This quantity is closely related to the conjugation operator. Before getting to this
though, we bring in an auxiliary operator dating back to as early as 1873 with the
work of Sokhotski. For \i G M, the Cauchy integral
\-iz
is defined for all z G D. At least formally, replace z G D with a boundary point
( G T and consider the integral
1
/
/
i-£C
dM(0.
Technically this integral is not always denned since the function £ h-> (1 — £C)_1
may not belong to !}{\l). Perhaps a more precise definition of the integral should
be the principal value (or singular) integral
RV- I ^Tc M() := lim+ / ^Tc M()
whenever this above limit exists. This is indeed the correct one.
Theorem 2.5.1 (Privalov8). For /i G M,
(2.5.2) P.V.|_^__d/i(0
exists for m-almost every £ G T and zs egtm/ £o
^((c^co + ^^eCO).
We will give a proof of the general Plemelj formula in a moment. But for now,
we give a very elementary proof in a special case. Suppose that / is analytic on D,
continuous on D~, and satisfies the Lipschitz condition
\f(z)-f(w)\^Af\z-w\a, z,w€B~
Theorem 2.5.1 is known as the Plemelj or the Sokhotski-Plemelj formula. Sokhotski [202]
in 1873 was the first to prove this formula when d// = fdC, where / is a Lipschitz function. Plemelj
[160] refined this result in 1908. The version here is due to Privalov as part of this thesis [168].
See also [169] where one can find a more general version of Theorem 2.5.i for Cauchy integrals
on general curves.

2.5. PLEMELJ'S FORMULA 57
- 9
for some a > 0 and Af independent of the points z and w in D . For \z\ ^ 1 let
and for |tu| = 1 and e > 0 let
2?n 7{|c-«.|»e}nT C - ™
We will show that
lim (Cf)(rw)+ lim (Cf)(w/r) = 2 lim (Cef)(w)
7H J
/(C)
C — w
LdC
which is a special case of the Plemelj formula. To see this, observe from the classical
Cauchy integral formula that for 0 < r < 1 and \w\ = 1, we have the following two
identities
Using the Lipschitz condition,
\f(C)-f(rw)\^Af\C-rw\a
and the dominated convergence theorem, one can prove the identities
taJC/Kn.J.jL^/ifl^ « + /(.)
r—l- 2m JT C — w
Notice, by the Lipschitz condition, that these two integrals are absolutely
convergent. Now observe that
(Cef)(w)
±[ /(O-/H^ , f(w) J dc
Again using the Lipschitz condition and the dominated convergence theorem, we
see that
/(c)~/(wV i/H.
-I
lim (CJ)(w) = — / J^' " 'd( +
Combine this with the two identities above to get Plemelj's formula.
To prove the Plemelj formula for the general case, use the power series
expansions
oo oo n
(CWiK) = £ fi(n)rnC, (C/i)e(C/r) = - £ m(-")^,
n=0 n=l ^
and eq. (1.8.4) to get
(Cm)<«) + (C/x)e(C/r) = i(Q/x)(rC) + m(T).
A theorem of Hardy and Littlewood [82] says that the above condition is equivalent to /
being continuous on D" and |/(Ci) - /(C2)| ^ A/|Ci - C2|a for all Ci,C2 G T.

58
2. THE CAUCHY TRANSFORM AS A FUNCTION
Thus for m-almost every £,
(2.5.3) lim ((C/i)i(rC) + (C/i)e(C/r)) = *(Qm)(C) + M(T),
where, by Theorem 1.8.10, (Q/i)(Q is given by the principal value integral
But since
2 ,,-CV^ + C
— 1 + z^y '
i-ft U-C
the principal value integral
exists m-almost everywhere and is equal to /x(T) + z(Q/i)(C). From eq.(2.5.3) we get
lim ((CnUrO + (CnUCM) = PV. [ -^ <MO
r—1~ J 1 — ^C
which is Plemelj's formula (Theorem 2.5.1).
2.6. Tangential boundary behavior
From our previous work, we know that % C Hp for all 0 < p < 1 and such
functions have finite non-tangential limits at almost every point of the circle.
Generally, this is about the best one can do since by Theorem 1.7.4, there are bounded
analytic functions, which are certainly Cauchy transforms by Proposition 2.2.1,
which do not have tangential limits at any point of the circle. If one is willing to
make some extra assumptions, there is something more to be said. First, let us be
clear by what we mean by tangential limits. Consider the following types of contact
regions:
^(O^I^GDilC-^K^l-l^l)1^}, (GT, c,7>0
1 x-Vt*
E7jC(C) := <z G D : |C - z\ < c ( log — ) [> , ( G T, c, 7 > 0.
The first types of regions A1:C(Q are the finite order of contact regions with contact
point C £ T, while the second type E1:C(Q are the exponential order of contact
regions. For example, the regions A\^c are the non-tangential approach regions and
are triangular-shaped regions with vertex at (. When 7 = 2, these regions become
(essentially) circles tangent to T at £ and are called oricyclic approach regions. We
say, for an analytic function / on D, that / has an A^-limit L at £ if f(z) —* L as
z ~* C within AliC(Q for every c > 0. The definition of an E^-limit is similar. We
state here without proof some results from [148, 223] concerning the tangential
limits of Cauchy integrals of the form
hF(z):=± T-^rdt,
2?r Jo 1 - e %tz
where F is an integrable function on [0, 2n].
Theorem 2.6.1. Suppose F is integrable on [0,27r] and 00 G [0,27r].
(1) If F'(0q) exists, then hp has an E\-limit at e%e°.

2.7. CAUCHY-STIELTJES INTEGRALS
59
(2) //
F(e0 + t)-F(00) = o(\t\a), t^O,
for some a G (0,1), then hp has an Ea-limit at e%e°.
If F is of bounded variation on [0, 27r], then F must be differentiable almost
everywhere and we have the following corollary.
Corollary 2.6.2. If F is of bounded variation on [0,27r], then hp has an
Ei-limit almost everywhere on T.
For other global theorems, we need a notion of capacity. We follow [223] and
refer the reader to [1] for a modern treatment of capacity. Define the functions
Ha : R \ {0} -> [0, oo) by
, . 1 , , 0 < a < 1;
Ha(x):=\ ^f
log | . i ,, a = 0.
I sm^x|
For a G [0,1), a Borel set E C [0, 2tt} is said to have a-capacity zero if there is a
positive measure of total mass one and carried by E1, i.e.,
[ d»= [
JE «/[0,2
d/x = 1,
»,2tt]
for which
sup / Ha(x — t)d/j,(t) < oo.
xER J[0,2tt]
It is a well-known fact that any set of ce-capacity zero has Lebesgue measure zero.
However, there are sets of Lebesgue measure zero but with positive ce-capacity. The
0-capacity is usually called the logarithmic capacity. Here is a global theorem from
[223] on tangential boundary limits of Cauchy transforms.
Theorem 2.6.3. Suppose F is of bounded variation on [0,27r].
(1) For each a G (0,1), there is a set Wa C [0, 2n] of zero a-capacity such
that hp has an Ea-limit at all points e%e for which 6 G [0, 2n} \ Wa.
(2) There is a set W of 0-capacity (or logarithmic capacity) zero such that hp
has A1-limits for every 7 > 1 at all points e%e for which 6 G [0, 2tt] \ W.
Finally, we mention that these results are best possible in that the types of
tangential approach regions can not be increased. See [223, Thm. 3] for an exact
statement of this.
2.7. Cauchy-Stieltjes integrals
For a measure \i G M, we call the function K\i the Cauchy transform of \i. In
the classical setting, for a function F of bounded variation on [0, 27r], the function
was called the Cauchy-Stieltjes integral of F. Equating a function F of bounded
variation with a measure \ip (see [99, p. 331]), these two integrals are the same.
Though we pose the main function theoretic properties of these integrals in terms
of measures, we mention that in this original setting, these integrals were studied

60
2. THE CAUCHY TRANSFORM AS A FUNCTION
by Cauchy, Morera [145], Sokhotski [202], Plemelj [160], and Privalov [168, 169].
Caiichy examined these integrals in proving the 'Cauchy integral formula' when
dF(t) = f(ett)dt and / was the boundary function of an analytic function on
a neighborhood of the closed unit disk. Sokhotskii and Plemelj examined these
integrals when dF(t) = (f)(elt)dt and </> was a Lipschitz function on the circle.
Privalov, examined the general case. Collectively, their theorems say (in the context
of Cauchy-Stieltjes integrals) that the limits
(CFUe*e) := lim (CF)(rew)
(CF)e(eie) := lim (CF)(eie/r)
exist for almost every 0 G [0, 2n] and
(CFUeie) - (CF)e(eie) = F'(9)
{CrUJO) + (CF)e(e*e) = P.V1- jf j^dF®
for almost every 6.

CHAPTER 3
The Cauchy transform as an operator
In the previous chapter, we explored the function theoretic properties (growth
rates, boundary values, etc.) of
(*»(*)= /-Vd/i(C),
J 1 - Qz
the Cauchy integral of a measure \i G M. In this chapter, we examine the properties
of the linear mapping
from the space of Borel measures M on the unit circle to the space of analytic
functions on the disk. In particular, we examine the question: For a given class of
measures IcM, what can be said about the functions in K(X) = {K\i : /i G X}?
We already know from Smirnov's theorem (Theorem 2.1.10) and the discussion
following that
K(M) £ p| Hp.
0<p<l
In this chapter, we take a closer look at K/i when \i <C m, that is to say, d/i = fdm
for some / G L1. For notational convenience we write
/+ := K(fdm)
whenever f £ L1.
Most of the results we plan to cover here are stated in the literature in terms
of the conjugate function (see Theorem 1.8.10)
/(c")= lim / cotf^WA
For / £ L1, recall from eq.(1.8.4) the Poisson and conjugate Poisson integrals
(P/)K):= /Prc(0/(0<M0= E f(n)rMC,
JJ n=-oc
/» OO
(£/)«):= Qrc(8f($)dm(Z) = -i J2 /(n)sgn(n)rNC.
JJ n=-oc
Since
—1^ = 1(1+ Pz(0 + zQz(C)), C^T, ZED,
l — C^ ^
and
lim (P/)«) = /(C), lim (Q/)(rC) = /(C)
61

62
3. THE CAUCHY TRANSFORM AS AN OPERATOR
for almost every £ G T, then
/+(C) = ^{/(0)+/(C)+ */(<)} a.e.
Thus, continuity questions about the operator / h-> /+ (on spaces of functions
defined on T) are equivalent to continuity questions about the conjugation operator
f„f = -2if+ + if(0) + if.
For example, by Smirnov's theorem (Theorem 2.1.10), /+ G Hp for all 0 < p < 1
whenever / G L1. The following result now follows.
Proposition 3.0.1 (Smirnov). If f £ L1, then j G Lp for allO < p < 1 and
ll/llp<cpll/lli.
Moreover,
Cp = 0(rb)' p^l~-
3.1. An early theorem of Privalov
Perhaps the earliest theorem concerning the mapping properties of the Cauchy
transform is due to Privalov and deals with the Lipschitz classes
AQ := {/ : T ^ C : |/(C) - f(0\ ^ ^/IC - £|a VC,£ G T} , 0 < a < 1.
The Lipschitz classes are clearly linear spaces of continuous functions and when
given the norm
imi . imi i ~11TJ 1/(0-/(01 .r^A
\\f\\Aa := ||/||oo+sup|—|>_^,Q : C^O,
Aa becomes a Banach space. There is also the associated space
K = {feAa:f(-n)=OyneN}
which can be considered as a space of analytic functions on D. We will see in a
moment that A+ is the space of functions / that are continuous on D-, analytic
on D, and such that /|T G Aa.
Theorem 3.1.1 (Privalov1). If f e Aa, then f e Aa. Consequently, iff G Aa,
then /+ G A+.
Proof. To prove this theorem, we let / G Aa and first show that
Indeed, a computation shows that
(/+)'(*)= /-/^-dm(C).
Jt (1 - (z)2
The original proof that the conjugation operator maps Aa to Aa is due to Privalov (1916)
[167]. The proof here is adapted from [82, p. 411], which ultimately comes from a theorem of
Hardy and Littlewood [88]: An analytic function /onD belongs to A J if and only if \f (z)\ =
0((1 — |2|)a_1) (see also [66]). Another proof of Privalov's theorem, as well as generalizations to
other spaces of smooth functions, is found in Chapter 7 of [234]. There is also a several variable
analog of this theorem in [181] (see also [232]).

3.1. AN EARLY THEOREM OF PRIVALOV
63
Moreover, if c is any constant, then
/
■dm(C) =0,
h (1 - C^)2
since the above integral represents the derivative of the constant function z \-+ c.
Letting z = relt and £ = el°, we take advantage of this observation to obtain the
formula
(/+)'(re«) = f
Jo
Now use the Lipschitz property
|/(eW) - /(e«)| < C/le'8 - e"|* = Cf\l - e'C^I
to estimate |(/+)'(relt)| as follows:
,27, (1_ei(t-e)|a d0
c2n (f(eie) ~ f(eu))e-ie d0
/0 (i _ rei(*-e))2 2tt'
Jo I1 —'
. rez(t-^)|2 27t'
The estimate |1 - el^-^\ ^ 2|1 - re^*-^| yields
(3X2) ,(/*)•(„«), < C,^" |t _„£.„,_. - C, jf ij-^
Lemma 1.12.2 gives us the estimate
i
2" dix C
/0 |1 -reiu|2-a ^ (1 -r)1-0^'
Combining this with eq.(3.1.2), we obtain
(3.1.3) \U+)\K)\^ ^ , CeT, 0<r<l.
We will use this inequality to show that the boundary function £ h-> /+(£)
satisfies the Lipschitz condition
|/+(Ci) - /+(C2)| < C/Ki " C2|a VCi,C2 G T.
First we show that this boundary function actually exists for every £ G T (By the
fact that /_|_ G #p for all 0 < p < 1, we already know that /+(C) exists for almost
every (GT). Indeed, for any ( G T, note that
(3.1.4) /+«) = /+(0) + f C(/+),«) ds.
Jo
Also observe from eq.(3.1.3) and the fact that 0 < a < 1, that the integral
^ \(f+Y(sQ\ds
I
Jo
converges for every £. Thus,
/+(C) = lim /+«)
exists for every ( G T. We conclude, from eq.(3.1.3) and eq.(3.1.4), that /+ is
bounded on D and hence is the Poisson integral of its boundary function £ h-> f+(Q-
We will show in a moment that the boundary function for /+ is continuous - and
even Lipschitz. With this assumption, it follows, from solving the Dirichlet problem
(Proposition 1.8.5), that /+ is continuous on D~~.

64
3. THE CAUCHY TRANSFORM AS AN OPERATOR
^
To establish the Lipschitz inequality, it suffices to show that
l/+(Ci)-/+(C2)KC/|Ci-C2r vCi,C2gt, |Ci-C2|<i.
To prove this, we observe that when h = 1 — |Ci — C2I, we obtain the formula
/+(&)- /+(<2)= f(f+)'(w)dw,
where 7 is the piecewise smooth curve consisting of the straight line segment from
Ci to /1C1 followed by the arc of the circle of radius h subtended by /i(j and h(2
followed by the straight line segment from h^ to (2-2 This integral can be estimated
in three parts as
|/+(Ci)-/+(C2)l
/ |(/+)'(r<i)|dr+ / h\(f+Y(heu)\dt+ |(/+)'(rC2)| dr
* ^II (d^ + 1TW=-*C'Cdt (by eq-(3-1-3))
<C/|Ci-C2|a.
D
Zygmund went on to show that the conjugation operator / i—> / maps
Ana := {/ : T - C : /("> G Aa}
to itself and hence /•-+/+ maps A™ onto (A£)+ := {/ G A£ : /(-/c) = 0 V k G N}.
The conjugation operator also maps the Zygmund space A J to itself, where A J is
the set of functions f(el°) so that /^ is continuous and the symmetric differences
satisfy
\f(n)/ei(e+t)\ _ 2f(n)(e2<9) + f(n)(e^e-t))\
SUp - '- —-y ~ - ~ < 00.
e,t \t\
Hence / h-> /+ maps A J onto (A™) + . All of the above spaces can be endowed with
appropriate norms that make them Banach spaces. With these norms, the maps
/ k-> / and / h-> /+ are continuous. See [234] for details.
3.2. Riesz's theorem
If / G L2 has Fourier series
00
/~ £ f(n)C,
n= — oo
Parseval's theorem says that
£ l/(n)|2.
Furthermore, by eq.(2.1.2),
/+(*) = £/(nK\
n=0
Notice, from the estimate in eq.(3.1.3), that this integral converges.

3.2. RIESZ'S THEOREM
65
Again, by Parseval's theorem,
ll/+ll! = 2J/(n)l'
n=0
and so
Il/+ll2<||/||2.
This makes the map / k-> /+ a projection of L2 onto H2. In fact, an orthogonality
argument shows that / >—> /+ is the orthogonal projection of L2 onto i72, often
called the Riesz projection.
In a similar way,
oo
7~-i y" /(n)sgn(n)C
and so
li = £l/(n)l2<ll/lli,
making the conjugation operator / h-> / continuous on L2. The following theorem
generalizes this to Lp for 1 < p < oo.
Theorem 3.2.1 (M. Riesz). If 1 < p < oo, the conjugation operator f h-> / is
a continuous map from Lp to Lp. Consequently, f >—> /+ zs a continuous onto map
from Lp to Hp.
Proof. 3 Let us first set some notation. For 1 < p < oo and / G Lp, let
F{z) = I ^/(C)dm(C) = (Hfdm)(z)
be the Herglotz integral of the measure fdm (see eq.(1.8.3)). Let
u(z) = (Pf)(z), v{z) = (Qf)(z)
be the Poisson and conjugate Poisson integrals of /. Note that v(0) = 0 and
F = u + iv. Recall that u(rQ —> f(Q almost everywhere and in the norm of
jjp (Proposition 1.8.5) and v(rQ —> /(C) almost everywhere as r —-> 1~
(Theorem 1.8.10).
Let us first assume that 1 < p ^ 2 and / G Lp is non-negative. Notice that u is
a positive harmonic function and |F| > 0. A somewhat tedious computation using
the fact that the Laplacian operator
^ — Oxx i ®yy
can be written as
shows that
and
A = 4dA,4
Aup = p(p-l)\Ff\2up-2
A\F\P =p2\F'\2\F\p~2.
The proof here is found in [65, p. 54] which was ultimately adapted from a proof due to P.
Stein [209]. Other proofs of Riesz's theorem are found in [36], [229, p. 256], or [234, p. 253].
See [174] for the original proof.
4dz := \(dx - idy),dz := \(dx + idy)

66 3. THE CAUCHY TRANSFORM AS AN OPERATOR
But since u/\F\ < 1 and p - 2 ^ 0, we see that up~2 ^ \F\P~2 and so
(3.2.2) A\F\P ^ -^—Aup.
p-1
Apply Green's theorem in polar coordinates
r f ^d0^ ff Acf)dxdy
Jo or JJ\z\<r
to eq.(3.2.2) to get
d_
dr
f |F(rC)|pdm(C) ^-E-± [ |U(rC)|pdm(C).
Jt P ~ *- ar Jt
Now integrate the previous inequality from 0 to r and use the facts that F = u + iv
and v(0) = 0 to see that
J |F«)|pdm(C) - «(0)* < -^—Y {^ |U(rC)|pdm(C) - «(0)"}
equently,
(3.2.3) / |F(rC)|pdm(C) ^ -?— f \u(rQ\pdm(C)-
Jt P ~~ l Jt
and consequently,
Thus
f \f(Q\pdm(Q= [ lim \v(r()\pdm(() (by Theorem 1.8.10)
lim f HrQ\pdm(C)
r-+i- Jt
lim /" |F(rC)|pdm(C) (since H ^ |F|)
r-+i- Jt
^ lim -?- [ \u(rQ\pdm(Q (by eq.(3.2.3))
r-i- P- ! Jt
^
£
<
P
-||/||£ (by Proposition 1.8.5).
P
For a general real-valued f e Lp we write / = f\ — /2, /i, /2 ^ 0, to see that
ll7ll^2"(||/1||?+||/2||P)
<^(ll/l|l? + ll/2||?)
= Ap\\f\\PP-
The proof for complex-valued / now follows. Thus, we have shown that for 1 <
p<2,
(3-2.4) \\f\\P^Ap\\f\\p, f&V.
We will now show that if eq.(3.2.4) works for some 1 < p < oo, then eq.(3.2.4)
holds for the associated conjugate index q (l/p+ 1/q = 1). Since we have already
shown eq.(3.2.4) for all 1 < p ^ 2, we will have completed the proof. To this end,
let g be any trigonometric polynomial
N
g(() = £ anC
n=-N

3.2. RIESZ'S THEOREM
67
and
TV
h = Q(gdm) = -i ]P sgn(n)an(n
n=-N
be its conjugate function. Notice that g + ih is an analytic polynomial. Apply the
mean value property for harmonic functions to the imaginary part of the analytic
function
(u(rz) + iv{rz)) (g(z) + ih(z))
to get
/■
Jj
(u(rQh(C) + v(rQg(Q) dm(() = u(0)h(0) + ^(0)^(0) = 0.
/T
Rearranging the terms and putting in absolute values, we conclude that
[v{rQg{Qdm(Q\ = \[ u(r()h(C) dm(C)
<\\h\\P\\u{r-)\\q
<Ap\\9\\p\Hr-)\\q.
In the last inequality, we are assuming that 1 < p ^ 2 and so, using eq.(3.2.4),
||/i||p ^ i4p||^||p. Now take a supremum over all trigonometric polynomials g with
||^||p ^ 1 and use the fact that trigonometric polynomials are dense in Lp (via
Cesaro means - Theorem 1.6.5) to get, via eq.(1.2.2),
\\v(r-)\\q^Ap\\u(r-)\\q
which, after taking limits as r —> l~ and using Fatou's lemma and Proposition 1.8.5,
shows that ||/||9 ^ .Ap||/||9. The proof is now complete.
D
Remark 3.2.5.
(1) Riesz's theorem says the Cauchy transform is a projection operator, the
Riesz projection, f \-+ f+ from Lp onto Hp'. Furthermore, when 1 < p < oo
and f e Lp, then / = fx + J2, where fx G #P(T) and f2 G ffJ(T). Also
note that by the F. and M. Riesz theorem (Theorem 1.9.7)
iP(T)n#£(T) = {0}.
In other words, HP(T) is complemented in Lp. When 0 < p < 1, we still
have the decomposition
Lp = Hp(T) + H^(T)
although
In fact Hp(T) n H%(T) is the closed linear span in Lp of {KSC : C e T}
[8, 10] [44, p. 116].
(2) There are several proofs of Theorem 3.2.1. The one presented here has
the advantage of keeping track of the constant in the norm inequality.
We were not too careful here - and often went from one line to the next
using the same symbol Ap - since there is a result of Hollenbeck and
Verbitsky (see Theorem 3.7.3 below) which computes the best constant in
||/+||p ^ i4p||/||p as Ap = l/sin(7r/p). If one is not particular about the
constant Ap in Riesz's theorem, there is an alternate proof in [36].

68 3. THE CAUCHY TRANSFORM AS AN OPERATOR
The endpoint cases p —- 1 and p = oo are different. For example [65, pp. 63 -
64], the f £ L1 whose Fourier series is
y> cos nO _ y> ein0 + e~ine
^ logn ^ 2 logn
n=2 to n=2 to
has Cauchy transform equal to
oo
1 Zn
n=2 to
which does not belong to H1. The reason for this is 'Hardy's inequality' (see [65,
p. 48]) which says that
OO OO I I
/ = ^an,"eJf/1^^^L<7r||/||1.
*-^ ^—' n + 1
n=0 n=0
Notice that the Taylor coefficients of /+ in eq.(3.2.6) do not satisfy Hardy's
inequality.
With a little bit more work, one can create examples of / G L1 for which /
(equivalently the boundary function for /+) is not integrable on any interval [234,
p. 257]. It is worth remarking here that not only does the Riesz projection / i—► /+
fail to be bounded from L1 onto H1, but there is no other bounded projection of
L1 onto H1 [152]. Equivalently, H1 is not complemented in L1. If the function
/ is slightly better than L1, there is the following theorem of Zygmund [233] (see
also [65, p. 58] and [234, p. 254]) which we state without proof.
Theorem 3.2.7 (Zygmund). If f £ LlogL, that is to say,
[ |/|(1 + log+ |/|) dm <oo,
Jj
then /+ £ H1.
The bounded function
(3.2.8) f(e-):=h/2-d/2 "' ^ < °'
V ) JK } I tt/2 - 6>/2 O<0^tt,
has Fourier series
oo
Esin nO ^ e ~ e
n ~ 2-j n2i
n—l n=l
and thus has Cauchy transform
1 °° 7U 1
2z ^—' n 2i
n=l
which is unbounded. Not only does the Riesz projection / i—► /+ fail to be a
bounded projection of L°° onto i7°°, there is no other bounded projection of L°°
onto H°° [101, p. 155]. Equivalently, H00 is not complemented in L°°.

3.3. BOUNDED AND VANISHING MEAN OSCILLATION
69
if
3.3. Bounded and vanishing mean oscillation
Definition 3.3.1. A function g G L1 is of bounded mean oscillation, or BMO,
\\g||* := sup < —— / \g — gi\ dm : / is a subarc of T > < oo,
where
is the 'average' of g on /.
With the norm
9i '-= -T7T I 9 dm
/■
+ \\9h
gdm\
one can check that BMO is a Banach space of functions on T. The functions in
BMOA := BMO n H\T) = {/ G BMO : f(-n) = 0 Vn G N},
the analytic functions of bounded mean oscillation, form a closed subspace of BMO.
Recall from our previous discussion (see eq.(3.2.8)) that
(L°°)+ = {/+:/€ L°°}
is not contained in H°°. However, the following theorem, discovered independently
by Spanne [203] and Stein [206], is true.
Theorem 3.3.2 (Spanne, Stein).
(1) The operator f >-* f is continuous from L°° to BMO.
(2) The operator f \-+ f+ is continuous from L°° onto BMOA.
There are some standard proofs of this theorem (see [79, 118]). The one we
wish to present here involves the Garsia norm. For / G L2, a routine computation
shows that
(3.3.3) (P\f\2)(z) - (P\f\)(z)2 = \ |" I" |/(e") - /(e«)|2p2(e»)Pz(e«)^^.
One also checks that
(3.3.4) (P|/|2)(z) - (P\f\)(zf = P{\f - (Pf)(z)\2)(z) > 0.
Define the Garsia norm of / G L2 to be
S(/):=sup{(P|/|2)(2)-(P|/|)(.)2}1/2
zeB
and observe the simple inequality,
(3.3.5) S(/) < CII/IU /6I°°.
Technically, the Garsia norm is not really a norm since it does not distinguish the
constants. However, it is equivalent to || • ||*.
Proposition 3.3.6. The Garsia norm is equivalent to the BMO norm. Equiv-
alently, there are constants c\,C2 > 0, independent of f G L2, such that
Cl||/ll*<S(/)<C2||/||*.

70
3. THE CAUCHY TRANSFORM AS AN OPERATOR
We will only prove one of the inequalities. It is the easier of the two inequalities
and it is the only one we really need to prove Theorem 3.3.2. The other inequality
depends on the John-Nirenberg inequality
m(\g -gi\>y)^ Cm(I) exp (—-) , y > 0, g £ BMO.
\\9\\
The interested reader can consult [79, 118]. The one direction we present below is
from [118, p. 222]. Indeed, let J = (-a, a), where 0 < a ^ n. When
we have
a i r i
r = 1 — sin — and t e [—a, a],
1 — r
1 + r2 — 2r cos t
(l + r)(l-r)
(l-r)2+4rsin20/2)
1
5sin(a/2)
2
^ 5a
Thus
(Pl/|2)W - (P|/l)W2 = ^ j f 1/(0 - f(e")fPr(e")Pr(e«) d.dt
^)22w/J/Jl/(e'*|-/(e")|2'"*
1/2
where I (J) is the length of the interval J. We observe that when
the average of / on J, we have the inequality
^//(e",-/jiM^/J|/(e")-/^tF
Combine this with our previous estimate to get
(3-3.7) ^|/(e«)-/.H«tt<yS(/).
Now take the supremum over all intervals J to see that
ll/ll* < y S(/).
Our next step, to prove that / —> / is bounded from L°° to BMO, is to show
that _
S(/) = 5(f).

3.3. BOUNDED AND VANISHING MEAN OSCILLATION
71
For a G D, let
a — z
iW :=
1 — az
and notice that ipa is a conformal self map of D with
(3.3.8) ^a(T) = T, ^a(O) = a, ^a(a) = 0.
Lemma 3.3.9. For g G L1,
(go ^a)Pz dm= gP^a{z) dm.
Proof. If g G C(T), then both sides of the above equation are harmonic
functions of z which tend continuously to g(ipa(Q) as z —> C- By the maximum
principle, these two harmonic functions must be equal. Hence the result is true
when g G C(T). Now use the density of C(T) in L1 to obtain the full result. □
Lemma 3.3.10. If f e L2, then
Q(foi/,a) = (Qf)o4a-(Qf)tya(0)).
Proof. From Lemma 3.3.9,
/ (/ o i/ja)Pz dm = / fP^a(z) dm
and so the harmonic function
(Qf)(M*)) = J fQi,a(z)drn
is a conjugate function for /o^a. It may not vanish at zero to be the conjugate
function Q(f o ipa)(z). We will fix that in a moment. Since conjugate functions
must differ only by a constant,
(Qf)(^a(z)) = Q(fo^a)(z) + C.
But Q(f o ^o)(0) = 0 and so c = (Q/)(^a(0)). D
Proposition 3.3.11. For f e L2,
5(f) = 5(f).
Proof. From Lemma 3.3.9 and he fact that ^a(^) = 0 we have the identity
(3.3.12) (ho^a)Padm= hdm
Jt Jt
for all h G L1. For g G L2, apply this identity to h = \g — (Pg)(a)\2 and use the
fact that ^a(O) = a to get
J\g- (Pg)(a)\2Padm = J\go^a- (P5)(^o(0))|2 dm.

72
3. THE CAUCHY TRANSFORM AS AN OPERATOR
This last identity applied to g = / shows that
J |/- (Pf)(a)\2Padm = J |/ o^Q - (P/)Wv(0))|2dm
|/° 1>a - (Q/)W>a(0))|2dm (since Q/ = Pj)
J
Jj
T
= \fo~^pa\2dm (Lemma 3.3.10)
= J\f^a~P(fo ^a)(0)|2 dm (Parseval)
= [\f-P(fo^a)(0)\2Padm (by (eq.(3.3.12))
= / 1/ - (P/)(a)!2p- dm (^ eq.(3.3.12) and ^ = ^a).
Now take a supremum (in a G D) over both sides of the above string of equalities
and use eq.(3.3.4) to get
SCO = S(/).
□
Proof of Theorem 3.3.2. If / g L°°, then
11/11* ^CS(/) (Proposition 3.3.6)
= CS(/) (Proposition 3.3.11)
<C1|/||oo (eq.(3.3.5)).
D
A natural subspace of BMO is the space VMO, the functions of vanishing mean
oscillation. These are the functions / G BMO satisfying
limn) sup ^Tn / \f-fi\dm\ =0-
A routine argument shows that VMO is a closed subspace of BMO.
As we did for BMO, we consider the analytic functions of vanishing mean
oscillation
VMOA — VMOnff^T).
If / £ C(T), then / may not be continuous [79, p. 127]. The following result of
Sarason [187] is the analog of Stein's (Spanne's) result for BMOA.
Theorem 3.3.13 (Sarason).
(1) The operator f i—► f is continuous from C(T) to VMO.
(2) The operator f i—► /+ is continuous from C(T) onto VMO A.
Though we will not present it here, a proof of this theorem depends on the
following alternate characterization of VMO using the Garsia norm: f £ L2 belongs
to VMO if and only if
lim {(P|/|2)(^)-(P|/|)(^)2}=0.
|z| —1

3.4. KOLMOGOROV'S THEOREM
73
In fact, one can see one direction from the proof of eq.(3.3.7). To see
Theorem 3.3.13, note that if / G C(T), then
(P\f\2)(z) - (P\f\)(z)2 = (P\f\2)(z) - (P\f\)(z)2
and so from the proof of Proposition 3.3.11,
lim Up\J\2){z) - (P\f\)(z)2} = lim {(P\f\2)(z) - (P\f\){zf)
|z| —1 ^ ) |z| —1
= 0
since / is continuous. The continuity of the operators / •—> / and f \-+ f+ follows
from Theorem 3.3.2.
3.4. Kolmogorov's theorem
Despite the fact that (L1)^ is not contained in L1, there is a well-known and
often revisited theorem about the Cauchy transforms of L1 functions due to Kol-
mogorov. In fact, we will prove a stronger result.
Theorem 3.4.1 (Kolmogorov). For /j, G M,
m(|iT/i|>y)aH 2/>0.
y
A similar result holds for the conjugate function
M«) = lim (Qm)K).
r—>-l_
Remark 3.4.2.
(1) We write, as is traditional in many probability books, m(\Kii\ > y) in
place of the more proper m({C £ T : \(Kfi)(Q\ > y}).
(2) We will treat the distribution function
y ■-► m(\K/i\ > y)
in greater detail in Chapter 7.
Proof of Theorem 3.4.1. 5 We will first prove that if fi e M+ and
(3.4.3) F(z):= [p^-drtO,
Jt s, — z
then
(3.4.4) m(\F\ > y) < -^-, y > 0.
IIMII > 2/
Indeed, for y > 0, the map
, . w — y
g(W) := 1 + £
maps {Sfai; > 0} to {\w — 1| < 1} and, since 9?F > 0, the function
F{z) - y
c/>(z) := 1 +
F(z) + y
For the original proof, in terms of the distribution function for the conjugate function, see
[116]. The proof here is modified from [118, p. 92].

74
3. THE CAUCHY TRANSFORM AS AN OPERATOR
is bounded and analytic on D. By the mean-value property for harmonic functions,
(3.4.5) 3R0(O) = /aftydra.
Jt
From the definition of F in eq.(3.4.3), we know that F(0) = \\/i\\ and so
Combining this with eq.(3.4.5) we obtain
2IMI
/
Jt
*3l(f)dm
, ,\F\2-y2
\F + y\*
It w\\ + y
Thus, since 3ft 0 ^ 0 we get
(3.4.6) m($R0 ^ 1) ^ [ R(j)dm= „ 2 ^ .
A IHI+2/
Using the identity
we see that
3ty«) > 1 ^ |F(C)| > 2/
and so by eq.(3.4.6)
This proves eq.(3.4.4).
Still for /i G M+, we notice, from the observation
^ = 2-L_-l,
C - * 1 - (z
that
(3.4.7) (Kvl){z) = ±(\\h\\+F{z))
and, by eq. (3.4.4),
m(|A>l>!/)<-||Mll-
Here we are using the fact that if / = /i + /2? then
{|/l^»}c{|/i|^y/2}u{|/2|>y/2}.
Hence
(3.4.8) m(|/| > y) < m(|/i| > y/2) + m(|/2| > j,/2).
Kolmogorov's theorem for general \i G M follows from the Jordan
decomposition of \i as
/i = (/xi - /i2) + i(/X3 - M4), Mj £ M+,
and the trick in eq.(3.4.8). D

3.4. KOLMOGOROV'S THEOREM
75
Remark 3.4.9. If we are willing to be a bit more careful, we can, in some sense,
keep track of the constant A in Kolmogorov's inequality
m(\K»\>y)^-M.
y
Start with the observation from eq.(3.4.7) that
|#Ml^lH
and apply it, for \i G M+, to the inequality
I^K^II + ^I
m(\F\>y)< 2M
M\ + v
from eq.(3.4.4) to get, for y > ||//||/2,
m(\Kri>y)<m(^\F\ + ±M>y
= m(|F|>2j,-||/x||)
^ 2||/x||
^ IImII + 2y- ||//||
= ]HI
y
Hence
mflityl > y) < M, ^eM^ y>||/i||/2.
Recall the analytic weak-type space i/1'00 from Chapter 1. From Theorem 1.10.4,
Theorem 3.4.1, and the fact that K\i G iV+, we get the following corollary.
Corollary 3.4.10. X c if1'00.
Thus we can refine the string of containments in eq.(2.2.2) a bit to
ij hp £ % £ if1-00 £ p| hp.
p^l 0<p<l
Proposition 3.4.11. For /x < m,
(3.4.12) m(|tf/z| > y) = o (-) ,
*Aa* 25 to say, ^(L1) C #o'°°.
Proof. Suppose d// = /dm, / £ L1. Let e > 0 be given and choose a
trigonometric polynomial p so that ||/ — p||i < e (Theorem 1.6.5). Notice that
ym(\f+\ ^ y) ^ ym(|p+| ^ y/2) + ym(|/+ -p+| ^ y/2).
For y sufficiently large, m(|p+| ^ y/2) = 0 (since p+ is a analytic polynomial and
hence bounded). By Kolmogorov's theorem (Theorem 3.4.1)
m(|/+-p+|£y/2)<^||/-p||1^C.
Thus
lim ym(|/+| ^ y) ^ Ce
and the result follows. D

76
3. THE CAUCHY TRANSFORM AS AN OPERATOR
More surprising however, is that the converse of this is true, namely
1
m(\Kii\ >i/) = o[-J<^>/i<m.
See [218] for details. We will not prove this here since we will be proving this, as
well as a stronger result, in Chapter 7.
One can work out the distribution function for K5\ — (1 — z)~l. Indeed,
iKS^e*
alO
2ieie/2 sin(0/2)
1
sin(0/2)
Thus
m(|^1|>2/) = -sin-1(^).
n Zy
Notice that this quantity is 0(1/y) but not o(l/y) and moreover,
lim ym(\KS1\>y) = ^ = ^\\S1\i
We will see later in Chapter 7 that
7T 7T
lim y7rm(\Kfi\ > y) = ||/x5||.
y-^oo
In Chapter 9 we will prove the stronger result
y7rm(\K/i\ > y) • m -> /x5
weak-* as y —-> oo.
3.5. Weighted spaces
What about Cauchy transforms of functions from weighted Lp spaces? The
question here is the following: given 1 < p < oo, what are the conditions on a
measure \i G M+ such that there is a constant C > 0 with the inequality
J\g+\"d^^cJ\g\"d^
holding for all trigonometric polynomials gl A theorem of Helson and Szego [97]
says that such a measure fi must satisfy /i <^ m and so the problem can be rephrased
as: given 1 < p < oo, what are the necessary and sufficient conditions on a non-
negative weight function w onT such that
(3.5.1)
I \g+\pwdm^C f \g\pwdm
for all trigonometric polynomials g? In the special case where the weight function
is w(e%e) = |1 — el0\a, a classical result of Hardy and Littlewood [89] says that when
p = 2, eq.(3.5.1) holds if and only if — 1 < a < 1. For general w, Helson and Szego
[97] proved that eq.(3.5.1) holds in the case p = 2 if and only if \ogw = u + v,
where it, v G L°°, |H|oo < tt/2, and v is the conjugate function for v. The definitive
theorem here is one of Hunt, Muckenhoupt, and Wheeden [106] (see also [79,
p. 253]) which says that when 1 < p < oo, eq.(3.5.1) holds if and only if the weight
w satisfies the condition
s™p(zrrxfw(Qdm(Q) (z^ Aw(0)"1/(p"1)dm(C)V < oo,
7CT
m(I)
m(I)

3.6. THE CAUCHY TRANSFORM AND DUALITY
77
where the supremum is over arcs / of the circle. We refer the reader to the references
in [106] for other sufficient conditions on w. See [150] for a shorter proof of this
result (at least when p = 2).
3.6. The Cauchy transform and duality
In this section we will use Riesz's theorem (Theorem 3.2.1) to equate the norm
dual of Hp with Hq. Here 1 < p < oo and q is the Holder conjugate index to p.
From the Riesz representation theorem for Lp, we know that every bounded linear
functional on Lp takes the form
/
f *-* fgdm
for some unique g G Lq. Moreover, the norm of the above linear functional is
i.e.,
(3.6.1) |M|, = sup<
fgdm
T
: / G ball(I7)
An application of the F. and M. Riesz theorem (Theorem 1.9.7) says that
(H?)1- := L G L« : J fgdm = 0 V/ G #P(T)1
is equal to Hq and so we can apply Theorem 1.4.6 to conclude the following.
Theorem 3.6.2. For 1 < p < oo, (Hp)* is isometrically isomorphic to Lq/H^.
We can use Riesz's theorem (Theorem 3.2.1) to identify, in an isomorphic (but
unfortunately not isometric) way, the dual of Hp with Hq when 1 < p < oo. One
can see this as follows. By Holder's inequality, the linear functional
/ *-* / /^dm
JT
is continuous on Hp for fixed g G Hq. On the other hand, if £ G (Hp)*, the
Hahn-Banach extension theorem says that
*(/)= f ffidm
for some g\ G Lq. Using the continuity of the Riesz projection operator h h-> h+
from Lq onto Hq and the identity
/ fg1dm= / /(#i)+dm
JT JT
(which follows from Proposition 1.8.5 and Theorem 1.9.6), one can replace the
above g\ G Lq with a unique function g := (#i)+ G Hq. Thus every £ G (Hp)* takes
the form
for some unique g G Hq. Hence we can identify (Hp)* with Hq when 1 < p < oo.
We can apply Riesz's theorem again to say something about norms. Clearly,
by Holder's inequality, the norm of the functional £g, that is,
sup -
fgdm
T
/ G ball(iF)

78
3. THE CAUCHY TRANSFORM AS AN OPERATOR
is no bigger than \\g\\q- For the other direction, let Ap be the norm of the Riesz
projection, i.e., the smallest constant Ap so that ||/+||p < i4p||/||p for all / G Lp,
and observe from eq. (3.6.1) that
sup
sup
U
fgdm
f+gdm
:/Gball(Lp)|
: / G ball(I7)
= Ap sup
^ Apsup<
= MtgW
\\[l±
\\jT Ap
gdm
Fgdm
:/Gball(27)}
Thus
\\9\\q<Ap\\£g\\<\\g\\q.
This makes the sesquilinear map g —> £g continuous and invertible from Hq to (Hp)*
and hence one identifies (Hp)* with Hq with comparable norms. If one changes the
dual pairing slightly to
L9{f):= //(Ck(C)dm(C)
JT
the mapping g h-> Lg becomes an isomorphism. We summarize this with the
following corollary.
Corollary 3.6.3. Let 1 < p < oo. A linear functional £ belongs to (Hp)* if
and only if there is a g G Hq such that
£(f) = jfgdm.
Moreover, this g is unique and satisfies
ci\\g\\q<\\e\\<c2\\g\\g.
When p = 1, we can still say that (i/1)* is isometrically isomorphic to L°°/H§°.
However, (i/1)* is not identified with H°° via the above dual pairing. Instead,
(i/1)* is identified with BMOA, the analytic functions of bounded mean oscillation.
Certainly, by the Hahn-Banach extension theorem, if £ G (i^1)*,
e(f)
I
fgdm
for some g G L°°. However, when g is replaced by #+, the above integral may
not converge since g+ may not belong to H°°. By the Spanne/Stein theorem
(Theorem 3.3.2), g+ G BMOA and moreover
£(f) = lim / frj^dm,
r-+l- J
where fr(z) = f(rz). Conversely if g G BMOA then
£g(f) = lim / frgdm
r-+l- J

3.7. BEST CONSTANTS
79
defines a continuous linear functional on H1 and
Ci||#||bMO ^ \\tg\\ ^ C2||^||bMO-
This is the Fefferman-Stein duality theorem [71, 70] (see also [79] and [118]).
Finally, we would like to mention an alternate and useful representation of the dual
pairing between H1 and BMOA involving truncations [211] (see also [44, p. 32]).
3.7. Best constants
In the previous sections of this chapter, we examined the mapping properties
of the operators fi ^ K/i and \i h-> Q/i on various subspaces contained in M. In
this section, we discuss, without proof, the norms of these operators. A nice survey
paper on this material is [158].
We begin with a relatively easy one. From Smirnov's theorem (Theorem 2.1.10)
we have
\\Kfx\\p^cp\\fxl
where cp x (1 — p)~l. This first result computes, at least for positive measures, the
smallest such constant cp. We thank Al Baernstein for showing us this proof.
Proposition 3.7.1. For fixed 0 < p < I,
sup{||^/i||p:/iGM+,|H| = l}
1
Proof. The function <j>(z) := (1 •
univalent. Furthermore, for /i G M+, ||/i
1
z) l maps D onto {'Rz > 1/2} and is
1, we have the containment
(Kfi)(B) C {Rz > 1/2}.
Thus K\i is subordinate to <fi and so by Littlewood's subordination theorem [65, p.
10]6,
1
\\KlA\v ^
1
The equality in the statement of the proposition is achieved when \i = 5\.
□
Remark 3.7.2. For any complex measure /i = (/ii — ^2) +^(^3 — M3), Hj £ M+,
one can use a slight variation of the above argument four times to prove Smirnov's
theorem: ||if/i||p ^ -Ap||/i||. However, the best constant Ap is not known for general
complex measures, only for positive ones.
The Riesz theorem (Theorem 3.2.1) says that for fixed 1 < p < 00, the operator
/ h-> /+ = K(f dm) (the Riesz projection) is a continuous operator from Lp onto
Hp. This next result computes the norm of this projection.
Theorem 3.7.3 (Hollenbeck and Verbitsky). For fixed 1 < p < 00,
1
supdlZ+Hj
O)}
sin(7r/p)
Littlewood subordination theorem: If /, g are analytic on D and / = g o w, with w analytic
on D and \w(z)\ < \z\, then M(r;f) < M(r;g) (see eq.(1.9.1)) for all 0 < r < 1.

80
3. THE CAUCHY TRANSFORM AS AN OPERATOR
A proof of this theorem can be found in [102] where they also discuss the norms
(and even essential norms) of some other classical operators.
For f £ L1, the conjugate function
dt
]2^
/(e*) := Urn / cot(^)/(e^
e~^0+ J\0-t\>e Z
\e-tfee
is defined almost everywhere and by Riesz's theorem, the operator / h-> / is
continuous on Lp for 1 < p < oo. Moreover, for 0 < p < 1, / h-> / is continuous from
L1 to Lp (Proposition 3.0.1). The norms of these operators are computable [159].
Theorem 3.7.4 (Pichorides).
(1) For fixed 1 < p < oo,
sup{||/||P:||/||„<l} =
tan(^) ifl<p^2,
7T
cot( —) if 2 < p < oo.
(2) For fixed 0 < p < 1,
(cos(W2))"1/p < sup{||/||p : H/lli ^ l} < 21/^1(cos(W2))-1/p.
We know, for p, G M, that the conjugate function
(QM)(^=y*5(^)dM(c)
is harmonic on D and has radial limits
Jl(ew) := lim (Qfi)(reie)
r—+l~
almost everywhere. Moreover, a variant of Kolmogorov's theorem (Theorem 3.4.1)
says that for fixed 0 < p < 1, Jl belongs to Lp and
NIp < cpIHI-
B. Davis7 computes the best constant cp, at least for real measures Mr.
Theorem 3.7.5 (Davis). For fixed 0 < p < 1,
sup{||/I||p:/iGMR,||/i|| <1} = ||i7||p,
Kolmogorov's theorem says that
v :=
ys that
m(\Kfi\
^
^2/)
>
^C
■i-
y
Though the smallest constant C is unknown here, it is known when Kfi is replaced
by the conjugate function Ji [53, 54].
The original proof is found in [55] and involves a Brownian motion argument. A more
analytic proof can be found in a paper of Baernstein [24]. See also [74].

3.8. THE HILBERT TRANSFORM 81
Theorem 3.7.6 (Davis). (1) For f £ L1,
m(l/>2/)^y, y>0,
where
1 _ 3-2 + 5-2 _ ...
(3.7.7) 9 =
1 + 3-2+ 5-2+ •••'
Moreover, 0-1 is the smallest possible constant.
(2) For /ig M+,
m(|/Z| ^ 2/) < 1 • M, j,>0.
Moreover, 1 zs £/ie smallest possible constant.
Notice that
_8_ [°° _t^_
7T2 J0 1 + e'
where C := 1 — 3~2 + 5~2 — • • • is the Catalan constant. It is unknown whether or
not the Catalan constant is rational.
We will say more about the distribution function for K\i in Chapters 7 and 9.
3.8. The Hilbert transform
For / G LX(R), the functions
(?/)(*) := - / 3 ( — ) f(s) ds, (Qf)(z) := i / K (— ) f(s) ds
n Jr \s-zJ n Jr \z-sJ
are harmonic on the upper-half plane C+ := {2 G C : 3z > 0} and
(?/)(*) + i(Qf)(z) = - [ —f(s) ds
7TI JR S - Z
is analytic on C+ and is often called the Borel transform of /. As it turns out,
the function 7/ plays the role of the Poisson integral Pf (f G L1(T)) in the disk
setting (it is in fact called the Poisson integral) in that
]im(?f)(x + iy) = f(x)
almost everywhere and when 1 < p < 00, (3>/)(- + iy) —> / in the norm of LP(R) as
y —> 0. The function Q/ is the harmonic conjugate of 7f and plays the role of the
conjugate Poisson integral Qf (f G I/X(T)) in the disk setting. One can show that
lim (Qf)(x + iy) = P.V. [ -^~ds
y-o+ JR x - s
almost everywhere. The above singular integral is called the Hilbert transform of
/ and is denoted by (!Kf)(x). Notice how the boundary function for the conjugate
function Q/ yields a singular integral similar to that of Qf.
Many of the theorems, for example Privalov's theorem (Theorem 3.1.1), Riesz's
theorem (Theorem 3.2.1), Kolmogorov's theorem (Theorem 3.4.1), Spanne's (Stein's)
theorem (Theorem 3.3.2), etc., have direct analogs for the Hilbert transform. In
summary,
/ G AQ(R) =► Dif G AQ(R)
/ G 27 (R), 1< p < 00 =► "Kf G Z7(R)

82
3. THE CAUCHY TRANSFORM AS AN OPERATOR
/ G L\R) =* rmaiXfl > y}) < C\\f\\iy-1
f g l°°(r) n l\r) => jc/ g bmo(R)
/ G C(R) D L^R) =>Hfe VMO(R).
In the above, mi is Lebesgue measure on R.
Just as in the conjugate function case, the best constants are known. For
example [159], if 1 < p < oo,
sup{p-C/||p:||/||p<l}H
tan(^), l<p<2;
. 7T .
cot( —), 2 < p < oo.
'2p'
The smallest constant C in the weak-type inequality
m1{{\Kf\>y})<iC\\f\Wy-1
is 0_1, where 6 is the constant denned in eq.(3.7.7). The best constant C above
for non-negative functions is one [53, 54].
As was the case on the circle, the mapping properties of the Hilbert transform
play a crucial role in determining the mapping properties of the Cauchy transform.
In Chapter 7 will be looking at the function
y \-+ mi(|J£/x| > y),
the 'distribution function' of the Hilbert transform
CKii)(x) = P.V. [ —— d/x(s)
J x - s
of a measure /i on the real line. We refer the reader to [207] for a thorough treatment
of the Hilbert transform as well as other singular integral operators.

CHAPTER 4
Topologies on the space of Cauchy transforms
In this chapter, we discuss several natural topologies one can place on the space
of Cauchy transforms X. For notational purposes, the reader might want to review
the basic functional analysis facts covered in Chapter 1.
4.1. The norm topology
For / G X recall from Definition 2.1.4 the set
Rf :={veM : / = Kv)
of 'representing measures' for /. From Proposition 2.1.5 we know that
/i, v G Rf => d/jL — dv = (/) dm,
where 0 G Hq (T) and Hq is the subspace of H1 consisting of functions which vanish
at the origin. As is customary, we will abuse notation slightly and let Hq denote
the following subspace of M,
#J:= {^dm-.&eHQ1}.
Since ||0dm|| = ||0||i and Hq is a closed subspace of L1, then Hq is a closed
subspace of M. Thus the quotient space M/Hq of cosets
is a Banach space with norm
\M\\ := dist(/x,i^) = inf {||d/i + ^dm|| : <j> G Hl0) .
By identifying a Cauchy transform with its set of representing measures and
then using the previous discussion, it makes sense to associate the Cauchy transform
K\i with the coset [/jl] in M/Hq. The mapping
is a vector space isomorphism from X onto M/Hq. If we equip X with the quotient
space norm
(4.1.1) \\Kri\:=\\[n}\\,
the map K/i h-> [/j] becomes an isometric isomorphism, making X a Banach space.
Let us say a few words about this norm topology on X.
Proposition 4.1.2. For f g X,
H/ll =inf{|H|:i/G/Z/}.
83

84
4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
Proof. Proposition 2.1.5 says that
Rf = {d/i0 + (pdm : (p G Hq}, /x0 G #/.
Now use the definition of ||/|| from eq.(4.1.1). D
Normally, computing the norm of a Cauchy transform is quite difficult.
However, we can compute some easy ones.
Corollary 4.1.3. If n e M+ and f = K\i, then \\f\\ = ||/x||.
Proof. For any v e Rf,
\\p\\ = Jd» = f(0) = Jdv = v(T) ^\\v\\.
Now use Proposition 4.1.2. D
Corollary 4.1.3 says that for certain Cauchy transforms /, there is a \i G Rf
such that ll/H = ||/i||. This turns out to be true in general.
Proposition 4.1.4. For each f G X, there is a unique /i G Rf such that
IHI = 11/11-
This proposition is really just a Cauchy transform version of the following result
from the theory of dual extremal problems (see Theorem 1.4.6 and Theorem 1.4.7).
Originally shown by Doob [63], we present the proof from [79] (see also [65]). The
papers [94, 95, 176] also relate Cauchy transforms to dual extremal problems.
Proposition 4.1.5. For a given f e L1, there is a unique g G H1 with
\\f-g\\1 = dist{f,Hl).
Proof. Our first step is to show that a best approximant g G H1 exists. Let
(<7n)n^i C H1 with ||/ — gn\\i —> dist(/, Hl) as n —> oo. This means that the
H1 norms of gn are uniformly bounded and so from eq.(1.9.3), (gn)n^i is a normal
family on D. Passing to a subsequence if necessary, we can assume that gn converges
to an analytic function g pointwise on D. Moreover, for any 0 < r < 1,
/"|5(rC)|dm(CK Mm / |5n«)| dm(C)
J n—+oo J
^ lim / \gn(Q\dm(Q (integral means increase in r)
n—->oo J
^ sup||#n||i
for some constant c independent of 0 < r < 1. Now take a supremum in r to
conclude that g G H1. If P(f — g)(rQ is the Poisson integral of / — g evaluated at

4.1. THE NORM TOPOLOGY
85
r£, note that P(gn)(r() = g<n(r() —* #(VC) as n —* oo and so
/V(/-s)«)|dm(CK lim /|P(/-5n)K)|dm(C)
J n—+oo J
dm(C)
= lim / /^rc(0(/(0-5n(0)dm(0
n—+00 J \J I
< lim /"/Prc(OI/(0-ffn(0|dm(Odm(C)
= J^y(y"Prc(Odm(C))|/(0-ffn(0|dm(0
= lim ||/-5n||i
n—>-oo
= dist(/,#1).
Observe from Theorem 1.8.6 (Fatou's theorem) that
Il/-5l|i< Mm /|P(/-5)K)|dm(C)
and so, using the obvious inequality dist(/, Hl) < ||/ — #||i, we have
(4.1.6) ||/-5||1=dist(/,^1).
Thus a best approximant g exists. We now argue that it is unique.
By the F. and M. Riesz theorem (Theorem 1.9.7), the annihilator of Hl in L1
via the dual pairing
Jfgdm, feL\ geL™
is ~H™. Thus, from Theorem 1.4.7,
: F G ball(#(
0°°)}.
distif.H1) = sup| f fFdm
Let Fn G ball(#0°°) with
J fFndm^distif.H1).
By the Banach-Alaoglu theorem, the sequence (Fn)n^i has a weak-* limit point
F G ball(#0°°).
With g being a best approximant in eq.(4.1.6), note, since F G Hfi0, that
/
gFdm = 0
and i
dist(/, H1) = J(f - g)Fdm < ||/ - 5||iII^IU < ||/ - <?||i = dist(/, H1).
This string of inequalities says that
J(f-g)Fdm = J\f-g\dm
and so
(4.1.7)
(f-9)F = \f-g\ a.e.

86 4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
If / £ jff1, clearly the best approximant g must be equal to / and it is unique.
Otherwise, the above equation says that F is the unique function in baling0)
satisfying
[ fFdm = dist(f,H1).
If g is any best approximant in eq.(4.1.6), we can use eq.(4.1.7) to say that
Qt(gF) = Qt(fF)
and so $s(gF) is unique. But since gF £ Hq°, this determines gF uniquely. Finally,
|F| > 0 almost everywhere and so g is uniquely determined. □
Corollary 4.1.8. For a given f G L1, there is a unique g G Hq so that
mi{\\f + h\\1:heHl} = \\f + g\\1.
Remark 4.1.9. Before proceeding to the proof of Proposition 4.1.4, we would
like to mention a particular extremal problem we will make use of later. If
p(z) = c0 + c\z + c2z2 H h cnzn
is an analytic polynomial, one can consider the extremal problem
L = iid{\\p-g\\1:gGH^}.
We know from the previous corollary that this extremal problem has a unique
solution go G Hq . What is interesting here is that one can actually compute go and
L. When Cj = 1 for all j, this problem was explored by Landau as far back as 1913
[121, 122] where he was able to compute L — Ln as
n
Ln = 2_^ |Aj| ,
3=0
where
A {2j)l
3 4W
A estimate using Stirling's formula yields
(4.1.10) ci log(n + 2) ^ Ln ^ c2 log(n + 2), ne N,
for some universal constants ci,C2 > 0. Putting this in another way, the Cauchy
transform norm of 1 + z + z2 + • • • + zn can be estimated as
||1 + z + z2 + • • • + zn\\ = Ln x logn.
See [79, p. 175] for a nice exposition of this.
Proof of Proposition 4.1.4. First note that for any v G Rf,
(4.1.11) H/ll = inf{||di/ + gdm\\ : g G H^} < ||z/||.
By the definition of the norm on X, there is a sequence [yn)n^\ of measures from
Rf such that ||z/n|| ^ ||/|| + 1/n. Since this sequence is uniformly bounded in total
variation norm, it has a weak-* cluster point v G M (Banach-Alaoglu theorem).
Hence, we can pass to a subsequence (z^n)n>i such that
lim [gdvn= [gdis Vg G C(T).

4.1. THE NORM TOPOLOGY
87
Thus, for each z G D,
f(z) = / =- dz/n(C) -* / =- dz/(C), n -* oo,
7 1 - (z J l-Qz
and so is e Rf.
We now need to prove the equality \\i/\\ = ||/||. One direction (||/|| < \\is\\)
comes from eq.(4.1.11). For the other direction, observe from Proposition 1.6.2
that
H < Urn KU < lim (||/|| + l
For uniqueness, first notice that when v G Rf with ||z/||
+ K
dm
inf{||i/ + /idm|| : h e H^}
Thus
li/JI+inf
dv
— h
dm
*}
• he Hi) (by Corollary 1.3.10).
dm
inf <
dv
dm
h
:heHl
and by Corollary 4.1.8, this infimum is achieved precisely when h = 0. Hence
du
(4.1.12)
If /i, v G Rf with |
(4.1.13)
and consequently
(4.1.14)
dm
<
— h
dm
heHl\{Q}.
||z/|| = ll/H, then by Proposition 2.1.5,
Ms = Vs
d/i
dm
li
dm
(4.1.15)
he Hi
Since n,v e Rf, Proposition 2.1.5 says that
d/i dv
dm dm
By eq.(4.1.14) and eq.(4.1.15) along with eq.(4.1.12) we know that h = 0 and so
d/i dv
dm dm
By eq.(4.1.13), fis = vs and hence /i = v. Thus the norm attaining representing
measure is unique.
□
Remark 4.1.16. One should be careful as to not over interpret this theorem.
It does not say, for a particular measure /i, that ||Zf/i|| = ||/i||, although this is the
case when \i G M+ (Corollary 4.1.3). Instead, it says that there is some (perhaps
other) measure v with K\i = Kv such that \\K/i\\ = \\i/\\.
This next result relates the growth near the boundary with the norm.

88 4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
Proposition 4.1.17. If f eOC, then
(4.1.18) 1/(3)1 ^ JJZL, XGD.
Proof. If / = K\x, then for all z G D,
l/(*)l
J 1 - Cz \ J \1-Cz\ 1
The desired estimate follows by taking /x to be the unique ^ £ Rf with ||/|| = ||/x||
(Proposition 4.1.4). D
The previous proposition implies that if (/n)n^i C 3C converges in norm to /,
then this sequence also converges to / uniformly on compact subsets of D (as is
the case with all the well-known Banach spaces of analytic functions on D). Thus
ball(3C) forms a normal family of functions on D.
By the Lebesgue decomposition theorem (Theorem 1.3.9),
M = Ma + M3,
where
Ma := {/x G M : /x < m}, Ms:={/iGM:/il m}.
The + sign here means that
M = {vx + i/2 : i/i G Ma, i/2 e Ms] and ManMs = {0}.
Moreover (Corollary 1.3.10),
IHI = llM«ll + ll^ll,
where
/i = /ia + /is, /ia ^ Ma, ns e Ms.
Thus
1 = 108^.
We can use the above decomposition to rewrite X as
X = 3Ca + 3CS,
where
Xa := {X/i : /x G Ma}, Xs := {X/x : /x G Ms}.
We will now see that in fact X = Xa®Xs.
Proposition 4.1.19. If v G MS7 tAen \\Kv\\ = ||z/||.
Proof.
||X/x||=inf{||^ + ^dm||:^G^}
= inf{||i/|| + ||^||i -gelll} (since 1/_L ra)
= IMI-
□
We leave it to the reader to generalize the above proof to obtain the following
proposition.

4.1. THE NORM TOPOLOGY
89
Proposition 4.1.20. Suppose \± = \ia + \±s. Then
||X/i|| = ||X/ia|| + ||X/is|| = ||X/ia|| + ||/is||.
In particular, Xa is isometrically isomorphic to L1 /Hq while Xs is isometrically
isomorphic to Ms.
The above says that Xa and Xs are indeed closed subspaces of X and that
Jv = Jva © Xs.
Proposition 4.1.21.
(1) Xa is separable. In fact, the analytic polynomials are dense in Xa.
(2) Xs is not separable.
Proof. For any f e L1, note that the Cesaro sums (JN(f) approximate / in
the L1 norm (Theorem 1.6.5) and so, by the definition of the norm on X,
|K(/)+ - /+|K IK(/) - /Hi - 0 as N -+ oo
and so the analytic polynomials are dense in Xa.
Since Xs is isometrically isomorphic to Ms and, by means of eq. (1.6.6), Ms is
not separable, we conclude that Xs is not separable. □
One can use the F. and M. Riesz theorem along with Theorem 1.4.7 to prove
the following.
Theorem 4.1.22. The dual of L1 jH\ is isometrically isomorphic to H°°. A
pairing between these two spaces is given by
([f],9)-=Jf9dm, [fjeL'/Hl, gGH°°.
As a consequence, the dual ofXa can be identified with H°° by the pairing
(f+,9) := / fddm-
Let us compute, as it will be used later, the X-norms of the functions
z i-> 3-, a GO .
1 — az
When a G T, then
and so, since 5a -L m, we can use Proposition 4.1.20 to get
||Ar*„|| = lkll = i.
When \a\ < 1,
1 — az \ 1 — a("
and so, using Proposition 4.1.20 again and the definition of the norm in L1/^, we
obtain
(*)

90
4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
Using the dual pairing in Theorem 4.1.22 along with the dual extremal setting of
Theorem 1.4.7, we conclude that
1
1-aC
9* Hi
infj
= sup|||/(C)r^dm(C)
= sup{|/(a)|:/Gball(ff00)}
= 1.
: / G baU(JJ°°)
Thus for a G
(4.1.23)
1
1.
1 — az\
Remark 4.1.24. One can avoid the above argument by noticing that for each
a G
and so
Pa(Q
1
1
<
1 - K l-a(
K(Padm)(z).
1 — az
But since Padm G M+, we can apply Corollary 4.1.3 to see that
1 " t\Padm\\ = \\Pa\\1 = l.
1 — az
We end this section with a remark about reflexivity.
Proposition 4.1.25. X is not reflexive.
Proof. Since X is isometrically isomorphic to M/Hq and reflexivity is
preserved under isometric isomorphisms, we just need to show that MjH\ is not
reflexive. By Theorem 1.4.11 (subspaces of reflexive spaces must be reflexive), we
reduce the problem to showing that L1 jH\ is not reflexive. By Theorem 4.1.22 we
can use Theorem 1.4.11 once again (the dual of a reflexive space is also reflexive)
to reduce the problem to showing H°° is not reflexive. Since L1 /'Hq is separable
(L1 is separable), we can use Proposition 1.4.13 to reduce the problem to showing
that H°° is not separable. For this last detail, let
/c(*)=exp(ji|):<eT}
be a family of atomic inner functions. A computation shows that
||/c - /5|U > lim |/c(r<) - fM)\ = 1 VC + S, X
r—► 1
and so by Proposition 1.4.12, H°° is not separable.
□
Actually, if one is willing to work a bit harder, one can show that ||/^ — /$ ||oo = 2. To see
this, note that fg is analytic at £ with |/f (C)| = 1 while f^ has D as its cluster set at £.

4.2. THE WEAK-* TOPOLOGY
91
4.2. The weak-* topology
Let us apply the basic functional analysis layed out in Theorem 1.4.6, i.e.,
X*/S±^S*,
to the case where X = C(T), the continuous functions on T. By the Riesz
representation theorem (Theorem 1.3.6), the mapping from M to C(T)* denned by
\i i—► L^, where
Mf) := J fdji,
is a conjugate linear isometric isomorphism. If A, the disk algebra, denotes the
functions / G C(T) which have continuous extensions to D~ which are also analytic
on D, one notices that
A± = j/iG M : f fdjl = 0 for all/ G-A j.
Letting /(£) = (n for n G No in the previous equation, we see, by the F. and M.
Riesz theorem (Theorem 1.9.7), that
Thus from Theorem 1.4.6,
A* ^ M/~Hl
2and the dual pairing between A and M/Hq is
(4-2.1) (/,M) = //d^.
By the discussion in the previous section, the map Kfi i—► [fj] is an isometric
isomorphism between % and M/Hq. Put this all together to obtain the following.
Theorem 4.2.2. A* is conjugate linearly isomorphic to % with dual pairing
(f,K») = Jfdii.
A computation with power series shows that this pairing can be written in the
more familiar Cauchy pairing
oo
(4.2.3) (f,K»)= lim Y,ftn)W)rn-
n=0
Indeed,
OO OO • /» \
lim J2 f(n)Wn)rn = lim_ £ f(n) ( / CM&) rn
= rlim /ff;/(n)K)nJdMC)
= lim ff(rQMO
r->l~ J
-J/3H
2The duality A* ^ M/tf* has an analog in higher dimensions [181, p. 202].

92
4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
We can use the above pairing to prove an interesting result about the norms of
Cauchy transforms. For / G Hp', recall from eq. (1.9.2) that when fr(z) := f(rz),
the integral means, ||/r||p increase as r —» 1~ and ||/r||p ^ ||/||p- The same is true
for the norms of Cauchy transforms.
Proposition 4.2.4. Iff = K\±, and 0 < n < r2 ^ 1, tfien ||/ri|| ^ ||/r2||-
Proof. We will first show that ||/r|| ^ ||/|| for each 0 < r < 1. If / = Kfi and
g G A, then
/r(2) = X>n£(nK\
n=0
oo
<?,.(*) = $>n£(ra)*n
n=0
and so by eq.(4.2.3),
(fr,g) = (f,9r)-
Furthermore,
||/r||=sup{|</r,ff)|:5GbaU(A)}
= sup{|(/,5r)|:«?eball(.4)}
< sup {||/||||ffr||oo : ff G baU(A)}
< 11/11-
Now suppose ri < r2 < 1. Then with s = ri/r2, the above estimate shows that
H/nlHII(/,2)sKI|/,2||.
□
Proposition 4.2.5. ^4 sequence (gn)n^i C 3C converges to g G 3C weak-* if and
only if gn —* g pointwise on D as n —> oo and the sequence (\\gn\\)n^i is bounded.
Proof. Let gn — K\in and g = if/i. Suppose gn —* g weak-*. This means
that (f,gn) —* (f,g) f°r each / G A. In particular, for each /G A, the sequence
((fi9n))n^i is bounded. By the principle of uniform boundedness (Theorem 1.4.2),
(||#n||)n>i is a bounded sequence. To show pointwise convergence, notice that for
each fixed 2 GB, the function £ i—► (1 — C^)_1 belongs to A and
(4.2.6) ( i \^gn~g) = / x _ ._ d(/xn - /x) =5nW - 0(2)
1
1 "cr
and so (since gn —* g weak-*), gn —* g pointwise in D.
Conversely, suppose that ||#n|| ^ c for all n and that #n(z) —* #(z) for each
2GO. Let B be the set of all finite linear combinations of the functions
and notice from eq.(4.2.6) that (h,gn) —* (/i,#) for each h £ B. Assume for a
moment that B is a dense subset of A (we will prove this shortly), and observe that

4.2. THE WEAK-* TOPOLOGY
93
if / G A and e > 0 are given, we can find an h G B with ||/ — /i||oo < e- With this
choice of /i,
\(f,9n -g)\ = \(f ~h,gn-g) + (/i,#n -#)|
<||/-ft||oo||^n-^|| + Kft^n-^>|
<€(c+|M|) + |(h,(;n-^>|.
Since \(h,gn — g)\ —* 0, the proof (except for the proof that B is dense in A) is
complete.
To show B is dense in A, note that a measure v belongs to B1- if and only if
/i °°
— d*/ = ]T znv(n) \/ze :
0
/ 1 - C z
n=0
By the F. and M. Riesz theorem, this takes place precisely when v G H\ = -Ax.
Thus Bx = .A1- and, by the Hahn-Banach theorem, the proof is complete. □
Remark 4.2.7. Another proof of Proposition 4.2.5, which can be applied to
other spaces of analytic functions, can be found in [34, Prop. 2].
In the previous section we saw that 3C, endowed with its norm topology, is not
separable (Xa is separable but Xs is not).
Proposition 4.2.8. X, endowed with the weak-* topology, is separable. In fact,
Xa and %s are each weak-* dense in %.
Proof. When M and M/Hq are endowed with their respective weak-*
topologies, eq.(4.2.1) shows that the natural map
7T : M -> M/lll ^ %
is weak-* continuous. The fact that Ma and Ms are weak-* dense in M
(Proposition 1.6.7) finishes the proof. □
Remark 4.2.9. (1) If g = K/i and crn(g) is the n-th Cesaro mean of g,
that is o-n(g) = crn(/i), then an(g) —» g weak-* in X as n —» oo. One can
see this directly by the identities
if,o-n(g)) = / /<7n(/x)dm= / crn(/)d/i,
where / G A [101, p. 20]. This last integral converges to
fdjl=(f,g).
/■
(2) From the proof of Proposition 4.1.21, we saw that if / G L1, then crn(/) + ,
the n-th Cesaro mean of /+, converges to /+ in the norm of Xa. However,
if (1 _L m and /i ^ 0, then an(Kfi), although it converges weak-* to
Kfi, does not converge in norm to anything. The reason for this is that
an(K/i) G Xa and Xa is a norm closed subspace of X (Proposition 4.1.20).
Thus if o~n(Kii) has a norm limit F, it must belong to Xa. However, it is
easy to check that an(Kfi) converges pointwise to Kfi and so K\i = F G
Xa which is a contradiction to the fact that Kji G Xs.

94 4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
4.3. The weak topology
A topological vector space y is complete if every Cauchy net (see [113, p. 190]
for a definition) in y converges to an element of y. If y is a Banach space, it is
complete by definition. On the other hand, if y is either (X, wk) or (X*, wk-*), that
is, a Banach space X endowed with its weak/weak-* topology, then y is complete
if and only if y is finite dimensional [142, p. 215, p. 226]. Thus in the weak and
weak-* topologies, questions about completeness are irrelevant. There is, however,
a more interesting notion of weak/weak-* sequential completeness.
Definition 4.3.1.
(1) A sequence (£n)n>i m a Banach space X is weak Cauchy if the numerical
sequence (£(xn))n^i is a Cauchy sequence for every IgX*. A sequence
(^n)n>i C X* is weak-* Cauchy if the numerical sequence (£n(x))n^i is a
Cauchy sequence for each x G X.
(2) X is weakly sequentially complete if every weak Cauchy sequence in X
converges weakly to some element of X. X* is weak-* sequentially complete
if every weak-* sequence converges to some element in X*.
As a consequence of the Principle of Uniform Boundedness and the Banach-
Alaoglu theorem, X* is always weak-* sequentially complete whenever X is a
separable Banach space. Thus the more interesting topic to explore is weak sequential
completeness.
The classical Lebesgue spaces Lp, 1 < p < oo, are weakly sequentially complete
since they are reflexive and so the weak and weak-* topologies coincide. For the
same reason, the Hardy spaces i7p, 1 < p < oo, are weakly sequentially complete.
The non-reflexive space L1 is weakly sequentially complete but for a different reason
[231, p. 140][60, p. 91]. From here one can argue that H1 is weakly sequentially
complete. In fact, an arbitrary L1(^,E,/i) space is weakly sequentially complete.
This will be important in a moment.
The space C[0,1], however, is not weakly sequentially complete. One can see
this by observing that the functions fn(t) = (1 — i)n satisfy
lim / /nd/x = /x({l})
n—>oo J
for each measure /i on [0,1] and so (/n)n>i is a weak Cauchy sequence. However
there is no continuous / such that
y"/dAi=M({i})
for every measure fi. When 0 < p < 1, the Hardy spaces are not weakly sequentially
complete [66]3.
Although the space of Cauchy transforms % does not have a readily identifiable
dual space (as a Banach space of analytic functions on D), it is weakly sequentially
complete. We would like to very briefly discuss this result, which is often called
Mooney's theorem. The key to this is the following.
Technically, these spaces are not Banach spaces. However, the dual of Hp can be identified
with a non-trivial algebra of analytic functions on D and from here, the weak topology can be
denned. The reader can find all of this done quite precisely in [66].

4.4. SCHAUDER BASES
95
Theorem 4.3.2 (Mooney). Let (0n)n^i be a sequence in L1 such that the limit
lim / (j)nJdm = L(f)
exists for every f G H°°. Then there is a function <fi G L1 such that
L(f)= f<t>Jdm V/e#°°.
JT
Mooney's theorem was discovered independently by Mooney [144] and Havin
[93]. Proofs can also be found in [79, pp. 206-209] or [118]. How does this
prove that % is weakly sequentially complete? The above theorem, along with
Theorem 4.1.22, is just the statement that L1 /Hq is weakly sequentially complete.
Now notice from Proposition 4.1.20 that
and so (Theorem 4.1.22)
x* ~if°°eiM;.
We use 0i to denote an exterior direct sum4. Suppose (0n)n^i is a weak Cauchy
sequence in %. Each <fin has a unique decomposition as <fin = ipn + z/n, where
ipn £ Xa and vn G %s. Because of the direct sum decomposition of X*, it is
evident that each of the sequences (^n)n^i and (Vn)n>i is a weak Cauchy sequence.
By Mooney's theorem, the first sequence converges weakly to some ip G Xa. On
the other hand, a theorem of Kakutani says that Ms is isometrically isomorphic
to L1(^,E,/i) for some abstract measure space (fi, £,/z) [110]. It follows that
Ms is weakly sequentially complete since every such space L1(^,E,/i) is weakly
sequentially complete [60].
The second theorem we wish to present on the weak topology in % is a deep
result due independently to Delbaen [59] and Kisljakov [115]. For each / £ 3C,
recall from Proposition 4.1.4 that /if is the unique measure such that / = K/if and
11/11 = llM/ll-
Theorem 4.3.3. Let W be a weakly compact set in X, and let
W={»f.f£W}.
Then W is relatively weakly compact in M, that is to say, the weak closure of W
is weakly compact.
A thorough discussion of this theorem can be found in [157, Ch. 7] or [231].
This theorem derives its significance from the Dunford-Pettis characterization of
the weakly compact subsets in Af: W C M is weakly compact if and only if it is
norm bounded and uniformly absolutely continuous (cf. [60]).
4.4. Schauder bases
A sequence (xn)n^i in a Banach space X is called a Schauder basis for X if
every x G X can be written uniquely as
oo
X — j ^ CnXn^
n=l
4i©i B = {(a,b) : a£ A,b£ B} with norm ||(a,6)|| = ||a|U + \W\b-

96
4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
where cn are complex numbers and the = sign means convergence in the norm of
lim
N
£■
n=l
0.
The sequence space £p, 1 ^ p < oo, has a Schauder basis (en)n^i, where
en(j) = Snj. The space of continuous functions C[0,1] has the classical Schauder
basis discovered by J. Schauder [189] (see also [142, p. 352]). Schauder [190] (see
also [142, p. 361]) also proved that the Lp spaces for 1 ^ p < oo have a basis, the
Haar basis. As a consequence of this, and the fact that Hp is isomorphic to Lp, one
can show that Hp, 1 < p < oo, has a basis. Maurey [138] proved that H1 has an
unconditional basis. Carleson [39] explicitly constructed such a basis. Wojtaszczyk
[231] provided further improvements.
Bockarev [29] discovered a basis (frn)n>i for the disk algebra A with the
additional property that
(4.4.1) JKVk*m = Kk.
We will make use of this in a moment. The existence of a Schauder basis is not
automatic since there are separable Banach spaces (even reflexive ones) without a
basis [67]. We refer the reader to [60, 142, 157, 231] for more on bases.
A sequence (£n)n^i C X* is called a weak-* Schauder basis if every £ G X* can
be written uniquely as
oo
n=l
where dn are complex numbers and = in the above equation means weak-*
convergence, i.e.,
N
lim S^ dn£n(x) = £{x)
n=l
for each x G X.
For a Schauder basis (xn)n^i, there is a natural sequence (a^)n^i of continuous
linear functionals defined by
x„(x) = cn, where x = ]P cnxn
n=l
(remember that the expansion of x is unique and so x^ is well-defined). The fact
that the functionals x*n are continuous is a deep theorem of Banach [60, p. 32].
Proposition 4.4.2. If (xn)n^\ is a Schauder basis for X, then (x* )n^i is &
weak-* Schauder basis for X*.
Proof. For £ G X*, let dn := £(xn) and for each N G N let
N
£n := ]Pdnx*.
n=l

4.4. SCHAUDER BASES
97
Then for any
oo
x y CjiXji kz A/,
n=l
N N N / N \
*n(x) = ]P dnXn(x) = Yl dnCn = ^ ^(Xn)cn = t I ]P XnCn I
n=l n=l n=l \n=l /
Thus £n -^ £ weak-* as N —» oo and so every £ G X* can be written as
oo
t = / v anxn.
n=l
The uniqueness follows from the identities
Hence (x*)n^i is a weak-* Schauder basis for X*. □
What is more difficult to prove (and we refer the reader to [60, p. 36] for the
details) is the following.
Proposition 4.4.3. If (xn)n^i is a Schauder basis for X, then (x* )n^i is &
Schauder basis for its closed linear span in X*.
We will now apply the previous two propositions to X = A and X* ~ X to
identify a weak-* Schauder basis for X and a Schauder basis for Xa. We will use
the Bockarev basis (bn)n^i for A. To clarify notation, we let
Certainly Bn = bn as analytic functions on D. We use this notation to avoid
confusing, bn, the element of the disk algebra A, with Bn, the element of X (in fact
Bn G Xa). The result here is the following.
Proposition 4.4.4. (Bn)n^i is a weak-* Schauder basis for X and a Schauder
basis for Xa.
Proof. From eq.(4.2.3), the dual of A can be identified with X using the
pairing
(f,Kfi) = Jfdii.
So if / G A is written in terms of its Schauder basis (the Bockarev basis)
oo
/ = ;>>&„,
n=l
then for each k G N, we can use the orthogonality in eq. (4.4.1) to see that
// oo \ oo «
I Y2 cnK I bk dm = ^2 cn / bnbk dm = ck.
\n=l / n=l ^
Note that passing the sum through the integral is justified since the sum converges
in the norm of A (i.e., uniformly). Thus, via our linear pairing, Bk can be identified
with the linear functional &£(/) = ck. Applying Proposition 4.4.2 we have shown
that (-Bn)n^i is a weak-* Schauder basis for X.
In order to prove that (Bn)n^i is a Schauder basis for Xa, we notice that
Bn = (frn)+ G Xa and so, using Proposition 4.4.3, it suffices to show that the closed

98
4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS
linear span of (Bn)n^\ is all of %a. From Theorem 4.1.22, 3C* can be identified
with H°° by means of the dual pairing
(/+,<?) = f /5 dm, /ei\ </e#°°-
Thus if # G i/°° and annihilates every Bn, then
0 = (Bn,g) = ((bn) + ,g) = Jbngdm Vn G N.
But since the closed linear span of the 6n's is the disk algebra A, we have
[Cngdm = 0 VnG N0
and so <?(n) = 0 for all n G No. This means that g = 0. An application of the
Hahn-Banach separation theorem completes the proof.
□

CHAPTER 5
Which functions are Cauchy integrals?
5.1. General remarks
Which analytic functions on D belong to the space of Cauchy transforms %1
Gathering up our observations from the previous three chapters, here are some
necessary conditions a Cauchy transform must satisfy.
Proposition 5.1.1. Suppose f = K/i for some /i e M. Then
(1) / satisfies the growth condition
(2) / has finite non-tangential limits m-almost everywhere on T and
™(l/l >!/)<—, y>0.
(3) / G HP for allO<p<l and
(4) If f = J2n>oanzn> then (an)n^o is a bounded sequence of complex
numbers.
None of the above conditions is sufficient. The above necessary conditions can
only be used to determine which analytic functions on D are definitely not Cauchy
transforms. Known necessary and sufficient conditions are difficult to apply and in
a way, the very question is unfair. For example, suppose that / is analytic on D
with power series
f(z) = a0 + a\z + a2z2 -\
and we want to determine whether or not / = K\i for some fi G M. Since
(X/i)(z)=/i(0)+/i(l)^ + /i(2)22-",
we would be trying to determine, by equating an with fi(ri) for n G No, the measure
li from only 'half its Fourier coefficients, the non-negative ones.
5.2. A theorem of Havin
If one is willing to settle for a functional analysis condition, there is an old
characterization of % [91], albeit difficult to apply.
99

100
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
Theorem 5.2.1 (Havin). Suppose
/ = £
akz
k=0
is analytic on D. Then the following statements are equivalent.
(1) There is some constant C > 0, depending only on f, such that
(5.2.2)
/ J Afcft/e
/c=0
^ C max •
£W
/c=0
:(€T
for any complex numbers Ao,..., An.
(2) / = K\i for some \i G M.
Proof. Recall that if / = K\i, then
—j- d/i(o = y, zk / cfcdM(o = E £(*)**
1 ^z k=o J k=o
and so the Taylor coefficients of / are equal to the (non-negative) Fourier coefficients
of /i. Also recall from eq.(4.2.3) that the dual of A (the disk algebra) can be
identified with % via the Cauchy pairing
OO
(g,Kfj)= lim V?(n)/i(n)rn, g G A.
r-+l~ f—-'
fc=0
More specifically,
(5.2.3)
lim Yjj(n)jl{n)rri
k=0
< cjg\\
To prove (2) => (1), let / = Kfi and Ao, Ai, • • • , An be given complex numbers.
With g(z) = Ao + \\z + • • • + Anzn, the inequality in eq.(5.2.2) follows from the
inequality in eq.(5.2.3).
To prove (1) .=> (2), the hypothesis imply that the linear functional £, defined
first on polynomials p(z) = Ao + \\z + A2z2 + \- Xnzn by
n
fc=0
extends to a bounded linear functional on the disk algebra A. Hence £(p) = (p, /)
for some / = K/jl. Thus
a^ = £(zk) = /2(k) V/cgNq
and the result follows.
□
5.3. A theorem of Tumarkin
Instead of asking whether or not an analytic function defined only on D is a
Cauchy transform, suppose we were to ask whether or not an analytic function /
on C\T is a Cauchy transform
(Cfx)(z)= f—L-d^O, zeC\7.
J 1 - Qz

5.3. A THEOREM OF TUMARKIN
101
This is a more tractable question since we would be comparing
oo oo
(f\m*) = Y,a»zn with (Cfi\mz) = J2^zn
n=0 n=0
and
OO OO ^/ x
(/l°e)W = E^ Wlth «»»)(*) = -£^
1 Z 1 Z
n=l n=l
which would involve knowing all of the Fourier coefficients of /i and not just the
non-negative ones as before. An early result which answers this question is one of
Tumarkin [220] (see [133] for a generalization).
Theorem 5.3.1 (Tumarkin). Let f be analytic on C\T with /(oo) = 0. Then
f = C\i for some \i £ M if and only if
(5.3.2) sup /|/(rC)-/(C/r)|dm(C)<oo.
0<r<lJT
/T
Proof. Writing
f(z) = J2anZU, ^D,
n=0
n=l
a power series computation shows that for any 0 < r < 1,
(5.3.3) //K)-_/(CA)dm(c)= /M, ,6D,
Assuming the hypothesis in eq.(5.3.2), the measures
<K = (/(rC)-/(C/r))dm(C)
are uniformly bounded in the norm of M and so, by the Banach-Alaoglu theorem,
some subsequence d/ir?i (where rn /* 1) converges weak-* to a measure d/i. Passing
to the limit in eq.(5.3.3) says / = C/i.
Conversely, if / = C\i, a computation shows that for any £ G T and 0 < r < 1,
f(rC)-f(CM = jPriMMw)
/l/«)-/(C/r)|dm(C)^ / /"PrcHd|/i|Hdm(C).
Now use Fubini's theorem along with the identity
[ PrC(w)dm(() = l
Jt
to obtain the result. □
and so

102
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
5.4. Aleksandrov's characterization
We now present a refinement of Tumarkin's theorem due to Aleksandrov [10]
which identifies the type of measure used to represent a Cauchy transform. Before
doing this, we need a definition and a few reminders.
For 0 < p < oo, let HP(C\T) denote the analytic functions on the disconnected
domain C \ T for which
>! i/P
|/(rC)|pdm(C) <oo.
ltfp(c\T) :~ j^7T
Recall from Smirnov's theorem (Theorem 2.1.10) that if / := C/jl for some \i G M,
then / G HP(C \ T) for all 0 < p < 1. Also recall the jump function
(J/)(C):= lim (/«) - /(C/r))
which exists for almost every £ G T and, via Fatou's jump theorem (Corollary 2.4.2),
is equal to the Radon-Nikodym derivative d/i/dm. In summary, a Cauchy transform
/ = Cfi on C\T satisfies the four conditions
(5.4.1) /(oo) = 0,
(5.4.2) /G f| #P(C\T),
0<p<l
(5-4-3) H/IIhp(c\t)=o(I^
(5.4.4) J/ei1.
Also recall from Proposition 2.1.15 that
M<^^ll/ll^(c\T) = 0(]fr^
and, from Fatou's jump theorem, that
li JL m => Jf = 0 m-a.e.
Theorem 5.4.5 (Aleksandrov). If f is an analytic function on C\T satisfying
the conditions in eq. (5.4-1) through eq. (5.4-4) above, then f = C\i for some \i G M.
Moreover, if the conditions in eq. (5.4-1) and eq.(5.4-2) are satisfied, then
(1) / = C\i for some \i <C m if and only if
1™ ll/ll//p(c\T)(1-rf = 0

Poland Jf G L1.
(2) / = Cfi for some /i JL m if and only if
I™ ll/ll//p(c\T)(1-rf <°°

Poland Jf = 0 m-almost everywhere.

5.4. ALEKSANDROV'S CHARACTERIZATION
103
The proof of this theorem requires a few preliminaries. We first need some
basic facts about subharmonic functions. Two good references for this are [79, 96].
A function u : ft C C —» [—oo, oo) is subharmonic if u is upper semicontinuous on
fi, that is,
u(z0) ^ lim u(z), z0 e fi,
z-^z0
and satisfies the following sub-mean value property: given z0 £ ^ and 5 > 0 with
A(zq,s)~ cfi, the following inequality holds
r27r ^
i/(z0) ^ / u(z0 + selt)—-.
Jo 27r
The mean-value property for harmonic functions says that u is harmonic if and
only if equality holds for every s. Here are two important examples of subharmonic
functions. If / is analytic on ft and p > 0, then \f\p is subharmonic onl]. If u is
subharmonic on C and / is analytic on H, then u o / is subharmonic on ft.
For a subharmonic function u on fi, we say that a function U on ft is a harmonic
majorant for ii if U is harmonic and ii ^ U. A harmonic function U on ^ is called
the least harmonic majorant of u if 17 is a harmonic majorant and U ^ V for
any other harmonic majorant V of it. The Perron construction [7, p. 248] says
that if u has a harmonic majorant, it has a least harmonic majorant. If we focus
our attention to the unit disk D, determining whether or not a harmonic majorant
exists and computing its least harmonic majorant, when it does, is not too difficult.
Indeed, if u is subharmonic on D, then the integral means
M(r;u) := / u(rQ dm(()
Jt
increase as r /* 1~. This next result is found in [79, p. 38].
Lemma 5.4.6. A subharmonic function u on D has a harmonic majorant if
and only if the increasing integral means M(r\ u) are bounded as r —» 1~. The least
harmonic majorant is then
U(z):= lim [ Pz(0u(r()dm(().
With these preliminaries about subharmonic functions in place, we begin the
proof of Aleksandrov's theorem by proving a few technical lemmas. Define, for
0 < p < 1, the following function Gp on C.
(,A7, r (A. j\z\" cos (p9(z)) ifzjLO,
(5-4.7) Gp(*).= |Q [iz = Q/
where
f arctan(2//|x|) x ^ 0,
(5.4.8) 6{z) := I tt/2 x = 0, y > 0,
[-tt/2 x = 0,y<0.
Note that \8(z)\ ^ n/2 and 0 < p < 1 and so
(5.4.9) Gp ^ 0.
The following technical lemma of Pichorides [159] was used to find the best
constants in the Lp estimates for the conjugation operator (see Theorem 3.7.4 presented
earlier).

104
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
Lemma 5.4.10. The function Gp is subharmonic on C.
Proof. Clearly Gp is continuous on C and so it suffices to show that given
any z G C, there is an r(z) > 0 such that
(5.4.11) Gp(z) ^ -1- / Gp(z + relt) it, for all r G (0, r(z)).
27r J-tt
Since
G (z)=l^zP) for|arg(z)|<^/2,
A } \&{-z)p for | arg(-z)| < tt/2,
Gp is harmonic on {Rz ^ 0} and thus the inequality in eq.(5.4.11) holds on the set
{Rz + 0}.
If z = 0, then
^ [* Gp(reu)dt = 2[±- [** rp cos (pt) dt) = ^sin(^)^0
27r7-7r \27TJ-7r/2 J pn 2
since 0 < p < 1. But Gp(0) = 0, by definition, and so the inequality in eq.(5.4.11)
is satisfied.
Letting Hp(z) = $lzp on {z : | arg z\ < 7r, z ^ 0}, we first notice that Gp — Hp ^
0 on this set. To see this, observe that
Hp(z) = R(zp) = Gp(z), z = re1*, \c/>\ ^ tt/2, r > 0.
If z = re^, where r > 0 and 7r/2 < (ft < 7r, then
Gp(z) — Hp(z) = rp cos (p (0 — 7r)) — rp cos(p0)
= 2rp sin (p ($ - -J J sin(p-)
If z = re^, where r > 0 and —7r < (ft < —tt/2, then
Gp(z) - Hp(;z) = rp cos (p (0 + n)) - rp cos(p0)
= -2rp sin (p (V + |) ) sinO^)
But i/p is harmonic on its domain, and so, for y ^ 0 and r G (0, |?/|),
GP(iy) = Hp(iy)
1 Z*71"
= 7T / Hp(iy + relt) dt
which proves the inequality in eq.(5.4.11). D
The next lemma is a standard real analysis exercise.
Lemma 5.4.12. Suppose q > 1 and (hn)n^i c Lq with \\hn\\q ^ C for all
n G N. If hn —> h almost everywhere as n —» oo, £/ien h £ Lq and hn —> h weakly
as n —» oo.

5.4. ALEKSANDROV'S CHARACTERIZATION
105
Proof. By Fatou's lemma,
f \h\qdm= f lim \hn\qdm^ lim [ \hn\q dm ^ Cq
< OO
and so h G Lq.
Let e > 0 and g £ Lp be given (1/p + 1/q = 1) and note, by basic properties
of the integral, there exists a 5 > 0 such that for any measurable set A C T with
m(A) < 5, we have
r r \1/P
J \g\pdm) <€.
By Egorov's theorem, there is a set E C T with m(T \ E) < S such that hn —> h
uniformly on E. With this in place,
/ (hn — h)gdm\ ^ / (hn — h)gdm
Jt \ \Je
<
<
/ {hn-h)g(
Jt\e
[ {hn - h)gdm\ + \\{hn - h)\\q I [ \g\pdm
JE I \JJ\E j
/ (hn -h)g(
JE
i/p
+ (C+\\h\\q)e.
The last integral converges to zero since hn —» h uniformly on E. Thus
lim
ldm
^(C+\\h\\q)e
\ (K - h)g
and, since e was arbitrary, the result follows.
The following technical lemma is the key to proving Aleksandrov's theorem.
Lemma 5.4.13. Suppose f € W for all0<p<l, /(0) € M, 5ft/ G L1, and
(5.4.14) lim (1-p)||/||p <oo.
D
Then
M
J
jt
C-z
dM(C)
for some real measure \i G M satisfying
MKP/IU> + ^ lim(l-rt||/||^.
P—1-
Moreover, if
then d/ji = 'Stfdm.
Iim(l-P)ll/Il? = 0,
Proof. Notice that Gpo/^0onD (since Gp ^ 0 - see eq.(5.4.9)). Moreover,
since Gp is subharmonic on C (Lemma 5.4.10) and / is analytic on D, Gp o f is
subharmonic on D .

106
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
Assume for a moment that 9ft/ = 0 almost everywhere on T. For q G (1,1/p)
and r G (0,1), the Lq integral means of Gp o / satisfy
f(GP o f)(rC)qdm(C) = [ |/K)P (cosp0(/(rC)))9 dm(C)
< / l/«)lP9dm
113
which is finite since 0 < pq < 1. Thus the function Gp o / is subharmoriic on D
with uniformly bounded Lq integral means. By Holder's inequality, these integral
means are uniformly bounded in L1. By Lemma 5.4.6, the function
a^ lim /pa(C)(Gpo/)(rC)dm(C)
/T
is the least harmonic majorant for Gp o f. Thus for all a G D,
(G„o/)(a)< lim [ Pa(()(Gpof)(rC)dm(().
Since /(r£) —» /(C) almost everywhere as r —» 1~ and Gp is continuous, (Gp o
/)(rC) —> (Gp o /)(C) almost everywhere as r —» 1~. This, together with the
uniform boundedness of the L9 means of Gp o /, say, via Lemma 5.4.12, that
(Gp o /)(r •) -> Gp o / weakly in L9. Thus
(GP o /)(a) ^ | Pa(C)(Gp o /)(C) dm(C)
^a(C)l/(C)lPcos(^)dm(C), aGD.
/
In the last integral above, notice that we are assuming 3£/(C) = 0 and so arg/(£) =
±7r/2.
Since cos(p7r/2) x 1 — p, we can use our assumption in eq.(5.4.14) to observe
that
Mm / |/(C)lpcos (^)dm(C)
< 00.
This means that for some sequence pn / 1, the measures
oVp„ := |/|"» cos (^) dm
satisfy
(5.4.15) lim \\vvJ = lim / |/(C)|pcos (?£■) dm(0-
Since these measures are uniformly bounded in total variation norm, they have a
weak-* cluster point dis. However, by the definition of GPn from eq.(5.4.7),
n—>oo
From the estimate
0 ^ lim (Gpn o/)(a) = |/(a)|cos(arg/(a)) = »/(a).
n—>oo
iate
0 < (GPB o/)(a) < |p„(C)|/(C)Ip"cos(^) dm(C),

5.4. ALEKSANDROV'S CHARACTERIZATION
107
we get, from weak-* convergence,
(5.4.16) 0^/(oK J Pa(C)cMC).
Moreover,
WW = lim \Wp
n—+oo
= lim f |/|p" cos (^-) dm
lim [ \f\pcos(~)dm
= lim^^d -p)||/||?
= \ Um(l-p)||/||?.
But, by eq.(5.4.16), 9ft/ is a positive harmonic function that is majorized by
the Poisson integral of a positive measure. Thus the sequence of positive measures
(5.4.17) {($R/)(rn.)dm:rn/l}
satisfies
||(»/)(rn.)dm||= /W)(r„C)dm(C) < / / Pr„c(u;)di/(«;)dm(C) = IMI
and hence is uniformly bounded in r. Letting d/i be a weak-* cluster point of the
sequence in eq.(5.4.17), we conclude that
(»/)(o) = lim (Stf)(ra)
r—>1~
lim [w)(r0Pa(0dm(()
I
T
Moreover,
l|/x|| ^ lim ||(9ft/)(rn-)dm|| (by Proposition 1.6.2)
n—>-oo
< IMI
<f Iim(l-P)ll/Il?.
The condition /(0) G R says that the harmonic conjugate of 9ft/ is
(3/)(a)= y~Qa (C)d/i(C)
and so
/(a) = K/(a) + »3/(a) = | (Pa(C) + »Qo(0) dM(C) = / ^ MO-
Notice how this last integral is the Herglotz integral H/i of the measure \i.
For the general case, when 9ft/ ^ 0, let
h(z):= [i±ljtf(Qdm(Q.

108
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
We claim that the function f — h satisfies the hypothesis of the theorem along
with the extra property that 3£(/(C) — MO) = 0 ^-almost everywhere. Indeed, by
Fatou's theorem,
m(C)= lim /"Prc(«(0dm(£) = a/(C)
for m-almost every £ G T. Furthermore, by using the fact, from Proposition 2.1.15
that
lim(l-p)||fo||£ = 0,
p-*l-
we see that
lim(l -p)\\f - fo||£ ^ lim(l -p)||/||J < oo.
p—>1 £>—>1
From our above analysis applied to the function / — h, we see that the function
f — h = Ha for the measure a which is a weak-* cluster point of the measures
»(/-ft)(r.)dm
and, as such, satisfies
IN < ~ M(l -P)\\f - h\\* < J Um(l -P)l|/||?.
Putting this together, we have / = H\i, where d/i := 3£/dm + da. Also,
H^IWIk+?lim(l-p)||/||£.
^ p—►!
Finally, if limf,_1(l — p)||/||^ = 0, then a = 0 and / = if(SR/dm). D
Proof of Theorem 5.4.5. Assume that / e HP(C \ T) for all 0 < p < 1,
Jf G L1, and /(oo) = 0. Without loss of generality, assume further, by subtracting
off a constant multiple of Cm, that /(0) = 0. Define the following analytic functions
/i,/2onOby
fi{*).= -i(m+707*)), f2{z)~f{z)-70jt)
and note that both fi and /2 satisfy the hypothesis of the previous lemma. Hence
there are real measures \i\ and /i2 such that
Set /i = i\i\ + /i2 and observe that
'<*> = s / ^ d"«) - / rns d"K» - s /"" - /1^5 d"(<)
since
/d/i = i/1(0) + /2(0)=«0 + 0 = 0.
/■
This says that / = Cfi.
If, in addition to conditions eq.(5.4.1) through eq.(5.4.4), we assume that
iim(i-rtll/li;p(£w = o,
p—»1~ v x '
then
Uffi (1-p)II/j;||£ = 0
P-*i-

5.5. OTHER REPRESENTATION THEOREMS
109
and by Lemma 5.4.13, we can take the measures /jlj above to be absolutely
continuous.
Assume / satisfies conditions eq.(5.4.1) through eq.(5.4.4) and also that Jf = 0
almost everywhere. From the first part of the proof, we know that / = C\i for some
\i G M. Now, by Fatou's jump theorem (Corollary 2.4.2),
-— = Jf = 0 m-a.e.
dm J
and so \i _L m. This completes the proof of Aleksandrov's theorem. □
There is a related characterization of Cauchy integrals, also due to Aleksandrov
[9], that involves the space L1,oc (see Chapter 1 for a definition).
Theorem 5.4.18 (Aleksandrov). Suppose f is analytic on C\T with /(oo) = 0.
Then f = C\i for some \i G M if and only if
sup||/(rC)|Ui.~ < oo.
5.5. Other representation theorems
Let us refine the problem further. Suppose that E is a closed subset of T and
/ is analytic onC\E with /(oo) = 0. When is / = C/jl for some \i G M that is
supported in El This next theorem of Havin [91] (see also [78, p. 52]) provides an
answer.
Theorem 5.5.1 (Havin). Let E be a closed subset ofT and let f be analytic
on C\E with /(oo) = 0. Then there is a \i G M supported in E such that f = C/jl
if and only if there is a constant C such that
y^^kfjdk)
k=i
^ C sup
V"^ ^k
f-^z-ak
k=l
zeE
whenever ai, • • • , an 0 E and Ai, • * • , An G
See [78] for equivalent versions as well as refinements of Havin's theorem.
Though we stated the Havin result for Cauchy transforms of measures on the
circle, the theorem solves the more general problem: when is a function / analytic on
C\E, where E is a compact subset of the plane, equal to a Cauchy transform
d/i(w)
I
w — z
for some finite measure \i supported in El
There is even a further refinement. Let X be a class of analytic functions on
D and E be a closed subset of T. Let jF(X, E) denote the functions / G X such
that / = K\±, where \i G M and has support in E. Under what conditions on E is
J(X,£)^(0)?
For the Hardy spaces Hp, the result is known. For 0 < p < 1, notice that
3(HP, E) ^ (0) for every non-empty closed set E (Theorem 2.1.10). For 1 < p ^ oo,
there is the following result of Havin [92].
Theorem 5.5.2 (Havin). For 1 ^ p ^ oo, 3(HP,E) ^ (0) if and only if
m(E) > 0.

110
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
Proof. Suppose m(E) = 0 and there is a measure \i supported in E with
K\i G Hp. Note, since /i is supported E and m(E) = 0, that /ilm. Since /dm,
where / is a boundary function for Kfi, is also a representing measure for Kfi
(Cauchy integral formula - Proposition 2.2.1), and all representing measures have
the same singular part (Proposition 2.1.5), then \i must be the zero measure.
Conversely, if m(E) > 0, then, by a theorem of Ahlfors [78, p. 6], there is a
non-trivial bounded analytic function g onC\E and, by subtracting off a constant,
we may assume g(oo) = 0. Define / m-almost everywhere on T by
/(C) := lim (g(r() - g((/r))
and note that the measure /dm is supported in E. A power series computation
shows that K(fdm) = g\D is non-trivial. □
Though the analysis is more complicated, Hruscev [103] provides an answer to
this question for other spaces of analytic functions on D. We state these results
without proof.
Theorem 5.5.3 (Hruscev).
(1) If A is the disk algebra, then 3(A,E) ^ (0) if and only if m(E) > 0.
(2) If A°° is the space of functions f which are analytic on D such that f^ G
A for all n = 0,1, 2, • • •, then 3(A°°,E) ^ (0) if and only if E contains
a closed subset F of positive Lebesgue measure satisfying the Carleson
condition
oo
]Pm(/n)logm(/n) > -oo,
n=l
where (in)n>i is the sequence of complimentary arcs of F.
5.6. Some geometric conditions
So far, we have investigated the question as to whether or not a particular
analytic function on D (or on C \ T) is a Cauchy transform in terms of growth
conditions near the boundary. We could also attempt to answer this question in
terms of some geometric conditions. For example, if /(D) is contained in a half-
plane, then / G X (Proposition 2.1.13). In this section, we provide some other
geometric conditions.
Lemma 5.6.1. If f eX and (p : D -> D is analytic, then f o 0 e X.
PROOF. If /x e M+, then ^R(K/i o <j>) > 0 on D and so by Proposition 2.1.13,
K\i o (j) e X. Write any \i G M as fi = (fii — /i2) + i{v>3 — M4), Mj ^ ^+> and apply
the above argument four times to conclude that K\i o (ft e X for any \i G M. □
This simple lemma yields the following corollary.
Corollary 5.6.2. Suppose that ft is a proper simply connected subset of C
and ift : D —> ft is a Riemann map. If ift G X (this may not always happen!), then
any analytic map f : D —> ^ 1 belongs to X.
PROOF. If ift = K/i, apply Lemma 5.6.1 to (ft := ift'1 o / : D —> D to see that
/ = K/i o (ft g X. □
Note that /(D) need not be equal to Cl.

5.6. SOME GEOMETRIC CONDITIONS
111
The following [31] is a nice geometric condition for membership in %.
Theorem 5.6.3 (Bourdon and Cima). Suppose f is analytic and f(W) is
contained in a region that omits two oppositely pointed half-lines (see Figure 1), then
f£X.
PROOF. Assume /(B) is contained in a region that omits two oppositely pointed
half-lines and let £1 be the complement of those two oppositely oriented lines. In
a moment, we will construct an invertible analytic i\) : D —» ft and show that the
harmonic function h := 3?^ satisfies
(5.6.4) sup / \h{rQ\dm{C) < oo.
0<r<lJj
By standard harmonic analysis [65, p. 2] (also see the proof of Theorem 9.1.1),
h = P/jl, the Poisson integral of a real \i G M. This will imply that
^ = Qv + C,
where Q\i is the conjugate Poisson integral of /i, and so
i/j = P/x + iQfi + iC = H/i + iC,
where H\i is the Herglotz integral of \±. It follows now that ip = Kv for some
v G M. Now apply Corollary 5.6.2.
We now construct i/;. Let
Af \ 1 + z
*(*) := —
be the usual analytic map from D onto {$lz > 0} and notice that
4>(elt) = iy, y > 0, whenever 0 < t < 7r,
(j)(elt) = iy, y < 0, whenever n < t < 2n.
For b > 0 and c G M, fixed, let
i/j(z) = </>(z) - —— - 2d log (2^(2;)).
(p(z)
A computation shows that
_/
ik(—y), n < t < 27r,
*(e«)={;
where
Notice that
it\ _ J *Hy) + 2c7r> 0 < t < 7r;
b
Kv) =V+ - -2clog|y|.
y
k(y) —» +oo as y —» 0+ or y —» oo,
&(2/) ~~* —°° as ?/ —> 0~ or ?/ —> —oo.
Let
ym = min{fc(2/) : 0 < y < oo}, yM = m&x{k(y) : -oo < y < 0}
and observe that ^ maps the unit circle to the two half lines
Li := {(2ctt, y) \ y ^ ym}, L2 = {(0, y) : y ^ yM}-
Since ^ covers each of these half lines twice, it follows from the argument principle
that ijj is univalent. The parameters b and c adjust the geometry of these two lines

112
5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?
(distance they are apart, where they start). Any map from D onto a region omitting
two oppositely oriented half-lines takes the form
*(z) = e^(z) + d,
where 6 is a rotation and d is a translation.
Finally, we leave it to the reader to show that
l-\z\2 1 - \z\2 /14-7
Hz) = WM = y^ - ^ - 2carg (^
The first two terms are constant multiplies of Poisson kernels and thus have
uniformly bounded integral means as in eq.(5.6.4). The last term is bounded since 0
maps onto a half-plane. □
/ ".?
s
G
/
A
/
t
/
f
./
/
'
/
^
X
%
Figure 1. A region G omitting two oppositely pointing rays.
Here is another Cauchy transform problem involving conformal maps. Let G
be an open connected subset of C which satisfies
leG and Gn(-G) = 0.
Such domains are called Gelfer domains and were explored by Gelfer [80]. Certainly
any half-plane is a Gelfer domain. If one arranges things correctly, certain spiral
domains are Gelfer domains. It is known [80] that if / : D —■> G is a Riemann map,
then / has bounded Taylor coefficients, / G Hp for all 0 < p < 1, and
_1_
Is / a Cauchy transform? Using a result in [130], the paper [41] constructs a
counterexample. The counterexample is a Gelfer domain for which / has an unbounded
argument. The question now is: if G is a Gelfer domain with bounded argument,
is / a Cauchy transform?
IIP = 0[^—:}, P-l-

5.6. SOME GEOMETRIC CONDITIONS
&
' * 'ft I f *
\
:***
Figure 2. A Gelfer domain. Note that w e G implies —w 0 G.
^ * / ST
1 ",.*-*, >^ ! , ^ ' - -
Figure 3. A Gelfer domain with bounded argument.

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CHAPTER 6
Multipliers and divisors
6.1. Multipliers and Toeplitz operators
An analytic function 0 on D is said to be a multiplier of the space of Cauchy
transforms X if
f eX=xpf ex.
We will denote the set of multipliers of X by 9Jl(X). If 0 G 9Jl(X), the operator
M^-.X^X, M$f:=(pf
is well denned.
Proposition 6.1.1. If<t>€ 9Jl(3C), then M^ is continuous on X.
PROOF. By the closed graph theorem (Theorem 1.5.2) it suffices to show that
the graph of M^ is closed. Suppose (/n)n>i C X with fn —> / and (j>fn —» g in
the X-norm. By the continuity of the evaluation functional £z(h) = h(z) on X (see
eq.(4.1.18)), /n —» / and </>fn —» # pointwise on D. Thus g = 4>f on D, proving the
graph of M^ is closed. D
Definition 6.1.2. For 0 e SDt(3C), we define the operator norm of Mj, via
eq.(1.5.1) by
||A^||:=sup{||0/||:||/||<l}
and call it the multiplier norm of (p.1
Since the constant functions belong to 3C, we know that 9Jl(X) C X. It turns
out that the multipliers of X have a certain amount of 'extra' regularity near T and
so jm(OC) £ X.
Proposition 6.1.3. If </> e 9Jl(3C), tfien 0 e H°° and H^U < ||M0||.
Proof. For each A e D, let ^a(/) = /(A) be the (continuous) evaluation
functional on X. For each n G N,
|0(A)|" = |0(A)"1| = \£x{M;\)\ ^ ||^||||MJ"||1||.
Thus
|0(A)|< nEdi^iniiiD^iiM^ii
n—+oo
andso Woo^HM^H. □
Here ||/||, without any subscripts on the norm, denotes the Cauchy transform norm of /
(see eq.(4.1.1)).
115

116
6. MULTIPLIERS AND DIVISORS
The multiplier theory goes well beyond X to general Banach spaces of analytic
functions X on the unit disk for which the evaluation functional / i—► f(z) is
continuous on X for each z G D. In this setting, 0 is a 'multiplier' of X if (ftX C X.
With nearly the same proofs, one can show that 9Jt(X) C H°° and that when
(ft G 9Jt(X), the multiplication operator M^f = (ftf is continuous on X. For certain
spaces, such as the Hardy spaces Hp, the multipliers are easy to describe. Indeed,
dJl(Hp) = H°°. To see this, note that whenever (ft G H°° and / G Hp,
[ \<KrQf(rQ\pdm(Q ^ HW^ [ \f(r()\p dm((), 0 < r < 1,
Jt Jt
implying \\(/>f\\p <: ||0||oo||/||P. Hence H°° c 9Jl(Hp). The other direction follows
from Proposition 6.1.3. For other spaces of analytic functions such as the Dirichlet
space V of analytic functions /onD which have finite Dirichlet integral
7T JO
dx dy,
the multipliers are more complicated [205]. By Proposition 6.1.3, 9Jt(D) C #°°
but, unlike i7p, this inclusion is proper. The multipliers of many spaces, even spaces
of differentiable functions, are cataloged in the rather encyclopedic book of Maz'ya
and Shaposhnikova [139].
Before getting into the function-theoretic behavior of multipliers on 3C, we
would like to connect the multipliers of X with the co-analytic Toeplitz operators.
For 1 < p < oo, the Riesz projection operator
is a bounded operator from Lp onto Hp (Theorem 3.2.1). Thus for each (ft G H°°,
the co-analytic Toeplitz operator
Tr.H*^H*, T?/:=(0/) +
is also bounded. Recall that if g G Lp has Fourier series
oo
n= — oo
oo
g+(z) = Y,g(n)zn.
then
n=0
Using the identity
4>f ~ E £(0/(*)C
/c-Z
i,kez
oo
= E <" E W~n)f(k)
n=-oo /c=-oo
oo oo
£ CnE £(*-")#*)'
we observe that
n=-oo k=n
oo oo
(6.1.4) (T?/)(z) = E ^ E ^fc - ")/(*)•
n=0

6.1. MULTIPLIERS AND TOEPLITZ OPERATORS
117
When p = 1 or p = oo, the situation becomes more complicated. For example,
the continuity of T-r on H1, which is not automatic since (L1)^ 2 H1 •> depends on
whether or not !R0 and ^s(ft are multipliers of BMO [204]. Since
(L°°)+ = BMOA 2 H°° and (C)+ = VMOA D A,
(see Chapter 3) the continuity of T-r on H°° or A is also not automatic. However,
we can determine which symbols (ft give rise to continuous T-r on H°° or A [227].
Proposition 6.1.5. For (ft G H°°, the following are equivalent.
(1) (fteWl(X).
(2) (ft e fBl(Xa), i.e., feXa^(ftfe Xa\
(3) T-t\A^A is bounded.
(4) T^: H°° ^ H°° is bounded.
Moreover, \\T^ : A \-+ A\\ = ||M^||.
PROOF. (1) => (2): For (ft G Wl(X) and / G 3Ca, note that 0/ G X. We need
to prove that (ftf G 3Ca. By Proposition 4.1.21, there exist a sequence (fn)n^i of
polynomials with fn —-> / in X-norm. But 0/n G H1 C Xa (Proposition 2.2.1) and,
since (ft G 9Jt(3C), (ftfn —> 0/ in X-norm. Thus, since Xa is a closed subspace of X
(Proposition 4.1.20), (ftf G Xa. It follows now that (ft G Tl(Xa).
(2) => (1): Assume (ft G 3DT(3Ca). Given / £ 3C, there is a sequence (/n)n^i
such that /n —> / weak-* in X (Remark 4.2.9). Then 0/n G Xa (by assumption) and
moreover, since weak-* convergent sequences are norm bounded (Proposition 4.2.5),
(ftfn is norm bounded. Passing to a subsequence if necessary, (ftfn converges weak-*
to some g G X (Banach-Alaoglu). Now use the fact that fn —> / and (ftfn —> g
pointwise in D (Proposition 4.2.5) to argue that g = (ftf and so (ftf G X. Thus
(ft G SDt(OC).
(3) => (1): By Theorem 4.2.2, the dual of A can be identified with X via the
pairing
lim [ fr(C)9(C)dm(0
(f,g)= lim / /r(C)$(C)dm(C), f e X, g G A.
If T-r : A —> .A is continuous, then by Proposition 1.5.4, T-£ : X —> X is continuous
and IIT-JII = ||T-r||. Now notice that T-£ = M^.
II <£ II II 011 ^ <^
(2) => (4): By Theorem 4.1.22, the dual of Xa can be identified with H°° via
the pairing
(f,9)
lim [ MQg{Qdm(0, feOCa, g e H°
r-+l JT
Thus, as before, if M<f> : 3Ca —> Xa is continuous, then so is M^—T^\ H°° —> H°°.
(4) => (3): For any # G A, let #n be the Cesaro polynomials for # and note that
gn —* g uniformly (Theorem 1.6.5). From eq.(6.1.4), we know that T-rgn is also a
polynomial and hence belongs to A. If we assume T-r : H°° —> H°° is continuous,
then T-rgn —> T-rg uniformly and so T-rg G A. It follows that Tr : A —* A is
continuous.
□
Recall from Chapter 4 that %a — {Kjjl : jjl <^. m}.

118
6. MULTIPLIERS AND DIVISORS
6.2. Some necessary conditions
A key in determining some necessary conditions for an analytic function to be
a multiplier of X is the following proposition.
Proposition 6.2.1. For a function (ft analytic on D, the following are
equivalent.
(1) </>efBl(X).
(2) For every £ G T, the function
belongs to X and
C-z
}
sup I
\\MJ =
1 ^
k-z
sup I
:Ce1
1 0 1
k-*l
:(€T .
Moreover,
(6.2.2)
Proof. (1) => (2): If (ft is a multiplier of 3C, then, by Proposition 6.1.1, the
operator M^f = (ftf is continuous on X and so for each £ G T,
1
Furthermore,
(6.2.3)
C-*
C-z
1
= M<*
M,
'C-*
C-z
^ \\M4
ex.
c-
= \\M^
since \\(z - C)_1|| = \\KSC\\ = 1 (Proposition 4.1.20).
(2) =>(!): Now assume that (ft is analytic on D and
a := sup
C-*
:Ce
t|
< 00.
For each / G X, we will show that (ftf £ X and
110/11 <«ll/ll-
By Proposition 4.1.4, we can choose a fi G M such that / = K/j, and ||/|| = \\/i\\.
Since the convex balanced hull of {d^ : £ G T} is weak-* dense in the ball of M
(Proposition 1.6.8 and Remark 1.6.9), we can find a sequence (crn)n^i C M such
that each o~n is a finite linear combination of point masses that also satisfy the two
conditions
Iknll ^ IImII
and
If
then
and for each z G D,
o~n —-> \i weak-*.
9n = Kan,
\\gn\\ ^ ||<7n|| ^ ||m|| = ll/H
9n(z) -> f(z) as n -> oo.

6.2. SOME NECESSARY CONDITIONS
119
By Proposition 4.2.5, gn —> / weak-* in X.
Clearly
9n eX,
\\<t>9n\\ < a|kn|| ^ a|MI =a||/||,
and
<t>{z)gn{z)^<l>{z)f{z), zeB.
Another application of Proposition 4.2.5 and the Banach-Alaoglu theorem says that
(ftf £ X and (ftgn —> (ftf weak-*.
Furthermore, if h G ball (A), then, from the inequality \\(ftgn\\ ^ all/||> we Set
|(0/,ft>|= lim |(^n,/i)Ka||/||
n—+ oo
and so
(6.2.4) H0/II = sup{\(<Pf,h)\ : ft G ball(A)} < a||/||.
To prove eq.(6.2.2), we combine eq.(6.2.3) with eq.(6.2.4).
a
Remark 6.2.5. In statement (2) of the above result as well as the expression
for ||M^||, the circle can be replaced by any set E that is dense in T. To see this,
use the same proof as above but replace T with E in the statement that the convex,
balanced hull of {^ : C G T} is weak-* dense in the ball of M. We will make use of
this in a moment.
We know that a multiplier must be a bounded function. This next set of
necessary conditions says a bit more.
Theorem 6.2.6. If (ft e 3DT(3C), then
(1) the Taylor sums of (ft are uniformly bounded, that is,
I N I
sup
(2) for each C G T,
exists.
k=o
< oo;
Z lim (ft(z)
Proof. To prove (1), observe from eq.(6.1.4) that for each £ G T and n G No,
k=0 k=0
Since (ft is a multiplier, we can apply Proposition 6.1.5 to see that T-r : A —-> A is
bounded. Thus
££(*)**
k=0
=
loo 1
n
\c^2mck
fc=0
= lir^lU
< ll^ll
= 1
MA.

120
6. MULTIPLIERS AND DIVISORS
To prove (2), recall from Proposition 6.2.1 that for each £ G T, there is a v G M
so that
P- = (*,)(*)■
Thus, by the dominated convergence theorem (see also eq.(2.1.8)),
lim 0«) = lim C(l - r)(AV)«) = C({C})-
r—+l~ r—+l~
This says that the radial limit exists for every £ G T. Note that (ft is bounded and
so we can apply Lindelof's theorem (Theorem 1.7.3) to see that the non-tangential
limit exists for every £ G T.
□
Remark 6.2.7.
(1) The existence of a non-tangential limit at all points of the circle is not
sufficient to be a multiplier. For example, the singular inner function
'z + V
cft(z) = exp
has non-tangential limits at all points of T but is not a multiplier (see
Theorem 6.6.11).
(2) The multipliers of % need not be continuous on D~. Consider the following
result of Hruscev and Vinogradov [105]: given a set E of first category
in T, there is a (ft G 9Jt(3C) such that (i) the partial sums of the Taylor
series of / are uniformly bounded, (ii) the Fourier series of (ft converges
everywhere on T, (iii) every point on E is a point of discontinuity of (ft.
6.3. A theorem of Goluzina
There are two papers of Goluzina [83, 84] that contain further information
about the relationship between the measures \i and v in the equation (ftKfi = Kv,
when (ft G 9Jt(3C). Here is one particularly interesting result from those papers.
Theorem 6.3.1 (Goluzina). If (ft G 2rt(9C) and (ftKfi = Kv, then
dvs = 0d/is,
that is to say
vs(E) = / (ftd/is
Je
for every Borel set EcT.3
Proof. Start with the identity
(Ku)(z) = (KM(z) + /•^)--f(0dMC)
Jj 1 - Qz
and observe that if we can show that the function
Jt 1 - O
From Theorem 6.2.6 and Proposition 9.1.17 (see below) the function cf)(Q = limr_^1_ (f>(r£)
exists everywhere and is a bounded Borel function. Thus 0d//s is a finite Borel measure.

6.3. A THEOREM OF GOLUZINA
121
belongs to 3Ca, then
Kv = K((j)fi + /m), / G Ll(m).
From Proposition 2.1.5, the singular parts of the representing measures must be
the same and so
dz/s = 0d/is.
For each C G T, let
(6.3.2)
vc(z) =
l-Cz
Since 0 is a multiplier, we see that v^ E X for each £ G T and moreover,
I <AW - 0(C) I
INI
l-<z
<
+ Wc
\l-Cz
< IIMJI + H0IU
i-<*
1
l-Cz
\\M4 + ||0|U (by eq.(4.1.23))
(by Proposition 6.2.1)
Thus
sup{K||:CGT}<oo.
This means that the vector-valued function
is a Bochner integrable function, that is,
yiKiid/x(o<oo
and so the vector-valued integral
y«CdAi(o
is norm convergent. If we can show that each vq belongs to 3Ca, then we can use
the fact that %a is norm closed to say that the above (Bochner) integral belongs to
%a and thus completing the proof. We will skip an in-depth discussion of Bochner
integrals and refer the reader to [60, 61] for thorough treatments of this.
To prove that v^ G %a for each ( G T, we will use Poltoratski's distribution
theorem (Theorem 9.7.1) which says that for any / G C(T) and any tj G M,
lim ny I f dm ■
y^°° J\K>n\>y
I
fdrjs.
We know from eq. (6.3.2) that
vc = KXC
for some A^ G M. If / is continuous and vanishes near £, we can use the fact that
vq is bounded on the support of / (since multipliers are bounded) to say that for
large enough y, the set
{\vc\ >y}nsupp(f)

122
6. MULTIPLIERS AND DIVISORS
is empty and so
/ /d(Ac)s = lim ny f dm = 0.
J y^°° J\v,\>y
This implies that the singular part of A^ is at most a point mass at £. Now we use
the identity
lim(l-r)(^)(rC) = 77({C}), V £ M,
r—>l~
(see eq.(2.1.8)) to prove that
Ac({C}) = lim(l-r)(iac)(rC)
= 7m(l-r)^-^
= lim 4>(r() - <A(C)
r—>l~
= 0
since (ft(rQ —> (ft(Q as r —-> 1~ (Theorem 6.2.6). This means that A^ G Ma and so
^C ^ ^-a as desired. □
6.4. Some sufficient conditions
In the earlier sections of this chapter we saw that an analytic function needs
to be sufficiently well behaved near the unit circle in order to be a multiplier of X.
We now discuss some sufficient conditions a bounded analytic function can satisfy
to be a multiplier of X. We follow [227, 228].
Theorem 6.4.1. Let (ft G H°°.
(1) //
oo
]T|0(n)|log(n + 2)<oc,
n=0
then (/) is a multiplier ofX and
oo
\W4 <C^|0(n)|log(n + 2).
n=0
(2) //
I w — C
then (ft is a multiplier of X and
IIa^ikihu + v
Remark 6.4.2. These conditions say that if (ft is sufficiently smooth, say (ft is
analytic in a neighborhood of D~, then (ft is a multiplier of X. For another example,
consider the function
oo 1
<f>(z) := (1 - z) log(l -*) = -*+£ j—r^z".
Condition (1) says that (ft is a multiplier.
L<£ := ess-sup
dm(w) : C e T } < oo,

6.4. SOME SUFFICIENT CONDITIONS
123
To prove Theorem 6.4.1, we need to review some Fourier analysis. For a
bounded function h on T, let
N
Sn(K e
i0\
E M'
inO
n=-N
be the iV-th partial sum of the Fourier series for h. Note that
/
SN(h,ete)= / h(elt)DN(0-t)dm(elt),
where
N
DN(t) := Yl eM
k=-N
is the Dirichlet kernel [234, p. 49]. It is a standard fact [234, p. 67] that the
'Lebesgue constants' ||.D/v||i satisfy the estimate
and so
\\DN\\1-\og(N + 2)
\SN{h,eie)\^C\\h\\QOlog{N + 2).
Proof of Theorem 6.4.1. To prove statement (1) of Theorem 6.4.1, we will
use Proposition 6.1.5 and show that the co-analytic Toeplitz operator TV is bounded
on the disk algebra A. To this end, let h G A and notice from eq.(6.1.4) that
\TzhWoo
oo oo
Y^zn^2h(k)$(k-n)
n—0 k=n
oo oo
=0
oo
^ znh(k + n)
k=0
oo
n=0
n=0
<Etoi
k=0
A routine computation shows that for each £ G T,
oo
Y, Ch(n + k) = Ck(h(() - Sfc_ihK)).
n=0
Thus, from the estimate
|5n(h, e^)| ^ CII^Hoo log(n + 2),
it follows that
oo
(6.4.3) Halloo ^ C||h||oo Yl 1^)1 lQg(^ + 2) < oo.
k=0
The above argument shows that TV is a continuous operator from A to H°°.
However, if p is a polynomial, then T-rp is also a polynomial. Moreover, polynomials

124
6. MULTIPLIERS AND DIVISORS
are dense in A (Theorem 1.6.5). Thus T-t : A —> A is continuous. Furthermore,
IIMJI = llT-rll
supjUT^IUi/iGball^)}
^ C ]T |0(fc)|log(fc + 2) (by eq.(6.4.3)).
k=0
To prove statement (2), let E be a set of full measure in T for which
0(C) := lim 0K)
exists and
For C e E,
L,
sup
4>{w) - 0(0
w — C
dm(w) : ( e E
}•
0-0(0 , 0(C) _ 0-0(0
+
x(C0(C)^c)-
z-C z-C z-C z-C
Clearly the last term belongs to X and by Proposition 4.1.19,
\\Kfy(06c)\\ = WOlll^cll = 1^(01 < Halloo-
The first term belongs to Hp for all 0 < p < 1 since it is the product of the bounded
function 0 — 0(C) and the Hp function (z — C)_1- Moreover, by hypothesis, it has
integrable boundary values. Thus, by Proposition 1.9.5,
<t> - 0(0
z-C
eH1.
By the Cauchy integral formula (Proposition 2.2.1), this function also belongs to
% and, by the definition of the norm on 3C,
^L*.
10 — 0(C) 1
1 z-C \
<
10-0(01
1 z-C 1
Combining these two facts we conclude that
z-C
for all C € E and
< ||0||oo + V
1^ — CI
By Proposition 6.2.1 and Remark 6.2.5,
□
Remark 6.4.4. The sufficient conditions of Theorem 6.4.1 are not necessary.
In particular, condition (1) says that 0 is continuous. In a moment, we will show
that certain infinite Blaschke products are multipliers and such Blaschke products
can not be continuous. See [228] for some other nicely worked counterexamples.
From Theorem 6.4.1, part (1), one can immediately see that
||Mz»||<C(l + logn).
The techniques developed in this chapter also give us the following lower estimate.

6.4. SOME SUFFICIENT CONDITIONS
125
Proposition 6.4.5. There is a constant c> 0 so that for each n eN,
\\Mzn\\ ^clogn.
Proof. Since
1
1
— z
= 1,
we have
(6.4.6)
1 zU
\\-z\
|^I|M,»|||
1 1
I- z\
We will now obtain a lower bound for
zn |
ll-
z
\MZ
Use the identity
to obtain the estimate
(6.4.7)
1-Z ~ 1-Z ^
k=0
l-z
>
k=0
l-z
n-1
k=0
-1
From eq.(4.1.10) (Landau's theorem), we know that
ln-1
£•
k=0
^ clogn
and the result follows.
□
Remark 6.4.8. Here is another proof of Proposition 6.4.5 shown to us by V.
Peller. For each n E N, let
— eike
qn(e*e):=c Yl ~V = 2lcYl
s'm(k6)
k= — n
k=l
which happens to be the partial sums of the Fourier series of a saw-tooth function
(see eq.(3.2.8)) and so has uniformly bounded sup-norm. The constant c is chosen
so that 11 gn ||oo ^ 1 for all n. Furthermore,
n l.
ZK
and so
Let
(qn) + (z) = cJ2k
k=i
n 1
||(tfn)+||oo > CYl~k ^ Cl°gn
k=l
Pn(Q := CV(C), C^T,

126
6. MULTIPLIERS AND DIVISORS
and notice that pn G A and ||pn||oo ^ 1 f°r aU n- Thus
||T*» : A -+ A\\ = supdlT^/ilU : h G ball(A)}
> \\Tz"Pn\\oo
= ||(CPn)+||oo
= ll(tfn)+||oo
^ clogn.
Thus
\\Mzn\\ = \\T^ : A ^A\\> clogn.
We can also use part (2) of Theorem 6.4.1 to get another estimate that will
become important later.
Proposition 6.4.9. Suppose </>,</)' G H°°. Then (/) G 9Jl(3C) and there is an
absolute constant c > 0 such that for each n eN,
IIM^H^c^ll^lloo+logCneJWc
Proof. From Theorem 6.4.1, part (2), there is a set E C T of full measure
such that
\M4 ^ Woo + sup /
<P(Celt) ~ 0(0
1
dt
2?r'
Let us now estimate the second term in the above expression. Fix n G N and write
the integral as
J-it J\t\<ir/n J\t\>ir/n
>\t\<ir/n J\t\>ir/n
For any £ G E, the first integral is estimated as
L
\t\<ir/n
<f>(Ceu) ~ <A(C)
1
i*^!*""-
where c is an absolute constant. The second integral is estimated as
/
\t\>n/r
</>(Ceu) ~ <A(C)
dt
2n
r dt
\4>U / — (sin(*/2) > t/n)
WL.f di
Jl/n u
I^Hoo log(n),
where, again, c is an absolute constant. Combine these two inequalities to obtain
the result. □

6.5. THE ^-PROPERTY
127
6.5. The jF-property
Suppose that X is a class of analytic functions contained in the Smirnov class
iV+. We say that X has the 3-property if whenever / G X and d is an inner function
which 'divides /', that is, //# G iV+, then f /d G X. From the factorization
theorem for Hp functions (see eq. (1.9.8)), Hp certainly has the ^-property, as do
other well-known classes of analytic functions such as BMOA, the disk algebra, and
the analytic Lipschitz and Besov classes [197]. The ^-property is not universal.
For example, Guraril [86] showed that the analytic Wiener algebra of / G A for
which
oo
£ \f(n)\ <oo
n=0
does not have the ^-property.
A theorem of Hruscev and Vinogradov [105, 227]4 says that both X and 9Jt(3C)
enjoy the ^-property.
Theorem 6.5.1 (Hruscev-Vinogradov). Suppose d is inner.
(1) IffeX and f/d G JV+, then f/d G X and \\f/d\\ ^ ||/||.
(2) If</> G m(X) and </>/# G H°°, then </>/# G SDt(3C) and ||M^-i|| ^ ||M^||.
Proof. We will use Theorem 5.4.5 to prove (1). Recall that for fi G M, the
function
(CW(*)= /-VdMC)
J 1 - (z
is analytic on C\T and (Cfi)(oo) = 0. Suppose that d is inner and G = Kfi/d G N+.
Note from Theorem 2.1.10 that G has Lp boundary values for each 0 < p < 1 and
so G G Hp with
\\G\\p = \\K»\\p = o(j^y p-i-.
On De, the function 1/d is 'inner'5 and so
IIC/V0ll/*p(De) ^ ||C/x||^p(De) = O
Thus C(1/d satisfies the Hp condition in Theorem 5.4.5. Since the radial limits of
d are the same almost everywhere from inside and outside the disk,
j(^f]= #J{Cfi) = d^ m-a.e.
\ v / dm
and so Cji/d satisfies the jump condition in Theorem 5.4.5. Hence Cji/d = Cv for
some unique v € M and so K(i/i) = Cv\3 € 3C.
We now prove the norm estimate
Kfi
■&
< \\K»\\-
See also [228] for some partial results.
For z EDe, fi(z) := l/#(l/z). This function is a pseudocontinuation of $ (the non-tangential
boundary values match those of #|D almost everywhere). Moreover, this function is an analytic
continuation of tf|D to the set C\{l/z : z G D~,limA__ |tf(A)| = 0}.

128
6. MULTIPLIERS AND DIVISORS
By Proposition 4.1.4 we can assume that the representing measure \i has been
chosen so that
(6.5.2) \\K»\\ = H/4
Use the Lebesgue decomposition theorem to decompose the measures \i and v as
(6.5.3) d/i = (j) dm + /is, dv = ip dm + vs
where, by Fatou's jump theorem and the identity
we get
0 = J(C/x), ^ = J(Ci/) = 00.
Hence
(6.5.4) l^lli = Mi.
A result of Hruscev and Vinogradov, which we will prove later in Theorem 7.4.4,
says that for A G M,
lim ym(\K\\>y) = -\\\s\\.
y-^oo 7T
Observing that \Kfi\ = \Ki/\ almost everywhere on T, we apply this Hruscev-
Vinogradov result to see that
(6.5.5) IImJHKII.
Hence,
\\K"\\ ^ H
= |Mli + IM (by eq.(6.5.3))
= ||<A||i + \\fis\\ (by eq.(6.5.4) and eq.(6.5.4))
= |H (by eq.(6.5.3))
= \\Kfi\\ (by eq.(6.5.2)).
For the proof of (2), we recall Proposition 6.1.5 which says that for ip G i7°°, the
co-analytic Toeplitz operator T^ is bounded on H°° if and only if ip is a multiplier of
X. Moreover, the multiplier norm of ip is equal to the operator norm of T^. Suppose
that 0 is a multiplier of % and d is inner and divides 0, that is, (p/d G H°°. Then
Tt# is a well-defined co-analytic Toeplitz operator. Moreover, for any h G i/°°,
Halloo = IKW + lloo
= II^WIU
<\\T^\\\m\oo
= \\T4\\h\u
Thus
ll^ll < ll^ll
and so (p/d is a multiplier with
l|M^-i|| = ||r^||<||r?|| = ||M^||.
Remark 6.5.6.

6.6. MULTIPLIERS AND INNER FUNCTIONS
129
(1) There is a result of Poltoratski (see Corollary 10.5.9) which says that if
Kn/d = Kv, then d has non-tangential boundary values /i-a.e. and dis
can be chosen to be fid/i.
(2) There is an alternate proof of the ^-property for X in [129].
6.6. Multipliers and inner functions
In this section, we discuss the question: when is an inner function a multiplier
of XI This question has a complete answer due to Hruscev and Vinogradov [105]
that we will present here. Recall from eq.(1.9.11) that any inner function (ft can be
factored as (ft = S^B, where SM is the singular inner factor
S„(z) = exp ( - / |±| dM(0
H
with \i G M+, \i _L m, and B is a Blaschke product
B(z) = zm Y[
CLn 1 CLnZ
n=l
whose zeros at z = 0 as well as {an} C ^{0} (repeated according to multiplicity)
satisfy the Blaschke condition
oo
]T(l-|an|) <oo.
n=l
Our first observation is that when trying to determine whether or not an inner
function is a multiplier, we can consider the singular inner factor and Blaschke
factor separately.
Proposition 6.6.1. An inner function (ft = S^B as above is a multiplier ofX
if and only if both S^ and B are multipliers of X.
Proof. If S^ and B are multipliers, then, since the multipliers form an algebra,
(ft = S/j,B is also a multiplier. Conversely, if (ft = S^B is a multiplier, then applying
Theorem 6.5.1 (the ^-property for multipliers), we conclude that both S^ and B
are multipliers. □
Let us set a bit of notation that will make some explanations easier later on.
For a sequence E C ID) \ {0}, such that
£(l-|a|)<oo,
let
B(z) = J]
a a — z
„ a 1 — az
a£E
be the Blaschke product whose zeros (repeated according to multiplicity) are
precisely E.
By Theorem 6.2.6, every multiplier must have non-tangential limits at every
point of the circle. We recall Frostman's theorem (Theorem 1.7.7) to help us
eliminate certain Blaschke products as possible multipliers.

130
6. MULTIPLIERS AND DIVISORS
Theorem 6.6.2 (Frostman). Let B be a Blaschke product with zero set E. A
necessary and sufficient condition that B and all its partial products have non-
tangential limits of modulus one at £ G T is that
aeE ls '
So certainly for a Blaschke product B to be a multiplier of 3C, the quantity
Vl-M
aeE ^ '
must be finite for each £ £ T. It will turn out that the stronger condition
crF(E) := sup > — r < oo,
C€Tfl^IC-a|
is the precise condition for B to be a multiplier.
Theorem 6.6.3 (Hruscev-Vinogradov [105]). An inner function (ft is a
multiplier for X if and only if (ft is a Blaschke product whose zeros E satisfy o~f(E) < oo.
Some remarks about the Frostman condition: At this point, we feel obligated
to give some examples of Frostman sequences. Before doing so, we present the
following theorem of Vasjunin [226] which shows that Frostman sequences must be
slightly better than Blaschke.
Theorem 6.6.4 (Vasjunin). If B is a Blaschke product whose zeros E satisfy
cff(E) < oo, then
£(1"|a|)logri<00-
aeE ' '
Proof.
]T(1 - \a\) log ^J— ^ c ]T(1 - \a\) f -^- dm(C) (Lemma 1.12.3)
aeE ' ' aeE ^T '^ '
•/Ta6£IC a|
v^ 1- |a| f ,
< c sup > t- r / dm
C6T^lC-a|yT
= co-p{E) < oo.
D
Specific examples of Blaschke products for which o~f{E) < oo are somewhat
difficult to come by. Here is an example from [228].
Lemma 6.6.5. Suppose (rn)n^i C (0,1) and (0n)n^i C (0,1) are such that
sup(%tI:neNl<l and V 1=^ < oo.
Then the sequence
E = {rnel6n : n G N}
satisfies o~f(E) < oo.

6.6. MULTIPLIERS AND INNER FUNCTIONS
131
Proof. Clearly
oo 1 _
\e\>\n=l\e rne |
since the points rnel0n only accumulate at £ = 1 and Xm(^ ~ rn) < oo.
We get a uniform bound on the quantity
oo 1
^ |e*'-rne*'«|
for |0| < 1/2 in two steps. When —1/2 < 0 ^ 0, we can use Lemma 1.12.1 to see
that
|e" - rneie"\ = |1 - rne^-^\ > c(0n - 0) > c0n
and so
oo 1 oo 1
(6.6.7) sup V" -— ^-. <. cV n < oo.
The uniform bound for 0 < 0 < 1/2 is a bit more delicate. Let
{¥-"*"}
g:=sup^ -^ :nGN < 1.
The above condition implies that 0n [ 0 and thus, for each 0 < 0 < 1/2, there is an
N = N(0) such that 0 G [0N-i,ON]. For any n> N,
0 — @n ^ @N — @n
^ 0n-l — On
?n-l _ 1
>0n(l-q).
Hence for n > iV, we can use Lemma 1.12.1 as well as the above estimate for 0 — 0n
to get
|e^-rne^| = |l-rne^^)|
= |l-rne^-'»>|
> C(0 - 0n)
^ C0n-
Thus
1 ~ rn / V^ 1 - rn
n>N |6 Vne ! n=l ^n
< OO.

132
6. MULTIPLIERS AND DIVISORS
In a similar way, for n < N — 1,
0n — 0 ^ 0n — On-
^ 0n — #n+l
^ 0n+i \ 2 *
As before, we get
oo
1 — rn.
< OO.
n<N |6 rn6 I n=l ^n
Finally, notice that when n = N — I or N, we get the obvious estimate
^ 1
\e%e -rne%e^\
and so, summing over the cases (n < N — I, n > N, n = N — 1, n = AT), we obtain
oo ^ oo 1
o<0<Jn=i|e r™e I n=i ^
Now use eq.(6.6.6), eq.(6.6.7), and eq.(6.6.8), to conclude that cff(E) < oo. □
Remark 6.6.9.
(1) Adding in the extra hypothesis that
sup 111 _^+1 : n € NJ < 1,
says, via Theorem 1.11.8, that E is also uniformly separated (i.e., 5(E) >
0).
(2) If 0 < m < c < 1, then Ok '•= ch and r^ := 1 — m^ is a specific example
of a sequence satisfying the hypothesis of the above lemma (including the
extra hypothesis mentioned above).
(3) Matheson [134] has recently shown that if B is a Blaschke product whose
zeros Zb satisfy gf(Zb) < oo, then Zb can accumulate only on a nowhere
dense subset of T.
(4) For a given closed nowhere dense subset L C T, an elaboration of the
above construction produces a Blaschke product B whose zeros Zb satisfy
(Jf(Zb) < oo and accumulate precisely on L. Moreover, given e > 0, B
can be constructed so that gf(Zb) < 1 + e [134].
Some partial results: The proof of the main theorem of this section
(Theorem 6.6.3) is quite beautiful but somewhat technical. For the reader to wants an
overview of some of the tools used here, we offer these two partial results from
[228]. We review the following definitions from Chapter 1. Let E be a sequence of
points in D. Define
s(E) := inf <J ^ J^ : a, b G E, a ^ b i .
\a-b\
\"-b\
\l-ab\
*(*)== n $
,_ ab\'
a,beE,a^b ' '

6.6. MULTIPLIERS AND INNER FUNCTIONS
133
M
a
a — z
1 — az
B(z)
If B is a Blaschke product with zeros E, repeated according to multiplicity, and
a G E, we write ba for the individual Blaschke factor
ba(z) =
and let
Oa{Z)
be the Blaschke product with one of its factors divided out. With this notation,
5(E) = inf \Ba(a)\.
Recall that the sequence E is separated if s(E) > 0 and uniformly separated if
6(E) > 0. Define the measure he on D by
He(A):= ]T (l-|a|), A CD,
aeEnA
and note that /ie is a Carleson measure if
7(E):=sup^M<oo,
Q m(I)
where the supremum is over all Carleson boxes Q with base /. Define
C(E):= sup inf{||/||0O:/G//oo,/|£; = 5}.
g:E-^C
\\9\\oo<l
The sequence E is interpolating if given any # G ^°°, there is an / G i7°° such
that /IE1 = g. The constant C(E') is called the constant of interpolation. Note
that E is interpolating if and only if C(E) < oo. Finally, recall Carleson's theorem
(Theorem 1.11.5) which says that the three conditions
(i) 6(E) > 0, (ii) s(E) > 0 and 7(E) < 00, (hi) E is interpolating
are equivalent.
Theorem 6.6.10. A Blaschke product whose zeros E satisfy both o~f(E) < 00
and 6(E) > 0 is a multiplier ofX.6
Theorem 6.6.11. The singular inner function
SSl (z) = exp I ——
is not a multiplier of%.
The proof of the first result depends on this interesting proposition that will
be used again later.
Proposition 6.6.12. For a Blaschke sequence E c O\{0} satisfying 6(E) > 0,
the corresponding Blaschke product B has the representation
(6-6-13) b(z) = -i__y; l-\f 1 1 ,
K) B(°) t^E N Ba(a)l~az
The above series converges uniformly on compact subsets of D. Moreover, if we
also assume that o~f(E) < 00, this representation holds pointwise for every £ G T.
Such E exist by Lemma 6.6.5 and Remark 6.6.9.

134
6. MULTIPLIERS AND DIVISORS
Proof. Let {En : n G N} be an increasing sequence of subsets of E with
card(E'n) = n and such that
oo
E = (J En.
n=l
For each n, let B^ be the Blaschke product with En as its zeros. If a G En, we
let Ba be the Blaschke product with En \ { a } as its zeros. By considering each
B(n)
as a rational function on C and then applying the Cauchy residue theorem, it
is easy to check that B^ has the expansion
(6-6.14) b(-)(z) = -* - y; *-lf 1 l .
Obviously,
£(n)U) " n,\, x -* B(^) " T^TT as n -^ oo
v y B(«)(0) V y S(0)
uniformly on compact subsets of D. Now define
1 — lal2 1 1
f \ III —7-^ -i - if a G ^n,
A{an\z)={ H BW(fl)l-^
0 otherwise.
Since
it follows that
Thus for fixed z G
Since
6(E) = inf |Ba(a)| > 0,
\BinHa)\>\Ba(a)\>5(E)>0.
\A^(z)\^c}~^. VneN.
1 - az\
lim A^jz) = l , l,a| —?——?—, ae£, ze:
»-«, |a| B„(a)l-a«
it follows from the dominated convergence theorem that the right-hand side of
eq.(6.6.14) converges to the right-hand side of eq.(6.6.13) whenever
Vl-H2
^, |l-az|
converges, and in particular for z in any compact subset of D.
If (Jp(E) < oo and ( G T, Theorem 1.7.7 (Frostman's theorem) says that
limB(r<) = B(C).
r—>1
On the other hand, we have already shown that B(rQ has the representation
Using the estimate
^2, z£B, (Kr < 1,
1 — rz

6.6. MULTIPLIERS AND INNER FUNCTIONS
135
it is easy to see that the terms of the series on the right-hand side of eq.(6.6.15) are
dominated by a universal constant times the terms of the convergent series
2-*L |1 - a.
cr
and so, as r —* 1 , the sum on the right-hand side of eq. (6.6.15) converges to the
sum on the right-hand side of eq. (6.6.13), with z replaced by £. □
Proof of Theorem 6.6.10. To show B is a multiplier of 3C, we will use
Proposition 6.2.1 and show that for each £ G T,
B(z) _ ^
and
(6.6.16)
For C e T
(6.6.17)
sup I
B(z) _
c-
1 B(z)
K-A
_B{Q
- t A
Z
: C £^1 < oo.
, B(z)-B(Q
C-z C-z C-z
The first term on the right-hand side of eq.(6.6.17) is
B(0 _ <B(<)
C-z l-(z
K (CB(C)6C) (z)
and so, via eq.(4.1.23),
(6.6.18)
B(C)
C-z
\\CB(QS<
'c|| = 1 for all (GT.
The second term on the right-hand side of eq.(6.6.17) can be written, using
Proposition 6.6.12, as
B(z)-B(C)
az
where
c-z
Aa(C):=
= £>^
a£E
l - lal2
|a|Ba(a)(l-3C)
From our assumption that (Tp{E) < oo and S(E) > 0 it is clear that
sup V|Aa(C)| = C<oc.
^T^E
From eq.(4.1.23),
and so for any £ G T,
\B-B(Q
1 — az
Kl
C-z
a£E
1 — az
a£E
Combining this with eq.(6.6.17) and eq.(6.6.18) we have shown eq.(6.6.16) and our
proof is complete. □

136
6. MULTIPLIERS AND DIVISORS
Proof of Theorem 6.6.11. We do this in three steps. We first claim that if
heX and W = {z:\z- 1/2| < 1/2}, then
(6.6.19) h! G HP(W) for all 0 < p < 1/4.
Here HP(W) is the set of analytic function h on W such that
(6.6.20)
sup
p27T
M ^ + sel<
dO < oo.
Note, as is the case with Hp, that if h G HP(W), then
lim h [ - + sev'
exists for almost every 0 and
f>G+H
dO < oo.
To show eq. (6.6.19) we observe that
ii
O-H2 VzeW
(just write every point z G W as z = 1/2 + se%9 where 0 < s < 1/2) and so if
h = K\i, then
CIMI
(6.6.21)
\h\z)\ ^
<
(l-k|)2" \l-z\v
zeW.
With z = \ + se'*9, we have for 5 > 1/4,
1
SCOS0 + S'
= [\2~S) +2ssin W2)
^ 4s2 sin4 (0/2)
>isin4(0/2).
Use this estimate along with eq.(6.6.21) to see that
sup /
0<s<i JO
h'i^+se*
dOCC
■I
d0
o (sin4(0/2))P
< OO
whenever 0 < p < \. Hence, from eq.(6.6.20), the definition of HP(W), we conclude
that h! G Hp(W) for all 0 < p < \.
We next claim that if h G X, then hi := (1 - z)2h' G Hl(W). Indeed, we
already know that h\ G HP(W) for all 0 < p < 1/4 so it suffices to show, by means
of an adaption of Theorem 1.9.12, that
To this end, note that
/
JdW
™=B
\hi(z)\\dz\ < oo.
w(l-z)2 .
■—r2 d/x(w .
u>z)2

6.6. MULTIPLIERS AND INNER FUNCTIONS
137
Hence
/ M*)||ds|^ 1(1 £-^\dz\)d\»\(w).
Jaw Jt \Jdw F - A )
We now use the fact that
\\-z\2 = \-\z\2 VzedW
to see that the above double integral becomes
Jt \Jdw \w ~ A2 J
%IM
For fixed wET, the function
i-\A2
\w — z\2
is harmonic on W and so, by the mean value theorem for harmonic functions, the
integral
low \w-z\2
is equal to n times the value of this function at the center z = \ of W and so
Jd\
I
Jew
l-N2,, , 3 1
I|2-
Prom here it follows that
/
Jaw
|/ii(z)||dz| < oo.
We now finish the proof by showing that the singular inner function
'z + r
S6l 0) = exp
z-\
is not a multiplier of X. If Ss1 is a multiplier of X, then h := (1 — z) 1S$1 belongs
to X. By the previous claim,
'z + V
(l-z)2K
z - 1 """ \z-l
exp
eH\W).
However, we can use the identity |1 — z\2 = 1 — \z\2 for z G dW', to see that for any
l-UF
z + i
z-l
exp
Z-l
^
exp
|l-*l:
which is not integrable on cW.
D
Some technical lemmas: This next section consists of several technical results
from [105] we will need as we make our way towards the proof of the main theorem
(Theorem 6.6.3). We start off with the following theorem from [141]. Recall from
Chapter 1, that for a Blaschke sequence E, we can form the finite measure he on
Dby
Ve(A):= ]T (1-H).
aEAHE

138 6. MULTIPLIERS AND DIVISORS
Proposition 6.6.22 (McKenna). If \±e is a Carleson measure, then E is a
finite union of interpolating sequences.
Proof. We present the proof from [140]. If Q is any Carleson square, we let
r(Q) = jzeQ:^< l-M <m(J)|
(see Figure 1).
Figure 1. The 'top half T(Q) of a Carleson square Q over the
arc J C T
Let <2i = Q, and for each n 6 N, we partition I into 2n congruent subintervals
J™, and let Q^ be the Carleson square over Jj\ This gives us the following dyadic
decomposition of Q (see Figure 2):
oo 2n
(6.6.23) Q=UU T(<#)'
n=0j-l
where, by carefully arranging the boundary points, we have a disjoint union.
Figure 2. The dyadic decomposition of a Carleson box

6.6. MULTIPLIERS AND INNER FUNCTIONS
139
By dividing up D into four equal sectors, we can assume that our sequence E
is contained in the Carleson square
Q:={re«:i<r<l,-j0^}
and we decompose Q as in eq.(6.6.23). Label the boxes
{T(Q?):neN0,jeN}
with the numbers one, two, three, and four, as in Figure 3, so that no two adjacent
rectangles are given the same number.
Figure 3. The numbered dyadic decomposition
Since he is a Carleson measure, there is an N e N such that
card(T(Q?) n E) ^ N Vn,j.
To see this, let
Mjtn := card(T(gjn) n E)
and note that since
mW) = ^™oo
and
-Lm(/) < (1 - |a|) < ^m(J) Va€T(QJ),

140
6. MULTIPLIERS AND DIVISORS
we see that
Mjtn-±Im(I)^nE(T(Q?)nE)
where
7(E):=sup^M
Q m(7)
is the Carleson measure constant for the Carleson measure he- Thus
Mhn^21{E) Vj,n.
We will now divide the sequence E into AN subsequences
Eir" >^4at,
each of which is interpolating.
For Ei, take one element of E from each of the boxes labeled one. For E2, take
a different element of E from each of these boxes, and continue until En (remember
that each box has at most N elements). Do the same for the boxes labeled two,
three, and four to get Ejv+i? • • • ^E^n- Since, for each j, the points from Ej are
uniformly separated in the hyperbolic metric7, each Ej is an interpolation sequence
(since Ej is separated and /z# is a Carleson measure - see Theorem 1.11.5). □
Proposition 6.6.24. Ifajr(E) < oo, then E is a finite union of interpolating
sequences.
Proof. By Proposition 6.6.22, it suffices to show that he is a Carleson
measure. Let Q be a Carleson square corresponding to the arc JcT, and let (q denote
the midpoint of /. Then
He(Q)
>;
aeQCiE
aeQHE
(1-
1-
ICq
l«l)
l«l
-a\
ICQ - a|
^ crF(E)cm(I),
where c is an absolute constant. The last inequality follows because
\z-Cq\ ^cm(I) \JzeQ.
□
We need a few more lemmas concerning the geometry of E. We let SQ(£)
denote the Stolz region at ( G T with opening angle 2ce, i.e., Sa(() is the convex
hull (with ( removed) of the circle of radius since centered at 0 and the point £.
When a = 7r/4, we write S(() for Sn/4 (().
Let
, v \ z-w
p(z,w) :=
1 — wz
If one thinks about this in the upper-half plane, then a sequence Wj = Xj + iyj is separated
if there is a constant s > 0 so that for fixed k, \wj — w^\ > sy^ for all j ^ k. If one puts the boxes
in this upper-half plane setting, the proof becomes easier to visualize.

6.6. MULTIPLIERS AND INNER FUNCTIONS 141
be the pseudo-hyperbolic distance between the points z and w in D. Following [79],
we define for zo G ID) and r > 0 the pseudo-hyperbolic disk
K(z0,r) := {z : p(z,z0) < ^}
and note that K(zo,r) is also the Euclidean disk
A(c,R) = {z: \z-c\ <R},
where
1-^ , „ l-No
|2
2;0 and R = r-
l-r2|z0|2 l-r2|z0|2'
This next fact depends on an argument with hyperbolic geometry and can be
found in [79, p. 299]. For the sake of completeness, we include a proof here.
Lemma 6.6.25. Suppose a sequence E is separated and E C S(Q for some
( GT. Let Q be a Carleson square with dyadic decomposition
oo 2n
Q = U U T(Q7)
n=0j=l
(see eq. (6.6.23)). Then there is an absolute constant c such that
card E n
for each n.
Proof. First note that S(Q intersects at most c\ of the T(Q™), where c\ is
an absolute constant (see Figure 4), and so it suffices to estimate card(E' C\ T(Q))
for an arbitrary Carleson square Q. Since s(E) > 0, the pseudo-hyperbolic disks
K(a, s(E)/2), a £ E, are disjoint, and their union is contained in the set S of points
z G D whose pseudo-hyperbolic distance to T(Q) is less than 1/2. Each pseudo-
hyperbolic disk K(a, s(E)/2) has Euclidean radius proportional to m(I)s(E), where
/ is the arc upon which Q sits. Hence
area! (J K(a,s(E)/2)
\aeEnT(Q)
is proportional to card(E' Pi T(Q))m(I)2s(E)2. On the other hand, the area of S is
proportional to m(/)2, and so
card(£ nT(Q))s(E)2 ^c
for some absolute constant c. □
Lemma 6.6.26. If a sequence E is separated and E c S(() for some £ G T,
then E is uniformly separated, and 5(E) depends only on s(E).
Proof. To prove that 5(E) > 0, it is enough to use Carleson's theorem
(Theorem 1.11.5) and reduce the problem to showing that \ie is a Carleson measure.
If Q is any Carleson square, we decompose Q using the diadic decomposition in
eq.(6.6.23).
By the geometry of 5(C), there is an absolute constant c\ so that for each n,
at most c\ of the sets T(Q™) intersect S(() (see Figure 4).

142
6. MULTIPLIERS AND DIVISORS
Figure 4. At any high enough 'level', S(Q intersects only two T(Q)'s.
Since E is separated, we see from Lemma 6.6.25 that for any n,
card (En ( \JT(Q])\ ) <
u=1
8{Ef
For a Carleson square Q, it follows from the facts that
and for each n, only ci of the boxes T(Q™) intersect 5(C), and of course E C S(Q,
that
oo 2n
n=0j=l
n=0
C
c m(I)
s(E)2 2n
,m(/).
8{EY
Thus the Carleson constant j(E) satisfies the inequality
7(^)<
*(£)
2 *
Using the inequalities
exp ( -C3^) < *(£) ^ *(£)>
from eq.(1.11.7), and the inequality i(E) ^ $i(s(E)), where $1(5) = c/s2, we
obtain
(6.6.27)
where
$2(s(£)) < 6(E) < s(£)
$2(s) := exp -c'

6.6. MULTIPLIERS AND INNER FUNCTIONS
143
is a non-negative increasing function on [0, oo). Hence S(E) depends only on s(E).
a
Lemma 6.6.28. If aF{E) < oo, then
cardan 5(C)) ^ caF(E)
for each £ G T, where c is an absolute constant.
PROOF. If z G S(C) and the segment from z to £ makes angle 6 with the
diameter through £, then the chord of the circle T passing through z and £ has
length 2cos# (see Figure 5).
Figure 5
On the other hand, the Law of Cosines yields
|z|2 = l + |z-C|2-2|z-C|cos#,
or, by rearranging the terms,
1 UI2
= 2cos0-iC-zl-
IC-*!
Since the circle determining S(Q has radius sin(7r/4), the portion of the chord
through z and £ lying in S(£) has length at most 2 cos 0 — (1 — sin(7r/4)). It follows
that
— r ^ 1 -sin-.
IC-*I 4
If E has iV points in S{C), then
aeEnS(c) ' '
D

144 6. MULTIPLIERS AND DIVISORS
Lemma 6.6.29. If E is a sequence in D such that
N :=supcard(£;n5f(C)) < oo,
CGT
then \±e is a Carleson measure and *y(E) ^ cN.
Proof. For a G P, let Ia consist of all £ e T such that a G S(Q (see Figure
6).
Figure 6. The arc Ia
We note that 1 — \a\ ^ c\m(Ia) for some absolute constant ci8. Next if £ G
Ia fl /&, then both a and 6 belong to 5(C). By the hypothesis of the lemma, no
( G T belongs to more than N of the arcs Ja, a G E. Finally, if Q is a Carleson
square, and a G Q, then m(Ia) ^ m(7), where J = Q~ Pi T. It follows that
MQ)= ]C (l-H)^ci ]T m(/a) < ciiVm(J),
and so \±e is a Carleson measure and ^(E) ^ cN. D
Before getting into these next lemmas, recall from the beginning of this chapter
that if (ft is a multiplier of X (written (ft G 9Jt(3C)), then the multiplication operator
M^ : X —> 3C, M^f = 0/ is bounded on X (Proposition 6.1.1) and its operator
norm
SUp{||0/|| : ll/H < 1}
is denoted by ||M^||. By Proposition 6.1.5, this norm is equal to the norm of the
co-analytic Toeplitz operator T-t : A —> A. Also recall from Proposition 6.1.3 that
if (ft G jm(OC), then (ft G H°° and H^ < ||Afy||.
8Indeed, tan0a x ^^J* and so since tan(0a) -+ tan(7r/8) as \a\ —> 1, the result follows.

6.6. MULTIPLIERS AND INNER FUNCTIONS
145
Lemma 6.6.30. Let E be a sequence in D, and let xa be a complex number for
each a G E. Suppose that
(6.6.31) ]T|£a|(l-|a|)<oc
and that the function
aeE
., . ^ 1 - \a\2
aeE
belongs to H°°.
(i) w
l-|a'2
C6T„tl |1-oC|
El II I L ~ \a\
\xa\\a\ < oo
aeE
then (/) e 2rt(3C) and
l-\a\
2^lx«llai
^oTe
(6.6.32) ||M,|| < Halloo + sup ]T kalH^L.
(2) If (/) £ 9Jt(3C) and E is interpolating, with interpolating constant C(E)
from eq. (1.11.4), then
,1-lal2
xa\m
aeE
Thus if E is an interpolating sequence, we have
l-|a|2
Xa\\Q>\
aeE
(6.6.33) sup Y, l*„||a|i-^L < 2C(£)||M^|| < oo.
€ 3Jl(0C) & sup V l^llaL1 ^ < 00.
Cextt |l-aC|
Proof. We begin the proof with a few preliminary comments. Since <j> £
H°°, the Toeplitz operator TV is well-defined operator from H°° to H2 and so the
corresponding Hankel operator
Hjf := (I - P+)$f)
is a well-defined operator from H°° to i72. Here we use P+ for the Riesz projection
operator. With
^ l-|q|2
aeE
we first claim that for / G H°°,
(6.6.34) (#_/)(^ = ^a^/(a)lI^L, ^ep.

146
6. MULTIPLIERS AND DIVISORS
To see this, observe that for z G D,
^) = ^xa(l-|a|2)I-^
aeE
oo
= ^^(1-^)^5^
aG£ fc=0
= Xyfe*»(1-i«ivY
fc=0 \aG£ /
Since we are assuming that (ft G #°°, we know that the almost everywhere denned
boundary function for (ft belongs to L°° and has a Fourier expansion equal to
oo
k=0
where
(6.6.35) 0(fc) = ]T xa(l - |a|2)a*, fc G N0.
aG£
For each n G No, we have
-k—n
k=0
and so
(/-p+)(C»~]T £(* + ")<
Z=l
Thus for each 2GD and n G N0,
1 = 1
= f>' ( E^1 " lal2)a'+" ) (^ eq.(6.6.35))
oo
aG£ Z = l
= ^^(l-|a|>"r^
aeE
Thus the formula in eq.(6.6.34) is valid whenever / is a monomial, and hence by
linearity, any analytic polynomial. To show that the formula is valid for a general
/ G H°°, approximate / in the weak-* topology of H°° with its Cesaro polynomials
fn = &nf (i-e-> pointwise on D with uniformly bounded sup-norms). Argue that
fn —> f weakly in H2 and so, since H-t : H2 —» i/g is continuous, H-rfn —» i^r/
weakly in i/,2 and hence pointwise in D. The dominated convergence theorem says
that for each z G D,
12 — .1-Jap
az
lim V axafn(a)z~ — = V axaf{a)z—~
n-^oo ^—' 1 — az z—' 1
aeE aeE

6.6. MULTIPLIERS AND INNER FUNCTIONS
147
and thus the formula in eq.(6.6.34) is valid for all / G H°°.
To prove (1), use the assumption that
l-lal2
El II I l ~ \a\
\xa\\a\v —i < oo,
aeE
along with the inequality
1 — ra(
^2
1-aC
and the dominated convergence theorem, to see that for each £ G T
(Htf)(Q = lim (/%/)«)
= J2 axaf(a)C-
Thus, for any / eH°°,
r->l-
lim V ax-af{a)rl\^-H: (by eq.(6.6.34))
1 - lal2
aeE - <
ll^/lloc < ll/lloosup^ \a\\xa\]—^ < oo,
^joTe l1-^!
OO
and so H-$ : H°° -> #0°° and its operator norm ||iJ^|| := \\H^ : H°° -> H°
satisfies
||^Ksup^H|xji^.
The estimate ||IV|| ^ ||0||oo + ||#dl> along with Proposition 6.1.5 shows that (ft G
SDT(3C). The identity ||M^|| = ||7V|| from Proposition 6.1.5 implies the estimate in
eq.(6.6.32).
To prove part (2), we first observe, that for fixed z G D, we can choose e =
{ea : a G E} with |ea| = 1 for all a and such that
(6.6.36) £ |*0||a||*|i^ = Y^x-aa^z1-^^.
a£E ' ' a£E
Since E is interpolating and e G ball(^°° (£?)), there is a function fe G H°° such that
fe(a) = ea for each aeE, and ||/e||oo ^ C(E), where C(E) is the interpolating
constant from eq.(1.11.4). But for such an /e, we know from eq.(6.6.34) that
H*fe(z) = ]P x^aeaz l_°a-.
aeE
However, we are assuming that (ft G 9Jt(3C), and hence H-r : H°° —» i7°° is bounded.
Hence
ll^/£||oo<||^HI|/e||0o<||^||C(£?).
Thus we have
(6.6.37) Y, ^at«~zT=^i = H*f^ < II^^Hoo ^ \\Hj\\C(E)
aeE

148 6. MULTIPLIERS AND DIVISORS
and so for each £ G T,
^ |sa||0|(l-|a|*) < Um y. K||a|r(l-|a|2)
£?E |!-<l "r^T-^ |l-arC|
< ||i^||C(£) (by eq.(6.6.36) and eq.(6.6.37)).
The estimate in eq.(6.6.33) now follows from the estimates
\\Hf\\ < Moo + II^H = Halloo + ||M,|| and ||^>||oo ^ ||M,||.
D
Lemma 6.6.38. Let 0 < e < 1/2, and suppose that B is a finite Blaschke
product with n zeros, all lying in the disk {z £ D: \z\ < e}. Then there is an
absolute constant c such that
i^ i ^_1 + Wmb\
1 < c ne + ■
logn
Proof. We write our finite Blaschke product B as
B{z) = fxM^rz
M a*. 1 -
fc=i afc l ~ akZ
Prom Proposition 6.4.5 and the triangle inequality,
(6.6.39) clogn ^ \\Mzn\\ ^ \\MB{1)zn_B\\ + \\MB\\.
Now use Proposition 6.4.9 to get,
(6.6.40) ||MB(1)2n_B|| < c (UnB{l)zn-1 - B'U + \og{en)\\B{l)zn - B\\c
To estimate the first term on the right, note that
(nS(l)2n-1 - B')\T = nS(l)Cn_1 - ^B
B

6.6. MULTIPLIERS AND INNER FUNCTIONS
149
Hence
\nB{l)zn-1 - B'lU ^ n\\B(l)zn - B||oc + £)
fc=i
^n\\B(l)zn-B\\oc + Y^
fc=i
l-\ak\2
\ak-(\2
l-\ak?
(1~M)2
rllRm,B nil , ^2|gfc|-2|afc|2
= n\\B(l)z -S|U + ^ (i_K|)2
^nllBa^-BllooH-^-^
< n||£(l)zn - BWoo + 8ne (since 1 - e > 1/2).
If bk is the Blaschke factor corresponding to the zero a^, we have B = bi — -bk and
so by the triangle inequality,
fc=i
||B(l)Cn-B||00<X)||CMl)-ftfclloo.
Now for each /c = 1, • • • , n,
|CWi)-WOI
a/c nlf _ ak~t>
1 -ak 1 - a/cC
l«/c|2(C - 0 + flfc - Ok + aifeC - a/cC
(1 -afc)(l -a/cC)
^
6e
(1-6)2
<24e.
Hence
115(1)^-51100 <24ne.
Combine the above inequalities and put them in eq.(6.6.40) (and re-adjust the
universal constant c) to get
\\MB{i)z"-b\\ ^ c(e +era logn).
Now bring in eq. (6.6.39) to see that
c\ logn ^ c2(e + enlogn) + ||MS||
and so, again re-adjusting the universal constant, we get
, e+\\MB\\\ ^ ( , 1 + ||MB|^
1 ^ c ( en H ^ c [en H .
logn ) \ logn )
D
Lemma 6.6.41. Let a G D and /e£
7(2) :=
// 0 G Wl(X), taen 0 o 7 g JWl(3C) and
1
a — 2;
1 — az
(6.6.42)
M^K||M^07K2||M^||.

150
6. MULTIPLIERS AND DIVISORS
Proof. Notice that 7 is an automorphism of D and so from Lemma 5.6.1 both
#07 and #o7~1 belong to % whenever g G X (i.e., the composition operator is well
denned on X). So let / G X and apply the previous line to see that / o 7_1 G X.
But since (ft is a multiplier, then <ft(f 07"1) G X. Compose with 7 to conclude that
(0 ° 7)/ £ K- Thus 0 o 7 is a multiplier.
We now prove the string of estimates in eq. (6.6.42). For /iGi,
W.) = /»*«l
and so, from Proposition 6.1.5 and the definition of the operator norm,
IIMJI = llT-rll
sup
sup
sup
4>(0H0
f\f4>(0H0
dm(C)
: /i G ball(A)
dm(C)
dm(C)
|z| = l,he ball(A) f
|z| = l,/ieball(.A)l.
Observe that by a change of variables,
rw^-i
(^o7)(c)7'(0(/i°7)(C)
1-7(0*
and, via a computation with partial fractions, that
V(0 z z
dm(C)
where
Notice that 1/a G .
1-7(C)2: l/a-C w-C
1 + a 2:
it = — .
a + z
3 and it G T (since z G T) and so the functions
2 ~z
O
and C
l/a-C " " u-£
are Cauchy transforms which are at most of unit norm (see eq.(4.1.23)). Thus
~z ~z
(6.6.43)
l/a-C u-C
^2.
From our earlier argument that (ft o 7 is a multiplier, we can use eq.(6.6.43) to
conclude that for each \z\ = 1,
(6.6.44)
(0o7)(C)7,(C)
1-7(0*
By the Cauchy dual pairing
< 2IIM,
cf)0^y I
Urn [(Kn)(rQg(Qdm(0, » G M, g e A,

6.6. MULTIPLIERS AND INNER FUNCTIONS
151
between A and X, we see that for all \z\ = 1 and h G ball(.A),
I/'
<A(C)MO
f <KQ>
dm(C)
<
(0°7)(C)7,(C)(^°7)(C)
l-7(0*
(0o7)(c)7'(C)
dm(C)
/lo7||c
1-7(0^
<2||AVJ (by eq.(6.6.44)).
Thus, again using the definition of the operator norm of ||XV
\\Tt\\ = HM^H, it follows that
and the equality
\\W*
< IIM*
°7 I
Finally, to prove the inequality
\\Mt
0O7|
^2||M*
we recall the argument at the beginning of the proof that 0 o 7 is a multiplier and
apply the above estimates to get
\\\M,
0°7 I
< IIM,
((^07)07
\M*
n
In a moment, we will need the following technical lemma relating pseudo-
hyperbolic disks and Carleson boxes. See [79, p. 299] for details.
Lemma 6.6.45. For a given e > 0, there is an integer M > 0 such that for any
Carleson box Q, the box T(Q) can be covered by at most M pseudo-hyperbolic disks
of radius e.
Lemma 6.6.46. Let B be a Blaschke product such that B G 9Jt(3C). If c is the
universal constant from Lemma 6.6.38, let N be an integer such that
and let
Then for any a G
ofB.
e :=
iV^exp(2C(l + 2||Ms||))
1 l ■exp(-2c(l + 2||MB|l
AcN Ac
the pseudo-hyperbolic disk K(a, e) contains less than N zeros
Proof. The automorphism
7(*) "
1 — az
maps K(a, e) onto the Euclidean disk A(0, e). Thus B has precisely as many zeros
in K(a, e) as B o 7 has in A(0, e).
Suppose £07 has at least N zeros in A(0, e). We now derive a contradiction as
follows: let B* be a sub-product of Boj with precisely N zeros in A(0, e). By the jF-
property for multipliers (Theorem 6.5.1), ||Mb*|| ^ ||A^T^o^H. Apply Lemma 6.6.41
to see that ||Mb07|| ^ 2||Mb||. Combining these two estimates yields
\\MB*\\^2\\MB\\.

152
6. MULTIPLIERS AND DIVISORS
Notice that N was chosen so that
1 + 2||MB||
C logiV
Now apply Lemma 6.6.38 to B* to see that
1 < c (Ne +
Hence
'4
log N J log N
- <: cNe = cN
2
1 1
4ciV ~ 4
< cNe +
1
2
which is a contradiction. Thus B has less than N zeros in K(a,e). D
Combine Lemma 6.6.45 and Lemma 6.6.46 to obtain the following important
corollary.
Corollary 6.6.47. If a Blaschke product B is a multiplier, then there is a
positive integer M such that for any Carleson box Q, the box T(Q) contains at
most M zeros of B.
Lemma 6.6.48. // a Blaschke product B is a multiplier, then
B = jE?i • • • Bn,
where for each j G {1, • • • ,n], Bj is a Blaschke product and the corresponding zero
sequence Zb is separated.
PROOF. We first note that if 0 < e < 1 is given, Corollary 6.6.47 produces a
positive integer M such that T(Q) can be covered by at most M pseudo-hyperbolic
disks of radius e. If each such disk contains at most TV zeros of jB, then each T(Q)
contains at most MN zeros of B. By partitioning the zero sequence into at most
MN subsequences, we may assume that each T(Q) contains at most one zero of B.
Now use the decomposition with the dyadic decomposition of Q (with the boxes
labeled 'one', 'two', 'three', 'four') in Proposition 6.6.22 to complete the proof. □
This next technical lemma is interesting in its own right. Let 03 denote the
collection of Blaschke products. An old theorem of Caratheodory [38] says that 03
is a weak-* dense subset of ball(i7°°). Our needed technical lemma is the following.
Lemma 6.6.49 (Tumarkin [222]). For each c>0, the set
03c := {B e 03 : aF(ZB) < c}
is a weak-* closed subset of H°°.
The proof of this lemma requires a few facts. The first is an application of
Jensen's formula.
Lemma 6.6.50. If B is a Blaschke product and n(p) is the number of non-zero
zeros of B in the disk {\z\ < p}, we have the formula
(6.6.51) /log|£(pC)|dm(C) = Alogp- / ^ dr,
JT J p r
where A is the order of the zero of B at the origin.

6.6. MULTIPLIERS AND INNER FUNCTIONS
153
Proof. For fixed 0 < p < 1, we can write
Bi(*)=(f) B(z).
Then B and B\ have the same modulus on the circle {\z\ = p} and -Bi(O) ^ 0.
Jensen's formula (see [5, p. 208]) applied to B\ yields
(6.6.52) log 1^(0)1 = - V log (/-) + /log \B(pQ\ dm(C).
However, from the definition of B\,
oo
£i(o) = paIIM
3 = 1
and so
oo
log|Bi(0)| = Alogp + ][>g|aJ-|.
Combine this last identity with eq.(6.6.52) to get
(6.6.53) Alogp + JTlogla^- V log (-^) + /log |B(pQ\ dm(().
With the observation that
]T log(m) =n(p)logp- ]C losl%l'
eq.(6.6.53) becomes
(6.6.54) Alogp + n(p)logp+ ]T logla^l = f log\B(pQ\dm(Q.
\aj\>p Jj
Now if pi ^ P2 ^ • • • denote the successive moduli of the zeros of B which are
greater than p, we observe that
n(pj) =n(p)+j,
and so
JP r JP r friJP} r
oo
n(p)log^+^n(ft)log^±i
P U Pi
OO
P U Pi
OO
= -n(p)logp + ]TlogpJ.
i=i
Combine this identity with eq.(6.6.54) to obtain the result. Note that the Blaschke
condition guarantees the convergence of all the infinite series. □
We also need the following characterization of Blaschke products [79, p. 56].

154
6. MULTIPLIERS AND DIVISORS
Proposition 6.6.55. A function U e H°° with ||£/||oo ^ I is a Blaschke product
if and only if
(6.6.56) lim / log \U(r()\ dra(C) = 0.
rtl J J
Finally, to prove Lemma 6.6.49 we will need another technical result of Tu-
markin [221]. Let (a^)j^i be the zeros of the Blaschke product Bk and let rik(p)
be the number of non-zero zeros of Bk in {\z\ < p).
Lemma 6.6.57. Let (Bk) be a sequence of Blaschke products which converges
uniformly on compact sets of'D to a function B. Then B is a Blaschke product if
and only if the following two conditions hold: (1) for each 0 < r < 1, the number
of zeros of Bk in {\z\ < r} is uniformly bounded in k; (2) for every e > 0 there is
an 0 < R < 1 such that
E a-i4i)<e
\a*\>R
for each k.
Proof. First suppose that B is a Blaschke product. Since for each 0 < r < 1,
Bk —> B uniformly on the circle {\z\ = r}, the principle of the argument (assuming
B has no zeros on {\z\ = r}) shows that for large enough /c, B and Bk have the
same number of zeros in the disk {\z\ < r}. Condition (1) now follows. Since B
is a Blaschke product it follows from eq.(6.6.56) that for every e > 0 there is an
0 < R < 1, such that
/
log|JB(JRC)|dm(C)>-e.
/T
Since Bk —> B uniformly on the circle {\z\ = R}, there is a positive integer K such
that
/log|jB,(^C)|dm(C)>-26
Jt
for k ^ K. From eq.(6.6.51)
Jr r
for k ^ K, since A& log r is negative. Finally,
1,1 2*£) dr = f1 nk{r) ~ nkiR) *- ' ^ nkiR)
ir r Jr
r^irldr=rnk(r)-nk(R)dr+r
Jr r Jr r JR
> f1 nk{r) -nk(
^ Jr r
dr
\a^\>R ' ^
Condition (2) now follows because
1 3
1 - x < log - < -(1 - x), 0 < x < 1.
x 2
Now suppose that conditions (1) and (2) hold. Because of (1), B is not the
zero function. Let e > 0 and let R be given by condition (2). Notice how there is

6.6. MULTIPLIERS AND INNER FUNCTIONS
155
an Ri with R < R\ < 1 and such that
nk(R)log—<e Vfe,
Hi
since nk(R) is bounded in k by condition (1). Thus
<
1 nk(r) -nk(R)
Ri r
1 nk(r) -nk(R)
dr +
f
JR.!
nk(R)
dr
dr + nk(R) log —-
5Z log-^+nfc(^)log
|a*|>* |a^
#1
< 2e + e
= 3e
for all fe. Choosing #2 > #i so that
A/clog— < e Vfe,
^2
it follows from eq.(6.6.51) that
/
Jt
log|Bfc(pC)|dm(C)
<4e
for R2 < p < 1. Since Bk ^ B uniformly on each circle {|z| = p}, we have
/
log|JB(pC)|dm(C)
<4e
for R2 < p < 1. Hence
lim /"log|B(rO|dm(0 = 0,
and so B is a Blaschke product. This ends the proof of the Tumarkin result. □
Proof of Lemma 6.6.49. To show that 03 c is weak-* closed, it suffices to
show that 03 c is weak-* sequentially closed9. Let (Bk)k^>i be a sequence of Blaschke
products which converges uniformly on compact sets to the function jB, and such
that
<jF(ZBk)^<j Vfe.
This sequence enjoys condition (1) of Tumarkin's result (Lemma 6.6.57). Indeed,
for any Carleson box Q and a^ G T(Q) we have the estimate
^ < 1 - 1^1 < m(I),
where / is the base of Q. From here, it follows from the proof of Proposition 6.6.24
that if N is the number of zeros of B in T(Q) then
V J ak3eT{Q)
Finally, any disk {\z\ < r} can be covered by a finite number of T(Q)'s.
This is a consequence of the fact that the weak-* topology on the unit ball of H°° is
metrizable.

156
6. MULTIPLIERS AND DIVISORS
Now Vasjunin's theorem (Theorem 6.6.4) gives us the estimate
oo 1
E(1H^i)iogr-—^
3 = 1 ' 3
where c is an absolute constant. The function
log-
1 — r
is increasing for 0 < r < 1 and so
E(1-^i)<^V E a -i4i) log ^
C<7
Since the last expression tends to zero as R f 1, condition (2) of Tumarkin's result
follows. Thus we know that the weak-* limit function of the sequence (Bk)k^i is a
Blaschke product. We now need to show that
ctf(Zb) ^ a.
Define the measures
Ik •--
E (x - M)*«
and
a£ZB
associated with the Blaschke products Bk and B respectively.
Notice from the inequality
that
d7/c
-j
^ 2sup / — d-fk(z)
CGT JB I1 _ Sz\
= ^F(ZBk)
^ 2cr.
Hence the sequence (7/c)/c^i forms a bounded sequence in the space of measures on
the closed unit disk D~~. By the Banach-Alaoglu theorem, there is a weak-* limit
point 7*. So, passing to a subsequence, we know that
7/c ->7*
weak-* as k —-> oo. We now show that
7* = 7s-
From condition (2) of Lemma 6.6.57 we know that 7*|T = 0. By condition (1) of
Lemma 6.6.57 we know that for any 0 < r < 1, the number of zeros of Bk inside
{\z\ < r} is bounded in k. Using this fact, along with two applications of Hurwitz's
theorem ([132, Vol II, p. 49]), and the fact that 7^ is discrete for every k (with

6.6. MULTIPLIERS AND INNER FUNCTIONS
157
its atoms at the zeros of Bk), one can show that (assuming that B has no zeros on
{\z\=r})
(6.6.58) 7/c|{kl ^ r} ~> 7b\{\z\ ^ r} weak-* as k —> oo.
It follows from the facts that 7fc|T = 7s|T = 7*|T = 0 and condition (2) of
Lemma 6.6.57 that 7^ —> 75 weak-*.
Finally, for each £ G T, choose an r such that B has no zeros on {\z\ = r}.
Then
1^ T7 1 = / T7 \d7B{z)
a£ZB
= lim / -l—d^kiz) (by eq.(6.6.58))
< a (since aF(ZBk) ^ a).
It follows now that ctf(Zb) ^ cr. D
Remark 6.6.59. There is an alternative proof of Lemma 6.6.49 in [105] using
a Green function argument.
The proof of the main theorem: After working through those technical details,
the reader is finally rewarded with the proof. Recall from Proposition 6.6.1 that the
inner function d = Bs^ is a multiplier if and only if both B and s^ are multipliers.
We now prove that B is a multiplier if and only if cff{Zb) < 00 and that s^ is
never a multiplier (unless /i = 0).
Claim 1: If ctf(Zb) < °°> then B is a multiplier.
Proof. By Proposition 6.6.24,
B = B\ - ■ ■ Bn,
where each Bj is a Blaschke product and Zbj is interpolating (and hence, by
Theorem 1.11.5, uniformly separated and thus Carleson). Notice also that (Jf(Zbj) ^
gf{Zb) < 00. Combine Proposition 6.6.12 along with Lemma 6.6.30 (especially
eq.(6.6.32)) to the Blaschke product Bj to conclude that Bj is a multiplier. But
since the multipliers form an algebra, B is a multiplier. □
Claim 2: If B is a multiplier, then gf{Zb) < 00.
Proof. By Lemma 6.6.48, we can write
B = B\ - - ■ Bn,
where each Bj is a Blaschke product and Z#. is separated. For a Blaschke sequence
AcO, let Ba be the Blaschke product with zeros A. Fix j and let E = Zj. We
will now show that (Jf{E) < 00.
For each C £ T, let
Ec :=EnS(().
We can apply Lemma 6.6.26 to see that E^ is uniformly separated and, more
importantly, from eq.(6.6.27),
S(EC) > $2(s(Ec))i

158
6. MULTIPLIERS AND DIVISORS
where <£2 is a non-negative increasing function on [0,oo). Moreover, since
s(E) < s(Ec),
we have
5(EC) > $2(s(£<)) > *2(s(£)).
Using the estimate
from eq. (1.11.6), we see that
C(EC) < $3(s(S)),
where
$3(^) = Ctt-t-t- ( 1 + log ——- ]
and is decreasing.
Use Proposition 6.6.12 along with Lemma 6.6.30 (especially eq.(6.6.33)) to get
*f(Ec)^cC(Ec)\\MBeJ
^ c<&3(s(E))\\MBe\\ (by Theorem 6.5.1).
Lemma 6.6.28 shows that
card(£c) ^ c$3(s(E))\\MBe\\ V( G T.
Hence by Lemma 6.6.29,
7(£) < csupcard(£c) < c$3(s(E))\\MBe\\.
C6T
Hence \±e is a Carleson measure. By Carleson's interpolating theorem
(Theorem 1.11.5) E is uniformly separated and so C(E) < 00. Finally, apply
Proposition 6.6.12 along with Lemma 6.6.30 (especially eq.(6.6.33)) to see that
aF(E)^cC(E)\\MBE\\< 00.
Finally, with B = B\ • • • Bn, we have
n
(tf(zb) < ]rvF(zSj)
3=1
n
^cY,C{ZB3)\\MBj\\
^cnmax{C(ZB,)||MB,||:j = l,... ,n}
< 00
which proves Claim 2. D
Claim 3. There is a non-negative increasing function $ on [0, 00) so that whenever
B is a multiplier,
of{Zb)^*(\\Mb\\).

6.6. MULTIPLIERS AND INNER FUNCTIONS
159
Proof. In the proof of Lemma 6.6.46 one can set
€ = exp(-c||AfB||)
and show that every pseudo-hyperbolic disk K(a,e) contains at most
exp(c'||MB||)
zeros (counting multiplicity). See [105, Lemma 4.3] for another proof of this fact.
Important note: As is the usual tradition in analysis, c, c', c", etc., are universal
positive constants that may change from line to line.
From here, follow the proof of Lemma 6.6.48 to show that when factoring B as
B = B\ — - Bn
into Blaschke products with separated zeros, the number of factors n is bounded
above by a function of the form
cexp(c/||Ms||), c,c > 0.
In that same proof, one can also show that the separation constant s(Ej), where
Ej are the zeros of Bj, for one of the factors Bj, satisfies
5(^-)^cexp(-c,||MSi||).
However by the ^-property for multipliers, ||Mb || ^ \\Mb\\ and so
5(^)^cexp(-c,||Ms||).
Now follow the proof of Claim 2 above to obtain the inequality
But since $3 is a decreasing function,
7(^) ^ c<f>3(c'exp(-c"\\MB\\))\\MB\\.
But now observe that the function
$4(x) := c$3(c/exp(—c"x))x
is an increasing function of x. Hence
7(^-) < *4(I|MB||).
Using the standard estimates (see eq. (1.11.6) and eq.(1.11.7))
and
5{E3) > exp [-CS{E.)2
along with our estimates
s(Ej) ^cexp(-c/||MSi|
and
7(^)<*4(||Mb||),

160
6. MULTIPLIERS AND DIVISORS
we see that
C(Ej) <: cec'^(llMBll)exP(-c"llMBll)$4(||MB||)
^cec'*4(||A/B||)$4(||MB||)
= *5(I|MB||),
where $5 is increasing.
The last line of the proof of Claim 2 says that
aF(E3) ^ cC(E3)\\MB\\
<c*5(||Mb||)||Mb||
= <J>6(||MB||),
where <£6 is increasing.
Finally,
n
aF(ZB)^J2a^)
<n$6(||MB||)
<cexp(c"||MB||)$6(||MB||)
= *7(I|MB||),
where $7 is increasing. D
Claim 4: 5 = 5^ is not a multiplier unless \i = 0.
Proof. Suppose that S is a non-constant multiplier. By a well-known theorem
of Frostman [79, p. 79]10, there is a sequence (cen)n^i C ID) with an —> 0 such that
D S - an
tin — ' =r^
1 - anb
is a Blaschke product. We can assume that |an|||Ms|| < 1/2 for all n. Observe
that
S-Bn = S-^-
1 - anb
_ S - a^S2 - S + an
1 - a^S
00
= (an-o^S2)Y,°^kSk.
Since |an|||S|| < 1/2, the multiplier norm of
00
k=0
is bounded by 2. Also observe by Proposition 6.1.3 that
I|MS|| £ ||5|U = 1
The theorem here is the following: suppose / is a non-constant inner function. Then for
all ujGD, except possibly for a set of logarithmic capacity zero, the function fw(z) = _
l-wf(z)
is a Blaschke product.

6.6. MULTIPLIERS AND INNER FUNCTIONS
161
and so the multiplier norm of an — anS2 is bounded above by 2|an|||Ms||2. Thus,
again using Proposition 6.1.3,
||S - Bnlloo ^ ||MS_BJK 4|an|||Ms||2.
It follows from here that \\S — -Bn||oo —-> 0 as n —-> oo and that for every n G N,
||MBJ| < ||MBn_s|| + ||MS|| < 4|on|||Ms|| + ||MS|| ^ 3||MS||.
But since each Bn is a multiplier, Claim 3 says that
<TF{ZBn)^Q(\\MBn\\)^c.
Now observe that Bn —> S weak-* and Bn G 03 c. Hence, by Lemma 6.6.49, S G 03c
which is clearly a contradiction. □

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CHAPTER 7
The distribution function for Cauchy transforms
7.1. The Hilbert transform of a measure
Kolmogorov originally proved his weak-type inequality (Theorem 3.4.1)
m(\f\>y)^% feL\
y
in order to show that the conjugate function / belongs to Lp for all 0 < p < 1. His
original proof in [116] is fairly complicated and uses the idea that the powers znk, for
an appropriate lacunary sequence {rik)k^i-> are nearly stochastically independent.
In this chapter, we examine this distribution function
y^m(\f\ > y)
more closely, as well as the corresponding one for the Cauchy transform, paying
special attention as to how one can obtain information about the singular part of
the measure just by knowing the behavior of m(\Kii\ > y) as y —-> oo.
To make some of our proofs easier, and since it is interesting in its own right,
we first look at the corresponding results for the Hilbert transform. Let M(R)
denote the finite, complex Borel measures on R and M+(R) the positive ones. Let
mi denote Lebesgue measure onM1. For e > 0 and \i G M(R), let
(Xe(i){x):= [ -^dfi(s)
J\X-S\>€ X S
be the truncated Hilbert transform, which is defined for every xEl, and let
CKfi)(x) := lim(JCc/x)(x)
e—>0
denote the Hilbert transform2, whenever this limit exists. The existence of this
limit for mi -almost every x G R, as well as the basic distributional inequality, is
well known [37, 128]3.
Theorem 7.1.1. For /i G M(R), the Hilbert transform (3<fi)(x) exists for m\-
almost every x G R. Moreover, there is a C > 0 such that
(7.1.2) m1(\Xp\>y)^C^ Vy > 0.
Using this notation, mi 0 M(M) (since it is not finite).
In many books, the Hilbert transform has an extra factor of 1/rc.
The Loomis proof [128] is for one variable. The well-known and often cited paper of Calderon
and Zygmund [37] looks at the Hilbert transform in higher dimensions.
163

164
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
As is convention here, when d/i = fdmi, we will use 3if in place of the
more cumbersome Jifdmi. This next result of Riesz is the Hilbert transform
version of Theorem 3.2.1. The proof is quite standard and found in sources such as
[79, 118, 207, 234].
Theorem 7.1.3 (Riesz). If I < p < oo and f £ LP(R), then *Kf e LP(R) with
W\\p^cp\\f\\p.
It is also worth pointing out, as it will be used later, that if / is infinitely
different iable with compact support, then 3if is bounded and infinitely different iable.
As was the case for the Cauchy transform (see Proposition 3.4.11) and with nearly
the same proof, if \i <C mi, then the basic distributional inequality in eq.(7.1.2) can
be improved to
m1{\'Kn\>y) = o{lly).
It is well known that if \i is compactly supported, say in the interval [0,27r],
then for mi-almost every x, the Hilbert transform (!K/i)(x) is equal to
The first integral is the conjugate function (Q/jJ)(etx) (see Theorem 1.8.10) while the
second integral is the integral convolution of a measure with the bounded function
2/x — cot(x/2), and hence is bounded. This means, as observed by Titchmarsh
[215], that the functions 3i/i and Qfi have similar types of singularities and mapping
properties.
7.2. Boole's theorem and its generalizations
For a non-negative singular measure /ionl, the distribution function
y h-> mi(JC/x > y)
of its Hilbert transform !K/i can be computed explicitly [104].
Theorem 7.2.1 (Hruscev and Vinogradov). If fi E M+(R) and is singular,
then for every y > 0,
mii'Kn > y) = ^ and mi(JC/x < -y) = —
y y
As a consequence,
mi(|M/x| >2/) = 2^.
To prove this, we begin with Boole's lemma [30] which was discovered in 1857
and computes the distribution function for
Ci_
T —
i=l
Ci
x — ai
the Hilbert transform of a finite linear combination of point masses. Boole's proof
was subsequently rediscovered by others, for example, [128].

7.2. BOOLE'S THEOREM AND ITS GENERALIZATIONS
Lemma 7.2.2 (Boole's Lemma). Let
g{x):=J2-*-,
^—' X - CLi
i=l
where q > 0 for i = 1,..., n, and a\ < a,2 < • ■ • < an. Then
1 n i n
11(9 > v) = ~YlCi and mi(9 <-y)= ~Y2Ci
165
mi
* £1 »i=i
/or ever?/ ?/ > 0.
Proof. Consider the graph of g(x) and assume that y > 0 is given (see Figure
Figure 1. Graph of the function g(x)
It is easy to see that g(x) is decreasing on each of the open intervals
(-00, ai), (ai, a2),..., (an, 00).
It follows that the equation g(x) — y has precisely n solutions si,...,sn, with
a^ < Si < az+i for z = 1,..., n — 1, and an < sn. Moreover,
n
™>i(9 > y) = Yl(Si ~a^'
2=1

166
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
The expression
f[(x-az)(l
when multiplied out (and redefined at az in the obvious way), turns out to be a
polynomial p(x) of degree n whose roots are si,..., sn. In fact,
^ i=l 2=1
'"-(tv + lt*)*"-1
p(x) = Y[(x - ai) - - Yl °i T[(x -a^
2=1
' +h(x),
where h(x) is a polynomial of degree less than n — 1. By Viete's theorem4,
n n 1 n
2=1 2=1 y 2=1
or equivalently
n 1 n
2 = 1 y 2=1
This completes the proof of the first assertion. The second follows in a similar
way. □
Proof of Theorem 7.2.1: First suppose that /i e M+(K), is singular, and,
in addition, is supported on a compact set F of Lebesgue measure zero. From
Boole's lemma, we may assume that F is an infinite set.
Notice that 3ifi is a differentiable function onl\F and
£^> = -/rf^
JC/z(x) = - / , PV ' xeR\F
Thus Jifj, is strictly decreasing on each interval complementary to F. It follows for
y > 0 that the set
{xeR\F: JC/x(x) = y}
is countable and so has Lebesgue measure zero. Let
a = min{ x G F } and b = max{xGF}.
Note that
oo
[a,b]\F= (J/n,
n=l
where In are disjoint open intervals.
We now go through the following approximation argument to produce a
sequence of discrete measures \in with finite support that approximate fi weak-* and
such that each /in places mass only at the end points of the complimentary intervals
(In)n>i of F5 Indeed, let
FM = [a,6]
Viete's theorem: if p(x) = xn + an-ixn 1 + • • • + ao is a monic polynomial with roots
si, • • ■ ,sn, then si + s2 H hsn = -an_i.
One can prove this using Proposition 1.6.10. However, for the sake of completeness, we
provide a direct construction.

7.2. BOOLE'S THEOREM AND ITS GENERALIZATIONS
167
and define
111 =/x(Fiji)Jai)1,
where ai5i = a. Write
[a, 6] = /iUF2,i UF2,2
as a disjoint union, where i<2,i and i7^ are disjoint closed intervals, and define
/X2 = M(^2,i)^2,! + M(^2,2)^a2,2,
where a^j is the left-hand endpoint of the closed interval Fij. Doing this again,
write
[a, 6] = h U h U F3ji U F3,2 U F3,3
as a disjoint union, where ^,1,^,2,^3,3 are disjoint closed intervals. Define
M3 = ^(^3,1)^3,1 + KF3,2)Sa3,2 + ^(^3,3)^3,3-
In general, write
[a, 6] = [\Jl3 U \\jFnd
KJ=1
U = 1
and define
By our construction,
We will now show that
fin —+ fi weak-*, n —* 00.
To this end, let / be a continuous function on R and note that /|[a, b] is uniformly
continuous and so there is a S > 0 such that whenever x, y G [a, 6] with |x — y\ < 5,
we have \f(x) — f(y)\ < e. Since m\(F) = 0, there is an n so that
ll/UHHI VnEN.
^i 1 U F^
<(J.
With this n,
/ /d/xn - / /d/x = V / /d/xn - / /d/x
J J I ,=1 kf-.j Jf^j
= £
n
= £
n
= £
/KjX^nj) - / /d/X
f(anj)fJ>(Fnj) - {f - f(anJ) + /(an>j)) d/x
/ (f ~ f(anj))dfi
^j|/x||.

168
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
Thus \in —* ii weak-* as n —* oo.
Using the fact that \in —* \i weak-*, it follows that !K/in converges uniformly
to 3i/i on any compact set that is disjoint from F. If \n denotes the characteristic
function of the set {3ipn(x) > y} and \ the characteristic function of the set
{!K/i(x) > ?/}, then %n converges pointwise and boundedly to \ onC F, except
possibly on the countable set {x: Ji/j,(x) = y}. Hence, since F has Lebesgue
measure zero,
mi(JC/x > y) = xdmi = lim / \n dmi = lim -||/xn||.
J n—>-oo j n—>-oo y
The last equality is from Boole's lemma (Lemma 7.2.2). However, our construction
says that ||/in|| = ||/i|| for all n and so
(7.2.3) ml{'Kii>y)=l-M
for singular fi G M+(R) with compact support.
Now let \i G M_|_(R) be singular but not necessarily of compact support. For
fixed e > 0, there are positive singular measures v and p such that \i = v + p, z/ has
compact support on a set of Lebesgue measure zero, and ||p|| < e2||/i||.6 Applying
the estimate in eq. (7.2.3) to v and the estimate in eq.(7.1.2) to p, we have
mi(JC/x > y) ^ mi (JCz/ > (1 - e)y) + m\(3ip > ey)
< IIHI , cIIpII
(1 - e)y ey
y V1-^
Let e —* 0, to get
(7.2.4) ym1(Kp.>y) ^ ||/x||.
To prove the reverse inequality in eq.(7.2.4), we start with the inequality
(7.2.5) mi (3-0/ > —^- ] ^ mi(JC/x > y) + mx (yip < -e- V
1-eJ v r "7 V 1
Now use the facts that
IHI = IH|-IHI and ||p||<62|H|
to prove
\W\\>\\n\\(l-e2).
From eq. (7.2.3) (applied to v which has compact support on a set of Lebesgue
measure zero) we know that
m* (^ > tti) = ^H > ^(i - e)(i - ^2).
From eq.(7.1.2) (applied to p) we know that
mi fMp < -6-^-) ^ mi (\Kp\ > e-^-] ^ C\\p\\— ^ Ce^^.
Indeed, since \i _L mi, there is a set FM so that At(FM) = ||/x|| (i.e., FM is a carrier for /z) and
'7ii(-^/x) = 0. Since // is regular there is a compact set F C FM so that //(F^ \ F) ^ e2||yu|| [182,
p. 48]. Let v := jjl\F and p := m|(Fm \ F).

7.3. A REFINEMENT OF BOOLE'S THEOREM
169
Apply these two facts to the inequality in eq.(7.2.5) to get
M(i _ €)(i _ €2) ^ mi(!K/x > y) + Ce(l - e)M.
y y
Now let e —* 0 to obtain
||/x|| ^ymi(!K/x>2/).
Combine this inequality with the one in eq.(7.2.4) to complete the proof that
m1(M/i>y) = M.
y
The other equality
mi(MM<-y) = M
y
is proved in a similar way. D
7.3. A refinement of Boole's theorem
By the Lebesgue decomposition theorem, every \i G M(R) can be written as
Theorem 7.3.1 (Tsereteli [218, 219]). For /x G M(R), the following are
equivalent
(1) mi(|lK/i| > y) = o(l/y) asy-+oo.
(2) /x = /xa.
Proof. The result stated here is true for complex measures but to avoid some
technical details, we prove it only for positive ones.
Suppose d/jL = gdmi for some non-negative g G L1(M). Given e > 0, choose an
infinitely differentiable 0 with compact support such that \\(/> — g\\i < e. For large
enough a, mi(\3i(/))\ > a) = 0, since J£(</)) is a bounded function7. Thus
m1(\Kfi\>y) < mi(|!K(/x-0dmi)|>2//2) + mi(|!K0|>2//2)
= mi(|!K(/x-0dmi)|>2//2)
for large enough y. From eq.(7.1.2) it follows that
mi(|!K/x|>2/)<C-.
From here one can show that
lim ymidJifil > y) = 0.
Conversely, suppose that m(|!K/i| > y) = o(l/y). Start with the inequality
mi(|!K/xa| >y)^ mi(|!K/xa| > j,/2) + mi(|!K/x| > j,/2)
and use the hypothesis that mi(|!K/i| > y) = o(l/y) and the above fact that
^i(|^Ma| > y) = °(l/y)? to conclude mi(|2£/zs| > y) = o(l/y). However,
{|!K/xa| > y} = {^/is > y}|J{M/xa < -?/}
and these sets are disjoint. Thus
miflJC/Xsl > y) = miCKfjLs > y) + mi(JC/xs < -?/).
Recall that when d/z = /dmi, we use 3K/ in place of the more cumbersome CK(/dmi).

170
7, THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
Apply Theorem 7.2.1 twice to obtain
mi(|!K/xa|>2/) = 2||/xa||/2/.
The fact that mi(\<Kiis\ > y) = o(l/y) implies that /is = 0 and so /i = fia. □
Theorem 7.3.2 (Hruscev and Vinogradov). For fi G M+(K),
lim 2/mi(|!K/x| > y) = 2||/xa||.
PROOF. By Theorem 7.2.1, this formula is certainly true if /i = /is. Also note
that mi(|IK/ia| > y) = o(l/y) as ?/ —* oo. If a,/? > 0 with a + /? = 1, we can apply
Theorem 7.2.1 to get
2/mi(|JC/x| > ?/) ^ 2/mi(|JC/xa| > 2/a) + ymi(\3i/^s\ > y($)
= ymi(\^a\>ya) + 2^.
Combining this with the above observations, we conclude that
i!5ymi(|M/i|>j/)^2%Ji.
But since 0 < f3 < 1 is arbitrary, we get
Mm" 2/mi(|!K/x| > y) ^ 2||/xa||.
In a similar way,
2/mi(|JC/xs| > y) ^ 2/mi(|JC/x| > ya) + 2/mi(|JC/xa| > (3y).
Hence
and so
2||/zJ ^- lim 2/mi(|!K/x| > j/)
^ y—+oo
2||/xs|| ^ lim 2/mi(|JC/x| > y) ^ lim ymiflft/xl > y) < 2||/xs
y-^oo
y-^oo
completing the proof. □
7.4. Measures on the circle
We state and prove the analogs of the Hilbert transform theorems of the
previous two sections for the Cauchy transform. The proofs of the results for the Hilbert
transform used 'real variable' techniques. The proofs we present for the Cauchy
transform results will rely heavily on complex analysis.
For \i G M+ recall from eq. (1.8.3) the Herglotz integral
(Hti(z) = j^MQ-
We saw earlier that H\i G Hp for all 0 < p < 1 and so has non-tangential boundary
values m-almost everywhere. Also recall that RH/i = P/i, the Poisson integral of
/i, and for m-a.e. C £ T,
(P/x)(C)=rlim_(P/x)(rC) = ^(C)
(Theorem 1.8.6 - Fatou's theorem). Here is the analog of Theorem 7.2.1 for the
Herglotz integral.

7.4. MEASURES ON THE CIRCLE
171
Theorem 7.4.1. For a singular measure \i G A/+,
m(\Hn\ > y) = - arctan ( \—
7r v y
Proof. Let us assume first that ||/i|| = 1. At the end we will see how to adjust
the proof if this is not the case. Since /i G M+, H/i maps the unit disk to the
domain {Rz > 0}. Since
z-\
z + 1
maps {$lz > 0} to D, the analytic function 0 defined by
maps the disk to itself. Moreover, observe that
1-0
Since (H/i)(0) = ||/i|| = 1, it follows that 0(0) = 0. Now observe that for m-
a.e. (GT,
K(ff/i)(0 = (^)(0 = ^(0-
But since /i _L m, this last quantity is zero m-almost everywhere. Thus the
boundary values of H/i are purely imaginary and, by eq.(7.4.2), |0| = 1 m-a.e. on T.
Hence 0 is an inner function.
We now claim that the map C l—> 0(C) 1S measure preserving on T. Indeed,
since 0(0) = 0, the mean value property for harmonic functions says that
(?-4-3) //"dm=<U; neZ\{0}.
Thus
ho (pdm = / hdm
7t Jt
whenever h is a trigonometric polynomial. Approximating by Cesaro polynomials
(Theorem 1.6.5) and using the bounded convergence theorem, we can get the same
identity when h is a bounded Borel function. Hence the map C l—> 0(C) is
measure preserving. We will give another proof of this fact using Clark measures (see
Remark 9.4.6).

172
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
With these preliminaries, we are now ready to compute m(\Hfi\ > y). Indeed,
i + <K01
m({CeT:|(tfM)(C)|>2/})=m(jceT:
;jC€T:
1 - <A(C)
i + C
i-C
>y
'})•
2tt
m1 [\0 e (-7r,7r) :
> y f ] , (p is meas. pres.
1 + eie >
1
0i<9
>y
})
—m1({^e(-7r,7r):cot(^/2)>y})
;mi({0€ (O,7r):cot(0/2)>y»
cot-1 y
arctan(l/j/).
D
To prove the result when ||/i|| ^ 1, observe that
Hfj, _ 1+0
for some inner 0 and follow the above proof as before.
The following is the analog of Theorem 7.3.2 for the Cauchy transform.
Theorem 7.4.4. For \i e M,
lim ym(\Kii\ > y) = -||/xs||.
y-^oo 7T
Remark 7.4.5. Our notation gets a bit easier if we work with a modified
definition of the Cauchy transform
(K'v)(z):=J-^—zMQ-
) and so if we can s
lim ym(\K'ii\ > y)
Notice that K\i = K'(£d/j,) and so if we can show that
In „
y-^oo
then
lim ym{\Kp\ > y) = lim ym(\K'(Cdfi)\ > y) = -||«d/x)s|| = -||m,||.
y-^oo y-^oo 7T 7T
The proof of this theorem depends on the following two technical lemmas.
Lemma 7.4.6. Let \i e M. For each e > 0, there are functions <t>\,<t>2 £ C°°(T)
such that
(7.4.7) Hd/x-M/xlll^e, ||d|/x| - 02d/x|| ^ €, ||0i||oo O, Halloo ^ 1.
Proof. The reader might want to review some basic facts about polars in
Banach spaces (in particular Proposition 1.4.5). For v G M+, the set
B = {/ G L°» : ll/IU < 1}

7.4. MEASURES ON THE CIRCLE
173
is a closed, convex, balanced subset of Ll(v). Clearly B is convex and balanced.
To see that B is closed in Ll(v), let (/n)n^i be a sequence in B with fn —» / in
L1(z/). If need be, we can pass to a subsequence and assume that /n —-> / z/-almost
everywhere. Thus, for z/-almost every £ G T,
|/(C)| = lim |/„(C)| < 1
n—+oo
and so f £ B.
Let
Woo = {/GC00(T):||/||00<l}
and VF be the Ll(v) closure of Woo- A similar argument as above shows that
W C B. We claim that 5cl^. Indeed, let (p £ W°, that is to say, 0 G L°°(» and
sup -
<Pfdv
few\^l.
Notice that
1 ^ sup
^ sup
= sup
\\</>du
\4>\&v.
From here it follows that
sup
{|/^/dz/
{|/^/dz/
(1/
<A/dz.
:/GW:
f G C(T),
< 1
/'
cPfdis
■■/eflki
h.
which says that 0 G B°. Since VF° C B°, Proposition 1.4.5 yields the containment
°(B°) C°(W°). Using the facts that °(B°) = B and °(W°) = W, we conclude
that B C W.
To prove the approximations in eq.(7.4.7) we apply the above equality (B = W)
with the measure v = \\i\ and the function
d/i
A little thought will show that /i G L°°(|/i|) and \h\ = 1 |/i|-almost everywhere.
Hence both h and h = 1/h belong to B. By the equality B = W and the fact
that W is the L1(|/i|) closure of VFqo, we can, given e > 0, produce two functions
0i, 02 G VFoo such that
yv-<Aii<%Ke, yi^-
The inequalities in eq. (7.4.7) now follow. □
Recall from Chapter 1 (especially Proposition 1.2.4) the distribution function
\f(y) = m(\f\>y), y>0,
for a measurable function / on T.
d|/i| ^e.

174
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
Lemma 7.4.8. For \i G M,
(7.4.9) lim y\K>v(y) ^ lim y\K'\n\(y),
y—+oo y—+oo
(7.4.10) lim y\K>v(y) > lim yXK^(y).
y—+oo y—+oc
Proof. For the proof of eq.(7.4.9) we let e > 0 be given and use Lemma 7.4.6
to produce a 0 G C°°(T) with H^H^ ^ 1 and
||d/i-0dH||^62.
On T, write
K'p = <l>K,\n\ + f + g,
where
/(o = K\4>d\n\m - m(K'\fi\)(o = J ^"f° di/xKo
and g = K'(&ii — 0d|/i|). Since <fi G C°°(T), we can apply Privalov's theorem
(Theorem 3.1.1) to see that / is continuous on T and so
(7.4.11) lim y\f(y) = Q.
y-^oo
By Kolmogorov's theorem (Theorem 3.4.1),
(7.4.12) ;
Since ||0||oo ^ 1, we have
\K'n\^\K'\n\\ + \f\ + \g\
almost everywhere on T.
This implies that
Ak>(2/) < *k'm((1 - e)2/) + A/ Qy) + Xg Qy)
and so
lim y\K>n(y) ^ lim y\K,{ |((1 - e)y) + lim yAp (%) (by eq.(7.4.11))
^ lim 2/AK%|((l - e)j,) + —e2 (by eq.(7.4.12))
1 HE(l-e)j/A^M((l-e)y) + 2Cc
(7.4.12) A9(y)^||d/i-*d|/x|||<^-C2.
1 — e 2/-*'
= lim 2/AK/|/i|(2/) + 2Ce.
1 — e y-^oo irM
One obtains the inequality in eq.(7.4.9) by letting e —» 0.
To prove eq.(7.4.10), let e > 0 be given and, via Lemma 7.4.6, choose <fi G
C°°(T) with H^Hoo ^ 1 and such that
||d|/i|-0d/iK62.
Now, on T, write
K'\fi\ = 4>K'fi + f + g,
where
and proceed as before. □

7.4. MEASURES ON THE CIRCLE 175
Proof of Theorem 7.4.4. By Lemma 7.4.8, it suffices to show
(7.4.13) lim y\K>M(y) =-\\Hs\\.
Observe that |/i| = \/ia\ + \f^s\ and so Kf\fi\ = Kf\fia\ + Kf\v>s\- Thus for a,f3 >
0, a + /? = 1, we have
2/AK'|M|(2/) ^ y*K>\na\(<xy) + 2/AK/t)Us|(^)
= -(ay)XK^al(ay) + -(/fy)AK1)Us|(/fy).
The first term goes to zero as y —* oo (Proposition 3.4.11) and so
lim y\K>M(y) ^ - lim ?/AK/|Ms|(?/).
Using the inequality |if'|/is|| ^ |if'|/ia|| + l^'MI and an analog of the previous
argument, we get,
Mm y\K'\ns\(y) <: ^ Mm y^K>\»\{y)-
y—+oo P y—+oo
Letting /? —-> 1, we obtain the two inequalities
lim yXK'wiy) ^ lim y\K'\na\(y)i
Mm y^K>\ns\{y) ^ Mm s/Ak^^).
y—+oc y—+oo
In light of these inequalities, to prove eq.(7.4.13), it suffices to show
lim y\K'\na\(y) = -llMsll-
Recall (Theorem 7.4.1) that for a positive singular measure z/,
n \ y
Also recall the formula
(Hi/)(z) = 1 + 2z(K'v)(z), zeB.
This previous formula says that m-almost everywhere on the circle, we have the
inequality
\K'\Hs\\<\ + \\H\ns\\.
Hence for a + (3 = 1,
y*K>\ns\(y) < yxi/2{ay) + y\^H^si(l3y)
= yx±H\ns\{0y) (since Xl/2{ay) = 0 when ay > 1/2)
= y\H\^\{Wy)
= ^(2Py)XHl^(2Py)
YpWv)\tan_1 (^) (byTheorem 7A1)-
Now take a lim as y —-> oo to get
HS y\K'\^s\{y) < —d|/xs||.

176
7. THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS
Letting /? —» 1 we obtain
lim y\K,\ {(y) ^ -||//s
To show the corresponding lim inequality, we note that on T,
|#|/is!lo + 2|xvn
and so for a + /? = 1,
2/A2K'|/za|(/?2/) (since Ai(ay) = 0 when ay > 1)
2/Ak'I^i f-y
= |(fy)^iM.i(fy
Use Theorem 7.4.1 and the above inequalities to obtain
2 2
-||/is|| ^ - lim yXK^sl(y).
/i P y—+oo
Let /? —> 1 to get
-||/is|| ^ lim 2/AK'|Ma|(2/).
7T y—+oc
Combine this with the above lim inequality to obtain the result. □
7.5. A theorem of Stein and Weiss
We end this chapter with the following well-known theorem of Stein and Weiss
[112, p. 71] [208] that computes the Hilbert transform (conjugate function) for
a characteristic function. What makes the conclusion of this result remarkable is
that the distribution function for "Kxe (or Qxe) does not depend on the geometry
of the set E1, just the its measure.
Theorem 7.5.1 (Stein - Weiss).
(1) If E C K is measurable with mi(E) < 00, then
mmXE\>y)=1-sin^(2^^
(2) If A C T is measurable, then
/.^ . n - i /sinM-4)/2))\
PROOF. We follow [79, p. 115] and only prove that for fixed y > 0, the quantity
^(IQXaI > y) depends only on m(A), and not actually compute the function. The
reader can get the exact formula by computing m(\QxA) > y) when A is an arc of
the circle.
Fix y > 0 and let h be the harmonic function on the strip {0 < $lz < 1} whose
boundary function is X\Sz\>y, i-e->
h(it) = h(l + it) = 0, \t\ ^y,
h(it) = h(l+it) = 1, |£| > y.

7.5. A THEOREM OF STEIN AND WEISS
177
The Herglotz integral H\a — PXa + iQXA is analytic on D and has the following
properties:
0 < MHXA(reie) < 1, ^HXA{ei9) = XA{eie), ffXA(0) = m(A).
The bounded harmonic function h o H\a on D has the property that
(hoHXA)(eie)=X\QXA\>y(eie)
almost everywhere. Hence
™>(\QXa\ > V) = l ho H\a dm
Jj
= h(HXA(0))
= h(m(A)).
a

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CHAPTER 8
The backward shift on H2
Much of the motivation for the ideas in the next two chapters comes from
attempts to understand various aspects of the backward shift operator
Bf=Ljm
z
on the classical Hardy space H2. In order to place the next two chapters in better
context, we spend some time here outlining the basic results on the backward shift.
The reader can consult [17, 44, 64, 153, 177, 178, 179] for further details.
8.1. Beurling's theorem
The shift operator S : H2 h-> H2, defined by
(Sf)(z) = zf(z),
is an isometry on H2. A classical theorem of A. Beurling [28] (see also [65])
characterizes the invariant subspaces of S. By 'invariant subspace' we mean a
closed linear manifold M C H2 for which SM C M. If d is an inner function, then
11$/11 = ll/H for all / G H2 and so dH2 is a closed linear manifold (a subspace) of
H2. It is also clearly S-invariant. Beurling's theorem says these are all of them.
Theorem 8.1.1 (Beurling). If d is an inner function, the set dH2 is an S-
invariant subspace of H2. Conversely, if M C H2, M ^ {0}, is an S-invariant
subspace, then M = dH2 for some inner function d.
Proof. The proof that dH2 is an S-invariant subspace of H2 was discussed
in our preliminary remarks. To prove the second part of the theorem, suppose M
is a non-zero S-invariant subspace of H2. First notice that SM ^ M. If this were
not the case, then f/z G M whenever / G M. Applying this k times we conclude
that
-4 € M Vfe G N.
But this would mean, since f /zk must be analytic on D, that / = 0, a contradiction
to the assumption that M ^ {0}.
Second, since SM ^ M, one observes that
MH(SM)±^{0}
and so M H (SM)1- contains a non-trivial function d. We now argue that \d\ = c
on a set of full measure in T. Indeed,
/,|$(C)|2Cndm(C) = (tf,Sntf) = 0 VnGN.
179

180
8. THE BACKWARD SHIFT ON H2
Taking complex conjugates of both sides of the above equation, we also see that
|tf(C)|2Cndm(C) = 0 VneN.
/
Jj
IT
This means that the Fourier coefficients of \d\2 all vanish except for n = 0 and so
|$|2 = c almost everywhere on T. Without loss of generality, we can assume that
|i9| = 1 almost everywhere on T and so d is an inner function.
Third, let [d] denote the closed linear span of the functions
and observe that
[d] = dH2.
To see this, notice that clearly [d] C dH2. For the other containment, let g = dG G
dH2 and let Gn be the iV-th partial sum of the Taylor series of G. Notice that
$Gn G [&] since Gn is a polynomial. From Parseval's theorem, Gn —» G in H2
and so, since d is a bounded function, $Gn converges to dG in H2.
Finally, observe that
[tf] = M.
Indeed, d G M and so [d] C M. Now suppose that / G M and / _L [#]. Since
/
Jj
/(C)tf(C)Cn dm(C) = (/, 5"tf) = 0 Vn G N0.
/T
But since d _L 5M, we also know that
/ f{QW)C dm(C) = (5"/, i?) = 0 Vn e N.
The previous two equations say that all of the Fourier coefficients of fd vanish and
so fd = 0 almost everywhere on T. But we have already shown that \&\ = 1 almost
everywhere on T and so / = 0. □
Remark 8.1.2.
(1) The key to proving Beurling's theorem is the fact that the invariant sub-
space generated by M H (SM)1- is equal to M. This idea extends to
other Hilbert spaces of analytic functions [18, 171, 196]. The papers
[119, 120, 135, 170, 172, 180, 197] characterize the 5-invariant sub-
spaces of some other Banach spaces of analytic functions.
(2) There is a Beurling theorem for the Hp spaces [65, 79]: suppose 0 < p <
oo and M is a non-zero subspace of Hp. Then M is 5-invariant if and
only if M = dHp for some inner function d.
8.2. A theorem of Douglas, Shapiro, and Shields
Using the inner product
/» OO
n=0
and the definitions of the forward and backward shifts on i72, one can quickly check
that
(Sf,g) = (f,Bg) Vf,geH2,

8.2. A THEOREM OF DOUGLAS, SHAPIRO, AND SHIELDS
181
and so 5*, the Hilbert space adjoint of 5, is equal to B. Using basic properties of
annihilators and adjoints, one can also check that
for any subspace M C H2. Thus, via Beurling's theorem (Theorem 8.1.1), every
S*-invariant subspace of H2 takes the form (dH2)^ for some inner function d. The
following theorem of Douglas, Shapiro, and Shields [64] describes ($H2)±.
Theorem 8.2.1 (Douglas-Shapiro-Shields). For f G H2, the following are
equivalent.
(1) / e (tf#V-
(2) The boundary function £ h-> /(£) belongs to H2(T) H $i72(T), i.e., there
is a g G H2 such that f = d^g m-a.e. on T.
(3) The meromorphic function f /d on D has a pseudocontinuation1 to a
function f# G H2(Be) with f#(oo) = 0.
Proof. We will first prove the statement
Indeed if / G H2 H $#o, then / = dh almost everywhere on T for some h G H2,.
Thus for every g G
(f,#9) = f{$h)(tig)&m = fhg~dm = ^(0) = 0
since gh € Hq and i? is an inner function. Hence / € ('dH2)±.
Conversely, if / belongs to
(#H2)± = {\J{zH:kenQ}Y,
then
ffICkdm = 0 V/cG N0.
This implies that /# G #o and so / G $#o- Hence
To finish, we must show that / G i72 fl $H2 if and only if f /d has a pseudo-
continuation belonging to H2(3e) which vanishes at infinity. If / G H2 D dH2 then
/ = dh almost everywhere on T for some h G Hq. Note that h(l/~z) G H2(3e) and
vanishes at infinity. Thus if
then (//#)(C) = /tf(C) almost everywhere on T (the non-tangential limit functions
are equal almost everywhere) and /# G i72(De) and vanishes at infinity. Conversely
if f /d has a pseudocontinuation /# G i72(De) with /#(oo) = 0, then the function
The meromorphic functions g (on D) and G (on De) are pseudocontinuations of each other
if the non-tangential limits of these two functions exist and are equal almost everywhere on T.
See [44, 64, 153, 178, 179] for more on pseudocontinuations.

182
8. THE BACKWARD SHIFT ON H2
belongs to Hfi. Hence
almost everywhere on T and this completes the proof. □
Remark 8.2.2. Using a slightly different proof [44, p. 87], one can even find
an integral formula for the pseudocontinuation /# of f /d. Indeed
MX) = -X [ /(C)^C)dm(C), A€D,
J j Z — A
We leave it as a simple exercise to the reader to show the following.
Proposition 8.2.3.
\J{Bn$:neN} = ($H2)±.
For our inner function $, let
a(D) := JagD" : lim |tf(z)| =0
be the spectrum of d. If d = bs^, where b is a Blaschke product with zeros (an)n^i
and s^ is the singular inner factor with positive singular measure /i, then
cr(d) = (an)^! Usupp(/i).
We can define d to be meromorphic on De by
tf(l/z)
and notice that d has poles at the reflected zeros of d. Moreover, d and d are
pseudocontinuations of each other. Using a version of Morera's theorem [79, p.
95], d is an analytic continuation of d across T \ o~(d) as long as cr(#) does not
contain all of T. It turns out that all functions in (dH2)^ enjoy this same property
[64] or [44, p. 84].
Theorem 8.2.4. If f G (dH2)^, then f has an analytic continuation across
T \ a($) to a function f analytic on the set
{zeBe: 1/z£<t(i?)}.
Furthermore, for A in this set, the linear functional f >—> /(A) is continuous on
(fill2)1-. Moreover, if z G Oe m£/& 1/z G cr(#), £/ien / has at pole at z whose order
does not exceed the order of the pole of d at z.
For any 0 < p < oo, the backward shift operator B on Hp is continuous.
When 1 < p < oo, we can use duality to characterize the ^-invariant subspaces of
Hp. Indeed, by Corollary 3.6.3, the dual of Hp can be identified with Hq, where
l/p+l/g = l, via the Cauchy pairing
/
Jj
fgdm.
T

8.2. A THEOREM OF DOUGLAS, SHAPIRO, AND SHIELDS
183
Thus if M is a ^-invariant subspace of Hp, then its annihilator M is an S-invariant
subspace of Hq and, via Beurling's theorem, takes the form $Hq. Using the Hahn-
Banach separation theorem we conclude that
M = ($Hq)^ = If eHp : f f^ddm = 0 V# G Hq\ .
The following theorem is the Douglas-Shapiro-Shields theorem in the Hp setting.
Theorem 8.2.5 (Douglas-Shapiro-Shields). For 1 < p < oo and f G Hp, the
following are equivalent:
(i) / g {m^
(2) The boundary function £ h-> /(£) belongs to HP(T) D $Hq(T), i.e., there
is a g G Hp such that f — dQg m-a.e. on T.
(3) The meromorphic function f /d on D has a pseudocontinuation to a
function % G Hp(pe) with Jo{oo) = 0.
Remark 8.2.6. To avoid having to make reference to the conjugate index g,
we will use the notation
d*(Hp)
to denote (i9Hq)-L (or one of its equivalent definitions via Theorem 8.2.5).
We mention a few other results. The first, a result of Aleksandrov [8] (see also
[44, p. 101]), is that every jB-invariant subspace of H1 takes the form ^(H1) :=
H1 n $Hq 2but the proof is more complicated due to some technical difficulties
when dealing with the functions of bounded mean oscillation and the fact that H1
is not reflexive.
The second fact to mention is that when 0 < p < 1, not all jB-invariant sub-
spaces take the form i/^n^i^Q. A complete description of the jB-invariant subspaces
of Hp', 0 < p < 1, is somewhat complicated, and was given by Alexsandrov [8] (see
also [44]). There is also the following analog of the analytic continuation result in
Theorem 8.2.4.
Theorem 8.2.7. If 1 ^ p < oo and f G $*(HP), then f has an analytic
continuation across T \ a(i9) to a function f analytic on the set
{z£Be:l/zgo-($)}.
Furthermore, for A in this set, the linear functional f h-> /(A) is continuous on
$*(HP). Moreover, if z G Oe with 1/z G cr(#), then f has at pole at z whose order
does not exceed the order of the pole of d at z.
Finally, there is the following duality result [44, p. 109].
Proposition 8.2.8. For 1 < p < oo, every £ G ($*(Hp)y takes the form
t(f)= [ ftdrn
JT
for some g G d*(Hq), where l/p-\- 1/q = 1. Thus, via the Cauchy pairing, the dual
of$*(Hp) can be identified with $*(Hq).
Furthermore, / G H1 belongs to H1 Pi-dH^ if and only if f/tf on D has a pseudocontinuation
to a function f# G ii/"1(De) with /#(oo) = 0.

184
8. THE BACKWARD SHIFT ON H2
8.3. Spectral properties
What are the spectral properties of the backward shift operator S* on H21 If
AGO, one can show by direct computation that (I — AS*)-1 exists and is given by
the formula
(7-A^)-1/="/"A{(A)-
z — A
Thus cr(5*), the spectrum3 of S* is contained in D-. Furthermore, since, for any
Ago,
l-Xz 1-Az'
then
ker(A/-S*)^{0}.
Hence
tr(5*) = D".
If d is an inner function and
T{d) :=5*|r(i72),
what is cr(T(tf))? First notice that if A G a(i3) n D, that is to say, i?(A) = 0, then
1
1-Az
,z*i?) = A*i?(A) = 0 VfcG
But since the closed linear span of {zkd : k G N0} is equal to tf#2 (Theorem 8.1.1),
we conclude that
However, from above,
and thus
<j($) C <7(T(i?)).
A proof of the following result can be found in [44, p. 95]. The original result is
due to Moeller [143] and independently by Livsic [125].
Theorem 8.3.1. For an inner function d and T(d) := S*|.#*(i/2), we have
v(T(#)) = *(#).
Moreover,
(1) A G <7(T(i?)) n O i/ and onij/ i/ (1 - Xz)-1 G i?*(H2).
(2) 7/(gT, £/ien £ 0 a(^(^)) */ anc^ on^/ */ ^ere zs an open neighborhood U
of C 5itc/i £/&a£ ever?/ / G d*(H2) has an analytic continuation to U.
"^See Chapter 1 (Proposition 1.5.3) for a review of the basic properties of the spectrum of a
bounded linear operator.

8.4. KERNEL FUNCTIONS 185
Notice that if A 0 T, and (I - AT(^))"1 exists, then
(I-\T(0)r1f=zf-Xcx}f\ fe#*(H2),
z — a
for some complex number c\(f). If A £ ID, then c\(f) = /(A) as above. If |A| > 1,
then
0 = ((/ - ATO?))-1/,*?) = lZi'7*C*(/)V
This says that
*</> = (^j,«) / (j4jf,«) - (^.#) / -J(iA).
It turns out that the function A h-> ca(/) on De (minus some poles) is a pseudo-
continuation of /. We refer the reader to [17, 44, 178, 179] where this technique
was used to explore the spectral properties of the backward shift on various Banach
spaces of analytic functions on D.4
8.4. Kernel functions
Let P$ denote the orthogonal projection of H2 onto d*(H2) and let
g^ g+ = K(gdm)
be the Riesz projection operator from L2 onto H2.
Proposition 8.4.1. For f g H2,
iV = /-W)+.
PROOF. Let Wf := / - #(#/)+ and observe that Wf G <d*(H2). Indeed, for
any n G No,
= {f,zn$)-{($f)+,zn)
= {f,znd)-{df,zn)
= {f,zn$)-{f,zn$)
= 0
and so Wf J. i?#2 (since {,2™$ : n G No} has dense linear span in dH2). Clearly
f-wf = W)+ g tf#2 = or (#2))x
and so (see [49, p. 9]) VF must be the desired orthogonal projection P$. □
For each A G O, define
Cx{z) = -Xr, zeB,
1 — Xz
and note that by the Cauchy integral formula,
f(X) = (f,Cx) V/G#2.
Hence, the Cauchy kernels {C\ : A G D} are called the reproducing kernels for H2.
The c\(f) that arise from various Banach spaces of analytic functions may or may not be a
pseudocont inuat ion.

186 8. THE BACKWARD SHIFT ON H2
Since P$ is the orthogonal projection onto the ^-invariant subspace $*(i72),
we have
/(A) = (P*/,Ca) = (/,P,,Ca>
for every / G $*(#2) and A G D. Thus the functions
kx{z) := (PoCx)(z)
are the reproducing kernels for d*{H2). Here is an explicit formula for k\.
Proposition 8.4.2.
1 — Xz
Proof. Using Proposition 8.4.1, it suffices to show that
1 ^ (*) - *<A>
1 — Xz J + 1 — Xz
Indeed,
1 - XzJ + Jj 1 - AC 1 - (z
m-TCdm(0
oo « oo
n=0 ^T /c=0
OO OO p
= J2rJ2zk W)C-kdm(Q
n=0 k=0
ErE^>-fc)
tf(A)
1-A*'
D
8.5. A density theorem
Recall that A, the disk algebra, is the space of continuous functions on D-
which are analytic on D. The norm on this space is, via the maximum modulus
theorem,
ll/lloo = sup{|/(C)| : C G T}
and we can identify A, by means of its boundary values and the F. and M. Riesz
theorem, with the space
{/GC(T):/(-n) = OVnGN}.
In several of his papers, Aleksandrov makes use of the following interesting density
theorem for $*(H2).

8.5. A DENSITY THEOREM 187
Theorem 8.5.1. If d is inner and
i?*(;4) :=An$*(H2),
then $*(A) is a dense subset of$*(H2).
Remark 8.5.2.
(1) At first glance, Theorem 8.5.1 does not seem to belong in a book about
Cauchy transforms. However, its proof uses several of the main results
in this book such as Theorem 5.4.5 (Aleksandrov's characterization of
Cauchy transforms), Theorem 6.5.1 (the ^-property for Cauchy
transforms), and Theorem 7.4.4 (the distribution theorem for Cauchy
transforms) .
(2) When d has no singular inner factor (i.e., d is a Blaschke product), then
(8.5.3) J—L- : A G D, i?(A) = o| C i?*(A),
since (1 — Xz)"1 = k\(z) G $*(i72), and one can use duality to check that
this set of kernels has dense linear span in $*(H2).
(3) When d has a singular inner factor, the set containment in eq.(8.5.3) is still
valid although the kernels do not have dense linear span in $*(H2). What
makes Theorem 8.5.1 remarkable is that i9*(A) is still dense in $*(i72).
(4) Even more remarkable is the case where d is a singular inner function.
Here it is not obvious why there are any non-trivial functions in &*(A).
Yet they are dense in $*(H2).
(5) Notice that the larger set H°° n d*{H2) contains
{S*nd : n G N}
which has dense linear span in $*(H2) (Proposition 8.2.3).
(6) The papers [114, 192] have further information about whether or not
d*(H2) contains functions from various smoothness classes.
Recall from Chapter 4 the space of Cauchy transforms % = {/ = K\i : \i G M}
with norm
\\f\\=wi{y\\:ȣRf},
where Rf = {fi G M : / = K\i\ is the set of measures that represent a particular
Cauchy transform. By identifying the dual of A with % via the Cauchy pairing
/oo
gdfi= lim V^(n)/2(n)rn, g G A, /x G M,
r-+l- *-^
n=0
recall that % can be endowed with a weak-* topology.
For an inner function $, let
t?(3C):={/6 3C://i?€Ar+}
and recall the 3"-property from Theorem 6.5.1: if / € i?(DC), then //# € % and
moreover, ||//t?|| < ||/||.
Proposition 8.5.4. The space •&(%) is weak-* closed in X.

188
8. THE BACKWARD SHIFT ON H2
Proof. As a consequence of the Krein-Smulian theorem [49, p. 165], it suffices
to show that d(X) is weak-* sequentially closed, i.e., if (fn)n^i is a sequence in $(X)
such that fn —» / weak-*, then / G d(X). By Proposition 4.2.5, fn —» / weak-*
if and only if ||/n|| ^ L for all n and fn —» / pointwise on D. Notice from the
^-property that ||/n/$|| ^ L for all n. Furthermore, fn/d —* f /d pointwise on D
and so f /d G X (Banach-Alaoglu). Hence / G d(X) as desired. □
For an inner function $, let
A* := i?(3C)± = {/GA:(/,5>=0V^ i?(3C)}
be the pre-annihilator of #(3C).
Proposition 8.5.5. With the above notation, we have the following:
(1) A$ is a norm closed subspace of A.
(2) ^ci>*(A).
Proof. The first item follows from the fact that pre-annihilators are norm
closed subspaces. Since H2 C X, we know that dH2 C d(X) and so
^ = tfpc) i. c (tf#2)± = r (#2) n a = i?*(A).
This proves the second item. □
The Hahn-Banach separation theorem says that (A?)1- = d(X) from which
i?*(A)x C(^)± = ^(^).
This yields the following corollary.
Corollary 8.5.6. If X e M such that KX e tf*(.A)\ then KX/d G iV+.
Let
ti*(H°°) :=d*{H2)nH°°
note that $*(i7°°) is a weak-* closed5 subspace of H°°. To see this, it suffices, via the
Krein-Smulian theorem [49, p. 165], to show that $*(i7°°) is weak-* sequentially
closed. Thus if (fn)n>1 C tf*(iJ°°) with fn -* / weak-* in #°°, then fn -* f
weakly in i72. But i)*(H2) is norm closed and hence weakly closed [49, p. 129] and
so / G r (H2). Hence f e H°° H $*(H2) = i?*(JJ°°).
This next theorem is the key to proving Theorem 8.5.1.
Theorem 8.5.7 (Aleksandrov [15]). d*(A) is weak-* dense in <d*(H°°).
Proof. For U C L°°, recall from Chapter 1 the pre-polar
°U = Ig G L1 : \Jjgdm\ ^ 1 V/ G u\
and for V C L1, the polar
V°=heL°°: \ffgdrn\ ^l\/gev\
If £/i C £/2 CL°°, then
and (°C/)° is the weak-* closure of the convex balanced hull of U.
Here the weak-* topology on H°° is the restriction of the weak-* topology on L°°, given by
the pairing f fg dm, / G L°°, g G L1.

8.5. A DENSITY THEOREM
189
Since
i?*(A) Ci?*(JJ°°),
it suffices to show that
°(ball(i?*(i4))) C °(ball(i?*(ff00))).
Hence it suffices to show that if g G L1 and satisfies
O V/Gball(i?*(i4)),
(8.5.8)
then
(8.5.9)
/ fgdm
/ Jg dm
O V/GbaH(i?*(ff°°)).
For such a g satisfying eq.(8.5.8), we can use the Hahn-Banach extension
theorem to produce a \i G M with ||/i|| ^ 1 such that
Jjgdm = Jjd^ V/Gball(r(,4)).6
Let A G M be denned by
(8.5.10) dX = d^-gdm
and note that KX G (ball(i?*(A)))_L and so by linearity, KX G $*(A)±. By
Corollary 8.5.6, we know that KX/d G N+.
For / G ball(i?*(/f°°)), note that / = dCji on T for some fx G ball(iJ°°) and
moreover, H/H^ = 11/i11oo. We let
(C\)(z) := [—L-d\(Q, \z\*l
J l-Cz
and define
G(*) := ( *M*)<j&WM> "-I < ^
I /(1/Z)(CA)(*), |*| > 1.
Since KX/d G iV+, then G is analytic on D and, for 0 < p < 1, satisfies the Hp
norm estimate
l|G||„ ^ ||/i||oc||^A||p = ||/||oo||^A||p ^ \\KX\\P.
Similarly, G|De is analytic and, for 0 < p < 1, satisfies the Hp(3e) norm estimate
l|G||#P(iD)e) ^ ll/lloo||C'A||J^p(]D)e) ^ IICAH/jpod^).
Finally, if JG is the jump function
(JG)(C)= lim(G(rC)-G(C/r)),
r—>-l_
(which exists for almost every (gT), one can check that
Notice that one can not necessarily choose d/j, = gdm since ||gdm|| may be greater than
one.

190
8. THE BACKWARD SHIFT ON H2
ra-a.e. on T and hence is integrable on T. Thus G satisfies the conditions of Alek-
sandrov's characterization of Cauchy integrals (Theorem 5.4.5) and so there is a
v £ M such that G = CV. By definition, G(0) = 0 and so
(8.5.11) 0 = G(0)
/d--
From Fatou's jump theorem and the identity J(G) = J(Cv) = /J(CA), we obtain
the useful identity
dm dm
Since |KA||/| = \Kv\ on T we have
and so
{\Kv\>y} = {\K\\\f\>y}
c{|KA|||/||00>y}
= {\K\\>y\\f\\-J)
ym(\Kv\ >y)^ ||/||ooTr^-m ( \KX\ > -£-
Now we use the Hruscev-Vinogradov asymptotic formula in Theorem 7.4.4, that is
lim 7rtm(\Kr]\ > t) = ||r/s||, 77 G M,
t—»oo
to show that
However, by the definition of the measure A in eq.(8.5.10), we see that
(8.5.13) K=^s
and so
(8.5.14) IKH ^ U/xJ.

8.5. A DENSITY THEOREM
191
It now follows that
\ Jgdm
i
/<u
tS-Jt
d/j,
dm
dfi
dm
dfi
dm
,d[i
-,d\_
dm
f-fdfi r dis
J dm J dm
(byeq.(8.5.10))
(byeq.(8.5.13))
(by eq.(8.5.12))
J dm J dm J
dv
(by eq.(8.5.11))
flt+J
dzA
^
^
djj,
dji
^1.
dm
dfi
dm
+ lk
Ikll
dm
+ \\tia\\ (by eq.(8.5.14))
Thus we have shown eq.(8.5.9) and so the proof is now complete.
□
Proof of Theorem 8.5.1. From Proposition 8.4.2, the kernel functions for
$*{H2) are
kx(z)
1 - i?(A)i9(s)
1-A*
and moreover, k\ G ^*(i7°°). If / G H2 annihilates $*(.A), we can use the fact that
$*(A) is weak-* dense in $*(i/°°) (Theorem 8.5.7) to conclude that
0 =(/,**>•
However,
</,*>
1-Xz
tf(A)(/,
1-Xz
f(X) - 0(A) /<>/, l
\ l-Xz
= /(A)-0(A)(0/)+(A).
Since (#/) + G H2, then / G tf#2 = {^{H2)^. An application of the Hahn-
Banach separation theorem says that
closer (A) Dr(i/2).
The reverse inclusion is obvious. □

192
8. THE BACKWARD SHIFT ON H2
8.6. A theorem of Ahern and Clark
When C £ T \ <r(tf), Theorem 8.2.4 says that every / G $*{H2) has an
analytic continuation across £ and the functional / i—» /(£) is continuous on i9*(H2).
Moreover, since $ is analytic near £, we can use the identity
<r(i?),
Mw) = = ,
conclude that the kernel function
M > 1,
w
^ J l-(z
belongs to H2. An application of the dominated convergence theorem will show
that krQ —> k^ in H2 norm as r —> 1~ and so $*(H2). This next result of Ahern
and Clark [2] discusses what happens when £ G <r($) Pi T.
Theorem 8.6.1. Le£ # = bs^ be an inner function factored as a Blaschke
product b with zeros (an)n^i C D and singular inner function sM. Then, for £ G T,
£fte following are equivalent.
(1) A:c er{H2).
(2) Every f G i!)*(H2) has a non-tangential limit f(Q at £ and £fte linear
functional f i—» /(C) is bounded.
(3)
£^ + /tkV*«>
< 00.
Remark 8.6.2. Observe from Corollary 1.7.14 that the above equivalent
conditions are also equivalent to the condition that $ has a finite angular derivative at
c
Theorem 8.6.3 (Cohn [47]). With # = bs^ as in Theorem 8.6.1, the following
are equivalent for p > 1 and £ G T.
(1) Every f G $*(HP) has a finite non-tangential limit at £.
(2)
E^W^V.«x~,
where q is the conjugate index to p.
We will say more about the radial limits of functions in i!)*(H2) in Chapter 10
when we talk about the 'normalized Cauchy transform'.
8.7. A basis for backward shift invariant subspaces
According to Clark [46], the inspiration for 'Clark measures' comes from the
following approximation problem: for what sequences (Xn)n^i C D do the
reproducing kernel functions {k\n)n^i span i)*(H2)?
Certainly the kernels (k\n)n^i span whenever the sequence (An)n^i has an
accumulation point in D. Indeed if / G ^(H2) and (f,k\n) = 0 for all n, then,
by the reproducing property of these kernels, /(An) = 0 for all n. But this would
mean that the zeros of / accumulate in D, implying that / = 0.

8.7. A BASIS FOR BACKWARD SHIFT INVARIANT SUBSPACES 193
Another easy class of spanning kernel functions (k\n )n^i are those for which the
An's have an accumulation point in T\cr(#)7. By Theorem 8.2.7, every / G $*(i72)
has an analytic continuation across T\<t(t9) and so if / _L h\n, then /(An) = 0 from
which / = 0 since its zeros have an accumulation point in its domain of analyticity.
So the real challenge are those sequences {k\n)n^i for which (An)n^i accumulate
ona(tf)nT.
One way to get at least some sufficient conditions on the sequence of kernels
is to use a Paley-Wiener type theorem. For example, we know from Parseval's
theorem that the functions
<t>n{eie) = ein\ neZ,
form an orthonormal basis for L2. The approximation problem examined by Paley
and Wiener [155] is the following: for a sequence (7n)nez C M do the functions
ipn(eie) = j-*"
span L2? The answer is yes if the ij;n are 'sufficiently close' to the orthonormal
sequence <fin in that
(8.7.1) max|7n - n\ < —.
neZ 7T2
There is a generalization of the Paley-Wiener theorem [173, p. 208] (see also [60,
Chap. 5] and [142, Prop. 4.3.4]) which says that if a sequence (^n)n^i m a Hilbert
space !K is 'sufficiently close' to an orthonormal basis (0n)n>i for ^ then (^n)n>i
spans !K. In our setting, !K = $*(i/2), the xpn will be the kernel functions k\n, and
the orthonormal basis (f)n will be certain normalized kernel functions cnk^n, where
(n GT and cn are constants that make ||cnfc^n|| = 1. Some obvious questions are:
(i) When do the kernel functions k^ actually belong to i!)*(H2) ? (ii) Why are they
orthogonal? (iii) When are there sufficiently many of them to form an orthogonal
basis for $*(H2)1
For first question, recall from Theorem 8.6.1 that if $ = bs^ is an inner function
factored as a Blaschke product b with zeros (an)n^i C D and a singular inner
function sM, then, for £ G T, the kernel k^ belongs to i!)*(H2) if and only if
Note that for k^ to be even defined, $(0 (the non-tangential limit of $ at Q must
actually exist. The condition in eq.(8.7.2) guarantees this.
For the second question, Clark produces a unitary operator Ua : $*(H2) —>
$*(i/2), where a G T, such that whenever £ G T satisfies the condition in eq.(8.7.2)
and #(£) = a, then k^ is an eigenvector for Ua with corresponding eigenvalue (.
Since Ua is unitary, these eigenvectors are orthogonal. By the spectral theorem
for unitary operators (Theorem 1.5.6), these eigenvectors will form an orthogonal
spanning set if and only if Ua has pure point spectrum (in that the spectrum of
Ua consists solely of the eigenvalues of Ua, or equivalently, the spectral measure
for Ua is discrete). With this in place, one can now apply the above mentioned
generalization of the Paley-Wiener theorem to give some sufficient conditions the
kernels (k\n)n^i must satisfy in order to span i!)*(H2) (see [46] for details).
It might be the case that T C <t(#), for example, when $ is a Blaschke product whose zeros
accumulate on all of T.

194
8. THE BACKWARD SHIFT ON H2
This kernel function approximation problem was also studied by Sarason [186]
in the special case where $ is the atomic inner function
v \ i 1+*
v{z) = exp
Here, by means of a certain unitary operator between L2 and $*(i/2), the kernel
functions k\n correspond to the functions e27r1^, where
.1 + A^
7n = % =.
1 - A„
Sarason then uses the original Paley-Wiener theorem in eq.(8.7.1) to get some
sufficient conditions. We also mention that this is only the beginning of the story.
There has been extensive work on when the kernels span i!)*(H2) and when they
form a Riesz basis [153].
As it turns out, the unitary operator Ua alluded to above will be a rank-one
perturbation of the compression of the shift operator S to $*(H2). The 'Clark'
measure o~a G M will be the spectral measure for Ua in that Ua is unitarily
equivalent to the operator 'multiplication by £' on L2(o~a). The next two sections set up
the compression as well as its unitary perturbation. The next chapter covers the
Clark measure o~a.
8.8. The compression of the shift
To avoid unnecessary technical details, we shall assume that
tf(0) = 0
for the rest of this chapter. With this technical assumption in force, let
(8.8.1) x := r{H2) e c- = r{H2) n (c-\ .
Recall that $*{H2) := {VH2^.
Lemma 8.8.2. A function g G H2 belongs to N if and only if both g and zg
belong to$*(H2).
Proof. Suppose g,zg G i!)*(H2). Then
0= (2tf,i?) = ^7,
and so g _L C$/z. Hence g £ N. Conversely, suppose that g G N. Then g _L znrd
for all n G No and g _L $/z. This means that zg _L znrd for all n G N0 and it follows
that z# _Ltf#2. □
Definition 8.8.3. Define the operator S# on ^(H2) to be the compression of
S to$*{H2), that is
S# = p#S\r(H2).
This operator plays an important role in operator theory in that it is the model
operator in the Sz.-Nagy-Foia§ functional model: if T is a contraction (||T|| $J 1)
on a Hilbert space such that
lim T*n = 0

8.8. THE COMPRESSION OF THE SHIFT
195
in the strong operator topology and the ranks of 1 — T*T and 1 — TT* are both
one, then T is unitarily equivalent to S$ for some inner function #. See [153] for
more. The following precise formula for S#f will be important.
PROPOSITION 8.8.4. 7/0(0) = 0, then
S»f = z(f-(f^)^) v/Gr(#2).
Proof. Write / e tf*(i72) as
Notice that f\ G Ji, since the operator / i—> (/, i?/z)#/z is the orthogonal projection
of ti*(H2) onto C-d/z, while /2 € Ctf/z. Since 2/2 6 tiH2, then Pe(z/2) = 0. By
Lemma 8.8.2, zf\ € $*(H2) and so P#(zfi) = zf\ and the proof is complete. □
The model operator S$ turns out to be a cyclic operator with an easily
identifiable cyclic vector. We discuss this in the following two propositions.
Proposition 8.8.6. For each n e N,
S£ = P#Sn\ti*(H2).
Proof. Let f,g€ iT(ff2). Then, since S*kg e -&*{H2) for all k e N,
(S^f,g) = ((P^S)(P^S)n-lf,g)
= (S(P#S)n-1f,g)
= ((P#S)n-1f,S*g)
= (f,S*ng)
= (Snf,g)
= ((PoSn)f,g).
The proposition now follows. □
Proposition 8.8.7. The constant function x = 1 ^ a cyclic vector3 for S#,
that is to say,
VWx:nGN0} = r(7f2).
PROOF. Suppose / G $*(i72) and / J_ S%\ for ali n ^ N0. Then for each
n G N0,
o = (/,s?x>
= (/,P^5nx> (by Proposition 8.8.6)
= (/,P^n>
= (/,zn) (since fe^(H2)).
This says that / = 0 which establishes the proposition. □
8There is a general result here: / G t9*(if2) is cyclic for 5$ if and only if the inner part of
/ is relatively prime to i9 [153, p. 82]. Also notice, since #(0) = 0, that \ G fi*{H2). Indeed, for
any n G N0, (^n^,x) = (^n^)(0) = 0.

196
8. THE BACKWARD SHIFT ON H2
Theorem 8.3.1 computes the spectrum of 5*|^*(i72). By using the well-known
fact that cr(A*) = o~(A) for a bounded operator A on a Hilbert space, as well as the
observation that S$ = 5*|^*(iJ2), we can compute the spectrum of S#.
Corollary 8.8.8. For an inner function $,
Moreover, i/(ET, then £ ^ cr(S#) if and only if there is an open neighborhood U
of C such that every f G i!)*(H2) has an analytic continuation to U.
8.9. Rank-one unitary perturbations
Definition 8.9.1. Recall from eq.(8.8.1) that
and so, for each ft E T, we can define the linear operator Ua : ^(H2) —> $*(i/2)
by
Uag = zg for g e'N,
Ua{-z)=a.
Notice from Lemma 8.8.2 that zg G i!)*(H2) whenever g G Ji. The fact that
constant function a belongs to $*(H2) follows from our standing assumption that
tf(0) = 0.
Using the decomposition in eq.(8.8.5) and Proposition 8.8.4, we observe that
(8.9.2) Uaf = S0f + Xaf,
where
Thus Ua is a rank-one perturbation of S#. One can easily show that
x;/ = 7^</,i>
and so, since 5* = 5*|0*(iJ2),
Theorem 8.9.3. For each a eT, Ua is unitary.
Proof. Notice that
Ua = S# + Xa and [/* = 5*+X*.
Thus
UaUa = S*S$ + X^Stf + S*Xa + X^Xa.
Routine computations with the definitions of these operators show that for each
/€i?*(ff2),
X*S*/ = S*Xaf = 0.

1.9. RANK-ONE UNITARY PERTURBATIONS
1?\ 1?
197
K**f = /.
Z / Z
Hence U£Ua = /. For the other direction,
UoJJa = S$S% + XaS$ + StiX^ + XaX^.
Again, straightforward computations show that for each / G $*(i/2),
5*S5/ = / - /(0),
XaS#f = StfX^f = 0,
xax;/ = /(o),
and so
It follows that C/q is unitary.
UaU* = I.
□
This next result says that {Ua : a G T} are a// of the unitary rank one
perturbations of S&.
Theorem 8.9.4 (Clark [46]). //#(0) = 0 and U is a rank-one perturbation of
S$ that is also unitary, then U = Ua for some a G T.
PROOF. Before beginning the formal proof, we make two observations. The
first is that
(8.9.5) ll^/H = ll/H =* / JL t?/z.
To see this, recall from Proposition 8.8.4 that
Z / Z
and so if ll/H = UStf/ll, then
ll/ll2 = l!^/ll2
+
i?\ 1?
'■!
-*</.</.*)!
/,
■&
Thus / 1 tf/z.
The second preliminary observation is that
(8.9.6)
Indeed, for / e tf*(ff2),
II55/II = ll/ll =* / -L 1.
s;f = s*f=f~f{0).

198
8. THE BACKWARD SHIFT ON H2
Thus if 1155/11 = 11/11, then
ll/H2 = ll^/||2
= ll/-/(o)ll2
= ll/H2 + l/(0)|2-2$K(/,/(0))
= ll/ll2-l/(o)|2.
Hence 0 = /(0) = (/,l).
We now proceed to the formal proof. If U is a unitary rank one perturbation
of S$, then there are vectors h, k so that
Uf = S#f + (f,h)k
and
\\Uf\\ = 11/11 = ||J7*/II
for all / G i!}*(H2). The goal here is to show that h = 'd/z and k is a unimodular
constant function.
If / _L /i, then Uf = S#f and so
11/11 = \\Uf\\ = \\S*f\\.
Now bring in eq.(8.9.5) to conclude that / _L 'd/z. From this, we observe that
.9.7) (Cft)1- C (c~\ .
where _L is the annihilator in ^(H2). Hence, taking annihilators in eq.(8.9.7), and
using the fact that the spaces are one-dimensional, we see that
C- = Ch.
z
This means that
h = C\ — .
z
Suppose that / _L k. Then
u*f = ssf + (f,k)h = s;f
from which
11/11 = \\u*f\\ = \\s;f\\
and so, from eq.(8.9.6), /-LI. An argument similar to the one used in eq.(8.9.7)
shows that k = C2I. Since
h = c\— and k = C2I,
z
then for all/ G tT(#2),
£// = S^/+(/,^)c
for some complex number c. To finish, we need to show that \c\ = 1 and thus
U = C/"c. One sees this by first observing that #(0) = 0 and so 1 G $*(#2). Hence
J7*l = S*l + c-(l,l) = c-.
z z

8.9. RANK-ONE UNITARY PERTURBATIONS
199
Finally, since U is unitary, we know that
1 = I|1|| = IIJ7*1|
c—
z
\c\.
D
Remark 8.9.8.
(1) Poltoratski [164] examined the finite rank perturbations of S#.
(2) There is a generalization of the operators S# and Ua in [75, 76].
For £ G T, Theorem 8.6.1 says that
kc er{H2)^\$'{()\ <oc.
This next theorem of Clark [46] determines the eigenvalues of Ua.
Theorem 8.9.9 (Clark). Suppose #(0) =0. A point £ G T is an eigenvalue
°f Ua if and only if |$'(C)| < °° and $(C) = a- Moreover, the corresponding
eigenvector is k^.
PROOF. Suppose |#'(C)| < oo (equivalently kc G $*(#2)) and #(£) = a.
Observe from Proposition 8.8.4 that
tf\ ${z)s
z I
zkc(z)-$(z)(k0^
S#kc(z) = z(kc(z) - (fcc,
= zkc(z)-$(z)(^
= zk^(z) — ^{zja^
and from eq.(8.9.2) that
Next, observe that
kc(z)
l-tf(O^) l-ati(z)
1-Cz l-(z
One can now use these identities to verify, by direct computation, that
Uakc(z) - (kc(z) = S#kc(z) + Xakc(z) - (kc(z) = 0.
Hence k^ is an eigenvector for Ua with eigenvalue £.
The other direction of the proof involves some technical details with
Theorem 8.6.1 and can be found in Clark's original paper [46]. □
We end this chapter with a few remarks about cyclic vectors. For each a G T,
the operator Ua is unitary. If we can show that Ua is also cyclic, the spectral
theorem for unitary operators (Theorem 1.5.6) will imply that Ua is unitarily equivalent
to M^ (multiplication by Q on L2(o~a) for some o~a G M. Let us focus on the issue
of cyclicity and leave a detailed discussion of the spectral measure aa for the next
chapter.

200
8. THE BACKWARD SHIFT ON H2
Theorem 8.9.10. //#(0) = 0, the constant function x = 1 ^ a cyclic vector
for Ua for every a G T, that is to say,
\J{UZx--n€N0}=r(H2).
PROOF. Recall from eq.(8.9.2) that
Uaf = Sef + Xaf,
where
Xaf = <*(/,*
Also observe that if a — atf^O), then
Finally, observe that
Kx = x,
UaX = S-ffX + ax,
U2aX = {So + Xa)\
= (S| + S<,Xa + XaS# + X2a)x
= Six + aS#x + c2X,
Ulx = {S^Xa)\
= (S# + S#Xa + S#Xa + Xa+ XaS# + XaS$)x
= Six + aSlx + a2S#x + czX,
From here, one can see that the linear span of
(O : n e No}
contains the linear span of
{S$x ■ n E No}.
By Proposition 8.8.7, this last set is dense in $*(i72). This proves the result. □

CHAPTER 9
Clark measures
In the previous chapter, we discussed the family of cyclic unitary operators
{Ua : a G T}. The spectral theorem (Theorem 1.5.6) says that Ua is unitarily
equivalent to M^ (multiplication by Q on L2(o~a) for some o~a G M. In this chapter
we focus on some of the remarkable properties of these spectral measures which are
called Clark measures due to work of D. Clark [46]
9.1. Some basic facts about Clark measures
The study of Clark measures begins with the following key fact.
Theorem 9.1.1 (Herglotz). If u is a non-negative harmonic function on D,
then there is a unique ii G M+ such that u = P/j>, the Poisson integral of [i.
Proof. The set {urdm : 0 < r < 1}, where ur(Q = u(rQ, is a collection of
positive measures satisfying
||urdra|| = / u(r()dm(() =u(0).
Jt
By the Banach-Alaoglu theorem (Theorem 1.6.3), this bounded set of measures has
a weak-* limit measure [i. It follows from the Poisson integral formula and the
definition of weak-* convergence that for each z G D,
[pz(Qd»(C)= lim [ Pz(()u(r()dm(()= lim u(rz) = u(z).
To prove that fi is unique, suppose that u = P[i\ = Pfi2 for some /ii,/i2 ^ ^+-
Then the measure v = \i\— \ii satisfies (Pv)(rQ = 0 for every 0 < r < 1 and £ G T.
However, using eq.( 1.8.4),
oo
n= — oo
and so
p(n) = 0 Vn G Z.
This happens only if v is the zero measure. □
Definition 9.1.2. For an analytic function <fi : D —> D1 and a point a G T, the
function
(9.1.3) ua(Z):=^'a + 4>{Z^-1-^2
a-4>(z)J \a-4>(z)\?
Such maps </> are often called analytic self-maps of I
201

202
9. CLARK MEASURES
is positive and harmonic on B. By Herglotz's theorem, ua = Pfia for some unique
fjba £ M+ • We let
A/> := {l^a - a eT}
denote the family of measures associated with the function 0. We will call A<f> the
family of Clark measures of <fi when 0 is an inner function. When 0 is a general
analytic self-map of the disk, we will call the family A^ the family Aleksandrov
measures of <fi.2
Remark 9.1.4. If /x is any measure in M+, the Herglotz transform of /x,
Jj c, - z
satisfies 5Rii/"/x > 0. One can use this, along with the fact that
w — 1
w h^ -
w + 1
maps {$iw > 0} onto D, to verify that the function
is an analytic self map of the disk. A little algebra shows that
and hence
Thus /x is an Aleksandrov measure for 0, that is to say, /x = /ii E *A</,. If /x is not only
a positive measure but also singular with respect to m, we can use Theorem 1.8.6
to conclude that
lim ]~^)}l = lim (PM)(rC) = ^(0 = 0 m-a.e.
r_>i- |1 - 0(rC)|2 r-1- M J dmK'
But since
lim |l-0(rC)| >0
r—+l~
for almost every £ G T (Theorem 1.9.4), it must be the case that
lim (1- |0«)|) = 0 a.e.,
r-+l_
that is to say, 0 is an inner function. Thus every positive singular measure is a
Clark measure for some inner function.
While we are on the subject of the Herglotz transform, here is a useful formula
for the Herglotz transform of an Aleksandrov measure.
Proposition 9.1.6. J//xa eA<f>, then
Jt(-z a-(j)(z) |a-0(O)|2
The literature has a variety of notation for this. Some call A^ Clark measures (even if cj> is
not inner) while others call A^ Aleksandrov measures (even if <j> is inner). Still others compromise
and call A& 'AC measures.

9.1. SOME BASIC FACTS ABOUT CLARK MEASURES
203
PROOF. For /? e T and w e D, the identity
j3 — w \j3 — w J \/3 — w
|/?-u>j2 |/? - iw|2
yields the equations
a + 4>(z) _ l-\4>(z)\2 , o. 3(a0(z))
■2zT
and
/ ^ dMa(C) = (Pl*a)(z) + i{Qna){z).
Jt ^ ~ z
(P/*a)(*)
But since
i-\4>{z)\2
\a-4>(z)\^
the functions {Q/J>a)(z) and
«Ma)(*) = 2^™^+c.
are harmonic conjugates of the same function and so
^s(~a(f)(z))
\<*-<Kz)\
Now plug in z = 0 to see that
= 3(750(0))
i«-0(O)|2
and so
" + #*) _ Rfa + ^)>\ , ^ {<* + <!>(*)
a — (j)(z) \a — 4>(z) J \a — 4>(z)
l«-^(0)|5
C + z , ... „. 9(75^(0))
= (Pm«) W + i(Q»a)(z) - 2i- _
= /'i±^d/ia(C)-2i7
7T
C-* ^w |a-0(0)1
a
There is also a nice formula for the Cauchy transform of an Aleksandrov
measure.
Corollary 9.1.7. If na eA^, then
7t 1 - C
1 ||/xa||-l . S(750(O))
C^w l-a<^) 2 + |Q-0(O)|2
Proof. Use the identity
P - w 1 - (3w
and Proposition 9.1.6. D

204
9. CLARK MEASURES
Proposition 9.1.8. If fia eA^, then
„ „ _ i-l^>(o)l2
Hence each iia is a probability measure whenever 0(0) = 0.
Proof. Since ua = P/xa, we have
ua(0) = / P0dfia = / dfia = \\fjLa\\.
On the other hand, by the definition of ua from eq.(9.1.3),
u^u>- |Q-0(o)|2-
Combine these two identities to prove the result.
We can also compute the Fourier coefficients of an Aleksandrov measure.
Proposition 9.1.9. 7/0(0) = 0 and na e A^, then
□
yVcwo
n «
fe=i "^
N ,
$>fc /0(<) C"dm(C), n<-l;
k_1 ^
fc=l
^ 1, n = 0.
PROOF. Proposition 9.1.6 says that whenever 0(0) = 0 and fia G A^, we have
(9.1.10) i±^=/£±iW0.
1 - a0(z) J C,~ z
For z G O and £ G T, observe that
c + *
c-z
1 + 2-
C*
l-C*
and so
Looking at the left-hand-side of eq.(9.1.10), we start with the identity
1 + a0(z)
1 + 2:
a0(z)
l + 2]Ta"0(z)n.
1 — acj)(z) ^ 1 — a0(z) _
Bringing in our computation with the right-hand-side of eq.(9.1.10), we get
/^Q(c) + 2 fy (7r dM«(o) = i + 2 f>>wn-
J n=l ^ ' n=l

9.1. SOME BASIC FACTS ABOUT CLARK MEASURES
205
By Proposition 9.1.8, ||/xa|| = 1 and so
OO / n \ OO
(9.1.11) E2" / C" <WC) = £ an<f>(z)n.
n=l ^ ' n=l
Applying the Cauchy integral formula to both sides of eq.(9.1.11), we get, for
each 0 < r < 1,
Since 0(0) = 0, the function
(f)(w)k dw
M=r£Ti ,Ci;'"" *'"'' ~1 JW=r WU+1 27Ti"
6(w)k
w »■ -
,,n+l
is analytic on D whenever k ^ n + 1. Thus, taking limits as r —> 1~ in eq.(9.1.12)
(it turns out to be a finite sum), we have the important identity
|rd^(C) = Eafc^^dm(C), neN.
Take complex conjugates of both sides of this identity to handle the case when
n< 0. □
Definition 9.1.13. For fia e A<j>, let
d/xa = hadm + d<ja, ha £ L1, aa _L m,
be the Lebesgue decomposition of /xa with respect to m. Observe from the Lebesgue
differentiation theorem (Proposition 1.3.8) that
ha = Dfj,a ra-a.e.
Proposition 9.1.14. For m-a.e. ( e T,
M0=1"WC)|a
MU |a-0(C)|2'
Proof. Use Fatou's theorem (Theorem 1.8.6) to see that for m-a.e. ( G T,
l-|0(rO|2
hm ua(r() = hm 2
r—i- r—i- |a - 0(rC)|2
= lim (PMa)(rC)
r—+1-
= (AO(C)
= MC).
a
We can use the fact that (Du)(Q = 0 m-almost everywhere implies v _L m,
along with Proposition 9.1.14, to prove the following corollary.
Corollary 9.1.15. If (f) is an inner function, then fia _L m for every fia G A^.
From Proposition 1.3.11 we deduce the following.
Proposition 9.1.16. The Borel set {D_fia = °o} is a carrier for aa.

206
9. CLARK MEASURES
It will be important for what follows to consider sets of the form (j)~l(B), where
B is a Borel subset of T. Since <fi(() is defined by means of its radial boundary
function
0(C) := lim <j>(rQ,
r—+\-
for ra-a.e. £ G T (and not necessarily every ( G T), we need to be more specific
about what we mean by <fi~l(B). Of course, we can say that (f)~1(B) is defined to
be the set of £ G T for which <fi(() exists and belongs to B. However, since we will
also be considering Borel measures on such sets 0_1(i?), we want to guarantee that
<f)~1(B) is not only clearly defined, but it is also a Borel set. To this end, we begin
with the following simple fact [48, p. 23].
Proposition 9.1.17. For an analytic cj> : D —> B, the set of points £ G T for
which
lim (j){rQ
r—+l~
exists is an Fa$ set.
PROOF. Let (en)n^i be a sequence of positive numbers with en —> 0 and
(^n)n^i be a sequence in (0,1) with rn —> 1. For each k,n G N let F(e^,rn)
be the set of points £ G T for which
|0(siC) - (p{s2Q\ < £k for all sx > s2 > rn
and notice by continuity that F(e^, rn) is a closed set. The set for which the radial
limit of (j) exists is the Fa$ set
oo oo
fc=ln=l
D
Let G be the set of points £ G T for which
lim 4>(rQ
r—+\-
exists and observe from Proposition 9.1.17 that G is a Borel set. With 0r(C) : =
0(r£), the functions XG&r are certainly Borel functions and so the limit function
<f (C) := lim Xg(C)MC)
r—*l~
is a Borel function. The above argument proves the following result.
Proposition 9.1.18. If B is a Borel subset ofB~, the set
is a Borel subset ofT.
By the above proposition, the sets
(9.1.19) £a:=(<n_1(M), «GT,
are Borel subsets of T and moreover, for a Borel set B C T,
(9.1.20) (0*)"1(S)= |J Ea.
aeB

9.1. SOME BASIC FACTS ABOUT CLARK MEASURES
207
Remark 9.1.21. In our analysis below, we will not get overly attached to this
notation and often just write {0 = a} for Ea and <fi~l(G) for (0*)_1(G?). We just
wanted to clarify notation to avoid any confusion later on.
Let us take a closer look at the positive harmonic function
Ua{Z) \a-4>{z)r
Proposition 9.1.22. The set of(eT for which
lim ua(rQ = oo
r—+ 1_
is a Borel set.
PROOF. For M G N and (rn)n^i C (0,1) with rn -> 1, let
F{M,rn):={CeT:ua{rnQ>M}
and notice this set is closed. The set of £ G T for which the radial limit is equal to
infinity is
oo oo oo
PI [jf] F(M,rn).
M = l fc=l n=k
D
The sets {ua = 00} and Ea (from eq.(9.1.19)) are Borel sets and
(9.1.23) {ua = 00} C Ea.
For /j,a G A$, we see from eq.(1.8.8) that
{D-Va = 00} C {ua = ex)}.
However, these sets need not be equal [118, p. 13]. In summary, we have the
following string of containments
{Hl^a = 00} C {ua = 00} C Ea
that yield the following useful corollary.
Corollary 9.1.24.
(1) For lie, G Atft, the Borel sets
{D_fia = 00} C {ua = 00} C Ea
are all carriers for o~a.
(2) cra J_ crp for a=£/3.
(3) For cra-a.e. (gT, 0(C) = a.
An obvious observation from this corollary is that if <fi has no radial limits of
unit modulus (e.g., <fi maps D to a compact subset of D3), then fia <C m.
Let us end this section with a comment about the absolutely continuous part
of /xa, that is,
h A 1-10(C)!2,
ha dm = dm.
\a-(j){Q\2
With somewhat more work, one can also construct an analytic self-map </> such that 4>(B) = D
but cf) has no radial limits if modulus one.

208
9. CLARK MEASURES
Since
^1({0}) = l<^r1({l}) ™-a-e.,
we know that for any a, j3 G T, the measures hadm and hp&m are mutually
absolutely continuous.
9.2. Angular derivatives and point masses
In this section, we examine the points of T where an Aleksandrov measure fia
has a point mass. The reader might want to review the basics of angular derivatives
from Chapter 1.
Theorem 9.2.1 (Nevanlinna [151]). If fj,a G A/> and C € T, then Mq({C}) > 0
if and only if
Z lim (f)(z) = a and |0'(C)| < °°-
Furthermore, in this case,
"a{{Q) = \m\-
Proof. We will make the simplifying assumption that 0(0) = 0 and leave it
to the reader to make the necessary adjustments to the proof if this is not the case.
Before getting started on the proof, we make a preliminary observation. By
Proposition 9.1.6 we have the formula
a + <j>(z) _ f £ + z
/f±>(«.
a — cf)(z)
For fixed ( G T, multiply both sides of this formula by (( — z) to get
(9.2.2) (a + cpiz))-^^ = |(£ + z)izJ.dfia(t).
Notice how for any (gT and for any z in a fixed Stoltz domain with vertex at £,
\c-z\
k-*l
for some c > 0 which only depends on the opening of the Stoltz domain. Since
IC-zl
1 - \z\
Zlim(e + ,)C^-/2C' ** = *
r 2c, if<
\ 0, ot]
' z-+C i~z I 0, otherwise,
we can apply the Lebesgue dominated convergence theorem to obtain
Z lim /(£ + z)^-d»a(Q = 2CMc({C}).
Now apply eq.(9.2.2) to see that
(9.2.3) Z lim (a + 4>(z)) C ~* = 20xa({C})-
z-+c ot - (p{z)
We are now ready for the proof. If /ia({C}) > 0> we use eq.(9.2.3) to see that
Z lim cf)(z) = a
z-+c
and _
<f)(z) — a a(
0'(C) = Zlim
"*-c z-c M(C})'
Hence |0'(C)| < oo.

9.2. ANGULAR DERIVATIVES AND POINT MASSES
209
Conversely, if |0'(C)| < °°> we see from Theorem 1.7.10 that </>'(£) ^ 0. If we
also assume that
Z lim (f)(z) = a,
we can use eq. (9.2.3) again to conclude that
□
Example 9.2.4.
(1) If <j) is any self map of the disk, we know that (/xa)d, the discrete part4 of
lia must take the form
MQ((C})>0
By what was said above in Theorem 9.2.1,
where E = {|0'| = oo}. Notice, since the measure is discrete, how the
sum in eq.(9.2.5) is at most a countable one. This implies the following
interesting fact: for any analytic self map 0 of D and any a G T, the set
Ea\E={CeT:4>(0 = a,\4>'(C)\<^}
is at most countable. This was observed before in [52, Thm. 8.1] and
[194, p. 385].
(2) If <p is & finite Blaschke product, then 0 is inner and so fia = aa. Moreover,
a carrier for aa is Ea = {<$ = a} which must be a finite set {Ci> • * * Xn}
since <fi has an analytic continuation across T. Thus, from eq.(9.2.5) and
the fact that angular derivative at Cj is equal to the ordinary derivative
at (j, we have
(3) If (j) is the atomic inner function
'1 + 2"
(f)(z) = exp
1- z
then iia = cra. Moreover, since <fi has an analytic continuation across
T \ {1}, the set Ea = {<fi = a} is countable and clusters only at the point
£ = 1. Thus the measure fia = aa is discrete and, since the angular
derivative is the ordinary derivative at all points of T \ {1}, we observe
that
\C- II2
Thus
v* = 2^ —2—c'
bSee Theorem 1.3.13 for precisely what we mean by the discrete part of a measure.

210
9. CLARK MEASURES
We end this section with a result of M. Riesz [175] that relates Aleksandrov
measures with the existence of angular derivatives.
Theorem 9.2.6 (M. Riesz). Let (p : D -
(1) If there is a j3 G T such that
be analytic and (GT.
/
< oo,
le-ci2
then \4>'{Q\ < oo.
(2) If
Z lim <f)(z) = a and \(/>'(C)\ < °°>
*-C
ther
JW*§<°° V^T\W.
(9.2.7)
PROOF. Suppose \<t>'{()\ < oo and <£(() = a. By Theorem 1.7.10,
1 - |0(rC)|2
|4>'(C)|= lim
r—+ 1_
l-r2
Since /xQ is an Aleksandrov measure for <f>, we have
(9.2.8)
i-|0K)l2
|a-</>«) |2
Let /3 eT\{a} and observe that
!l
<\-
■d(i,a(£).
dMO
/^<JS"-/i^«—-
lim
i-I^K)!2
^r-i-r2|/3-0K)|2
I0'(C)I
(from eq.(9.2.8))
|/3 -a|2
Conversely, suppose that
< oo (from eq.(9.2.7)).
/
ie-ci2
< oo
for some /? E T. Then, since
we get
K - Kl2
l-r2)
^2,
/
c-c
< 4(1- r^
d/xp(0
l£-CI:
rd/x/3(0
l^-CI2'
Combine this with the identity in eq. (9.2.8) to see that
l-|</»(rC)|2
(9"2,9) l/3-0K)l2
The above inequality says that
<C(l-r2).
</>(Q = lim 4>{rC)
r—>l~

9.3. ALEKSANDROV'S DISINTEGRATION THEOREM
211
exists and \(j>(C)\ = 1- Since
z-+C l-\Z\ r-1" 1~r
it suffices to show, via Theorem 1.7.10, that the right-hand side of the above
inequality is finite. To this end, we estimate
1-|0«)| l-|0(rC)|l + |0K)|l + r
1-r 1-r l + |0(rC)|l + r
tl-|0(rQ|2
1-r2
l-|0(rQ|2|/?-0(rC)|2
1-r2 |/^-0(rC)|2
l-|0(rQ|2(l + |0(rC)|)2
1-r2 |/?-0(rC)|2
1-I0K)!2 1
<2
= 2
^2
^8
1-r2 |/?-0(O|2
<C by eq.(9.2.9).
Now take a lim as r —> 1 to see that |0'(C)| < oo. □
Combining Theorem 9.2.1 and Theorem 9.2.6, one can prove the following
corollary.
Corollary 9.2.10. A fia e A^ has a point mass at C eT if and only if
Z lim (p(z) = a and / < oo for some (3 G T \ {a}.
*-< J If-CI
9.3. Aleksandrov's disintegration theorem
We now focus on a very beautiful and useful disintegration theorem for the
family of Aleksandrov measures. Suppose for a moment that <fi is an analytic self
map of D and 0(0) =0. If g is any continuous function on T and g^ is its iV-th
Cesaro approximation, Proposition 9.1.9 says that
(9.3.1) J (J9N(C) d/ia(C)) dm(a) = JgN(0dm(().
The function
a«-> / 0Jv(C)d/xa(C)
is a trigonometric polynomial in a (in fact of degree less than or equal to the degree
of g^)- Moreover, as N —> oo, the fact that the Cesaro means (see eq.(1.6.4))
approximate g uniformly (Theorem 1.6.5) shows that this function approaches the
function
a^-> / #(C)(WC)
uniformly in a (since \\fia\\ = 1 f°r aU &)• Thus this limit function is indeed
continuous and so the double integral
/(/
5(C)d/xQ(C) dm(a)

212
9. CLARK MEASURES
makes sense. Taking limits as N —> oo in eq. (9.3.1) and again using the fact that
9n —► 9 uniformly, we conclude the following.
Theorem 9.3.2 (Aleksandrov's disintegration theorem). For a continuous
function g on T,
(9.3.3) J (Jg(0 <WC)) dm(a) = Jg(Q dm(<).
Remark 9.3.4. One often sees this disintegration theorem written as
m = / fia dm(a).
By this we mean
m{E) = fia{E)dm{a)
for every Borel set E. There are some delicate points to consider here. For example,
how do we know that a i—» /^a{E) is a Lebesgue measurable function? We will
address these finer points in a moment.
We would like to present another proof of the disintegration theorem where we
can remove the technical condition 0(0) = 0. For any z G D, we have
= / P^)(a)dm(a)
= 1
= [pz(C)dm(Q.
This says that the disintegration theorem works whenever the continuous function
g in eq.(9.3.3) is a Poisson kernel Pz. The result now follows from the facts that
(i) the closed linear span of the Poisson kernels {Pz : z G D} is dense in C(T) (see
the proof of Theorem 9.1.1); and (ii) the total variation norms of /j>a are uniformly
bounded in a (Proposition 9.1.8).
9.4. Extensions of the disintegration theorem
We now generalize the formula
J (jg(0dna(()) dm(a) = Jg(Qdm(0,
initially valid for continuous g, to g G L1. Two technical difficulties appear. First,
the integrals
/
s(C)<W0
do not seem to make sense since a general g G L1 may not be measurable with
respect to the Borel measures /ia. Secondly, it is not obvious that the function
a~J
<?(0<WC)

9.4. EXTENSIONS OF THE DISINTEGRATION THEOREM
213
is Lebesgue measurable. Let us put these technicalities off for a moment and first
show that the disintegration formula is valid for bounded Borel functions. This
requires a few technical preliminaries.
For a bounded Borel function /onT, let
(G/)(Q):=|/(C)dMa(C).
Under the assumption that 0(0) = 0, we know from Proposition 9.1.8 that ||/xa|| = 1
and so
(9.4.1) HG/IU < H/IU
PROPOSITION 9.4.2. The function Gf is continuous whenever f is continuous.
Proof. It suffices to show that if an —> a as n —> oo, then
M<*„ -> Ha weak-*.
For each z G D,
f 1 1
lim / ^d^ari{()= lim _ (by Corollary 9.1.7)
n-+°° Jj 1 - QZ n-+oo 1 - an0(2j
/
1
d/xa(C).
The measures /xan and fia are positive and so, by taking complex conjugates of the
above, we also see that for each z G D,
lim / —:d^an(C)= / —d^a(Q.
Since the linear span of
f 1 1
: z G
u-c* i-c*
is dense in C(T)5, it follows that fian —> /xa weak-* as an —> a. D
We ultimately want to prove that Gf is a bounded Borel function whenever
/ is a bounded Borel function. So far this is true whenever / is continuous. To
extend this result, we will use the following version of the monotone class theorem
[230, p. 37].
Theorem 9.4.3 (Monotone class theorem). Let 6 be a vector space of bounded
real-valued functions on T such that
(1) C contains the constants;
(2) If (/n)n^i i*s a sequence of non-negative functions in C that is pointwise
increasing to a bounded function f, then f G C;
(3) xi £ 6 for every half open arc I C T.
Indeed, if v G M and annihilates this set, then Kv and KV are identically zero on D. From
Proposition 2.1.5 if follows that both v and u belong to Hq. This can only happen if v = 0. The
Hahn-Banach separation theorem completes the proof.

214
9. CLARK MEASURES
Then C contains all of the bounded Borel functions on T.6
Corollary 9.4.4. If f is a bounded Borel function, then Gf is also a bounded
Borel function.
Proof. Note from eq.(9.4.1) that Gf is a bounded function whenever / is a
bounded Borel function. It remains to show that Gf is a Borel function. Let C
be the linear space of real-valued bounded Borel functions / such that Gf is also
a Borel function. Clearly C contains the constants and so 6 satisfies condition (i)
of the monotone class theorem. By the monotone convergence theorem, C satisfies
condition (ii) of the monotone class theorem. If / is any arc of the circle, one
can find a uniformly bounded sequence of continuous functions (/n)n^i such that
fn —► Xi pointwise. From Proposition 9.4.2, Gfn is continuous and by the bounded
convergence theorem, Gfn —> Gxi pointwise. Thus Gxi is a Borel function. Thus
C satisfies condition (iii) of the monotone class theorem and so C contains all the
real-valued bounded Borel functions. For complex-valued /, one handles Sft/ and
9/ separately. □
If / is a bounded Borel function, then by Corollary 9.4.4, Gf is also a bounded
Borel function and so the integral
f(Gf)(a)dm(a)
JT
makes sense. Our first generalization of Aleksandrov's disintegration theorem is the
following.
Theorem 9.4.5. If f is a bounded Borel function, then
//(C)dm(C)= f(Gf)(a)dm(a).
JT JT
Proof. Let C be the collection of bounded real-valued Borel functions / such
that
//(C)dm(C)= f(Gf)(a)dm(a).
JT JT
JT JT
Notice that C is a vector space and by Theorem 9.3.2, C contains the continuous
functions, and in particular, the constant functions. Thus condition (i) of the
monotone class theorem is satisfied.
Now suppose that (/n)n^i C C is a sequence of non-negative, uniformly bounded
functions such that fnff- By the monotone convergence theorem, Gfn / Gf
and, again by the monotone convergence theorem,
//(C)dm(C)= lim [ fn(() dm(()
JT n~*°° JT
= lim (Gfn){a)dm(a) (since fn G C)
n-*oo JT
(G/)(a)dm(a).
JT
it
Thus / G 6 and condition (ii) of the monotone class theorem is satisfied.
There is an associated theorem of Baire that obtains the Borel functions through an iterative
transfinite limiting process [90, 100].

9.4. EXTENSIONS OF THE DISINTEGRATION THEOREM
215
Now let / be any arc of T and let (/n)n^i be a uniformly bounded sequence of
continuous functions (which belong to C by Theorem 9.3.2) that converge pointwise
to xi- The sequence (G/n)n^i is uniformly bounded (since (/n)n^i is) and, by the
dominated convergence theorem, Gfn —> Gxi pointwise. It follows that
[Xl(C)dm(0= lim //„(C)dm(C)
= lim f(Gfn)(a)dm(a)
n—oo/j.
(GXi)(a)dm(a).
/<
7T
/T
Thus C satisfies condition (iii) of the monotone class theorem and so C contains all
of the bounded real-valued Borel functions. One handles bounded complex-valued
Borel functions by taking real and imaginary parts separately. □
Remark 9.4.6. We can use the disintegration theorem to prove that an inner
function $ with #(0) = 0 is a measure preserving map from T to itself. See eq.(7.4.3)
for another proof of this. For any Borel set A C T, we know, with / = xa, m
Theorem 9.4.5 that
f ( f xa(C) daa(C)) dm(a) = / Xa(C) dm(C) = m(A).
This says that
/ cra(A) dra(a) = m(A).
Jt
Apply this formula with A = ,d~l{B) for some Borel subset 5cTto get
(9.4.7) [ aa{$-1{B))dm{a) = m{$-1{B)).
Jt
If Ea = 7T1 ({a}), recall from eq.(9.1.20) that
aeB
Hence, using the fact that Ea is a carrier for aa, the formula in eq.(9.4.7) becomes
/ ||aa||dm(a) = m($-1(B)).
JB
Under our simplifying assumption that #(0) = 0, we know that ||aa|| = 1 for all
a G T (Proposition 9.1.8) and so the above formula becomes
m{B) = m{$-1{B)).
Remark 9.4.8. The operator / —> (Gf) o t9, where t9 is inner and #(0) = 0, is
the 'conditional expectation operator' for the a-algebra generated by $ [12].
To extend the Aleksandrov disintegration theorem even further to not only
bounded Borel functions but to Lebesgue measurable functions, we need the
following technical lemma.
Lemma 9.4.9. Let E be a subset ofT with m(E) = 0. Then fia{E) = 0 for
m-a.e. a G T.

216
9. CLARK MEASURES
PROOF. By basic measure theory, there is a Borel set E' D E with m(E') = 0.
Apply the extended version of the Aleksandrov disintegration theorem
(Theorem 9.4.5), to the Borel function \E' to get
0= [xE>{C)dm(0= [{GxE>){<*)dm(a)= [ »a{E')dm(a).
J JT JT
Thus for m-almost every a G T,
0 = fJLa(E') >[JLa(E)>0.
D
Corollary 9.4.10. Suppose $ is inner and let
£7={C€T:|i?'(C)|=oo}.
If m(E) = 0, then aa is a discrete measure for m-almost every a G T.
PROOF. By Lemma 9.4.9, o-a(E) = aa(E D Ea) (since cra is carried by Ea)
must be zero for all a belonging to some set Q C T with m(Q) = 1. For a G Q,
observe that
o-a = va\{Ea HE)+ aa\{Ea \ E) = aa\{Ea \ E).
By eq.(9.2.5),
aa=aal{EaXE)=£XE\ms<
and so o~a is a discrete measure since E \ Ea is a countable set. □
If / is a non-negative function in Ll(m), let ft, be a Borel function with f = fb
m-a.e. The integral (Gfb)(a) makes sense although it may not be finite for every a.
Let (/n)n^i be a sequence of non-negative bounded Borel functions with fn /* fb
everywhere. By the monotone convergence theorem, Gfn / Gfb m-a.e. By two
applications of the monotone convergence theorem as well as an application of
Theorem 9.4.5,
//(C)dm(C)= lim [ fn(C) dm(C)
= lim (Gfn)(a)dm(a)
n-+ooJj
(Gfb)(a)dm(a)
7T
d/J>a{() ) dm(a) (by Lemma 9.4.9).
Extending this, in the obvious way, to general complex valued / G L1, we have
our final generalization of Aleksandrov's disintegration theorem.
Theorem 9.4.11 (Aleksandrov's disintegration theorem). For f G L1,
//(C)dm(C)= / ( //(C)dMa(0) dm(a).
jt Jt \jt /
For fjLa G Atf), write the usual Lebesgue decomposition d{ia = ha dm + daa.

9.4. EXTENSIONS OF THE DISINTEGRATION THEOREM
217
Corollary 9.4.12. For g e L1,
J (jg(Qdaa(C^j dm(a) = J ^(C)dm(C).
f\<f>\
Proof. By Proposition 9.1.14,
If C e T is such that \<j>(Q\ < 1, then
MO =^(C)(a)-
For such a point £,
(9.4.13) [ ha(C)dm(a)= ( Pm{a)&m{a
JT JT
On the other hand, for fixed a G T, the set Ea = {0 = a} has Lebesgue measure
zero and so
(9.4.14) ha = 0 ra-a.e. on \<j>\ = 1.
With these preliminary remarks, we get
) = i-
JT \JT
g(Odaa(0)dm(a)
T
= 1(15(0 dMa(C)) dm(a) - f ( f <?(C)MC) dm(C)) dm(a)
7T \Jt J JT \JT J
= [g(()dm(0- I ( [ g(C)ha(Odm(())dm(a) (by Theorem 9.4.11)
JT JT \JT /
= fg(C)dm(C) -1(1 g(()h«(()dm(()) dm(a) (by eq.(9.4.14))
JT JT \J\4>\<1 )
= I g(() dm(C) - / g(() ( I MO dm(a)) dm(C)
7t -M</>I<i Vt /
'\<t>\<
I g(Q dm(C) - / 0(C) dm(C) (by eq.(9.4.13))
JT J\(f>\<l
I g(0 dm(C).
7101=1
a
Finally we want to say one last word about the transformation
(Gf)(a)= //(C)dMa(C).
JT
Not only do we have GC C C, GL°° C L°°, GL1 C L1, but G preserves many
other classes of functions. In fact GLP C Lp (1 < p < oo), G(BMO) C BMO,
G(VMO) C VMO, G£*g C B8vq, where 5^ are the Besov classes. See [13] for
details. The paper [136] discusses other properties of G.

218
9. CLARK MEASURES
9.5. Clark's theorem on perturbations
For an inner function #, with the simplifying assumption that #(0) = 0, recall
from the previous chapter the compression operator
s# = p#s\r(H2)
and the family of rank-one unitary perturbations
Also recall that these are the only unitary rank-one perturbations of S&. The main
theorem of this section is one of Clark [46] (see Theorem 9.5.5 below) which says
that Ua is unitarily equivalent to the multiplication operator
Z : L2(aa) - L2(aa), (Z/)(C) := </(<),
where o~a is the Clark measure associated with the inner function $ and the point
a G T. Since Ua is a cyclic unitary operator (Theorem 8.9.10), the spectral theorem
(Theorem 1.5.6) says that Ua can be represented as 'multiplication by z1 on some
L2(n) space. Clark's theorem identifies this spectral measure \i as the Clark measure
It will turn out that the unitary operator that intertwines Ua and Z is
Va : L2(aa) -> Hol(D), (Vag)(z) := (1 - a&(z))K(gd<ra)(z).
Using the formula
from Corollary 9.1.7, the operator Va can be written as
_ K(gdaa)
The function Vag is a special example of the normalized Cauchy transform which
will be explored in greater detail in Chapter 10. Clark's theorem says that Va is a
unitary operator from L2(aa) onto ^(H2) and
vaz = uava.
Our presentation of this result follows [188] and will take several steps. One of
them is, of course, to prove that Vag is not only an analytic function D, but belongs
to7T(#2). To this end, let
CA(20 = —L-, MeD,
1 - Xz
denote the Cauchy kernels (which are the reproducing kernels for H2). Recall from
the previous chapter (see Proposition 8.4.2) that the kernels
kx(z) := {P*Cx){z) = V ; V ;
1-Xz
are the reproducing kernels for $*(H2).
For fixed A G D, the function C\(Q = (1 — ^C)_1 is continuous on T and so
certainly belongs to L2(o~a).

9.5. CLARK'S THEOREM ON PERTURBATIONS 219
Lemma 9.5.1. J/tf(0) = 0 and A,/x € B, then
Proof. For complex numbers /?i,/32 a computation reveals that
1 +_/| l+/?2 = 2_ 1 - A/32
l-/?i l-/32 (1-/30(1-/32)
Thus
(C^Cu),*, v = / —i= ^daQ(C)
2(1-A/x)7tVi-AC 1-Cm/
l /l+m?(A) l + m?(/i)\ ,_ .i# ni„,
+ z _a; ; (Proposition 9.1.6)
2(1 - A/x) \, 1 - m?(A) 1 - cw?(m)
1 1 - 0(A)0(ai)
(1 - cm?(A))(1 - c«?(/x)) 1 - A/x
(l-m?(A))(l-m?(/i)) M
/ fcA fcM \
\l-atf(A)'l-a^)/H2'
Corollary 9.5.2. 7/i?(0) = 0 and A G D, tfien
yacA = —^=fcA.
1 - m?(A)
Proof. Recall that
(Vag)(z) = (1 -a#(z))K(gdaa)(z)
and so
(V«Ca)W = (1 - a#(z)) f -^-= -Xrda^C)
Jt 1 - CA 1 - C,z
= (l-ai>(z))(Cx,Cz)LHaa)
= (1 - atf(z)) /—%=, 7-*=) (Lemma 9.5.1)
\l-atf(A) l-crd{z)/H2
(k\,kz)H2
□
1 - cn?(A)
1 - m?(A)
D

220
9. CLARK MEASURES
We know that Va maps C\ to a constant multiple of k\. Moreover, if
n
3 = 1
is a finite linear combination of the Ca's, then
H#llWQ) = (9,9)L*(<ra)
n
3,1=1
= \^ CjCi ( 3 , l ) (Lemma 9.5.1)
n
= ^^(14^,14^,)^ (Corollary 9.5.2)
= (Vaff, Vag)H2
= \\VagtfH*.
Thus Vq, is an isometry on the linear manifold of L2(o~a) generated by the C\ to
the linear manifold of $*(i/2) generated by the kernels k\. The latter manifold is
dense in $*(H2).
Lemma 9.5.3. The linear manifold generated by {C\ : A G B} is dense in
Proof. Suppose that g G L2(aa) and
(0,Ca>lv«)=° VAgD-
Working out the inner product says that the Cauchy transform K(gdaa) vanishes
on D. However, by Corollary 9.1.15, the measure gdo~a is singular with respect to
m and so, by Proposition 2.1.5, is the zero measure. Thus the manifold generated
by {C\ : A G D} is dense in L2(aa). □
We leave it to the reader to fill in the details to prove the following.
Proposition 9.5.4. If $(0) = 0, the operator Va is a unitary operator from
L2{aa) onto$*{H2).
We are now ready for Clark's theorem. Recall the bounded multiplication
operator
Z : L2K) - L2(aa),
(Zg)(C) = Cs(C)
and notice that
(z*5)(c) = Cs(0-
Theorem 9.5.5 (Clark). When -0(0) = 0, VaZ = UaVa.
Proof. It suffices to prove that
vnz*v: = u*

9.6. SOME REMARKS ON PURE POINT SPECTRA
221
For an analytic function /ionD (it need not belong to any special class of functions),
let
h - h{0)
Bh:=
z
Recall that 5* is the adjoint of the forward shift (Sf)(z) = zf(z) on H2 and that
5* =B\H2.
For g G L1(aa), let us use the shorthand
Kag := K{gdaa).
A routine computation shows that
(9.5.6) B(hf2) = hBf2 + h(0)Bh
and
(9.5.7) KaZ* = BKa.
Also remember that #(0) = 0 and so
B*=±.
z
Thus for g G L2(aa),
{VaZ*g)(z) = (1 - ad{z))(KaZ*g)(z)
= (1 - a#(z))(BKag)(z) (eq.(9.5.7))
= B((l - m(z))Kag)(z) + aB$(g, l>La(ffa) (eq.(9.5.7))
= (S*Vag){z) + a(g,l)LHaa)-z
= (S*Vag)(z)+a(Vag, Val)H'- (Proposition 9.5.4)
z
= (S*Vag)(z) + a(Vag,l)H2- (note Val = 1).
z
Now use the function g = V*f for an / G $*(i72) in the above identity to see that
VaZ*VZf = S*f + a(f, 1)H2- = U*f.
z
D
A nice identity relating Clark measures, unitary perturbations, and Cauchy
transforms is the following formula.
COROLLARY 9.5.8. Suppose #(0) = 0. Then for z G D and a G T,
((l-zUZ)-1l,l) = (K»a)(z).
9.6. Some remarks on pure point spectra
One of the problems we considered in Chapter 8 was the kernel function
approximation problem: for a sequence (An)n^i C D with |An| —> 1, does the sequence
of kernel functions {k\n)n^i have dense linear span in $*(iif2)? As discussed earlier,
the approach one takes to this problem is to observe from Theorem 8.6.1 that for
CeT,
kc e 0*(ff2) «■ |0'(C)| < oo.

222
9. CLARK MEASURES
Moreover, from Theorem 8.9.9,
{kc : |tf'(C)| < oo,i?(<) = a}
are all the eigenvectors for Ua. But since Ua is unitary, we know that
From our discussions in Chapter 8, we saw that if
{fcc:|f?'(C)|< 00,0(0 = a}
has dense linear span in $*(i/2), we could apply a Paley-Wiener type approximation
theorem to our original problem. By Theorem 9.5.5, the Clark measure o~a is
a spectral measure for Ua. If aa happens to be discrete, the spectral theorem
says that Ua would have pure point spectrum (see Definition 1.5.8) in that the
eigenvectors k^ form a spanning set for $*(i/2). The following theorem is one
particular instance of when this occurs. The proof follows from the above remarks
and Corollary 9.4.10.
Theorem 9.6.1. Suppose #(0) = 0 and the set
{CeT: |^(C)| = 00}
has Lebesgue measure zero. Then for m-a.e. a £T, the Clark measure o~a is discrete
and so Ua has pure point spectrum.
There are examples of families of Clark measures A# for which o\ is a discrete
measure but cra is singular continuous for all a G T \ {1} [62].
9.7. Poltoratski's distribution theorem
We know from Theorem 7.4.4 that for any fi e M,
lim 7rym(\Kfi\ > y) = ||/xj.
y-+oo
In this section we use Aleksandrov measures to prove Poltoratski's generalization
of this result [163].
Theorem 9.7.1 (Poltoratski). If fie M, then
nyx\Ki_i\>y • rn -> \fji8\ weak-* as y -> oo.
The key to proving this is the following.
Theorem 9.7.2. If fi e M+, then
^VXQii>y • m -> lis
nyXQv<-y • rn -> \i8
^VX\Qix\>y "m-+2fis
weak-* as y —> oo.
Before getting to the proof of Theorem 9.7.2, let us show how one can prove
Theorem 9.7.1 from Theorem 9.7.2 at least for /x G M+7. We already know that
2{Kfj){z) = {Pfj){z) + i{Qfj){z) + 1, z £ D.
To avoid some technical details, we will only prove Poltoratski's theorem for positive
measures.

9.7. POLTORATSKI'S DISTRIBUTION THEOREM
223
Taking non-tangential limits we get
2(*7x)(C) = g({) + WXC), rnrSL.e. C G T,
where
Since /x G M+, g and Q/i are real-valued and so
\Qn\ < 2|A>| < M + IQ^I ™-a.e.
Thus for y > 0 and / G C(T) with / ^ 0, we get
ny / /dm < 27r- / /dm.
J\Q»\>V 2 J\Kn\>y/2
By assuming the conclusion of Theorem 9.7.2, we know that
Mm ny f dm = 2 / /d/xs,
and so
(9.7.3) / /d/xs < lim ?ra / f dm.
For any e > 0
{2|KM| > y} c {M > ey} (J {|QM| > (1 - C)y}
and so for any / G C(T), / ^ 0,
7n/ / / dm ^ ny f dm + ny f dm
= ny f dm + th/(1 - e) / / dm.
J|5|>ey 1-e JlQ/xIXl-c)!/
Since
™(M > ey) = o(l/y),
8it follows that
2 lim 7ra / / dm ^ / /d/xs
u-+°° y|K-/x|>u i - e y
and hence, letting e —> 0,
lim ny f dm ^ f dfis.
y^°° J\Kri>y J
Combining this inequality with the one in eq. (9.7.3) yields
(9.7.4) lim ny [ f dm = [ f dMs, / G C(T), / ^ 0.
Now write any complex-valued / G C(T) as
/ = ((»/)+ - (»/)_) + i((3/)+ - (3/)_)
Indeed, if At := {\g\ > £}, then m{At) —*• 0 as £ —*• oo (Proposition 1.2.4). Moreover, by
basic measure theory, / |^|dm —*■ 0 as £ —*• cxd [149, p. 148]. Finally, tx^t ^ l^lx^t and so
£ra(At) ^ / |<7|dra.
J At

224
9. CLARK MEASURES
and apply eq.(9.7.4) four times to get
lim ny [ f dm = //d/is, / G C(T),
that is to say,
lim 7ryx\Ki_i\>y • m = \is weak-*.
The proof Poltoratski's theorem (Theorem 9.7.2), requires a few preliminaries.
Suppose that \i is a positive, singular, probability measure. By Remark 9.1.4,
\i = <Ji G A$ for some inner function $ with #(0) = 0. Moreover, each o~a G *Atf is
also a probability measure. Let
tu:R->T, w(x):=^-^.
x — %
The following technical lemma is a version of Aleksandrov's disintegration theorem
(Theorem 9.3.2).
Lemma 9.7.5. For g e C(T) and y > 0,
I g(C) dm(C) = / ( I 9(C) daa(C)) dm(a).
jQai>y Jw(y,oo) VT /
PROOF. By Corollary 9.1.24, each measure aa G A? is carried by the Borel set
Ea = {$ = a}. Notice that
l + #(z)
1 - i?(z)
Thus, since ai _L m,
(ff<7l)(*) = (P(7i)(z) + 2(Q<7i)(*), * G :
lim (Pai)(rC) = 0 m-a.e. (Gl
i—>i-
Using this and taking radial limits above, we see that
1 + 0(0
i - 0(c)
Let us make the observation that if
i{Q(Ti){C), m-a.e. C £ T.
:i + ^(0 ___i,
then
y = (^i)(C) = -^TT^gy = ™ WO),
tf(0 = ~ = ™(y)-
y-l
Hence, the two sets
{C e T : (Q<7i)(C) > y} and {CGT:0(0 £%, oo)}
are equal m-a.e. and so
(9.7.6) {Qax >y}= \J Ea m-a.e.
a£w(y,oo)
Let /i be the characteristic function of the Borel set9
U Ea.
a£w (y,oo)
By eq.(9.1.20), this set is equal to $ 1(w(y, oo)) and hence is a Borel set.

9.7. POLTORATSKI'S DISTRIBUTION THEOREM
225
Then for any g G C(T),
/ 9(C) dm(C) = / g(()h(() dm(C) (by eq.(9.7.6))
JQ<?i>y Jt
= J (J s(OMC)d<ra(C)) dm(a) (by Theorem 9.4.5)
= / f / <?(C)dMC)) dm(a) (^ Corollary 9.1.24).
Jw{y,oo) \Jj /
D
We now proceed to the proof of Poltoratski's theorem.
Proof of Theorem 9.7.2. We will prove, for any \i e M+ with ||/x|| = 1,
that
weak-*.
Let us first prove this when 11 is also a singular measure. From Remark 9.1.4
and eq.(9.1.5), [i — o\ G A$ for some inner function $ with #(0) = 0. From
Proposition 9.4.2 it follows that
°a —► 0"i> weak-* as a —> 1.
Routine estimates show that
1
Try
m(it;(?/,oo))
and so for any g G C(T),
lim 7ry / g dm = lim —-—- — / g dm
y^°° JQ<r1>y y^°° rn{w{y, oo)) Jq^^
when these limits actually exist (we will show they do). Now recall from
Proposition 9.4.2 that
(Gg)(a):= [ g(()daa(()
JT
is continuous. Use Lemma 9.7.5 to see that
—-— r- / gdm=—-— — / (Gg)(a)dm(a).
m(w(y, oo)) JQ<Tl>y m(w(y, oo)) yw(l/j0o)
However the last integral is the average of (Gg)(a) over the arc w(y,oo) and this
arc approaches the point w(oo) = 1. Thus
lim —-— — / (Gg)(a)dm(a)
y-oo m{w{y, oo)) Jw{yi0o)
= ]im(Gg)(a)
= (Gg)(l)
= f g(C)dMC)
JT
IT
since Gg is continuous. Hence we have shown that
^yXQa1>y -m-xTi
weak-* as y —> oo.

226
9. CLARK MEASURES
To prove that
^VXQii>y • ra -> \la
weak-* as y —> oo for any positive measure [i, note that if
fJL = fJLa + Pa
is the usual Lebesgue decomposition, then
Qli = Qua + QMs,
m{\Qfjba\ >y) = o(l/2/),
and
lim 7ryxQ^s>y -m = fji8 weak-*.
y-*oo
For any e > 0, observe that
{Qfi > y} C {Qfia > ey} U {Q/^s > (1 - e)j/}.
The argument used to prove that Theorem 9.7.1 implies Theorem 9.7.2 can be
applied here to show that for any positive g G C(T),
lim ny / g dm ^ / g d/xs
For the other direction, notice that
{QHs > y} C {0^ > (1 - e)y} U {Q^a < -ey}
and again observe that the argument used to prove that Theorem 9.7.1 implies
Theorem 9.7.2 can be applied here to show that for any positive g G C(T),
/ g d/j^s ^ lim ny / g dm.
JT y-+oo JQu>y
/J y-+oo JQii>y
The result follows as before. □
We make a final remark that Aleksandrov measures have made their way into
many areas of analysis. For example, they have been used to study composition
operators on Hp, rigid functions, perturbations of unitary operators, and the Nehari
problem (to mention a few). See the expository papers [137, 166] for extensive
bibliographies.

CHAPTER 10
The normalized Cauchy transform
10.1. Basic definition
When # is an inner function with #(0) = 0 and a G T, we know from
Theorem 9.5.5 that the operator Va : L2(cra) —> $*(H2) denned by
K(fdaa)
(10.1.1) Vaf
Kan
is a unitary operator that intertwines the rank-one unitary perturbation Ua on
i!)*(H2) with the multiplication operator (Zg)(Q = C#(C) on L2(aa). In this
chapter, we examine a generalization of Va.
For n;GlD notice that
1 - sft(w) 1
\l-w\2 ^ 2
and so for [i G M+,
(10.1.2) R(/fy)(*) = / \~*!f*l MO > ^ Vz G D.
yT |i — C^r 2
Thus for each / G Ll(ii), we can define the function
__ K(fdy)(z)
By eq.(10.1.2) this function, called the normalized Cauchy transform, is analytic
on D and, since VM is the quotient of two Hp functions (Theorem 2.1.10), has
m-a.e. defined boundary values given by
(VM/)(C) := lim (V„/)K).
r—+1-
When # is inner, #(0) = 0, and /j, = aa e A#, then
Thus VM is a generalization of the unitary intertwining operator Va.
10.2. Mapping properties of the normalized Cauchy transform
The first result in this chapter is a minor but useful extension of Kolmogorov's
theorem (Theorem 3.4.1).
Proposition 10.2.1. For \i G M+ there is constant C > 0 such that for any
m(rV|>y)<^||/||LiW Vy>0.
227

228
10. THE NORMALIZED CAUCHY TRANSFORM
PROOF. Since \Kfi\ > IMI/2 on T, we know that for any y > 0,
{\vl*f\>y} = {\KfM>\K»\v}
c{\KfM>^v}.
Thus, by the Kolmogorov weak-type estimate in Theorem 3.4.1,
m(|VM/| > yK m (V/d/x| > ^V)
□
Remark 10.2.2. The previous result says that VPiL1(fi) C H1'00. One can see
this by observing that V/LlL1(^,) C N+ (since it is the quotient of an Hp function
and an outer function) and has L}'°° boundary values. Now use Theorem 1.10.4.
One concludes from this and Proposition 1.10.1 that
0<p<l
Proposition 10.2.3 (Aleksandrov [14]). For fi G M+, the operator VM is a
contraction from L2(/j>) to L2(m), that is to say,
l|VM/||L*(m) < ll/IU^M)
for all f eL2(v).
Proof. Without loss of generality, we can assume that /i is a probability
measure. If/x is not only a probability measure, but also singular, then, by Remark 9.1.4,
\i — o\ G A® for some inner function # with #(0) = 0. In this case, VM is the unitary
operator V\ from eq.(l0.1.1) and so
(10.2.4) l|VM/||L2(m) = ||Vi/||La(m) = ||/||i2(M).
For a general probability measure /i G M+ (not necessarily singular), use
Proposition 1.6.10 to approximate /i in the weak-* topology with a sequence of singular
probability measures /in, that is,
lim f gd^n= (' gdn V# G C(T).
Again by Remark 9.1.4, each /xn is a Clark measure for an inner function $n with
#n(0) = 0. Thus from eq.(10.2.4) we know that for each n G N,
l|vMB/||La(m) = \\f\\m,n) v/eC(T).
Now use the weak-* convergence of the /xn to /x to conclude that for each z G D,
lim (VMn/)(z) = (VM/)(z).

/
10.2. MAPPING PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 229
Thus for each 0 < r < 1 and / G C(T),
/l(VM/)(rC)|2dm(C)= / lim |(VMrJ)K)|2dm(C)
< lim / l(VMn/)K)|2dra(C) (Fatou's lemma)
n—>oo JT
^ lim / |(V^/)(C)|2dm(C) (VMn/ e #2)
n—+00 JT
M / |/(C)|2dM„(C) (eq.(10.2.4))
n-+oc JT
|/(C)|2d/x(C) (weak-* conv.)
Since V„f G #p for all 0 < p < 1 (Remark 10.2.2), we know that (VM/)(rC) ->
(VM/)(£) ra-a.e. as r —> 1~, and so can use Fatou's lemma along with the above
string of inequalities to conclude that
l|VM/|U2(m) ^ \\fh>M v/eC(T).
To complete the proof, use the density of the continuous functions in L2(ii). □
COROLLARY 10.2.5 (Aleksandrov). For \i G M+ and l<p^2,V^isa
continuous operator from Lp(fi) to Lp(m). Consequently, VM is a continuous operator
from L*V) to Hp.
PROOF. Since Lp(n) C L1^) for p ^ 1, the operator VM is denned on Lp(n).
Using the fact (Proposition 10.2.1) that VM : L^/x) —> L°(ra) satisfies
m(|VM/|>y)<^||/||L1(M),
and that VM : L2(/x) 1—> L2(m) is continuous (Proposition 10.2.3), we can apply the
Marcinkiewicz interpolation theorem [79, 85, 207], to complete the proof of the
first part of the theorem.
To see the second part, observe that for fixed p G (1,2] and / G Lp(ii), the
analytic function VM/ belongs to Hs for every s G (0,1) (Remark 10.2.2) and has
Lp(m) boundary values. Theorem 1.9.12 implies that VM/ G Hp. □
Remark 10.2.6. If /x G M+ and is singular, then /x = /xi G ^1$ for some
inner function 7?. By Clark's theorem (Theorem 9.5.5), VM(L2(/x)) = tf*(H2). If
/j G M+ but not necessarily singular, then VM(L2(/x)) is the deBranges-Rovnyak
space associated with /x [188].
If fi = aa, then VM = Va and we can say much more. Indeed, as in Lemma 9.5.3,
the linear span of the Cauchy kernels
Cx(0 *
1-AC
forms a dense subset of Lp(aa). As was the case with p = 2, the kernels
1 - Xz

230
10. THE NORMALIZED CAUCHY TRANSFORM
belong to ^*(iJp)1 and have dense linear span. From Corollary 9.5.2, assuming
#(0) = 0, we have the formula
VaCx = ^=—kx.
l-ai?(A)
From these facts, together with the continuity of Va : Lp(aa) —> Hp when 1 < p ^ 2
(Corollary 10.2.5), we obtain the following corollary.
Corollary 10.2.7. Suppose $ is inner and #(0) = 0. If 1 < p < 2 and
aa G A#, £/ien
ya(LP(aQ)) c r(#p).
For p ^ 2, we look at V~l on $*(HP). Recall that V^ is a unitary operator from
L2(o~a) onto $*(#2). But since p ^ 2, we have, via Holder's inequality, the obvious
containment $*{HP) C $*(H2) and so V^1/ G L2(aa) whenever / G $*(HP).
Corollary 10.2.8. Suppose $ is inner and #(0) = 0. If p ^ 2 and aa e A$,
then
PROOF. Since the dual of $* (#p) can be identified with ^(iJ9), where 1/p +
1/q = 1, via the pairing
/
7T
fgdm,
T
(see Proposition 8.2.8) and Va : Lq(aa) -> 7?*(iJ9) (since 1 < q ^ 2), the result
follows by noting that the adjoint of Va : L«{aa) -> TT(i^) is V^1 : $*(#?) ->
£p(t7tt). □
We end this section by mentioning some related results. When 1 < p ^ 2
and /i G M+, the operator VM maps Lp(aO continuously to Hp. Moreover, when
\i = cra G ^ (tf inner with tf(0) = 0), then VM = Va and VaLp{aa) C $*{HP). Is
this map ever onto? When p = 2, the statement that VaL2(aa) = $*(H2) is part
of Clark's theorem. For other values of p, there is this result of Aleksandrov [14].
Theorem 10.2.9 (Aleksandrov). If 1 < p < 2 and VaLp{aa) = $*(HP) or if
2 < p < oo and V~x ($* (Hp)) = Lp(o~a), then o~a is a discrete measure.
Can the continuity of VM : Lp(fi) —► Hp for 1 < p < 2 be extended to p > 2?
Theorem 10.2.10 (Aleksandrov). Suppose p, G M+ ana7 is singular. If p > 2
ana7 VMC(T) C i/p, £/ien p is a discrete measure. Thus if VM is a continuous
operator from Lp(p) to Hp, then p is discrete.
10.3. Function properties of the normalized Cauchy transform
If
oo
3 = 1
Recall the definition of tf* (#p) from Remark 8.2.6.

10.3. FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 231
is a discrete measure in M and / G L1 (/x), observe, using the dominated convergence
theorem, the two formulas
lim(l-r)(i^)(r0)=c„
\im(l-r)(KfdMrQ = cjf(CJ).
These two formulas say that
lim (VM/)(rC) = lim K£*$™ = /(C) /i*e.
A theorem of Poltoratski [162] extends this observation to general measures.
Theorem 10.3.1 (Poltoratski). If fj, e M and f G L^/x), then
lim(VM/)(rC) = /(C) »s-a.e.,
r—*l~
where /is is the singular part of /i with respect to Lebesgue measure m.
Remark 10.3.2. Notice in the theorem that fi need not be a positive measure
and so VM/ is a meromorphic function on B. However, since VM/ is the quotient
of two Cauchy transforms, it is a function of bounded type and as such has non-
tangential limits m-a.e. The significance of Poltoratski's result is that VM/ has
non-tangential limits /i-a.e. and that /is-a.e., these limits equal /.
The proof we present here, due to Jaksic and Last [107], takes a slightly
different form than the one presented by Poltoratski. Recall from Chapter 7 that M(R)
denotes the finite Borel measures on M, M+(M) denotes the positive ones, and mi
denotes Lebesgue measure on IR. For fi G M(IR), let
F^z)= J j^dfjL{t), zeC\R,
be the Borel transform of [i and note that FM is analytic onC\l. For convenience
of notation, let
Ff^z)=J^-zf(t)dv(t)
whenever / G Ll(n). The Jaksic and Last version of Poltoratski's theorem is the
following.
Theorem 10.3.3. For \i G M(R) and f G Ll(ii),
FffJL(x + iy)
lim —Erv r = fix)
y-o+ F^x + iy) M ;
for iis-a.e. xGR, where iis is the singular part of n with respect to Lebesgue measure
m\.
Before we get to the proof of this, let us see how Theorem 10.3.3 proves
Theorem 10.3.1. We only provide an outline and leave some of the details to the reader.
The function
iz + 1
^(z) = -
iz — 1
maps the upper half plane {^sz > 0} onto D. For simplicity, we assume that the
point C = 1 is not in the support of/iGM and so the measure v := fi o i/j belongs
to M(M) and has compact support. With the substitutions,
C = if>{x) eT and w = il>(z) G D, x G M, Qz > 0,

232
10. THE NORMALIZED CAUCHY TRANSFORM
the Cauchy transform
1 -MO
I
1-Cw
becomes
{iz - 1) / = {iz - l)F{_lx_1)u(z).
J X Z
Hence
( )(w) = F( 1)ifo^(z)
and the result will follow.
To prove Theorem 10.3.3, we need a few preliminaries. Let
9z(t):=Q , zgC+, teR
be the Poisson kernel for the upper half-plane C+ := {$sz > 0} and notice that
For (i e M+(R),
(^)(z):=y"T2(i)dM0
will denote the Poisson integral of ii. A version of Fatou's theorem (Theorem 1.8.6)
says that whenever ii G M+(M) and (Dfi)(x) exists (and we include the possibility
that (Dfi)(x) = oo), then
lim CPfiMx + iy) = 7r(Dfi)(x).
y—o+
An upper-half plane analog of Proposition 1.3.11 says that {x : 0 < (Dp)(x) < oo}
is a carrier for fia and {x : (Dp){x) = oo} is a carrier for fis. Here
is the Lebesgue decomposition of ii. Thus, on a set of full //-measure we have
(10.3.4) 0 < lim (9^){x + iy) ^ oo.
y->o+
When fi G M+(M) and / G Ll(ii) is real-valued,
9tf>M(*) = {Wv){z).
We begin with some technical lemmas.
Lemma 10.3.5. For /x G M+(M), fis is carried by
x: lim \F^(x + iy)\ = oc
y->0+
Proof. Observe that
\F,(z)\>\ZF,(z)\ = (^)(z).
Thus
<x: lim \Fu(x + iy)\=oc> D <x: lim (CP^)(£ + zy) = oo >
I y->o+ J ^ y->o+ J
D {x : (Dii)(x) = oo}.

10.3. FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 233
From the discussion above (or Proposition 1.3.11), this last set is a carrier for
Ms- □
For fi G M+(M) and h G Ll(n), let
(Mh)(x):=sup \ f 6 \h(t)\dn(t).
e>0 fj,(x-e,x + e) Jx-e
This next inequality is a generalization of the famous Hardy-Littlewood
inequality for maximal functions [207, p. 5] (see also [182, p. 137] for a related
result).
Lemma 10.3.6. There is a constant N independent of any h G L1(/x) and y > 0
so that for any bounded interval [a,b],
N
fi([a,b]n{Mh>y})^ -\\h\\LHfl).
PROOF. For fixed y > 0, each point of the set {x : (M^)(x) > y} is the center
of a closed interval Ix for which
/.
\h\dfj, > yfi(Ix
For a compact interval [a, 6], consider the family of intervals
7:={Ix:x£ [a,b}C){Mh>y}}.
Apply the Besicovich covering theorem2 to extract a countable covering of [a, b] D
{Mk > y} by intervals of the form
N oo
U U *»>
k=lj=l
where N is some universal integer (a dimensionality constant) and for each k =
1, 2, • • • , N, the intervals
belong to J and are disjoint. Finally, we have
N oo
M([a,6]n{A//l>2/})<^^M^,fc)
k=u=i
N oo
k—i -. —1 & J li.k
N
^ —||^||l1(/x) (since Ij^ are disjoint for each fixed k).
D
Lemma 10.3.7. Let \i G M+(M) and h G L1{jjl). Then for each x eR,
Besicovich covering theorem [68, p. 30]: There is a universal constant N with the following
property: if jF is any collection of non-degenerate closed intervals in R with sup{diam(7) : / G
jF} < oo and if A is the set of centers of the intervals in jF, then there exists Gi, • • • Gjy G $" such
that each Gi is a countable collection of disjoint intervals from jF and A C U™=1 U/ec^ /.

234 10. THE NORMALIZED CAUCHY TRANSFORM
Proof. For v G M+(M) and non-negative / G Ll{v), there is the distributional
equality [123, p. 26] (see also Proposition 1.2.4)
(10.3.8)
j f(t)dv(t) = J v(f>s)ds.
We will apply this to the function
f(t) — -CPx+2y(£) —
2
y"*ws (t-xy + yf
A computation shows that whenever y > 0 and s G (0,1/y2),
Y^R: (t _ ^2 + yi >sj = (x-q,x + q), q = q(s, y) = yj - - y
Using the distributional equality in eq.(10.3.8), we get, for any v G M+(M),
1 f1^2
(10.3.9) -{9v){x + iy)= / v(x - q{s,y),x + q{s,y))ds.
y Jo
Let /(a, r) = (a — r, a + r) and notice from the definition of (M^)(x), that for
any s and y
(Mh)(x)> \ I |ft|dM
V>(I{x,q{s,y))) Jl(x,q(s,y))
and so
M(/(x,g(s,z/)))(M^)(x)^ / |ft|dAx.
7/(x,g(s,y))
Now integrate to get
ri/y2 ri/y2 ( r \
(Mh)(x) / »(I(x,q(8,y)))ds > / \h(t)\dfi(t) ds.
JO JO \Jl(x,q(s,y)) J
Using eq.(10.3.9) twice, once with v = ii and again with dz/ = \h\ d/i, we get
D
Proposition 10.3.10. For \i G M+(R) and real-valued f G Ll(n),
v (yfv)(x + iy) f( ,
lim ,^ w — = fix)
for ii-a.e. xGM.
Proof. To prove the result, it suffices to show that
M|W-/(xM» + iy)|=0
y->0+ (?^)(X + ^)
To do this, let (gn)n^i be a sequence of continuous functions that approximate /
in the norm of L1(/x) and let hn = / — gn. Thus ll/inllL1^) -> 0- Without loss of
generality, we can also assume, by passing to a subsequence, that hn —> 0 /i-a.e.
The continuity of the gn's imply that for every x G M,
lim {9(gn - gn(x))ii)(x + iy) = 0.
2/-+0+

10.3. FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 235
From eq.(10.3.4),
lim (?ii){x + iy) > 0 fi-a,.e.
y->0+
and thus
(10.3.11) lim .^ . . =0 zx-a.e.
Hence for /x-a.e. ieM and every n € N,
Now use Lemma 10.3.7 to see that this last quantity is bounded above by
(10.3.12) (Mhn)(x) + \hn(x)\.
Let
To establish
\i (< x : lim W(a: + zy) > 0 i J =0,
it suffices to show that for each fixed t > 0 and fixed compact interval [a, 6],
\i ( [a, 6] p| J x : lim W(z + iy) > t i j = 0.
For each fixed t > 0, it follows from eq.(10.3.12) that for any n G N
(10.3.13) | I5i+W(z + zy) >t| C {M^ > t/2}(J{|/in| > t/2} /x-a.e.
Let e > 0 be given. Since /in —► 0 /x-a.e. we can use Egorov's theorem3, to
produce a set A C R with /x(IR \ A) < e/2 and such that /in —► 0 uniformly on A.
Thus for n ^ if, \hn\ < t/2 on A and so
(10.3.14) /x(|/in| >t/2) <e/2 for all n ^ if.
Since ||^n||L1(/x) ~~> 0> we can> by making X even larger, assume that
IIMli(/x) ^ 4^7 for a11 n^K^
where N is the constant from Lemma 10.3.6. By Lemma 10.3.6,
N
(10.3.15) »([a,b]n{Mhn >t/2}) < ^\\hn\\LHtl) < e/2
for n ^ X. From eq.(10.3.13), eq.(10.3.14), and eq.(10.3.15),
/x ([a,6]n I fim W(x + n/) > tij < e/2 + e/2 = e
and the result follows. □
'Note that \i is a finite measure.

236
10. THE NORMALIZED CAUCHY TRANSFORM
Corollary 10.3.16. For any \i, v e M+(M),
lim^±M=0
y-+o+ (0»(x + n/)
for almost every x G K. with respect to the part of ii that is singular with respect to
v.
Proof. Let
be the decomposition of /x with respect to z/ (see Remark 1.3.12). Since \ivs and z/
have disjoint carriers, there is a function / with
(10.3.17) / = 1 ^-a.e and / = 0 z/-a.e.
By Proposition 10.3.10,
(For typesetting purposes, we are suppressing the x + iy.) Hence by eq.(10.3.17),
lim -W- = lim *££±J£ = 1 tf-a.e.
Consequently, the identity
lim \+M = ! ^.a.e.
implies that
lim^=limf^+^-lUo^-a.e.
Finally, observe that since all measures involved here belong to M+(M), we have
„ OV 3V
from which
OV 3V
0 < lim —— < lim —— = 0 u^-a.e.
D
One of the keys to proving the main theorem (Theorem 10.3.3) is a certain
resolvent formula. For this we need the spectral theorem for self-adjoint operators
on a Hilbert space (Theorem 1.5.7). For a \i G M+(M) with compact support let
if,9) = / f(x)g(x)d^(x)
be the usual Hilbert space inner product on L2(fi). Let
A:L2(M)-^L2(M), (Ag)(x)=xg(x)
and notice that the constant function
X(x) = l, xeR
is a cyclic vector for A in the sense that
\/{AnX : n e N0} = L2{»).

10.3. FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 237
One can see this last fact by observing that An\ = xn and so the linear span of
{Anx • n G No} contains the polynomials. Now use the Stone-Weierstrass theorem
and the density of the continuous functions with compact support in the space
L2(n). Notice also that (A*g)(x) = xg(x) and so A is self-adjoint. Define a rank
one perturbation A\ of A by
Ax : L2(M) -+ L2(M), A1=A+(;X)x
and notice that A\ is self-adjoint. Also notice that
A\X = X,
A\x = x + c1:oX,
A\x = ^2 + c2,iX + c2,oX,
and so
\]{AnlX ■ n E N0} D \/M"x : n 6 N0} = L2(/x).
Thus Ax is a cyclic self-adjoint operator with cyclic vector x- By the spectral
theorem for self-adjoint operators (Theorem 1.5.7), there is a /xi G M(M) and a
unitary operator
U : L\ii) - L\ni)
with
C/x = X and CMiET = Mx,
where (AfJ)^) = x/(x) on L2(^\). Furthermore,
Uxn = UAnx
= UAU*Uxn~l
= U(Al-(;X)xW*Uxn-1
= {UAxU*-U{;x)xU*Wxn-1
= MxUxn-1-U(;X)xxn-1
= xUxn-l-{xn-\X)x
and by induction on n, we see that U takes polynomials with real coefficients to
polynomials with real coefficients. Hence, by approximation, Uf is a real-valued
element of L2(^i) whenever / G L2(fi) is real-valued.
We will also need the following formulas that relate the Borel transform to
certain resolvent operators:
F^ = j T^zMt) = ({A ~ zI)~lx'x^'
Ff^ = J rb/(i) Mt) = ({A"z/)-1/'*)'
F(Uf)^) = j ^r-z(um)&nx(t)
= ((Mx-zI)-1Uf,x)L2(lil)
= (U*(Mx-zI)-lUf,U*X)
= ((A1-zI)-lf,X).

238
10. THE NORMALIZED CAUCHY TRANSFORM
From the basic operator identity
A~l -B~l =A~1{B-A)B-1
one shows that for any h,g G L2(/i)
(((A-zir'-iAi-zir^^g)
= ((A - zl)~l ((A, - zl) -(A- zl)) (Ai - zl)-lh,g)
= ((A - zI)-l(A1 - A)(A1 - zl)-lh,g)
= ((A- zI)-1(;X)x(A1- ziylh,g)
= ((A-zi)-l((A1-zi)-1h,x)x,g)
= ((A, - zl)~lh,x) ((A- zI)-lX,g) ■
This yields the identity
(10.3.18)
((A, - ziylh,g) = ((A - ziylh,g) - ((A - ziylX,g) {{A, - ziylh,X) .
We are now ready to begin the proof of Theorem 10.3.3.
Proof of Theorem 10.3.3. We will first prove the result for /x G M+(M)
with compact support and real-valued / G L2(/i). Setting g = h = 1 in eq.(10.3.18),
we have
(10.3.19) F|ii(z) = _^|_.
Let
X^:= \x : lim |FM(x + iy)\ = oo \
and recall from Lemma 10.3.5 that XM is a carrier for jjls (the singular part of jjl with
respect to mi). If x G XM, we can use eq.(10.3.19), to see that \Ffll(x + iy)\ —> 1
and so x ^ XMl. In a similar way, if x G XMl then x ^ XM. Thus, by Lemma 10.3.5,
lis and iiiiS (the singular part of [ii with respect to mi) have disjoint carriers and
so
Take imaginary parts of both sides of eq.(10.3.19) and use the identity ^sFu(z) =
(3V)(z), for v real, to get
(10*20) <*«><'> = HTW
Using eq.(10.3.18) again, this time with g — \ and h = /, we get
(10.3.21) Fffl(z) = (1 + F^(z))F(uf)^ (z).
Take imaginary parts of both sides of eq.(10.3.21) and apply the identity
Q(ab) = (9a)(K6) + (R6)(9*), a, 6 G C,
to get
(ia3-22) ^K^ = rf(^)mi(2) + L^>
where
(10.3.23) L(z) = ^l±teH(T(t//V1)(2).

10.3. FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM 239
In the above computations, we are using the fact that Uf is real valued whenever
/ is real valued and so S(F^/Ml) = l3)(Uf)fii. To estimate c?(Uf)/j>i we use the
Cauchy-Schwartz inequality (in a clever way) to get
y(Uf)(t)dm(t)\
\(?(Uf)l*i)(x + iy)\ =
/
(t - X)2 + yl
—-"'f|f
'([ y^dt) \ ([
\J (t-xy + y*J \J
y/{7m)(x + iy)(y(Uf)2^)(x + iy)
\{?{UfYn{){x + iy)
(10.3.24) = WnKx + ivWMx + iy^ {^){x + ly) •
Since /xs _L /x1? then fis _L (Uf)2fii. Thus by Corollary 10.3.16,
(10.3.25) lim 775-77——r = 0 /xs-a.e.
Now observe from eq.(10.3.20) that
(10.3.26) (0Vxi)(*)(fy)(*) = |/f(y^|f)|2 < 1,
and so eq.(10.3.24) and eq.(10.3.25) imply that
(10.3.27) lim \{'P{Uf)iJL1){x + iy)\ = Q /xs-a.e.
y->0+
By using eq.(10.3.24) and the equality in eq.(10.3.24) to estimate \{y(Uf)ni)(z)\
in eq.(10.3.23), we have
|r/T,M. < |8g(l + i^(a; + ty))| \{9y){x + iy)\ (7(Uf)2^)(x + ty)
1 l "1"WI ^ |(^)(^ + iy)| |l + ^(x + iy)|V CPn)(x + iy)
(nUfFmHx + iy)
CPfj)(x + iy)
Hence from eq.(10.3.25),
(10.3.28) lim \L(x + iy)\ = 0 ns-a.e.
Combining eq.(10.3.22), eq.(10.3.27), eq.(10.3.28) along with Proposition 10.3.10
implies that
5+ F«">«(* + iy) = ?%* (3>M)(* + fr) = /(X) Ms"a-e
Divide both sides of the identity in eq.(10.3.21) by F^(z) and apply Lemma 10.3.5
(which will say that \F^(x + iy)\ —> oo /xs-a.e.) to see that
(10.3.29)
lim+ FliXll]T = lim+ f it r\ ^ + 0 FWf)^x + ^) = /(*) Ms-a.e.
This completes the proof of Poltoratski's theorem in the special case where [i G
M+(R) with compact support and / is real-valued and belongs to L2(/j>). One can
remove the assumption that fi G M+(R) has compact support by making some

240
10. THE NORMALIZED CAUCHY TRANSFORM
technical adjustments coming from the fact that the operators A and A\ used to
prove the resolvent formula become unbounded.
To finish, we follow Poltoratski's original proof in [162]: Let / G Ll(p) be
positive and \i G M+(M). Set
and v — (1 + /)/x.
a i + /
Observe that g G L2(u) and the measures /x,s and vs have the same carrier. Apply
eq.(10.3.29) to obtain the identities
r F^x + iy) Fgi/(x + iy) 1
lim ——-—; = lim —*— — = q(x) — —— us-a.e.
y-+o+ F{1+f)fl{x + iy) y^o+ Fv(x + iy) yv 1 + f(x)
and
Hm F^x + iy) = Hm Wfr + fr) 1 =
y-+o+ F^x + iy) y->o+ F^x + iy)
This proves the result for /x G M+(IR) and positive / G Ll(ii). Write a complex-
valued / G Ll(p) as a complex linear combination of four positive functions and use
the linearity of the Borel transform to get the general result (still for /x G M+(M)).
For the general case where /x is a complex measure, we note that /x = p|/x|,
where g G L1(|/x|) with \g\ — 1 (recall the definition of the total variation measure
from eq.(1.3.4)). Hence
iim f/m(^ + ^) = lim Ffg\»\(x + iy)
y-+o+ F^x + iy) y->o+ FgM(x + iy)
y-*o+ F\»\{x + iy) F^x + iy)
f(x) fji8-a,.e.
and the proof is now complete.
□
Remark 10.3.30. In Poltoratski's theorem (Theorem 10.3.1), radial limits can
be replaced, via an argument using Harnack's inequality, with non-tangential limits.
See [162] for details.
We would like to end this section by mentioning (without proofs) a few related
results of Poltoratski. The main theorem of this section shows that for /x G M+
and/el,1^),
(V,f)(():=ZWm(V,f)(z)
exists /x-a.e. and is equal to / /is-a.e. Since VM/ is now /x-measurable, one can ask
the natural question as to whether or not VM/ belongs to any Z/p(/i) class. The
following result of Poltoratski [165] sheds some light on this.
Theorem 10.3.31 (Poltoratski). For /x G M+ and 1 < p < 2, the operator VM
is continuous from Lp(ii) to Lp(p).
Remark 10.3.32. This result is no longer true when p = 1 or when p > 2. In
fact, if /x = m + <r, where a is a non-discrete singular measure, then for any p > 2,
there is an / G Lp(/x) for which VM/ ^ Lp(fi).

10.4. A FEW REMARKS ABOUT THE BOREL TRANSFORM
241
10.4. A few remarks about the Borel transform
The theory of Clark measures was developed to study the unitary operators C/a,
the unitary rank-one perturbations of the model operator S#. We have discussed the
spectral measures aa for Ua, which turn out to be the Clark measures corresponding
to the inner function $ via the formula
In particular, the spectral theorem shows that
d<7a(C)
{{i-zK)-H,i)=Jf
We also discussed Aleksandrov's disintegration theorem
/ aa dm (a) = m
Jj
as well as the carriers of o~a and when the operators Ua have pure point spectrum.
As it turns out, there is a parallel and independent study of 'Clark measures'
involving the Borel transform and perturbations of self adjoint operators. This
work started with Aronszajn and Donoghue [22, 23, 62] and continued in a more
physical setting with Simon and Wolff [198, 199]. The following is a brief and
selective survey of some of these ideas.
If A is a bounded self-adjoint operator on a Hilbert space *K with cyclic vector
v, i.e.,
\J{Anv :nGN0} = J(,
we can apply the spectral theorem for self-adjoint operators (Theorem 1.5.7) to
produce a unitary operator U and a real measure ii with compact support so that
U : *K —> L2(/x), Uv = 1, and U*RU = A, where (Rg)(x) = xg(x) is the operator
'multiplication by x\ This yields the formula
{(A-zI)-1v,v)x = j^, z€C\]
As we did before, we denote the above function, the Borel transform of /x, by FM.
By a conformal mapping argument [198], we have the following.
Proposition 10.4.1. For \i e M(R),
lim F^(x + iy)
y->0+
exists and is finite for mi-a.e. x G M.
Proof. If \i e M+(M), it is easy to show that 3FM = ?/x > 0 on {%z > 0}. If
g is a fractional linear map from {^z > 0} to D, the function FM := g o FM o g~x
is an analytic self-map of D. The fact that the radial limits of FM exist ra-a.e. will
say that the 'radial' limits of FM = g~l o FM o g will exist mi-a.e.
For a general /x = (fii — fiz) + i(/J>3 — ^4) G M(IR), fij G M+(IR), apply the above
argument four times. □
There is also the following version of Fatou's jump theorem (see Corollary 2.4.2).

242
10. THE NORMALIZED CAUCHY TRANSFORM
Proposition 10.4.2. For^eM(R),
lim (FM(x + iy) - F^x - iy)) = 2m- (x)
yio ami
for mi-a.e. xEM.
One can also discuss the finer properties of the boundary values of FM. For
\i G M+(M) and x G R, let
d/x(2/)
G»(x):=J-
(x - y)2
and notice that G^x) G (0, oo].
Theorem 10.4.3. Suppose \i G M+(M) and GM(x) < oo. T/ien
d/x(2/)
/
\x-y\
and
< oo
yio M ^ 7 x-y
For our cyclic self-adjoint operator A with cyclic vector v, consider the following
family of rank-one perturbations of A,
AA := A + A(-,v)v, A GM,
and notice that each Ax is self-adjoint. The identities
A°xv = v,
A\v = Av + ci5o^,
A^ = A2?; + c2,iAv + c2,o^,
show that
\J{A$v : n G No} D \/{A% : n G N°> = M
and so Ax is also cyclic and self-adjoint. Thus, by the spectral theorem for self-
adjoint operators (Theorem 1.5.7), there is a spectral measure fix for Ax in that
z
We write
^a = Ff_iX
and let jFo denote the Borel transform of /xo :=z M- One of the first results is a
disintegration theorem similar to Theorem 9.3.2.
Theorem 10.4.4.
^Adrax(A) = mi.
((Ax-zI)-\v)x = J^
J-
The above integral formula must be understood in the weak sense as in Aleksan-
drov's disintegration theorem (Theorem 9.3.2). There is also the resolvent formula.
Proposition 10.4.5.
7, %
1 + AJo

10.5. A CLOSER LOOK AT THE ^-PROPERTY
243
As we did with the Aleksandrov measures in Corollary 9.1.24, we can determine
the carriers of ii\. With the notation /xq '•= fa set
d/x(2/)
*"-/(!
2*
Theorem 10.4.6. Define the following sets
X := (x:lim^J0(^ + n/) >ol , Y := {x : S(x) < oo}, Z:=R\(Xuy).
For any A ^ 0, Ze^ /x"c, /x^p, and /x^c denote the absolutely continuous, the discrete,
and singular continuous parts of fi\ respectively with respect to rai. Then
(1) /x^c is carried by X.
(2) fj^ is carried by Y.
(3) /x^c is carried by Z.
Furthermore, for A ^ 0, fi\({xo}) > 0 if and only if
S(x) < oo and lim9ro(xo + iy) = — — .
yio A
In fact
Ma({^o})
A2S(x0)*
Corollary 10.4.7. If X ^ 0 and
:=\xeR: lim %(x + iy) = -T\n{x: 9(x) < oo},
I y->o+ A J
£/ien £fte se£ £a is at most countable and
x€£a
As we discussed earlier with the operators Ua, we wish to know when the
operators A\ have pure point spectrum (see Definition 1.5.8).
Theorem 10.4.8. The following are equivalent.
(1) For rai-a.e. A G R, A\ has pure point spectrum.
(2) For mi-a.e. x eR, S(x) < oo.
This result has been generalized in several directions by Poltoratski [57, 58,
164].
10.5. A closer look at the J-property
Recall the ^-property for the space of Cauchy transforms (Theorem 6.5.1): if
$ is inner and K^i/'Q G Hp for some p > 0, then
for some v G M. What is the relationship between the measures /x, v and the
inner function #? Poltoratski's theorem on normalized Cauchy transforms
(Theorem 10.3.1) can be used to prove the following result.

244
10. THE NORMALIZED CAUCHY TRANSFORM
Theorem 10.5.1 (Poltoratski [162]). Suppose i? is inner and
KfJL
for some [i,v G M. Then
* K»
#(C) = lim tf(rC)
r-H-l-
exists [i-a.e. Moreover, the measure v can be chosen to satisfy
dz/ = #d/x.
The proof of this theorem requires a few preliminaries. To avoid confusion, we
will use the notation Kfi to denote the Cauchy transform regarded as an analytic
function on the disk, while C/x will denote the Cauchy transform regarded as an
analytic function on C \ T.
If # is an inner function, recall that $ is defined as an analytic function on
£\{l/z:zecr{$)},
where
a(tf) - Iz G ©- : lim |tf(A)| - ol .
Recall from Chapter 8 that if g is a meromorphic function on D and G is
meromorphic on De, then g and G are pseudocontinuations of each other if the
non-tangential limits of g and G exist and are equal almost everywhere. When
g and G are both of bounded type (quotient of two bounded analytic functions),
non-tangential limits can be replaced by radial limits in the definition of a pseudo-
continuation. Throughout this section we will consider an inner function # as an
analytic function on D and a meromorphic function on De. From the identity,
0(*) = =5"^v zeBe\{l/z:zeo-(0)},
one concludes that the functions g = #|B and G — $|Be are pseudocontinuations of
each other. Furthermore if z G D is a zero of # of order n, then 1/z, the reflection
of z across T, will be a pole of # of order n. Also notice that l/# G i7°°(De) and is
'inner' on Pe (i.e., has boundary values of unit modulus almost everywhere). This
next result of Aleksandrov is the key to proving Theorem 10.5.1.
Theorem 10.5.2 (Aleksandrov [10]). Let # be inner and g G iJ°°(De) with
g(oo) = 0. Suppose f G iJ°°(D) has a pseudocontinuation equal to $g and that for
some jjl G M, fiC/d, G Hp(De) for some p > 0. Then there is a v G M such that
fCfj, = Gv
onC\T.
Proof. Let
F{z) = t f(z)(Kv)(z), ze:
ti{z)g{z){Cii){z), zeBe.
One can quickly check, using Fatou's jump theorem (Corollary 2.4.2), that F is
an analytic function on C\T with /(oo) = 0 which satisfies the hypothesis of
Aleksandrov's representation theorem (Theorem 5.4.5). Thus F — Gv for some
v G M. □

10.5. A CLOSER LOOK AT THE ^-PROPERTY
245
Corollary 10.5.3. Under the assumptions of Theorem 10.5.2,
du _ d/x
dm dm
Proof. If
(J70(C)= lim (ft(rC) - MCA))
r-H-l-
is the jump function for an h analytic on C \ T, it follows from Fatou's jump
theorem (Corollary 2.4.2) and the fact that / has a pseudocontinuation, which we
also denote by /, that
f& = WW = «&) = £.
□
Corollary 10.5.4 (Goluzina, [83]). Under the assumptions of Theorem 10.5.2,
Proof. For a measurable function F on T and a Borel subset EcT, let
L{F,E) = lim 7rym({C G E : |F(C)| > y})
whenever this limit exists. From Theorem 7.4.4, L{Kfi,T) = ||/xs||.
Let 7 be any arc of T for which
(10.5.5) \v\(d7) = \v\(d>y)=0.
If
°y '.= nyX{\Kn\>y} " m,
Poltoratski's distribution theorem (Theorem 9.7.1) says that
ay —> \\is\ weak-* as y —> oo.
From the fact that cry(E) —> |^s|(£?) for every Borel set with |^s|(<9£?) = 0
(Proposition 1.6.1), we see that
(10.5.6) L(A>>7) = I^I(7) and L{Kv,i) = |i/fl|(7).
Since /K/x = if z/, we also have
(10.5.7) L(/K-/i,7) = H(7).
A calculation with the definition of L yields the estimate
L{fKn,i) < sup{|/(C)| : C e 7}^(-^M,7)
which by eq.(10.5.6) and eq.(10.5.7) gives us
N(7) < ll/JUKIM-
Combine this with Corollary 10.5.3 to get
(10.5.8) M(7) < ||/||oc|/x|(7)
for any arc 7 with |/x|(97) = |^|($7) = 0. Using the fact that the atoms of \v\ and
|/x| are at most countable, along with a limit argument using eq.( 10.5.8), one can
show that
M({<}) < H/llooM({C}) VCeT.
From here it follows that eq.(10.5.8) is valid for all arcs 7 C T. It now follws from
basic measure theory that \v\ ^C |/i| and hence, by Remark 1.3.12, v <^ \i. D

246
10. THE NORMALIZED CAUCHY TRANSFORM
Corollary 10.5.9 (Poltoratski, [162]). Assuming the hypothesis of
Theorem 10.5.2, we have the following.
(i)
lim /«) = /(C)
exists for fi-a.e. £ G T.
(2) In the identity fC/j, = Cv onC\T, the measure v can be chosen to satisfy
du — f d/j>, that is to say,
fCn = C(fn).
PROOF. By Corollary 10.5.3, we know that
du _ d/j^
dm dm
By Corollary 10.5.4, du — gd/j, and so g = f m-a.e. To finish, we need to show that
/ has radial limits /is-a.e. and that these limits are equal to the boundary values
of g ^s-a.e. Observe that for z G D (minus some poles),
(^K£) = (KgMz) =
HZ) (A>)(*) (KMz) {^9)(Z)
and so from Theorem 10.3.1, / has radial limits /i-a.e. that are equal to the boundary
values of g ^s-a.e. □
In order to prove Theorem 10.5.1 we require the following version of
Theorem 10.5.2 and Corollary 10.5.9. This version can be obtained from the earlier
versions by making the change of variables z i—> 1/z.
Theorem 10.5.10. Let if; G iJ°°(Bc) be inner and g G #°°(B) with g(0) = 0.
Suppose that f G iJ°°(De) has a pseudocontinuation equal to ipg and that for some
a G M, ipKa G HP(D) for some p > 0. Then there is an r\ G M such that
fCo = CV
on C \ T. Moreover, for o-a.e. £ G T,
lim /(C/r)
r—+l~
exists and in the identity fCa = Crj, the measure r\ can be chosen to satisfy
drj = f da.
Proof of Theorem 10.5.1. Suppose
KfJL
* Kv
on H>. In Theorem 10.5.10, let
f l i l
Note that / G iJ°°(De), ip is an inner function on De, and / has a
pseudocontinuation equal to ipg.
Let
// := -/x(-l)ra + CM
and observe that
zKfi = Kfi

10.5. A CLOSER LOOK AT THE ^-PROPERTY
247
and
i/>K}jl = \Kp, = —«zK\i = ^=Kve Hp{B).
ZV ZV V
Thus by the above theorem, there is a v\ G M with
Furthermore,
-Cp. = Cv1 onC\T.
exists // almost everywhere. Using the identity
W = ^—, H>i,
we conclude that
lim #(r£)
r-H-l-
exists /i-a.e. By standard iJp theory, this limit already exists ra-a.e. Moreover,
(fi)s — C/j^s and so the above limit exists /xs-a.e.
To finish the proof, we need to show that in the identity
on D, we can choose v to be v — #/x. To this end, use Theorem 10.5.10 to show
that in the identity
-CfjL = Cui onC\T,
v\ can be chosen to be
(10.5.11) i/i=7?/i.
On D, we have that
(10.5.12) Kvx = -Kji = -zKn = zKv = Ki>,
where
v — —P( —l)m + (V.
For a G M, let [a] denote the coset in M/Hq represented by a. From eq.(10.5.11)
eq.(10.5.12), and Proposition 2.1.5, we get
= [!/].
From this and the definitions of // and i>, we get
7?(/x(-l)ra + C/x) = 0m + 2?(-l)ra + <>, 0 G #0\
and so
£#/!( —l)ra + t9/x = C0m + C^(—l)m + z/-
Hence
IM = H
A final application of Proposition 2.1.5 yields K^ji) = Kv. D

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CHAPTER 11
Other operators on the Cauchy transforms
11.1. Some classical operators
On the Hardy spaces Hp, there are a variety of operators worthy of study.
Several classical ones are:
(Sf)(z) = zf(z) (forward shift)
(Bf)(z) = ^ ~ ^ (backward shift)
(T0/)(z) - (<l>f) + (z) (Toeplitz operator)
(C<f,f){z) = f((f)(z)) (composition operator)
(Cf)(z) = - IW-dt (Cesaro operator).
z Jo 1 — t
In this brief chapter, we mention how these operators act on the space of Cauchy
transforms %. We limit ourselves to an exposition and do not include the proofs.
Let us first review some classical results about these operators on Hp'. The forward
and backward shifts on Hp have already been discussed in Chapter 8.
For 0 G L°°, define the Toeplitz operator X^ : H2 -> H2 by
T*f = (0/) +
and note that T^ is bounded. By Riesz's theorem (Theorem 3.2.1), T^ is also
bounded on Hp whenever 0 G L°° and 1 < p < oo. When p = 1, Riesz's theorem
is no longer available to us and, as to be expected, T^ is not always bounded. For
example, a result of Stegenga [204] says that whenever i\) G H°°, the co-analytic
Toeplitz operator IV : H1 —> H1 is bounded if and only if Sfo/; and 3-0 are multipliers
of BMO. For a general symbol <p G L°°, there is a definitive result of Janson, Peetre,
and Semmes [108] which says that T^ : H1 —> H1 is bounded if and only if 0 — 0+
is of 'logarithmic mean oscillation'. Here a function g G L1 on T is of 'logarithmic
mean oscillation' if the supremum of
4tt \ 1
over all arcs I C T, is finite.1 Compare this to the definition of bounded mean
oscillation in Definition 3.3.1. When p = oo, we know from Proposition 6.1.5 that
the co-analytic Toeplitz operator IV : H°° —> iJ°° is bounded if and only if -0 is a
multiplier of %.
When 0 < p < 1, the discussion of T^ on Hp becomes more delicate. First
of all, the definition of T^ on Hp as T^f = (0/)+ does not make sense since the
lo£ ( IT77T ) ^77T \9 ~ 9i\ dm,
Note that qj = —-—- / a dm is the mean of g on /.
m(I)Jjy
249

250
11. OTHER OPERATORS ON THE CAUCHY TRANSFORMS
product (j)f belongs to Lp, 0 < p < 1, and the Riesz projection is defined on the
smaller space L1. Thus we need to ask whether or not the densely defined operator
T^ on H2 can be extended to be bounded on Hp. Aleksandrov [10] (see also [44,
p. 189]) showed that whenever x/; G i^°° and 0 < p < 1, the co-analytic Toeplitz
operator T-^r can be extended to a bounded operator on Hp if and only if ip is
Lipschitz of order 1 — 1/p.2 The definitive result is again one of Janson, Peetre, and
Semmes [108] and says that if 0 < p < 1 and 0 G L°°, then T$ can be extended to
a bounded operator on Hp if and only if <fi — <fi+ belongs to the Lipschitz class of
order 1 — 1/p.
If <j) is an analytic self map of the disk, an application of the Littlewood
subordination theorem [65, p. 11] shows that the composition operator C^ : / i—> / o cf>
maps Hp to itself. The closed graph theorem implies that C^ is bounded.
Composition operators are a rich source of results and we won't try to survey them here
but refer the reader to [51, 195] for two treatments of this subject.
The Cesaro operator, as it originally appeared in the operator setting, was
simply the map defined on the sequence space (£2)+ by
(&n)n^0 >-> (&n);V>0>
where
bN := jj—- (a0 + ai + • • • + aN), N e N0,
is the AT-th Cesaro mean of the sequence (an)n^o- It is easy to see, equating the £2
sequence (an)n->o with the H2 function / = ^anzn, that this operator, denoted
by C, can be viewed on H2 as the integral operator
The papers [16, 19] consider a generalization of the Cesaro operator on Hp by
observing that the kernel (1 — z)~l in the definition of C is the derivative of — log(l —
2), which belongs to BMOA. They show that this kernel can be replaced by any g'
with g G BMOA. In this setting, the operator assumes the form
1 fz
/'->-/ f(w)g'(w)dw
z Jo
and is known to be bounded on all Hp if and only if the symbol g belongs to BMOA.
The next four sections examine these classical operators on %.
11.2. The forward shift
Notice that
'fT±r*MQ = -/cd*K)+/r4<W0
= K(cdm + (dfi)
and so the forward shift (Sf)(z) = zf(z) is a well-defined operator from X to itself.
Proposition 11.2.1.
(1) S : (X, || • ||) -> (X, || • ||) is bounded with \\S\\ = 2.
See also [66] or [44, p. 38] for a precise definition of 'Lipschitz of order 1 — 1/p'.

11.2. THE FORWARD SHIFT
251
(2) 5 : (X, *) —> (X, *) 25 continuous.2"
PROOF. The continuity of 5 : (X, || • ||) —► (X, || • ||) follows from a routine
argument using the closed graph theorem. To see that ||5|| = 2, notice by the
duality A* ~ X (Theorem 4.2.2) via the pairing
„ oo
(11.2.2) (f,Kn)= lim //(C)(^)K)dm(C)= Urn £/(n)£fa)rn,
that 5 is the Banach space adjoint of B where B is the backward shift operator on
A. One quickly sees that
l|S/||oo = ||/-/(0)||oo<2||/||oo V/eA
Furthermore, if 0 < r < 1 and
f ( \ z + r
Mz) = TTFz>
then
HMIoc^l and ||/r-/r(0)||oo = l + r.
It follows that \\B:A-+ A\\ = 2 and hence, by Proposition 1.5.4, \\S : X -> DC|| = 2.
Duality and the continuity of B on A also shows that 5 : (X, *) —> (X, *) is
continuous. □
Beurling's theorem (Theorem 8.1.1) says that every 5-invariant subspace of Hp
is equal to $i/p for some inner function #. What is the analog of Beurling's theorem
for XI Endowed with the norm topology, X is non-separable (Proposition 4.1.21)
and so characterizing its norm closed 5-invariant subspaces is troublesome.
However, the subspace Xa — {/+ : / G L1} is indeed separable (Proposition 4.1.21).
Since not all inner functions are multipliers of Xa (see Theorem 6.6.3 and
Proposition 6.1.5), then $3Ca is not always a subset of Xa. However, the subspace
#(3Ca) := {feXa: //# G Xa}
does make sense and is clearly 5-invariant. Moreover, using Theorem 6.5.1, one
can show that #(3Ca) is norm closed. A theorem of Aleksandrov [11] is our desired
'Beurling's theorem' for Xa.
Theorem 11.2.3 (Aleksandrov). For each inner function ft, #(3Ca) is a norm
closed S-invariant subspace ofXa. Furthermore, ifM is any non-zero norm closed
S-invariant subspace ofXa, then there is an inner function #, such that M = #(3Ca).
Note that (X, *) is separable (Proposition 4.2.8) and so characterizing its weak-*
closed 5-invariant subspaces is a tractable problem. Define the 5-invariant subspace
i?(3C) := {/ G X : //# G X}
and note from Proposition 8.5.4 that #(3C) is weak-* closed. 'Beurling's theorem'
in the setting (X, *) is the following.
Theorem 11.2.4 (Aleksandrov [11]). For each inner function ft, #(3C) is a
weak-* closed S-invariant subspace ofX. Furthermore, ifM is any non-zero weak-
* closed S-invariant subspace of X, then there is an inner function $, such that
M = ${X).
We remind the reader that (3C, || • ||) denotes the space of Cauchy transforms % endowed
with the norm topology while (3C, *) denotes 3C endowed with the weak-* topology. See Chapter 4
for details.

252 11. OTHER OPERATORS ON THE CAUCHY TRANSFORMS
11.3. The backward shift
The backward shift
W)W - ^)
is a well-defined operator from X to itself. Indeed,
™ - K/^§>(0)
f CMC)
J l-(z
= (K(dv)(z).
Proposition 11.3.1. The backward shift operator B : (X, || • ||) —> (3C, || • ||) is
bounded and \\B\\ = 1.
PROOF. An application of the closed graph theorem says that B is bounded.
The backward shift B is also the Banach space adjoint (under the pairing eq. (11.2.2))
of the forward shift S on A. Since \\S : A -> A|| = 1, then ||£ : X -> 0C|| = 1
(Proposition 1.5.4). D
This next theorem of Aleksandrov [11] (see also [44, p. 99]) is the Douglas-
Shapiro-Shields theorem (Theorem 8.2.1) for Xa.
Theorem 11.3.2 (Aleksandrov). If W is a norm closed B-invariant subspace
of Xa, then there is an inner function # such that f G M if and only if there is a
G G N+(De)4 with G(oo) = 0 and such that
lim {«)= lim G(C/r)
r—>1_ V i—>1-
for m-almost every £ G T.
For a weak-* closed ^-invariant subspace N of X, the dual pairing eq.(11.2.2)
tells us that Nj_ (the pre-annihilator of N) is an 5-invariant subspace of A. Since
A is a Banach algebra and polynomials are dense in A, Nj_ is a closed ideal of A.
A result of Rudin [180] (see also [101, p. 82]) characterizes these ideals by their
inner factors and their zero sets on the circle. The authors in [43] use the Rudin
characterization to describe the corresponding N in terms of analytic continuation
across certain portions of the circle. This result also makes connections to an
analytic continuation result of Korenblum [119]. See [43] for some partial results
about the 5-invariant subspaces of (X, || • ||).
11.4. Toeplitz operators
For what <p G L°° can we define a meaningful Toeplitz operator T$ : X —> XI
Certainly when 0 is a multiplier of X, then T^ is a bounded multiplication operator
(Theorem 1.5.2). For other symbols, the situation is very much unknown. For
example, if ip is an analytic polynomial, then
T^Kn = K(V*dM),
4JV+(De) = {/(l/z) : / € N+}.

11.6. THE CESARO OPERATOR
253
where ^(z) = ip^z). Now approximate any function ip in the disk algebra A
uniformly on D~ by its Cesaro polynomials (Theorem 1.6.5) to show that T^K/j, =
K(ilj*dfi) is bounded on X. Are these the only co-analytic Toeplitz operators on
X? Is there a definitive characterization, like the one in [108], of the symbols that
yield continuous Toeplitz operators on XI
11.5. Composition operators
For an analytic map <p : D —> D, we know from Lemma 5.6.1 that the
composition operator C^ : / i—> / o 0 maps X to itself. Without too much difficulty, one
can show that Ca> has closed graph and so Ca, is bounded on X. Bourdon and Cima
[31] proved that
2 + 2y/2
ll°4 ^ 1-|0(O)|
which was improved to
iai<1+wl
1 - |«0)|
by Cima and Matheson [42]. Moreover, equality is attained for certain linear
fractional maps cf). This same paper also discusses compactness properties of Ca,. In
fact, Caj : X —> X is compact if and only if for every a G T, the Aleksandrov
measure /xrt, corresponding to 0, satisfies /xa <C m.
11.6. The Cesaro operator
Cima and Siskakis [45] proved that the Cesaro operator is bounded on X as
follows: write the duality between the disk algebra A and X as
</, (?>„:= lim /<?(C)/(rC)dm(C), / G 3C, jeA.
A short computation yields {Cf,g)0 = (f,Lg)0 where L is the operator acting on
A by
-1 oo / oo \
(Lg)(z):=Jo g(tz + i-t)dt = y£\S21^1\zn.
It is clear from the integral expression that L is bounded on A and the identity
C = L* along with Proposition 1.5.4 implies that C is bounded on X. Are any of
the generalized Cesaro operators
1 fz
/>-+-/ f(w)g'(w) dw, g £ BMOA,
on iJp considered in [16, 19] bounded on XI

This page intentionally left blank

List of Symbols
A (disk algebra) p. 91
A(f) (Aleksandrov measures associated with 0) p. 202
BMO, BMOA (bounded mean oscillation) p. 69
C (complex numbers) p. 11
C (Riemann sphere) C U {oo} p. 11
C+ (upper half plane) p. 81
C(T) (continuous functions on T) p. 14
Cfi p. 54
C(E) (interpolation constant for a sequence ^cO) p. 38
5(E) (uniform separation constant for a sequence £cB) p. 37
D (unit disk) p. 1
De (extended exterior disk) p. 54
Dfi (symmetric derivative of a measure /x) p. 15
Ea p. 206
/* (decreasing rearrangement of /) p. 13
FM (Borel transform of a measure /x) p. 231
S(/) (Garcia norm of a function) p. 69
j(E) (Carleson constant for a sequence ^CB) p. 37
Jifi (Hilbert transform of a measure /x) p. 163
Hfi (Herglotz integral of a measure /x) p. 30
Hp (Hardy space) p. 32
Hp(Be) (Hardy space of the exterior disk) p. 54
HP(T) p. 33
Hi (the set of / e H1 such that /(0) - 0) p. 34
iJ1'00, H^°° (analytic weak L1) p. 35
% (space of Cauchy transforms) p. 41
%a (Cauchy transforms of \i <C m) p. 88
%s (Cauchy transforms of /x _L m) p. 88
Kfi (Cauchy transform of a measure /x) p. 41
k\ (reproducing kernel for $*(H2)) p. 186
^ P. 15
Lp (Lebesgue spaces on T) p. 12
L1'00 (weak L1) p. 35
Xf (distribution function for /) p. 13
Aa (Lipschitz class) p. 62
m (Lebesgue measure on T) p. 12
mi (Lebesgue measure on M) p. 163
M (Borel measures on T) p. 14
M(R) (finite Borel measures on M) p. 163
255

256
LIST OF SYMBOLS
M+ (resp. M_|_(M)) (positive measures on T (resp. M)) p. 14
Ms (absolutely continuous measures) p. 16
Ms (singular measures) p. 16
M/Hl p. 83
m{%) (multipliers of X) p. 115
M^ (multiplication by 0) p. 115
fia (Aleksandrov measure) p. 202
^e P- 37
N (natural numbers) {1, 2, 3, • • • } p. 11
N0 (natural numbers along with zero) {0,1, 2, • • • } p. 11
N+ (Smirnov class) p. 35
Pfi (Poisson integral of a measure /i on T) p. 30
?/i (Poisson integral of a measure /x on M) p.232
P$ (orthogonal projection of H2 onto rdH2) p. 185
Pz (Poisson kernel) p. 30
Qfi (conjugate Poisson integral) p. 30
Qz (conjugate Poisson kernel) p. 30
Rf (representing measures for a Cauchy transform /) p. 42
<jn (/x) (xV-th Cesaro sum of a measure /x) p. 24
<ja (singular part of an Aleksandrov measure) p. 205
(Jf(E) (Frostman constant for a sequence ^ClD) p. 130
s(E) (separation constant for a sequence ^CB) p. 37
S (shift on H2) p. 179
S$ (compression of the shift) p. 194
T (unit circle) p. 1
T-t (co-analytic Toeplitz operator) p. 116
ua p. 201
Va p. 218
VM (normalized Cauchy transform) p. 227
VMO, VMOA (functions of vanishing mean oscillation) p. 72
!?*(#*) p. 183
/+ (Riesz projection of /) p. 61
||/||p (LP (or HP) norm) p. 34
/x(n) (n-th Fourier coefficient of a measure /x) p. 24
f(ri) (n-th Fourier coefficient of an L1 function /) p. 24
||/x|| (total variation norm of a measure /x) p. 14
Jl p. 80
Z (non-tangential limit) p. 33
Z (integers) p. 11

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Index
A-integral, 48
absolutely continuous measure, 15
Adams, D., 59
adjoint, 21
Ahern, P., 27, 30, 192
Ahlfors, L., 28, 103, 110
Aleksandrov
measure, see also Clark measure
disintegration theorem, 212, 216, 242
Aleksandrov, A., 1, 4, 6, 8, 36, 48, 49, 67,
102, 109, 183, 188, 215, 217, 228-230,
244, 250-252
Aleman, A., 179, 180, 185, 253
algebra, 11
cr-algebra, 11
Aliev, R., 54
analytic self-map, 28, 201
Andersson, M., 36
angular derivative, 28, 192, 208, 211, 216
annihilator, 18
Aronszajn, N., 9, 241
atoms (of a measure), 17
backward shift, see also Clark measure
H2
analytic continuation, 182
basis, 192
density theorem, 187
Douglas-Shapiro-Shields theorem, 181
kernel function, 186, 192
pseudocontinuation, 181
spectrum, 184
HP, 183, 192
X, 252
other spaces, 185
Baernstein, A., 79, 80
Bagemihl, F., 26, 43
balanced hull, 18
Banach-Alaoglu theorem, 19, 24
Bary, N., 54
basis, 95, 192
Bell, S., 9
Besicovich covering theorem, 233
best constants, 79, 82
Beurling's theorem, 179, 251
Blaschke condition, 27
Blaschke product
Caratheodory's theorem, 152
definition, 27
Frostman's theorem, 27
multiplier, 130
Tumarkin's theorem, 152
Bochner integral, 121
Boole's lemma, 165
Boole, G., 6, 164
Borel
algebra, 12
function, 12
measure, 14
sets, 12
transform, 231, 241
bounded mean oscillation, 69
bounded operator, 20
bounded type, 34
Bourdon, P., Ill, 253
Bockarev, S., 96
Brennan, J., 9
Brown, L., 93
Burkholder, D., 36
Calderon, A., 65, 67, 163
capacity, 59
Caratheodory, C, 152
Carleson
interpolation theorem, 38
measure, 37, 133
square, 37
Carleson, L., 5, 38, 96
carrier (of a measure), 16, 232
Cauchy
A-integral formula, 49
integral formula, 47
Stieltjes integral, 1, 59
Cauchy transform
A-integral formula, 49
Aleksandrov's characterization, 102, 127,
190, 244
and C(T), 72
267

268
INDEX
and L\ 68
and L°°, 68, 69
and LP, 65
and duality, 78
and weighted Lp, 76
boundary behavior, 42, 58
Cauchy integral formula, 47
Clark measure, 203
definition, 41
distribution function, 172, 222
F-property, 127, 243
Fatou's jump theorem, 55
geometric characterization, 111
Havin's characterization, 99
Lipschitz classes, 62
M. Riesz's theorem, 65
multiplier, 115
non-tangential limit, 44
norm, 83
normalized Cauchy transform, 227
Plemelj formula, 56
pointwise estimate, 87
principal value integral, 56
representing measures, 42
space of Cauchy transforms, 41
backward shift, 252
basis, 97
composition operator, 253
duality, 89, 91
forward shift, 250
Lebesgue decomposition, 88
multiplier, 115
reflexive, 90
separable, 89, 93
Toeplitz operator, 252
weak topology, 95
weak-* topology, 91
weakly sequentially complete, 95
Tumarkin's characterization, 101
Cauchy, A., 1, 46, 60
Cesaro
operator, 250
sum, 24
Choquet, G., 25
Cima, J., 26, 67, 111, 112, 181-183, 185, 250,
252, 253
Clark measure
Aleksandrov's disintegration theorem, 212,
216, 242
angular derivative, 208, 211, 216
carrier, 207
Cauchy transform, 203
composition operator, 253
deBranges-Rovnyak space, 229
definition, 202
Fourier coefficients, 204
Herglotz integral, 202
Lebesgue decomposition, 205
norm, 204
normalized Cauchy transform, 227
point mass, 208, 211, 216, 222, 230, 243
Clark, D., 1, 6, 7, 27, 30, 192, 193, 197, 199,
201, 220
closed graph theorem, 21
Cohn, W., 192
Collingwood, E., 26, 27
composition operator, 250, 253
compression, see also forward shift
conditional expectation operator, 215
conjugate
Poisson integral, 30
function, 32, 62, 65, 69, 72, 73, 80
continuous
measure, 17
operator, 20
convex
balanced hull, 18
hull, 18
Conway, J., ix, 9, 17, 20
coset, 18
Cowen, C, 28, 209, 250
cyclic, 21, 195, 200, 236
Davis, B., 80, 82
Day, M., 12
deBranges-Rovnyak space, 229
decreasing rearrangement, 13, 49
del Rio, R., 243
Delbaen, F., 95
Denjoy, A., 48
derivative (of a measure), 15
Diestel, J., 94-97, 121, 193
discrete measure, 17
disintegration theorem, see also Aleksandrov's
disintegration theorem, 242
disk algebra, 91, 117
distribution function, 13, see also decreasing
rearrangement
Boole's lemma, 165
Cauchy transform, 172, 222
conjugate function, 73, 80, 222
Herglotz integral, 170
Hilbert transform, 163, 176
Hruscev-Vinogradov theorem, 164, 170
normalized Cauchy transform, 227
Poltoratski's distribution theorem, 222
Stein-Weiss theorem, 176
Tsereteli's theorem, 169
Donoghue, W., 9, 222, 241
Doob, J., 84
Douglas, R., 181, 182
dual extremal problems, 84
duality
A, 91
H1, 78
HP, 78

INDEX
269
X, 91, 95
3Ca, 89
ti*(HP), 183
Duren, P., 27, 31, 32, 36, 41, 45, 65, 68, 84,
94, 111, 179, 180, 250
Dyakonov, K., 187
Enflo, P., 96
Evans, L., 11, 15, 16, 233
F-property, 127, 129, 151, 157, 243
F. and M. Riesz theorem, 34
factorization
bounded analytic function, 27
functions of bounded type, 34
Hardy space functions, 34
Fatou's theorem
jump theorem, 55
on non-tangential limits, 26
on Poisson integrals, 31
Fatou, P., 2, 26, 31, 55
Fefferman, C, 79
Fefferman-Stein duality theorem, 79
Fejer, L., 24
Fomin, S., 11
forward shift
H2
Beurling's theorem, 179
compression, 194
X, 250, 251
perturbations, 196
Fourier coefficient, 24
Frostman's theorem
on angular derivatives, 29
on radial limits, 27, 130
Frostman, O., 27, 29, 130, 160
Fuentes, S., 243
Gaier, D., 86
Gamelin, T., 9, 80
Garcia, S., 2, 54, 199
Gariepy, R., 11, 15, 16
Garnett, J., 9, 32, 36, 44, 69, 70, 72, 76, 79,
84, 86, 95, 103, 109, 141, 153, 164, 176,
180, 182
Garsia norm, 69
Gelfer domain, 112
Gelfer, S., 112
Goldstine, H., 20
Goluzin, G., 57, 62
Goluzina, M., 120, 122, 124, 130, 245
Grafakos, L., 13
Gundy, R., 36
Gurarii, V., 127
Holder's inequality, 12
Hahn-Banach
extension theorem, 17
separation theorem, 17
Hankel operator, 145
Hardy space, see also forward shift,
backward shift, Toeplitz operator
classical operators, 249
definition, 32
Riesz factorization, 34
Smirnov class, 35
standard facts, 33
Hardy's inequality, 68
Hardy, G., 36, 57, 62, 76
harmonic majorant, 103
Hausdorff, F., 214
Havin, V., 95, 99, 109, 122
Havinson, S. Ja., 84
Hayman, W., 38, 103
Hedberg, L., 59
Helson, H., 76
Herglotz
integral, 30, 170, 202
theorem, 32, 201
Herglotz, G., 32
Hewitt, E., 16, 17
Hilbert transform, see also distribution
function, 163, 164, 169, 170
Hobson, E., 214
Hoffman, K., 31, 32, 38, 68, 93, 252
Hollenbeck, B., 3, 67, 79
Hruscev, S., 3, 5, 6, 110, 127-130, 137, 164,
170, 190
Hunt, R, 76
inner function
angular derivative, 29, 192
Clark measure, 202, 216, 222
definition, 27
kernel function, 192
measure preserving, 171, 215
multiplier, 129
non-tangential limits, 27
spectrum, 182
interpolating sequence, 37, 133
Jaksic, V., 231
Janson, S., 249, 253
John-Nirenberg inequality, 70
Jordan decomposition theorem, 14
Julia-Caratheodory theorem, 28, 209-211
Kahane, J., 42
Kakutani, S., 95
Kalton, N., 35
Katznelson, Y., 176
Kelley, J., 94
Kennedy, P., 103
kernel function, 185, 192, 199
Khavinson, D., 187
Kisljakov, S., 95
Kolmogorov, A., 3, 5, 11, 48, 73, 80, 163,
227

270
INDEX
Koosis, P., 32, 36, 69, 70, 73, 79, 95, 164,
207
Korenblum, B., 180, 252
Landau, E., 86, 125
Last, Y., 231
Lebesgue
decomposition theorem, 16
and space of Cauchy transforms, 88
differentiation theorem, 15
measurable functions, 12
measure, 12
Lebesgue, H., 24
Lieb, E., 234
Lindelof, E., 26
Lipschitz class, 62, 250
Littlewood subordination theorem, 79, 250
Littlewood, J., 26, 36, 41, 57, 62, 76, 79, 250
Livsic, M., 184
Lohwater, A., 26, 27, 43
Loomis, L., 163, 164
Loss, M., 234
Lotto, B., 129
MacCluer, B., 28, 250
MacGregor, T., 9, 112
Markushevich, A., 101
Matheson, A., 132, 180, 217, 226, 252, 253
Maurey, B., 96
maximal function, 36, 233
Mazur's theorem, 19
Maz'ya, V., 116
McDonald, G., 138
McKenna, P., 137
measure
absolutely continuous, 15
atoms, 17
Banach-Alaoglu theorem, 24
Borel, 14
carrier, 16, 232
Cesaro sum, 24
continuous, 17
derivative, 15
discrete, 17
Fourier coefficients, 24
Jordan decomposition, 14
Lebesgue, 12
Lebesgue decomposition, 16
positive, 14
Radon-Nikodym derivative, 15
Riesz representation theorem, 15
singular, 15
support, 16
total variation, 14
Megginson, R., 17, 96, 193
Minkowski's inequality, 12
Moeller, J., 184
Monotone class theorem, 213
Mooney, M., 95
Morera, G., 1, 60
Muckenhoupt, B., 76
multiplier
HP, 116
BMO, 117
definition, 115
Dirichlet space, 116
F-property, 127, 129, 151, 157
Frostman condition, 130
inner function, 129
multiplier norm, 115
necessary conditions, 118
non-tangential limits, 119, 120
sufficient conditions, 122
Toeplitz operator, 117
Muskhelishvili, N., 9
Naftalevic, A, 38
Nagel, A., 58
Natanson, I., 11
Nazarov, F., 77
Nevanlinna class, 34
Nevanlinna, R., 208
Newman, D., 38, 68
Nikol'skii, N., 179, 181, 194, 195
non-tangential limit
Hp functions, 33
Cauchy transform, 44
definition, 25
Fatou's theorem, 26
Frostman's theorem, 27
Lindelof's theorem, 26
multiplier, 119, 120
normalized Cauchy transform, 231
Privalov's uniqueness theorem, 26
non-tangential maximal function, 36
norm
LP, 12
Cauchy transform, 83
operator, 20
total variation, 14
normalized Cauchy transform
definition, 227
distribution function, 227
mapping properties, 228-230, 240
non-tangential limits, 231
operator
adjoint, 21
bounded, 20
norm, 20
spectral theorem, 22
spectrum, 21
oricyclic limit, 58
outer function, 27, 34
Pajot, H., 9
Paley, R., 193
Parthasarathy, K., 23

INDEX
271
Peck, N., 35
Peetre, J., 249, 253
Peller, V., 125
perturbations
Clark's theorem, 220
of self-adjoint operators, 242
unitary, 196, 197, 199
Peiczyriski, A., 79, 96
Pichorides, S., 3, 80, 82, 103
Piranian, G., 26, 43
Plemelj's formula, 56
Plemelj, J., 1, 2, 56, 60
Poincare, H., 43
Poisson integral, 30, 232
Poisson-Stieltjes integral, 31
polar, 18
Poltoratski, A., 1, 3, 6, 8, 199, 222, 226, 231,
240, 243, 244, 246
Pommerenke, C, 209
pre-polar, 18
principle of uniform boundedness, 17
Privalov's theorem
on Lipschitz classes, 62
principle value of Cauchy integrals, 56
uniqueness theorem, 26
Privalov, I., 1, 3, 9, 26, 56, 60, 62
pseudo-hyperbolic distance, 37
pseudocontinuation, 181, 244
pure point spectrum, 22, 222, 243
Putinar, M., 199
quotient space, 18
radial
limit, 25
maximal function, 36
Radon-Nikodym
derivative, 15
theorem, 15
reflexive, 20
space of Cauchy transforms, 90
representing measures, 42
Richter, S., 179, 180
Riesz
projection, 65, 67
representation theorem, 12, 15
Riesz, F., 20, 34, 193
Riesz, M., 3, 29, 34, 65, 164, 210
Roberts, J., 35
Rogosinski, W., 84
Romberg, B., 94
Ross, W., 26, 67, 179, 181-183, 185, 250, 252
Rudin, W., ix, 11, 15-17, 20, 31, 58, 62, 91,
180, 233, 252
Rybkin, A., 54
Ryff, J., 13
Sarason, D., 2, 54, 72, 129, 194, 218, 226,
229
Schauder basis, 95
Schauder, J., 96
second dual, 19
Seidel, W., 26, 43
Seip, K., 36
self-adjoint operator, 22
spectral theorem, 22
self-map, 28, 201
Semmes, S., 249, 253
separable, 20
space of Cauchy transforms, 89, 93
space of measures, 24
separated, 37, 133
Shapiro, H. S., 36, 84, 179, 181, 182, 185,
187
Shapiro, J., 28, 58, 209, 250
Shaposhnikova, T., 116
Shields, A., 36, 93, 94, 181, 182
shift operator, see also forward shift,
backward shift
Shimorin, S., 180
Shirokov, N., 127, 180
Silverstein, M., 36
Simon, B., 9, 241
singular inner function, 27
singular measure, 15
Siskakis, A., 250, 253
Smirnov class, 35
Smirnov, V., 2, 34, 35, 43, 45, 62
Smithies, F., 46
Sokhotski, Y., 1, 56, 60
Spanne, S., 69
spectral theorem, 22, 218, 236, 241
spectrum
backward shift, 184
compression, 196
inner function, 182
kernel function, 192
operator, 21
pure point spectrum, 22, 222, 243
restriction of backward shift, 184
spectral theorem, 22, 218, 236, 241
unitary perturbations, 222
Stegenga, D., 116, 117, 249
Stein, E., 69, 79, 82, 164, 176, 233
Stein, P., 65
Stessin, M., 226
Stoltz region, 25
Stromberg, K., 16, 17
Stroock, D., 23
subharmonic function, 103
Sundberg, C, 79, 138, 180
support (of a measure), 16
symmetric derivative, 15
Sz.-Nagy, B., 20, 193
Sz.-Nagy-Foia§ functional model, 194
Szego's theorem, 22
Szego, G., 22, 76

272
INDEX
tangential boundary behavior, 58
Thomson, J., 9
Titchmarsh, E., 48, 164
Toeplitz operator, see also multiplier
A, 117
H1, 117, 249
H°°, 117
HP, 116, 250
X, 252
Tolsa, X., 9
topology
weak, 19, 95
weak-*, 19, 91
total variation, 14
Treil, S., 77
Tsereteli, O., 3, 5, 6, 76, 169
Tumarkin, G., 4, 101, 152, 154
Twomey, J., 58, 59
Uhl, J., 121
IjTyanov, R? 2, 48, 49, 54
uniform boundedness principle, 17
uniformly separated, 37, 133
unitary operator, 21
spectral theorem, 21
unitary perturbations, see also perturbations
vanishing mean oscillation, 72
Vasjunin, V., 130
Verbitsky, I., 3, 67, 79
Vinogradov, S., 3, 5, 6, 117, 122, 127-130,
137, 164, 170, 190, 249
weak topology, 19, 94
weak-* Schauder basis, 96
weak-* topology, 19, 91
weak-L1, 35
weakly sequentially complete, 94, 95
Weiss, G., 176
Wheeden, R., 11, 13, 65, 76
Wiener algebra, 127
Wiener, N., 193
Williams, D., 213
Wojtaszczyk, P., 17, 94-96
Wolff, T., 9, 241
Zhu, K., 62
Zygmund, A., 11, 13, 32, 42, 62, 64, 65, 68,
123, 163

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