The Cauchy Transform (Mathematical Surveys and Monographs 125)

Mathematical

Surveys

and Monographs Volume 125

The Cauchy Transform Joseph A. Cima Alec L. Matheson William T. Ross

American Mathematical Society

Mathematical

Surveys

and MonOiraphs Volume 125

The Cauchy Transform Joseph A. Cima Alec L. Matheson William T. Ross

American Mathematical Society

CONTENTS

vi

4.4.

Schauder bases

Chapter 5. Which functions are Cauchy hitegrals? 5.1. General remarks 5.2. A theorem of Havin 5.3. A theorem of 'I\llnarkin 5.4. Aleksandrov's characterization 5.5. Other representation theorems 5.6. Some geometric conditions

95 99 99 99

100 102 109 110

Chapter 6. Multipliers and divisors 6.1. Multipliers and Toeplitz operators 6.2. Some necessary conditions 6.3. A theorem of Goluzin~ 6.4. Some sufficient conditions 6.5. The !T-property 6.6. Multipliers and inner functions

115 115 118

Chapter 7. The distribution function for Cauchy transforms 7.1. The Hilbert transform of a measure 7.2. Boole's theorem and its generalizations 7.3. A refinement of Boole's theorem 7.4. Measures on the circle 7.5. A theorem of Stein and Weiss

163 163

120 122 127 129

,164

169 170 176

Chapter 8. The backward shift on H2 8.1. Beurling's theorem 8.2. A theorem of Douglas, Shapiro, and Shields 8.3. Spectral properties 8.4. Kernel functions 8.5. A density theorem 8.6. A theorem of Ahern and Clark 8.7. A basis for backward shift invariant subspaces 8.8. The compression of the shift 8.9. Rank-one unitary perturbations

179 179 180

Chapter 9. Clark measures 9.1. Some basic facts about Clark measures 9.2. Angular derivatives and point masses 9.3. Aleksandrov's disintegration theorem 9.4. Extensions of the disintegration theorem 9.5. Clark's theorem on perturbations 9.6. Some remarks on pure point spectra 9.7. Poltoratski's distribution theorem

201

Chapter 10.1. 10.2. 10.3. 10.4.

227 227 227 230

10. The normalized Cauchy transform Basic definition Mapping properties of the normalized Cauchy transform Function properties of the normalized Cauchy transform A few remarks about the Borel transform

184

185 186 192 192 194 196 201

208 211 212 218 221 222

241

CONTENTS

10.5. Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6.

A closer look at the

~-property

11. Other operators on the Cauchy transforms Some classical operators The forward shift The backward shift Toeplitz operators Composition operators The Cesaro operator

vii

243

249 249 250

252 252 253 253

List of Symbols

255

Bibliography

257

Index

267

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

cima~email.unc.edu

matheson~ath.lamar.edu

wross~richmond.edu

Overview Let X denote the collection of analytic functions on the open unit disk {z E C : Izl < I} that take the form (KI-')(z):=

j[]) =

dl-'«(), iTr~ 1- (z

where I-' belongs to M, the space of finite, complex, Borel measures on the unit circle 'Jl' = aD. In the classical setting, as studied by Cauchy, Sokhotski, Plemelj, Morera, and Privalov, the Cauchy transform took the form of a Cauchy-Stieltjes integral

r

1_'8 dF(O), e • z where F is a function of bounded variation on [0,211"]. i[O,21f]

1

-

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-' 1-+ KI-'; the functional analysis on the Banach space X (norm, dual, predual, basis); the representation of analytic functions as Cauchy transforms; the multipliers (functions cP such that cPX c X); the classical operators on X (shift operators, composition operators); and the distribution function y 1-+ m(l KI-' I > y) (where m is Lebesgue measure on 1['). 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 1-'. For example, a Cauchy transform f = K I-' satisfies the growth estimate

I f( z)j~. . ., ~ l-Izl'

z E D,

(111-'11 is the total variation norm of J.I.) as well as the identity lim (1- r)f(r() = I-'({(}),

r ..... l-

(E 'Jr.

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 1

OVERVIEW

2

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 f Kp. and 0 < p < 1,

=

sup f If(r()IPdm«() < 00, o 2, while a relatively recent theorem of Hollenbeck and Verbitsky identifies the best constant Bp as 1/ sin(7rp). This chapter also covers the important weak-type theorem of Kolmogorov

> Y)

= O(I/y),

y -+ 00, that gives an estimate of the distribution function for KJl. It will turn out, quite amazingly, that one can recover information about the measure from this distribution function. For example, Tsereteli's theorem says m(IKJlI

> y)

y -+ 00. 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. Jl« m # m(IKJlI

= o(l/y),

In Chapter 4 we treat the Cauchy transforms X = {K Jl : Jl E M} as a Banach space. Since KJll = KJl2 # Jll - Jl2 E HJ, where HJ are the measures fJ dm : 1 E Hl, 1(0) = O}, X can be identified in a natural way with the quotient space Mj HJ, by means of the mapping KJl ~ [JlJ. Here [Jll is the coset in M / HJ represented by Jl. One defines the norm of K Jl to be

OVERVIEW

4

the norm of the coset lJ.t] in the quotient space topology of M j HJ. Equivalently, the norm of an 1 E X is

11/11 =

inf {11J.t1l : 1 = KJ.t}.

Equipped with this norm, X becomes a Banach space and furthermore, the previous growth estimate can be improved to

I/(z) I ""~ JltlL 1 -izi ' z E

1n\ JUl.

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 X? 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

1=

K J.t. An interesting and still open question is to determine whether or not a given analytic function 1 on the disk takes the form 1 = K J.t. From what was said

above, certain necessary conditions hold. For example, a Cauchy transform 1 must have bounded Taylor coefficients; must satisfy the growth condition I/(z)1 0«1Izl}-l); the boundary values ofthe function 1 must satisfy the £P condition II/IILI' 0«1 - p)-l) for < p < 1; the boundary values for 1 must also satisfy the weaktype inequality m(l/l > Y) = O(ljy). Unfortunately, none of these conditions is sufficient. A more tractable question is: if 1 is not merely analytic on ID> but instead is analytic on the larger set C\ T with I(oo} = 0, when is I equal to

=

°

=

f~

dJ.t«() , z E C\ T, 1-(z for some measure J.t on the circle? Tumarkin answered this question with the following theorem: if I is analytic on C\ T with 1(00) = 0, then 1 is the Cauchy integral of a measure on the circle if and only if sup [ II(r() - 1«(jr)1 dm«() < 00. Of)(z):= ("¢(()f(() dm(() = ("¢f)+(z) 111'

1- (z

is a bounded operator from the space of bounded analytic functions to itself. Kolmogorov's weak-type estimate m(lKp.1 > y) = O(l/y) has been re-examined recently yielding some fascinating results on how this distribution function y 1-+ m(lKp.1 > y) can be used to recover the singular part of the measure p.. Chapter 7 is devoted to these ideas. For example, it is relatively easy to show that when p. « m, the Kolmogorov estimate can be improved from

m(IKp.1 > Y) = O(l/y) to

m(IKp.1 > Y) = o(l/y)· Tsereteli proved the converse, namely, m(lKp.l > y) = o(I/Y)

¢:>

p. «m.

OVERVIEW

6

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 1rym(IKpl

1/-+00

> y) = I1PslI,

where Ps is the singular part of p with respect to Lebesgue measure. Notice that when p « m, or equivalently Ps = 0, we obtain Tsereteli's theorem. The other more striking, and more recent, theorem of Poltoratski says that

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 f-+ m(IQpl > y) and y f-+ ml(\!Kp\ > y) of the conjugate function

(Qp)(ei8 ) =

P.V.! (0; t) dp(e cot

and the Hilbert transform

(!Kp)(x) =

p.v.l-

1-

J[l

x-t

it )

dp(t),

where p is a finite measure on JR. Some. of these distribution theorems are quite classical. For instance, an 1857 theorem of Boole says that if n

g(x):=L~' j=1

x - aJ

aj

E

JR,

Cj

> 0,

which is just the Hilbert transform of the positive discrete measure n

p:= LCjda;, j=1

then 1 '!-

LCj,

ml({x E JR: g(x) > y}) = Y j=1

where ml is Lebesgue measure on lR. Though the material in the first !Several chapters is certainly both elegant· and important, our real inspiration for writing this monogra.ph is the relatively recent work beginning with a seminal paper of Clark which relates the Ca.uchy 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 1JH2, where 1J 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

S*f = 1-1(0)

z are of the form (7J H2) l. . The functions

k>.(z) := 1 -19(307J(z) ,

A, Z E Jl)),

1 - AZ

are the reproducing kernels for (7JH2)1. in the sense that k>. E (7JH2)1. and

f(A) = (I, k>.)

V f E (7JH2)1..

Here we are using the usual 'Cauchy' inner product

(I,g)

:=

h

f«()g«)dm«)

on H2. Clark's work was inspired by the question as to whether or not a given sequence of kernel functions (k>'n)n~l has dense linear span in (7JH2)1.. Clark showed that for certain ( E T, the kernels kC; belong to (iJH2)1. and are eigenvectors for an associated unitary operator Uo. on (7JH2) 1. . Using the spectral properties of Uo., Clark determined when these eigenvectors kC; form a spanning set for (iJH2)1. and then used a Paley-Wiener type theorem to say when the k>'n's were 'close enough' to the kC;'s to form a spanning set. The unitary operator U0 mentioned above is the following: let Su be the compression of the shift S to (iJH2)1.; that is, Su := PuSI( iJH2)1.

where Pu is the orthogonal projection of H2 onto (iJH2)l.. All possible rank-one unitary perturbations of Su, under the simplifying assumption that iJ(O) = 0, are given by Uof:=

SuI + (I,~)

0:,

0:

E

T.

It turns out that U0 is also cyclic and hence the spectral theorem for unitary

operators says that Uo is unitarily equivalent to the operator 'multiplication by z', (Zg)() f--t (g«), on the space L2(170.), where 170. is a certain positive singular measure 011 T. It is quite remarkable, as we shall discuss in a moment, that 170. can be computed from the inner function 7J. The unitary equivalence of Z on L2(170.) and Uo on (iJH2)1. is realized by the unitary operator :Ja : (iJH2)1. ~ L2(170.), which maps the reproducing kernel

k>.(z)

:= 1 -19(307J(z)

1- AZ

for (iJH2)1. to the function

OVERVIEW

8

in L2 (ua) and extends by linearity and continuity. Clark uses this unitary equivalence, as well as the structure of the associated space L2(U a), to further examine whether or not the kernels (k.\.. )nfll fortn a spanning set for (t1H2).!.. This spectral measure U a for Ua arises as follows: for each fixed a E T the function

Z

1-+

R (ex + D(Z»)

ex - D(z) is a positive harmonic function on D, which, by Herglotz's theorem, takes the form !R (a + D(Z») a-D(z)

= [1 -lzl2 dua «(), JT\(-zI 2

where the right-hand side of the' above equation is the Poisson integral (Pua)(z) of a positive measure U a on T. Without too much difficulty, one can show that the measure U a is carried by the set {(. E T: D«() a} and hence U a ..1. m. Furthermore, U a .l. U(3 for a I- {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 cP E ball{HOO) to create a family of positive measures {J.'a : ex E T} associated with cP. It is becoming a tradition to call this family of measures the 'Aleksandrov measures' associated with cP. A beautiful theorem of Aleksandrov shows how this family of measures provides a disintegration of normalized Lebesgue measure m on the circle. Indeed,

lrJ.'adm{a)

= m,

where the integral is interpreted in the weak-* sensej that is,

Ir (i

f«() dJ.'a«(.») dm(a) =

1r f«(.)

dm«()

for all continuous functions / on T. The identity

1

(Kua)(z)

= 1- aD(z)

produces the following formula for ~

: L2(ua } -+ (DH2).!. I

in terms of the 'normalized' Cauchy transform

T./ = K(fdua ) a

Kuo

·

Poltoratski showed that several interesting things happen here. The first is that for uo-almost every (. E T, the non-tangential limit of the above normalized Cauchy transform exists and is equal to f«(.). On the other hand, for 9 E (DH2).!., the non-tangential limits certainly exist almost everywhere with respect to Lebesgue measure on the circle (since (DH2).!. C H2). But in fact, for ua-almost every (, the non-tangentialliInit of 9 exists and is equal to (~ag)«(). The compression S-{J 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

OVERVIEW

9

theorem

> y). m = j.ts, weak-*,

lim 7rym(IKj.t1

y-oo

are covered in Chapter 9. Poltoratski's theorems on the normalized Cauchy transform 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

AA

:=

A + AV ® v,

A E JR.,

of a self-adjoint operator A with cyclic vector v. Here, the Borel transform dj.t(t) JJR t-z' a close cousin to the Cauchy transform, comes into play.

r

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

J

w

~ z dj.t(w)

of a measure j.t 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 1216, 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, 169J as well as the survey papers [32, 33J are a good place to start. There is also a notion of fractional Cauchy transforms [131J.

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= C U {oo} (Riemann sphere) • R (real numbers) .1IJ)={zEC:\zi y) = O} .

(1.2.1)

As is customary, we equate two measurable n,.nctions that are equal almost every" where. Holder's inequality 1 1 Ilgl dm ~ 1I/1I,,1I9I1q , p ~ 1, -p + -q = 1,

i

Minkowski's inequality

III + gil" ~ as well as the associated inequality

Ilfll" + 11911", p ~ 1,

111+ 911: ~ 11/11: + 11911:, 0 < p < 1, imply that for 1 ~ p ~ 00, the quantity II/lIp defines a norm on V

which makes it a Banach space (complete normed linear space) while for 0 < p < 1, the quantity d(f,g) := III - gll~ defines a metric on L" that makes it a complete, translation invariant, metric space. A classical representation theorem of F. Riesz says that for p tinuous linear functionall : V - C takes the form

l(f) = for some unique 9 E Lq (1/p+ l/q (1.2.2)

sup

~

1, every con-

if9dm

= 1). Moreover, the identity

{Ii f9dml :I

E £P,

II/lIp ~ 1} = 11911q

implies that the norm of this functional is IIgliq. Thus when p ~ 1, one equates (V) "', the set of continuous linear functionals on L", with Lq. When 0 < p < 1, we have (U)'" = (0) [56].

1.2. LEBESGUE SPACES

13

We will now review distribution functions and rearrangements. Two nice references for this are [85, 229]. For I E LO, the function AI: [0, (0)

(1.2.3)

-+

Af(Y):= m(1/1 > V),

[0,1],

is called the distribution function for I and certainly plays an important role in analysis and probability. One can see that Af is a decreasing right-continuous function on [0, (0). There are also the following LP results. PROPOSITION

(1)

1.2.4. For p

11/11: = p 10

00

> 0,

yP-l Af(Y) dy.

(2) Af(Y) ~ y-PII/II~· For

I E LO, the function

(1.2.5)

r :[0,1]

-+

!,,(x) := inf{y

(0,00),

> 0: Af(Y)

is called the decreasing rearrangement of One can check that if

I.

~ x}

If Af is one-to-one, then

f* is A,l.

n

I(() = L)jXAj((), j=l

where the Aj's are pairwise disjoint measurable subsets of 'lr and

b1 > b2 > ... > bn , then n

AJ(X) =

L B jX[b;+l,b )(X) j

j=O

and n

!,,(x) = LbjX[B;_l,B;)(x), j=l

where j

B j :=

L m(Ai)' i=1

Note that bo := 00, bn +1 := 0, Bo := O. The first important fact about

r

is that

(1.2.6)

where Ar (x) at least for I that

= ml(f* > x) and ml is Lebesgue measure on R. ~

The second is that, -+ [0,1] so

0, there is a measure preserving transformation h : ']['

(1.2.7)

I=!"oh.

See [185] for details2 • 2In our presentation here, we are really considering the decreasing re-arrangement of III. If one is willing to expand the definition of decreasing re-arrangement, one can prove eq.(1.2.1) for general real-valued J.

1. PRELIMINARIES

14

1.3. Borel measures A (finite) Borel measure IL on 'll' is a function which assigns to each Borel set A c 'll' a complex number IL(A) such that IL(0) = 0 and

IL

(Q1 An) ~IL(An)' =

whenever (An)n~l c 'll' 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 'll' by M. A measure IL E M is positive (denoted IL ~ 0) if IL(A) ~ 0 for all Borel sets A c 'll'. We set M+ := {/l EM: /l ~ o}. THEOREM

1.3.1 (Jordan decomposition theorem). Any /l E M can be written

uniquely as (1.3.2)

For /l EM, define the total variation of IL to be the number (1.3.3)

II/lil := sup

{t {t

1/l(Aj) I : {AI," . , An} is a Borel partition of 'll'} .

3=1

For a measure IL, define the total variation measure l/ll by (1.3.4)

1/lI(A) := sup

1/l(A;)1 : {A 1 ,··· , An} is a Borel partition of A} .

3=1

Note that

l/ll('ll') = II/lil and that if /l is real with /l = /l1 - /l2, /lj E M+, then l/ll = /l1

+ /l2·

For a general /l E M with

/l = (/l1 - /l2) + i(/l3 - /l4), /lj E M+, it follows from the inequality . a+b la ± 2bl ~ ..j2' a, b > 0, that for all Borel sets A C 'll',

~ {t,~;(A)} " I~I(A)" t,M;(A). PROPOSITION

1.3.5. The space M, endowed with the total variation norm 11·11,

is a Banach space. Let G('ll') denote the Banach space of complex-valued continuous functions on 'll' endowed with the supremum norm

1111100 = sup {II(()I : ( E 'll'}. The identification of G('ll')* (the dual space of G(11')) 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 f a unique J.L EM such that f = fIJ' where

tJ.l(f) :=

Moreover, II f J.lII

= sup

{II I

E

C(1l') * . Then there is

1

I dJ.L.

dJ.L1 : I E C(1l'), 11/1100

~ I} = IIJ.LII·

The Riesz representation theorem implies that the map J.L 1-+ fJ.l is an isometric isomorphism between M and C(1l')* and one often identifies C(1l')* with M. A measure J.L E M is absolutely continuous (with respect to Lebesgue measure m), written J.L « m, if JL(A) = 0 whenever A is a Borel set with m(A) = O. A measure JL is singular (with respect to m), written J.L .1 m, if there are disjoint Borel sets A and B such that AU B = 1l' and J.L(A) = m(B) = O. THEOREM 1.3.7 (Radon-Nikodym theorem). A Borel measure J.L E M is absolutely continuous with respect to Lebesgue measure m il and only il dJ.L = I dm lor some I E Ll, that is to say,

J.L(E) = L/dm, lor all Borel sets E c 1l'. The function I in the above theorem is called the Radon-Nikodym derivative of J.L and is often denoted by dJ.L ._ -.-f. dm It is a standard fact that the Radon-Nikodym derivative of J.L 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 ( E 1l' and t > 0 (sufficiently small), let I((,t):= {(e is : -t < s < t} be the arc of the unit circle subtended by the points (e it and (e- it . If J.L E Mis real, define, for each ( E 1l',

~t(() = J.L(I((,t)) m(I((, t)) and note that (

1-+

~t(()

is a Borel function on 1l'. Define

(l2Jt)(():= lim

~t(()

t-O+

(DJ.L)():= lim

t-tO+

~t().

When (l2Jt)(() = (DJ.L)(() < 00 we say that J.L is differentiable at ( and we write (DJ.L)() := (l2Jt)() = (DJ.L)((). For a complex measure J.L = J.Ll +iJ.L2, where J.Ll,J.L2 are real measures, we say that (DJ.L)() exists if both (DJ.Ll)(() and (DJ.L2)(() exist. Here is a collection of important properties of DJ.L. PROPOSITION

exists lor m-a. e. (

1.3.8 (Lebesgue differentiation theorem). For J.L E 1l' and

(DJ.L)() =

:~ ()

m-a.e.

E

M, (DJ.L)(()

1. PRELIMINARIES

16

THEOREM 1.3.9 (Lebesgue decomposition theorem). Any I-" E M can be decomposed uniquely as

where 1-"0,,1-"8 E M W'ith 1-"0,

«

1-"=1-"0,+1-"8' m and 1-"8 .1. 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.

II I-" = 1-"0, + 1-"8 is the Lebesgue decomposition 01 J.L, then 1IJ..t1l

= 111-"0,11 + 111-",11·

Define

Ma:={I-"EM:I-"«m} Ms:={I-"EM:I-".1.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,

111-"11 = 111-"0,11 + 111-"811, J..ta E Ma, 1-"8 EM, and so Ma and M, are closed subspaces of M. For J.L EM, consider the union 'U of all the open subsets U c T for which I-"(U) = O. The complement T \ 'U is called the support 01 1-". A Borel set H c T for which I-"(H n A) = I-'(A) for all Borel subsets AcT is called a carrier 01 J..t. Certainly the support of I-' is a carrier but a carrier need not be the support and need not even be closed. For example, if I is continuous and dl-' = I dm, then a carrier of J..t is T \ I-I({O}) (which is open) ~hile the support of I-" is the closure of this set. The following facts are found in [68, 182]. PROPOSITION 1.3.11. II I-' E M+ and J.L = l-"a+J.L, is the Lebesgue decomposition of 1-', then (1) DJ..t, = 0 and DJ..t = DI-"a I-"a-a.e. (2) 1-"0, is carried by {O < !J.p < co}. (3) 1-'8 is carried by {l2J.t = co}.

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 v,J..t E M, we say that v is absolutely continuous with respect to 1-", written v « J.L, if

II-"I(E) = 0 =? veE) = o. If v = (VI -v2)+i(V3 -V4), Vj E M+, is the Jordan decomposition of v, the following are equivalent: (i) V « 1', (li) Vj « 1-", j = 1,2,3,4, (iii) Ivi « 1-', (iv) Ivi « 11'1. The Ra.don-Nikodym theorem becomes: if v « 1-", then there is an fELl (II-'I) such that

YeA) =

LI

dp.

for all Borel subsets AcT. We say 1-", v E M are mutually singular, written I' .1. v, if there are disjoint Borel sets A and B with Au B = T and II-'I(A) = Ivl(B) = O. The following are equivalent: (i) I' .1. v, (ii) 11-'1.1. 1"1, (iii) J.Lj .1. "k for j, k = 1,2,3,4.

1.4.

SOME ELEMENTARY FUNCTIONAL ANALYSIS

17

The Lebesgue decomposition theorem says that for J.L, /I E M,

/I = /I!: + /If, where /I!:

«

J.L and

/If .1 J.L.

Furthermore, this decomposition is unique.

For J.L E M+ and n E N, let Fn:= {( E 'll': J.L({O) > lin} and observe, since J.L is a finite measure, that Fn is a finite set. Also observe that 00

{( E 'll': JL({O)

UFn

> O} =

n=1

and so the set of atoms of a measure (Le., those ( E 'll' for which JL( {O) > 0) must be at most a countable set. A measure J.L E M is a discrete measure if it has a carrier that is at most countable. A measure J.L E M is continuous if J.L( {O) = 0 for all ( E 'll'. There is the following refinement of the Lebesgue decomposition theorem [99, p. 337]. THEOREM 1.3.13. If JL E M, then J.L = J.La

+ J.Lc + JLd,

where J.La « m, J.Lc, J.Ld .1 m, J.Lc is continuous, and J.Ld is discrete. Furthermore, J.La, J.Lc, J.Ld 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 11·11, let X" denote the dual space of continuous linear functionals '- : X t-+ C. Note that X" is a Banach space when endowed with the norm 11'-11 := sup {ll(x)1 : x E X, IIxll ~ I}.

(1.4.1)

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 E X, sup{ll(x)1 : l E!J} < 00, then sup{ 1I11l : l E :f}
... ,Xn E X and an € > 0 such that

n n

{i

E

X* : I(i - io)(xk)1 < €}

c

U.

k=l

The space (X*, *), X* endowed with this weak-* topology, is a locally convex topological vector space. A sequence (in)n~l c X* converges to i weak-* if and only if in(x) -. i(x) for each x E X. An application of the uniform boundedness principle (Theorem 1.4.2) says that a weak-* convergent sequence (in)n~l is uniformly bounded, that it to say, sup{IIinil : n ~ I} < 00. 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*) := {i E X* : IIill ~ I} is compact in (X*, *).

REMARK 1.4.9. If X is also separable (Le., 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 (in)n~l is a sequence in ball(X*), then there is an i E ball(X*) and a subsequence in" -. i 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(1:*) and i belongs to the weak-* closure of E then there is a sequence (in)n~l C E converging weak-* to i. 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 i in the weak-* closure of E, there is a net in E converging to i weak-*. The above argument using the Banach-Alaoglu theorem says there is a sequence in E converging to i weak-*. THEOREM 1.4.10. If Y c 1:*, then (oY)O is the weak-* closure of the convex, balanced hull of Y. If X is a Banach space, then so is X* and hence one can consider its second dual X** := (1:*)*. For x E X, let Q(x) be the element of X"* defined by

(Q(x))(i)

= i(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 V', for 1 < p < 00, is reflexive while L1 is not. We point out some basic facts about reflexive spaces. 1.4.'i1. For a Banach space X, the following are equivalent. X is reflexive. X* is reflexive. Every subspace of X is reflexive. Every quotient space of X is reflexive. The closed unit ball {x EX: IIxll ~ 1} is compact in the weak topology.

THEOREM

(1) (2) (3) (4) (5)

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 V', 1 ~ p < 00, spaces are all separable (the trigonometric polynomials are dense) while Loo is not. A topological vector space lJ (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. If X is a Banach space and {xa : a E A} is an uncountable subset of X satisfying

IIXa -

xbll

~ 1,

a,b E A,

a =F b,

then X is not separable. PROOF.

The hypothesis says that the open balls ~(a, 1/2) :=

{x EX:

IIx - all < 1/2}

are disjoint. H J were a countable dense subset of X then each ball ~(a, 1/2) would contain at least one element of J, making J uncountable. 0 For example, to see that Loo is not separable set where Ia

:= {e it :

xa«) := XI.. «), 0 < a < 271", 0 < t < a} and use the previous proposition.

A few results relating separability and reflexivity are the following. 1.4.13. (1) Let X be a Banach space. If X* 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.

PROPOSITION

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, lJ, a linear operator A: X -+}I is bounded if (1.5.1)

8up{IIAXII~

: IIxlix ~ I}
J-L(A) for each Borel set A C 1f' with J-L(8A) = O. 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 (fLn)n;:'l eM converges to J-L weak-*, then

< 00

sup IIJ-Lnll n

and IIJ-LII ~ lim IIJ-Ln II· n--+oo

PROOF. By the Principle of Uniform Boundedness, we know that sup IIfLn II n

< 00.

Let n--+oo

and choose a subsequence (J-Lnk)k~l so that lim IIJ-Lnk II = L.

k--+oo

Given

E

> 0, there is a KEN so that IIJ-Lnk II ~ L

+E

't/ k ~ K.

Since IIJ-LII =sup{lf gdJ-L1 :gEball(G(T))}, there is agE ball(G(T)) such that IIJ-LIIBut since J-Lnk

-->

E


0, where h is the arc of the circle subtended by e- ih and eih and observe that "h -+ 61 weak-* (Lebesgue differentiation theorem). Now use the density of Ms in (M, *) as argued in the first part of the proof. 0 We will also make use of the following. PROPOSITION 1.6.8. The convex balanced hull

{6, : ( is weak-* dense in the ball

0/

E 1'}

0/ M.

PROOF. If Y = {6, : (E 1'}, one can easily show that oy = ball(C(1'» and so = ball(M). Now use Theorem 1.4.10. 0

(oy)O

REMARK 1.6.9. We can combine Proposition 1.6.8 with Remark 1.4.9 to prove the following: given I' E M, there is a sequence (J.I.n)n;;>.1 c M such that each I'n is a finite union of point masses, IIl'nll ~ 111'11 for all n, and J.I.n -+ I' weak-*. There is also the following refinement (see [40, p. 221]) PROPOSITION 1.6.10. Suppose I' E M+ with support on a closed set F c 1'. Then there is a sequence J.I.n -+ I' weak-* such that for each n, J1.n E M +, is supported in F, is a finite linear combination 0/ point masses, and IIl'nll = 111'11.

1.7. Non-tangential limits and angular derivatives For an analytic function / on D and ( E 1', we say that / has a mdiallimit L at (, if lim f(r() = L. r-l-

For ( E l' and a

(1.7.1)

> 1, let

r a«() := {z ED: Iz -

(I < a(l -Izl)}

be a non-tangential approach region (often called a Stoltz region). Note that r a«() is a triangular shaped region with its vertex at ( (see Fig. 1). We say that / has a non-tangential limit value A at (, written

L lim f(z)

z-,

= A,

1. PRELIMINARIES

26

FIGURE

1. Non-tangential approach region with vertex at

(E

'lI'

if I(z) -+ A 88 Z -+ ( within any non-tangential approach region r a«)' Let us mention a few well-known results about non-ta.zi.gentia11imits. We refer the reader to [48] for the proofs. THEOREM 1.7.2 (Fatou). II I is a bounded analytic function on)[Jl, then the non-tangential limit of I exists and is finite lor almost every '" E T.

For bounded analytic functions, the existence of radial and non-tangential limits are the same. THEOREM

I(z)

-+

A as z

I is a bounded analytit; function on D. and along some arc lying in)[Jl and terminating at," E T, then

1.7.3 (Lindelof). If -+ (

L lim I(z) = A. z-C

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 < f) < 21[", let CfJ be the rotation 01 C through an angle (J about the origin. Then there is a bounded analytic function I on D which does not approach a limit as z approaches any point eifJ from the right or the left along CfJ.

Littlewood [1241 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 analytic on D and L lim I(z) = 0

I

is

z--+C

lor ( in some subset ofT of positive Lebesgue measure. Then f

== O.

Non-tangential limits are important in the statement of Privalov's theorem since there are non-trivial analytic functions on D which have r.adiallimits 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 'lI' [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 ( E ']f, we need the following factorization theorem [65]. THEOREM

1.7.6. If f is a bounded analytic junction on]D), then

f = 1JF, where 1J is a bounded analytic junction that has boundary values of unit modulus almost everywhere and F is a bounded analytic junction that satisfies

h

log IF(O)I =

log IF(()I dm(().

The function 1J is called the inner factor of f and the function F is called the outer factor of f. We can factor {} further as

{} = bSI!' where b is a Blaschke product

b(z)

=

zm

IT lanl an - z n=l

an 1- anz

whose zeros at z = 0 as well as {an} C ]D)\{O} (repeated according to multiplicity) satisfy the Blaschke condition

n=l

(which guarantees the convergence of the product) and sl! is the (zero free) singular inner factor SI!(z)

where

~ E

= exp ( -

/

~ ~ ; d~(()) ,

M+ and is singular. Furthermore, the outer factor F can be written as

F(z) = ei'r exp

(i ~ ~; log IF(()ldm(()) .

Note that log IFI E £1 (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 (a n )n;;:'l. A necessary and sufficient condition that b and all its partial products have nontangential limits of modulus one at ( is that

~ l-Ia n l

~ I( -

ani


l-

1.8.7. (1) From Proposition 1.3.8, Dp, = dp,/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 IL 1. m, or equivalently Dp, = dp,/dm = 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,

REMARK

L.lim(Pp,)(z) = (DIL)«() z-+(

whenever (Dp,)«() exists. (4) If ( E 1r and p, is a real measure, then [182] (1.8.8)

(QJ.L)(()

~

lim (Pp,)(r()

r->l-

~

lim (Pp,)(r()

~

(Dp,)«().

r->l-

5Fatou's original proof in terms of Poisson-Stieltjes integrals is in [69). The references [65, p. 39) or [101, p. 34) have modern proofs.

1. PRELIMINARIES

32

If J.I. E M+, then certainly PJ.I. ~ 0 on D. Also note that HJ.I. is analytic on D with ~HJ.I. = PJ.I. ~ O. This following theorem of Herglotz 6 is the converse. 1.8.9 (Herglotz). (1) If u ~ 0 on D and harmonic, then u = PJ.I. for some J.I. E M+. (2) II I is analytic on D, ~I ~ 0, and 1(0) > 0, then I = HJ.I. for some J.l.EM+.

THEOREM

From Fatou's theorem (Theorem 1.8.6), we know that PJ.I. has finite nontangential boundary values m-almost everywhere and we will see in the next chapter (Lemma 2.1.11) that HJ.I. does as well. Since HJ.I. = PJ.I.+iQp., then QJ.I. has boundary values and the m-almost everywhere defined boundary function

(QJ.I.)«():= lim (Qp.)(r() r-+l-

is called the conjugate junction. At least formally (replacing z with ei6 and ( with eit in the eq.(1.8.2», this boundary function (QJ.I.)(e i6 ) is equal to

(QJ.I.)(ei6)

= 121r ~ (::: ~ :::) dJ.l.(eit) = 12ff cot

(9; t) dJ.l.(eit).

Unfortunately, for fixed (), the function cot«({.... t) may not belong to L 1 (J.I.), making the integral possibly undefined. In terms of principal value integrals, we do have the following standard fact. THEOREM

1.8.10. II J.I. E M, then lim (QJ.I.)(re''6 ) = P.v.

r-+l-

127r cot (()-2- t) dJ.l.(e

:= lim E-+O+

0

1,

16-tl~E

t't }

(() - t) .

cot -2- dJ.l.(e't).

for m-a. e. ei6 . 1.9. The classical Hardy spaces For 0 < p < 00, let HP, the Hardy space?, denote the space of functions analytic on D for which the £P integral means

(1.9.1) remain bounded as r

Mp(r; f) :=

i

t£

If(r()IP dm«()

f

riP

1-. This definition can be extended to p =

00

by

Moo(rj f) := sup{lf(r()1 : ( E 'f} and so Hoo is the set of bounded analytic functions on D. The function r~

Mp{r;f)

is increasing on the interval [0,1), that is,

(1.9.2)

M(rl; f)

~

M(r2; f),

0 ~ rl

~

r2 < 1,

6The reference [98] contains the original proof while [101, p. 34] or [65, p. 2] have more modern proofs. 7We 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

II/IIHP:= sup Mp(r; f) = lim Mp(r; f) O 0,

where p(a, b) :=

la-bl

11- abl

is the pseudo-hyperbolic distance between a and b, and unilormly sepamted if

(1.11.2)

6(E) := inf IBa(a)1 > O. aEE

Let I be an arc on the unit circle, and define the Carleson square on I to be the set

(1.11.3)

Q = { z E lD>:

j:l

E I and 1 -

Izi < m(I) }

(see Figure 2). A positive measure J1. on lD> is a Carleson measure if there is a constant cp. depending only on J1. such that

J1.(Q)

~

cp.m(I)

for each Carleson square Q. We define "Yp. to be the infimum of all such constants Cp.o We say that E is a Carleson sequence if JLE is a Carleson measure and we set

"Y(E) := "YP.E' The sequence E is an interpolating sequence if, whenever g E too (E), the bounded functions on the sequence E, there is a function I E H oo such that liE = g. By the open mapping theorem, there is a constant C such that for each 9 E lOO(E), a function I E Hoo can be chosen so that

(1.11.4)

11/1100 ~ Csup{lg(z)1 : z E E}.

1. PRELIMINARIES

38

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 sepamtedj (3) E is sepamted and f..LE is a Carleson measure. In case any of these conditions hold, we have the following relationships between the constants s(E),6(E),'Y(E), and C(E):

(1.11.6)

1 6(E) ~ C(E) ~

(1.11.7)

s(E) ~ c5(E),

where Cl, C2, C3

> 0 are absolute constants.

Cl

'Y(E) 6(E) ~

1

C2 c5(E)

1 s(E) ~ C(E) ,

(

1 ) 1 + log 6(E) ,

6(E) ~ exp

(

'Y(E) )

-C3 s(E)2

'

Interpolation sequences actually exist [101, p. 203]. THEOREM 1.11.8 (Hayman-Newman). A sequence (Zn)n)1 C D such that

sup

IZn+11 : EN} < 1 {I-l-IZnI n

is an interpolating sequence. COROLLARY 1.11.9. If (rn)n)l C (0,1) with rn polating sequence if and only if

sup {

1- rn+1 } 1 _ r.. : n E N

i

1, then (rn )n)l is an inter-

< 1.

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~1 C (0,1) satisfies

n=l then there is a sequence of angles (9n)n~1 C [0,211") such that (rnei9n )n~1 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 Clo Cl > such that cl«1 - r)2 + ( 2)1/2 ~ 11 - rei9 1 ~ c2«I- r)2 + ( 2)1/2

for all r E (!, 1) and all 8 E [0,11"]. PROOF. Note that 11 - re i9 1= (1- 2r cos 8 + r2)1/2

= «1- r)2 + 4rsin2 (8/2»1/2.

Using the estimate

!!.11" ~ sin(9j2) ~ 9

V9 E [0,1I"j,

we get 9: ~ sin2(9/2) ~ 82 V8 E [0,11"]. 11" . Hence we obtain constants Cl, C2 > so that

°

Cl (1- r)2

+(

~ 1- 2rcosO + r2 ~ C2 (1- r)2

2)

+ ( 2)

for all r E (~, 1) and () E [0,11"]. LEMMA 1.12.2. Given p

°

> 1, there is a positive constant c > so that 1

~

dO ~ 1_~ -;----·;-;;"9:11- ret Ip

c -;-:---:---:(1- r)p-l

for all r E (!, 1). PROOF. Observe that

1_~~ 11 - d(}rei9 p = 1~_~ (1 1

Thus by Lemma 1.12.1,

l_~ 11 1r

d8 re i9 1P

dO 2r cos 0 + r 2)p/2

r

r

= 2 10

( [1-r

dO

~ c 10 «1 - r)2 + 92)p/2

d9

(1- 2r cos 0 + r 2)p/2'

=

C

10

r) .

+ 11-r

Estimating these two integrals, we get

[l-r

10 and

r

dO «1 - r)2 + (}2)p/2

r

[l-r

~ 10

d() «1 - r)2)p/2

1

= (1 -

r)p-l

d9 dO 1 C 11- r «1 - r)2 + (}2)P/2 ~ 11-r (02)p/2 = C+ (1 - r)p-l ~ (1 - r)p-l .

o

1. PRELIMINARIES

40

o 1.12.3. There are constants C1,C2 > 0 independent ofr E (~, 1) such

LEMMA

that 1 cllog 1- r ~

From Lemma 1.12.1,

PROOF.

71"

Cl

171" 1 1 -71" 11- reiBl dB ~ C2log 1- r'

1o

d()

v'(1 - r)2 + ()2 By integrating, we get

1 71"

~

171" 1 1'11" . '81 dB ~ Cl 0 -71" I1 - ret

d()

v'(1 - r)2 + ()2

•

d()

--;:;:=~=:; 0, then F = IFleit/J, where -Tr/2 < t/> < Tr/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 lD and ,

~

FP =

IFIP (cos~l + i sin(ptJ») .

For 0

0 given,

let

N

h(n)C

+ Lh(n)(n,

(E 'lI',

n=O

be a trigonometric polynomial with IIg-hll l < E (an appropriate Cesaro polynomial of 9 will work - see Theorem 1.6.5). Observe that N

K(hdm)(z) = Lh(n)zn. n=O

2. THE CAUCHY TRANSFORM AS A FUNCTION

46

Thus, N

P

IIK(hdm)ll~ = ~h(n)zn n=O

P P

I

N

(

~lh(n)12

~ ~h(n)Zn N

=

•

2 ) P/ 2

~ IIhll~ ~ IIhll~

and so certainly

1 = 0(-1-), -p Using Smirnov's theorem (Theorem 2.1.10), IIK(hdm)lI~

p

-+

1-.

1 IIK(hdm) - K(gdm)lI: ~ 01--lIhdm - gdmll P

-p 1

=0 1- pllh - gill ~

Finally, Letting

€ -+

1 0-1-€P, -p

(1 - p)IIK(gdm)lI~ ~ 0(1) + O€p. 0 yields the result.

o

2.2. Cauchy integrals and HI, The classical Cauchy integral formula4 says that if I is analytic in a neighborhood of 0-, then

I(z) = ~ 1 1«(,) d(,. 21rt .11'1=1 (, - z Making the observation that d(, 211'i(, = dm«(,) ,

we will write the Cauchy integral formula

I(z)

=f

88

1('2 dm«(,).

JT 1- (,z,

. The question now is: what is the 'largest', of analytic functions on III that can be written via the Cauchy integral formula? For I E HI, Theorem 1.9.4 says that

class

1(') == r-+llim I(r(,) exists for almost every ( E or and defines an integrable 'function. Thus for Hl functions, the integral on the right. hand' side 'of the Cauchy integrSJ. formula makes sense. It turns out that the left-hand side is equal to the right-hand side. 4See [201] for a historical overview of Ca""chy.

2.2.

CAUCHY INTEGRALS AND

HI

47

PROPOSITION 2.2.1 (Cauchy integral formula). For I E HI,

I(z) that is to say,

J.f 1- (z

dm«(),

1= K{fdm).

PROOF. For 0 slightly larger disk

< r < 1, let Ir(z)

= I(rz) and note that Ir is analytic on the By the classical Cauchy integral formula,

{Iz\ < l/r}.

J

Ir(z) = Note that Ir

1«(1

= (

--+

Ir('.) dm«().

1- (z

I in L1 (Theorem 1.9.4) and so for each z E ID>,

J

Ir('.) dm«)

J 1«(1

--+

1- (z

1- (z

dm«) as r -d-.

Clearly Ir --+ I pointwise in ID>. Combine these two limits to obtain the Cauchy integral formula. 0 By Holder's inequality, HP C HI for all p ~ 1, and so we can combine Theorem 2.1.10 and Proposition 2.2.1 to see that (2.2.2) O,

I(z)

=

J 1«(1

1- (z

dm«).

PROOF. If I E HI, then the two conditions hold by Theorem 1.9.4 and Proposition 2.2.1. Conversely, suppose the two conditions hold. Then I = K{fdm) E HP for all 0 < p < 1 (Theorem 2.1.10). But since the boundary function belongs to Ll, then I E HI (Proposition 1.9.5). 0

2. THE CAUCHY TRANSFORM AS A FUNCTION

48

2.3. Cauchy A-integrals In this section, we prove a generalization of the Cauchy integral formula

I(z) =

j 1(9.

1-(z

dm«)

involving the theory of A-integrals as studied by Denjoy, Titchmarsh [215], Kolmogorov, Ul'yanov [225J, and Aleksandrov [9J. A Lebesgue measurable function 9 on 11' is A-integrable if the following two conditions hold. The first is that 9 E L~'oo, that is to say,

m(lgl > y) =

(2.3.1)

o(l/y),

y

-+ 00.

The second is that

(A)jg«)dm«):= lim f g«()dm«() y.... oo J1g1 0, let I(y,() :=

{/«), 0,

if I/«)J < y; otherwIse.

and observe that

(A) 1 I«) dm«) The functions

(2.3.3)

= yl~ 1

I(y, () dm«).

U + g)(y, () and I(y, () + g(y, () equal 1(') + gee) off'the set {II + gl > y} u {III> y} u {Igl > y}.

But since

{II + gl > y} c {If I > y/2} u {Igl > y/2}, and m(I!1 > y) and m(lgl > y) are both o(l/yK the measure of the set in eq.(2.3.3) is o{l/y). Also observe that for all ( E 11',

IU + g)(y, ()I

~

y, I/(y, () + g(y, ()I ~ 2y.

Therefore,

1(1 I(y,()dm«) + 1 g(y,()dm«») - 1(J+g)(y,()dm«()1

~

f

3ydm«()

J{If+gl>Y}U{I/I>y}u{lgl>lI}

(

3y o(l/y)

-+

0

Swe thank A. Poltoratllki. for showing us this proof.

49

2.3. CAUCHY A-INTEGRALS

as Y - 00. This proves the first identity of the proposition. The second identity is obvious. 0 The growth condition in eq.{2.3.1) is essential for linearity (see Remark 2.3.18 below). For J.I. E M with J..L « m, the Cauchy transform I = K J.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), I satisfies the growth condition in eq.{2.3.1). However, this Cauchy transform may not belong to Hl (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 (UI'yanov). For J.I. E M with J.I.

«

m, the function

I

= KJ.I. is

A-integrable and

I(z) = (A)

(2.3.5)

1(0.

(

iT 1- (z

dm{() , zED.

The version of Ul'yanov's theorem we wish to prove is a generalization due to Aleksandrov [9] concerning the weak-type class H6'00. See Chapter 1, especially Theorem 1.10.4, for a reminder of the definition of H6'00. THEOREM 2.3.6 (Aleksandrov). II IE HJ'oo then

I(~) =

1{0.

(A) (

iT 1- (z

dm{(),

zED.

PROOF. We begin our proof by showing that

I{O)

(2.3.7)

= (A)

J

Before computing

(A) Jldm

= Llim ..... oo

I dm.

1

Ifl~L

Idm,

we make some preliminary observations. Recall from eq.(1.2.3) the distribution function A: [0,00) - [0,1], A(Y) = m{1/1 > y) for I and, from eq.(1.2.5), the decreasing re-arrangement

¢: [0,1}- [0,00),

¢(x):= inf{y > 0: A{Y)

~

x}

of f. Notice that m is normalized Lebesgue measure on 'the circle and so m(T) = 1. We will also make several uses of the inequality (2.3.8)

¢(x)

=0(;) ,

X-+O+.

Indeed, a geometric argument shows that when y

x¢{x)

~

= ¢(x),

ym(¢ > y) = YA{y),

where the last equality follows from eq.(1.2.6) which states that the distribution functions for ¢ and III are the same. The estimate in eq.(2.3.8) now follows from the assumption that YA(Y) == 0(1) as Y -+ 00.

2. THE CAUCHY TRANSFORM AS A FUNCTION

50

From eq.(1.2.7), there exists a measure preserving'map h: 'll' -. [0,1] such that

4> 0 h = III ..

(2.3.9)

Since h is measure preserving, it must therefore satisfy

!

(2.3.10)

10

G(h«(» dm«() =

1

G(x) dx 1

for every G E Ll[O, 1]. For large L > 0 and our measure preserving map h, let

, {O

log ~ig

gL«):= Notice that

iT( gLee) dm«) =

if h«() ~ '\(L), if h«)

1h'~(L)

log

~ '\(L).

~«2) dm«)

which by eq.(2.3.10) equals (>'(L)

10

(2.3.11)

x

log >.(L) dx

= ->.(L).

Thus 9L satisfies -00

< gL

~ 0 and 9L ELI.

Let 9L = PgL , 9L =:= QgL be the Poisson6 and conjugate Poisson integrals of gL and note that 9L(0) = O. The function

FL

= exp(gL + i9L)

is analytic on lD, IFLI ~ 1 (since 9L ~ 0), and is outer7 • Furthermore, I FL E HI. To see this last fact, first observe that I FL E HP for all 0 < p < 1 (since I E H~'oo c HP for all 0 < p < 1 and FL is bounded) and so, by Proposition 1.9.5, it suffices to show that I FL has integrable boundary values. Using the identities eq.(2.3.9)"and eq.{2.3.1O), ,

!

IIFLI dm =

1h'~{L) I/«)I ~((2)

dm«)

+(

lh>>'(L)

If«)1 dm«()

= lh'~(L) ( 4>(h«)) ~«2) dm«) + ( 4>(h«»dm«() lh>>'(L) (>'(L)

= 10

(I , 4>(x) >.(L) dx + l~(L) 4>(x) dx. x

.

Now use eq.(2.3.8) to show that the first integral converges. The fact that 4> is decreasing shows that the second integral also ,converges. Th~s f FL E HI. 6We abuse notation a little here and identify gL on the circle with its harmonic extension to the disk via the Poisson integral. Notic.e that both functions are the same almost everywhere on ~~

70bserve

that!

log IFLI dm =

!

,

9L dm

=(P9L)(O) =log !FL(O)I (see eq.(1.9.9)).

51

2.3. CAUCHY A-INTEGRALS

Since fFL E Hi,

J

f FL dm = f(O)FdO)

(2.3.12)

= f(0)e 9dO ) = f(O)e->-"(L).

This last equality follows from the identity

=

9L(0)

J

9L dm

= -"\(L)

and eq.(2.3.11). We are now ready to compute lim

L~oo

1

fdm.

III~L

We have

1

III~L

fdm=

rfFLdm+lI/I~L f(l-FL)dm-lI/I>L fFLdm

JT

= (I)

+ (II) -

(III).

By eq.(2.3.12), the first quantity (I) is equal to

f(O)FL(O)

= I(O)e->-"(L) -

since '\(L) = o(l/L) by our assumption that For the second quantity (II); (2.3.13)

r

JI/I~L

Ifill - FLI dm

~ (1

I/r~L

f(O) as L -

f

E

00,

HJ·oo.

1/12 dm) 1/2

(1I/r~L

11 _ FLI2 dm) 1/2

Let us estimate each of the factors in eq.(2.3.13). To this end,

r

JI/I~L

Ifl2 dm

={

¢(h)2 dm (from eq.(2.3.9»

=r

¢(X)2 dx (from eq.(2.3.1O»

J4>(h)~L J¢(Z)~L

~

1

r ¢(x)2dx. J>-"(L)

This last inequality follows from the containment {x: ¢(x) ~ L} C ("\(L), 1]

(2.3.14)

which follows from the definition of ¢. Apply the distributional equality (with k ~ 0)

10

1

k(X)2dx

=

1

00

tml(k > t)dt

to the function X¢, where X = X[>-"(L).lj, to get 1

¢(x?dx= rOOtm1(X¢>t)dt. r J>-"(L) Jo

2. THE CAUCHY TRANSFORM AS A FUNCTION

From the definition of 4>, it follows that 4>(.~(L)) ~ L. But since 4> is decreasing, it also follows that x4> ~ L and so

1

00

tml(X4>

> t)dt =

lL tml(X4> > t)dt

~ fo\m 1(4) > t)dt =

foL t .x(t) dt

(byeq.(1.2.6»

~ Lsup{t.x(t): t E ~

[O,oo)} cL (since .x(t) = o(l/t), t

-+

(0).

Combine all of this to get (2.3.15)

, We will come back to this inequality in a moment. Now let us estimate the second factor in eq.(2.3.13}. indeed,

1

11-FLI2dm=1

IIIE;L

Il-e9Lei9LI2dm

IflE;L

=

1

IflE;L

'

11- ei9L 12 dm

(9L =

0 on {If I ~ L}, eq.(2.3.14))

~ ill-ei9Ll2dm

~i ~.

IgLf2 dm

hIg~12

dm

(since

11- eizl ~ Ixl)

(by Parseval and eq.(1.8.4»

=lE;L (lOg ~~2r dm«() =

fo>'(L) (lOg ~tL») 2 dx

(byeq.(2.3.1O»

= 2.x(L). Putting this together with eq.(2.3.13) Bll:d eq.(2.3.15), ~ get

1

IJI~L

which goes to zero as L

-+ 00.

Ifill -

FLI dm

~ c';L>.l,L)

I

,

.'

2.3. CAUCHY A-INTEGRALS

For quantity (III),

1f1fl >L Ifl IFL Idm =

1

If Ilog A(~) dm

Ifl>L

4>(h«()) ~«f» dm«()

=f

1q,(h»L

=

53

(byeq.(2.3.9»

4>(x) A(~) dx

f

1';(a:»L

x

. = l0 4>(x) >.(L) dx. Using the fact that 4>(x) = o(l/x), one shows that this quantity approaches zero as ~(L)

L-

00.

Thus

(2.3.16)

=

(A)jf dm

lim

1

L-oo Ifl~L

fdm

= f(O).

To prove the Cauchy A-integral formula .

fez) = (A)

f f('l

1T 1- (z

dm«(), ZED,

fix zED and assume for a moment that fez)

= O.

Apply eq.(2.3.16) to the function

= wf(w)

F(w)

w-z

to get

0= fez)

(2.3.17)

= F(O) = (A) f

1T

F{() dm{()

= (A) f f«(2 dm«(). 1T 1- (z

When fez) is not necessarily equal to zero, apply eq.(2.3.17) to the function

G(w) = few) - fez) to get

G(z) = (A) ( G(C] dm«()

1T 1- (z

= (A) f fee) = (A)

1T

fez) dm{() 1- (z

f

f«(l dm«() - j{z).

J.r 1- (z

o

The result now follows.

2.3.18. (1) The little-oh condition in eq.{2.3.1) is important in the statement of Aleksandrov's theorem (Theorem 2.3.6) and the theorem fails without it. For example, consider the function

REMARK

fez)

=i 1+ z

1-z

and observe (see Theorem 7.4.1) that

m(lfl

~ L) =

; arctan

(~)

54

2. THE CAUCHY TRANSFORM AS A FUNCTION

which is not 0(1/L) as L

-+ 00.

Furthermore, for all. (E 'Jr \ {I},

-2

I( -112 ~«() E R..

I«() = Thus the quantities

1

I«)dm«()

IJI~L

are real-valued for every L. However, 1(0) = i and hence the formula in eq.(2.3.5) is not valid for this I. (2) Theorem 2.3.4, and various other results about conjugate functions, were explored by Ul'yanov [224, 225]. For example, if IE L1, then the nontangential boundary function j of the conjugate function (Qldm) , although defined almost everywhere, need not be integrable, making the quantity (Qjdm) undefined as a Lebesgue integral. However,

(Qjdm)(z)

= (A)

I

Qz«)f«)dm«), zED.

A good treatment of this is found in {26, Vol II]. (3) See [20, 184] for other A-integral theorems. (4) The paper [771 contains an applica.tion of A-integrals to the study of real outer functions. 2.4. Fatou's jump theorem So far I we have been considering the, Cauchy transform K J1. as a function on the unit disk. However I the function

JfT ~dJ1.«) 1- (z

(CJ1.)(z):=

is analytic on C \ 'Jr with an analytic continuation across the complement of the support of J.t. Let us use the notation

(C/l)i := CJ.tID,

(CJ.t)e:= C/lIDe ,

where De := {z E C : 1 < Izi ~ oo} is the extended exterior disk. Of course one notices that (C/l)i = KJ.t. One can check, using an analog of Smirnov's theorem Theorem 2.1.10, that

(C/l)e E

n HP(lIJ)e) ,

0 0 and AI independent of the points z and w in l!)-.9 For Izl '# 1 let (Cf)(z) = ~

r f«() d(

27r~ JT ( - Z

and for

Iwl =

1 and to

> 0 let

r

~

(Cd)(w) =

de.

f«()

2n J{I'-wl~E}nT ( - w

We will show that lim (Cf)(rw)

r-+l-

+ r-+llim (Cf)(w/r)

=

2 lim (Cd)(w) E-+O+

= P.v.~ 7r~

f

f«() d(

(- w

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 Iwl = 1, we have the following two identities (Cf)(rw) = ~ f«() - f(rw) d( + f(rw) 27rt

(Cf)(w/r)

=

r

JT

(- rw

rf«() - w/r f(rw) d(.

~

JT

27rt

(-

Using the Lipschitz condition,

If«() - f(rw)1

~

AII( -

rwlQ

and the dominated convergence theorem, one can prove the identities lim (Cf)(rw) =

~ 27rt

r-+l-

lim (Cf)(w/r) =

rf«() - wf(w) d( + f(w)

JT

~

(-

r f«() -

f(w) d(.

JT (- w Notice, by the Lipschitz condition, that these two integrals are absolutely convergent. Now observe that 27rt

r-+l-

(Cd)(w)

=~ 27rt

r

J{I'-wl~E}nT

r

f«() - f(w) d( + f(w.) ~. ( - w 27rt J{IC-wl=E}nD ( - w

Again using the Lipschitz condition and the dominated convergence theorem, we see that lim (Cd)(w) = ~ f«() - f(w) d( + f(w). E-+O+ 27rt T 2 ( - w Combine this with the two identities above to get Plemelj's formula.

r

J

To prove the Plemelj formula for the general case, use the power series expansions 00

(CJLMr() =

L jL(n)rn(n, n=O

00

(CJL)e«(/r)

=- L

n

jL( -n) ~n'

n=l

and eq.(1.8.4) to get

(CJLMr()

+ (Cf.L)e«(/r) = i(Qf.L)(r() + JL(T).

9A theorem of Hardy and Littlewood [82] says that the above condition is equivalent to being continuous on][)- and 1/«1) - /«2)1 ~ A/I(l - (21" for all (1,(2 E T.

1

2. THE CAUCHY TRANSFORM AS A FUNCTION

58

Thus for m-almost every (, (2.5.3) where, by Theorem 1.8.10, (Q",)«) is given by the principal value integral

P.V.

J~ (~ ~ ~)

But since

d",(e)·

(e+()

2 . --= 1 +~~ c-;: , l-e( ... -."

the principal value integral

P.V.J~d",(e)

l-e
0

(ET,

c,'Y>O.

The first types of regions .A.y,c«) are the finite order of contact regions with contact point ( E T, while the second type E.y,c{() are the exponential order of contact regions. For example, the regions A1,c are the non-tangential approach regions and are triangular-shaped regions with vertex at (. When 'Y = 2, these regions become (essentially) circles tangent" to T at ( and are called oric1lclic approach regions. We say, for an analytic function J on D, that J has an A-y-limit L at if J{z) --+ Las z --+ ( within: .A.y,c«) for every c> O. 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


0, independent 01 I E L2, such that

c111/11. ~ 9(f) ~ ~lIfll*·

3. THE CAUCHY TRANSFORM AS AN OPERATOR

70

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

~ Om(I) exp C~~),

m(19 - 911 > y)

y

> 0, 9 E BMO.

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 ~ 71'. When r

=1-

sin ~

and t E [-a, a],

we have

p. ( it) r

e

1 - r2

= l+r2 -2rcost (l+r)(l-r)

= (1-r)2+4rsin2(t/2) 1

~ 5sin(a/ 2) 2

~ 5a' Thus

(PlfI 2)(r) - (Plfl)(r)2

= 8:2 ~ =

i: i:

If(e is ) - f(eit)12pr(eiS)Pr(eit)dsdt

12 2ja, ja l'f(eiS ) 5071' a - a - a

(5~Y 21(~)2

ii

_

f(e it )12 ds dt

If(e iS ) - f(e it )1 2 dsdt,

where I(J) is the length of the interval J. We observe that when

fJ

= l(~)

i

f(e it ) dt,

the average of f on J, we have the inequality

1f

it

l(J) J)f(e ) -

fJl dt ~

{I

f

it

l(J) JJ If(e ) -

~ {2l(~)21 i

fJl

If(e it )

2dt }1/2 -

f(e iS )12 dtdS} 1/2

Combine this with our previous estimate to get

(3.3.7) .

1 f it I 5'11' ( ) l(J) JJ If(e ) - fJ dt ~ 2 9 f .

Now take the supremum over all intervals J to see that

571'

IIfli. ~ 2 Our next step, to prove that

f -

9(f).

1is bounded from Loo to BMO, is to show

that

9(1) = 9(f).

3.3. BOUNDED AND VANISHING MEAN OSCILLATION

71

For a E 11)), let

a-z tPa(Z) := --az 1 and notice that tPa is a conformal self map of II)) with (3.3.8) LEMMA

3.3.9. For 9

E

L1,

i(gotPa)Pzdm =

!

gP"'a(z)dm.

PROOF. If 9 E G(T), then both sides of the above equation are harmonic functions of z which tend continuously to g(tPa «» as z -+ (. By the maximum principle, these two harmonic functions must be equal. Hence the result is true when 9 E G(1'). Now use the density of C('JI') in L1 to obtain the full result. 0 LEMMA

3.3.10. If f

E

L2, then

Q(f 0 tPa) = (Qf) 0 tPa - (Qf)(tPa(O». PROOF.

From Lemma 3.3.9,

and so the harmonic function

(Qf)(tPa(Z» =

!

fQ"'a(z) dm

is a conjugate function for f 0 tPa. It may not vanish at zero to be the conjugate function Q(f 0 'I/la)(z). We will fix that in a moment. Since conjugate functions must differ only by a constant,

(Qf)(tPa(Z»

= Q(f

0

tPa)(z) + c.

o

But Q(f 0 'I/la)(O) = 0 and so c = (Qf)('I/la(O». PROPOSITION 3.3.11. For

f

E L2,

9(1) PROOF.

(3.3.12)

= 9(f).

From Lemma 3.3.9 and he fact that 'I/la{a)

= 0 we have the identity

i (h 0 tPa)Padm = i hdm

for all h E L1. For 9 E L2, apply this identity to h = fact that 'I/la(O) = a to get

Ig - (Pg)(a)1 2

and use the

3. THE CAUCHY TRANSFORM AS AN OPERATOR

72

This last identity applied to 9 =

! 11-

I shows that

! 11 = ill f 1~12 JII = ill f II -

o'I/Ja - (Pl)('l/Ja(0»1 2dm

(Pl)(a)1 2Pa dm =

0

'l/Ja - (Qf)('l/Ja(0»1 2dm (since QI = pl)

= =

0

dm (Lemma. 3.3.10)

'l/Ja - P(f 0 'l/Ja)(O) 12 dm (Pa.rseval) P(f 0 'l/Ja) (0) 12Padm (by (eq.(3.3.12»

(PI)(a)1 2Pa dm (by eq.(3.3.12) and 'I/J;;1 = 'l/Ja).

=

Now take a supremum (in a E D) over both sides of the above string of equalities and use eq.(3.3.4) to get 9(1) = 9(f).

o PROOF OF THEOREM 3.3.2. If 1 E Loo, then

11111. ~ 09(1)

(Proposition 3.3.6) = 09(f) (Proposition 3.3.11) ~

011/1100

(eq.(3.3.5».

o A natural subspace of BMO is the space VMO, the functions of vanishing mean oscillation. These are the functions 1 E BMO satisfying lim {sup

a-+O

m(I)";:a

m

if II - /II dm} = O.

1(1) {

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 oscilla.tion VMOA:= VMO n H1(1l').

1

If f E 0(1l'), 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 1 t-+ (2) The operator I t-+

1 is continuous from 0("Jl.") to VMO. f+ is continuous from 0("Jl.")

onto VMOA.

Though we will not present it here, a proof of this theorem depends on the following alternate characterization of VMO using the Garsia norm: 1 E L2 belongs to VMO if and only if lim {(PI/12)(z) - (PI/I)(z)2} = O.

1%1-+1

3.4.

KOLMOGOROV'S THEOREM

73

In fact, one can see one direction from the proof of eq.(3.3.7). rem 3.3.13, note that if IE C('ll'), then

To see Theo-

(PliI 2)(z) - (Plil)(z)2 = (PI/12)(z) - (PI/I)(Z)2 and so from the proof of Proposition 3.3.11, lim {(PI11 2)(z) - (Pll1)(z)2} Izl ..... 1

= 1zlim {(PI/12)(z) I"'" 1

(PI/I)(z)2}

=0 since I is continuous. The continuity of the operators from Theorem 3.3.2.

I

~

i

and

I

~

1+ follows

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 Kolmogorov. In fact, we will prove a stronger result. THEOREM 3.4.1 (Kolmogorov). For"" EM,

~ AJJcl, y

m(IK",,1 > y)

y > O.

A similar result holds lor the conjugate function

/1(0 =

lim (Q",,)(r().

r ..... 1-

REMARK 3.4.2. (1) We write, as is traditional in many probability books, m(IK",,1 > y) in place of the more proper m({( E 1': I(K",,)«()I > y}). (2) We will treat the distribution function y ~ m(IK",,1

> y)

in greater detail in Chapter 7. PROOF OF THEOREM 3.4.1. (3.4.3)

5

We will first prove that if "" E M+ and

F(z) :=

l

(+z

-r-

'll',,-z

d",,«(),

then (3.4.4)

m(lFI

> y) ~

211",,11 11",,11 +y'

y> O.

Indeed, for y > 0, the map

w-y g(w):= 1+-w+y maps {!Rw > O} to {Iw -11 < I} and, since RF > 0, the function <j)(z) := 1 + F(z) - Y F(z) + y 5For 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)

h

iRq,(O) =

'iRq, dm.

From the definition of F in eq.(3.4.3), we know that F(O) q,(O) = 1 +

IIJLII -

y =

+y

/IJL/I

= IIJLII and so

211JLII •

IIJLII + Y

Combining this with eq.(3.4.5) we obtain

f

ir Thus, since SRq,

~

lRq,dm =

211JLII . IIILII + Y

0 we get m(SRq, ~ 1) ~

(3.4.6)

iTf lRq,dm =

211JLII IIJLII + Y'

Using the identity

we see that lRq,«() ~ 1 {:}

IF«()I ~ Y

and so by eq.(3.4.6)

m(IFI ~ y) ~

211ILII /IJLII + y'

This proves eq.(3.4.4). Still for JL E M+, we notice, from the observation

(+z =2~-1, (- z

1- (z

that (3.4.7)

(KJL)(z) =

1

2 (IIILIl + F(z»

and, by eq.(3.4.4),

~ y) ~ cy IIJLII. Here we are using the fact that if I = h + h, then m(IKILI

{III

~ y}

c {Ihl ~ y/2} U {Ihl ~ V/2}.

Hence (3.4.8)

m(1/1 ~ y) ~ m(lh I ~ y/2) + mOhl

~

V/2).

Kolmogorov's theorem for general JL E M follows from the Jordan decomposition of JL as JL =

(JLl -

and the trick in eq.(3.4;8).

JL2)

+ i(JLa -

JL4),

ILj

E

M+, 0

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(IKI1I

~

> y)

:!.111111. y

Start with the observation from eq.(3.4.7) that

1

IKI1I ~ '2"1111

1

+ '2IFI

and apply it, for 11 E M+, to the inequality

m(WI > y) from eq.(3.4.4) to get, for y

~

2111111 111111 +y

> 111111/2,

m(IKI1I > y)

~ m (~IFI + ~III1I1 > y) = m(IFI

> 2y - 111111)

211JLII ~ 111111 + 2y - 111111

&

=M y Hence

m(IKJLI > y)

~~, y

11 E M+,

y>

1I1L1I/2.

Recall the analytic weak-type space H1,OQ from Chapter 1. From Theorem 1.10.4, Theorem 3.4.1, and the fact that KI1 E N+, we get the following corollary.

H1,OQ.

COROLLARY 3.4.10. Xc

Thus we can refine the string of containments in eq.(2.2.2) a bit to

UHP ~ X ~ H 1,OQ ~ p~l

PROPOSITION 3.4.11.

that is to say, K(L1)

HP.

For 11« m, m(IKI1I > y)

(3.4.12)

n

Oy)=o(t)

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 K61 = (1- Z)-l. Indeed, IK61(e i9 )1 = 11_1

ei9 1= 12ie!9/2 sin(~/2) 1=

Thus

m(IK61 1 > y) =

~ 1sin(~/2) I·

~ Sin- 1 (21 ). Y

'Tr

Notice that this quantity is O(l/Y) but not o(l/y) and moreover, lim ym(IK61 1 > y)

y-+oo

=.!.'Tr = .!.'Tr 1161 II·

We will see later in Chapter 7 that lim y'Trm(IKI-'I

y .....oo

> y) = III-'Bli.

In Chapter 9 we will prove the stronger result

Y'Trm(IKI-'I > y) . m weak-* as y

-+

I-'B

-+ 00.

3.5. Weighted spaces What about Cauchy transforms of functions from weighted £P spaces? The question here is the following: given 1 < p < 00, what are the conditions on a measure I-' E M+ such that there is a constant C> 0 with the inequality

!

Ig+IPdl-'

~C

!

IglPdl-'

holding for all trigonometric polynomials g? A theorem of Helson and Szego [97] says that such a measure I-' must satisfy I-' « m and so the problem can be rephrased as: given 1 < p < 00, what are the necessary and sufficient conditions on a nonnegative weight function w on 'lr such that

!

(3.5.1)

Ig+IPwdm

~C

!

IglPwdm

for all trigonometric polynomials g? In the special case where the weight function is w( ei9 ) = 11- ei9lQ, 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 logw = u + V, where u, v E Loo, IIvll oo < 'Tr /2, and 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 < 00, eq.(3.5.1) holds if and only if the weight w satisfies the condition

v

~~~ (m~I)

h

w(()dm((»)

(m~I) h(W((»-1/(p-1)dm((»)P-1 < 00,

77

3.6. THE CAUCHY TRANSFORM AND DUALITY

where the supremum is over arcs I 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 < 00 and q is the HOlder conjugate index to p. From the Riesz representation theorem for J;P, we know that every bounded linear functional on J;P takes the form

I

f-4

f

[gdm

for some unique 9 E Lq. Moreover, the norm of the above linear functional is 1!91!q, i.e., (3.6.1) An application of the F. and M. Riesz theorem (Theorem 1.9.7) says that

(HP)J.. := {9 E Lq :

i

Igdm = 0 VI E HP('lr)}

is equal to Hg and so we can apply Theorem 1.4.6 to conclude the following. THEOREM

3.6.2. For 1 ~ p
and 1= go w, with w analytic on lD> and Iw(z)1 ~ Izl, then M(r; f) ~ M(r; g} (see eq.(1.9.1}) for all 0 < r < 1.

3. THE CAUCHY TRANSFORM AS AN OPERATOR

80

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

1 E L1, the conjugate function

f

1<ei8 ):= lim e-+O+

JI9-tl~f

cot(9 -2 t)J(eit )2dt 7r

is defined almost everywhere and by Riesz's theorem, the operator J t-+ 1 is continuous on LV for 1 < p < 00. Moreover, for 0 < p < 1, 1 t-+ is continuous from L1 to £P (Proposition 3.0.1). The norms of these operators are computable [159].

1

3.7.4 (Pichorides). (1) For fixed 1 < p < 00,

THEOREM

(2) For fixed 0 < p < I, (cos{p7r/2»-1/V ~ sup {1IJ1lv We know, for

: 1I/11t ~ I} ~ 2 1/V- 1(cos(p:n"/2»-1/V.

I-" EM, that the conjugate function (QI-")(z)

=

J;}(~ ~

:)dl-"«()

is harmonic on JI) and has radial limits

l1{ei8 ):= lim (QI-")(re i8 ) r-+l-

almost everywhere. Moreover, a variant of Kolmogorov's theorem (Theorem 3.4.1) says that for fixed 0 < p < 1, 11 belongs to LV and

111111, ~ CpIlI-"I1· B.

Davis7

Cp, at least for real measures MR. (Davis). For fixed 0 < p < 1, sup {lIl1l1p : I-" E MR, 111-"11 ~ I} = IIi/lip,

computes the best constant

THEOREM

3.7.5

where

1 1 v:= 261 - 26-1.

Kolmogorov's theorem says that

m(lK1-"1

~ y) ~ eM. y

e

Though the smallest constant is unknown here, it is known when K I-" is replaced by the conjugate function 11 [53, 54]. 7The original proof is found in [55) and involves a Brownian motion argument. A more analytic proof can be found in a paper of Baernstein [24J. See also [14).

3.8.

THE HILBERT TRANSFORM

THEOREM 3.7.6 (Davis).

(1) For

m(ll1

81

IE L1,

~ y) ~ ~ 1I~11,

y

> 0,

where 1 - 3- 2 + 5- 2 - ••. . 1 + 3- 2 + 5- 2 + ... Moreover, 9- 1 is the smallest possible constant. (2) For 1-£ E M+,

8=

(3.7.7)

m(liLl

~ y) ~ 1· 111-£11 , y

y > O.

Moreover, 1 is the smallest possible constant.

81

Notice that

9=2" 1f

0

00

tet dt 8 -1 2t =2"C,

+e

1f

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 1-£ in Chapters 7 and 9.

3.8. The Hilbert transform For

I

E L1(lR), the functions

('J'f)(z):=

~ f ~ (_I_)'/(S)d8, 1f

JR

8- Z

(Of)(z):=

1(8)d8 JR !R (_1_) Z - 8

~ f 1f

are harmonic on the upper-half plane C+ := {z E C : ~z

('J'f)(z)

> O} and

+ i(Of)(z) = ~ f _1_/(s) d8 1fZ JR. 8 - z

is analytic on C+ and is often called the Borel transform of I. As it turns out, the function 'J'I plays the role of the Poisson integral PI (f E L1 (T)) in the disk setting (it is in fact called the Poisson integral) in that lim ('J'f)(x + iy) = I(x)

y--+o+

almost everywhere and when 1 < p < 00, ('J' f)(. + iy) - I in the norm of LP(lR) as y - O. The function 01 is the harmonic conjugate of 'J'I and plays the role of the conjugate Poisson integral QI (f E L1(T)) in the disk setting. One can show that lim (Of)(x + iy) = P.V.

y--+o+

f 1(8) d8 JR. X - 8

almost everywhere. The above singular integral is called the Hilbert translorm of I and is denoted by (~f)(x). Notice how the boundary function for the conjugate function 01 yields a singular integral similar to that of Q I. Many of the theorems, for example Privalov's theorem (Theorem 3.1.1), Riesz's heorem (Theorem 3.2.1), Kolmogorov's theorem (Theorem 3.4.1), Spanne's (Stein's) heorem (Theorem 3.3.2), etc., have direct analogs for the Hilbert transform. In ummary,

I E Aa(lR) =} ~I E Aa(lR) f

E

LP(lR), 1 < p

< 00 =} ~I E V(lR)

82

3. THE CAUCHY TRANSFORM AS AN OPERATOR

Ll(JR) =? ml({I~'1 > y}) ~ Cllflhy-l , E LOO(JR) n Ll(JR) =? ~, E BMO(JR) , E C(JR) n Ll(JR) => 'J(, E VMO(JR). In the above, ml is Lebesgue measure on JR. , E

Just as in the conjugate function case, the best constants are known. For example [159], if 1 < p < 00,

sup{II'J('"p: "'"p

~ I} =

tan( ~ ), {

1 < p ~ 2;

: cot(2p)'

2 < p < 00.

The smallest constant C in the weak-type inequality

ml({I'J('1 > y}) ~ C"'"ly-l is a-I, where e is the constant defined 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 playa crucial role in determining the mapping properties of the Cauchy transform. In Chapter 7 will be looking at the function y

1-+

ml(I~1-'1

> y),

the 'distribution function' of the Hilbert transform

('J(I-')(x) = P. v.

f

_1_ dl-'{s)

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

I

E X recall from Definition 2.1.4 the set Rj := {v EM:

of 'representing measures' for

I.

I

= Kv}

From Proposition 2.1.5 we know that

I-L, v E Rj => dl-L - dv = 4>dm, where ¢ E HJ ('lI') and HJ is the subspace of H1 consisting of functions which vanish at the origin. As is customary, we will abuse notation slightly and let HJ denote the following subspace of M,

HJ := {"¢dm : 4> E HJ} . Since II4>dmll = 114>111 and HJ is a closed subspace of £1, subspace of M. Thus the quotient space M / HJ of cosets [ILl:= I-L+ HJ

then

HJ

is a closed

is a Banach space with norm II[ILlll := dist(l-L. HJ) = inf {lldl-L + "¢dmll : 4> E

HJ}.

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-L with the coset [ILl in M / HJ. The mapping

KI-L

t-t

[ILl

is a vector space isomorphism from X onto M / HJ. If we equip X with the quotient space norm (4.1.1)

IIKI-LII:=II[l-LllI,

the map K IL t-t [ILl 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

I

E

X,

11/11 = inf{llvll : v E Rj}. 83

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

84

PROOF. Proposition 2.1.5 says that -

1

Rf = {dJ-to + ¢>dm: ¢> E Ha},

J.Lo E Rf·

o

Now use the definition of IIfll from eq.(4.1.1).

Normally, computing the norm of a Cauchy transform is quite difficult. However, we can compute some easy ones. COROLLARY 4.1.3. If J.L E M+ and PROOF. For any

II

E Rio

I!ILII =

!

I

= KIL, then 11111

= I\J.LII.

"'"

=

dJ.L = f(O)

!

dv = v('II')

~ IIvll·

o

Now use Proposition 4.1.2. Corollary 4.1.3 says that for certain Cauchy transforms such that IIfil = IIILII· This turns out to be true in general. PROPOSITION 4.1.4. For each IIILI! = IIfll·

I

E

f,

there is aILE R,

X, there is a unique J.L E Rf such that

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, 176J also relate Cauchy transforms to dual extremal problems. PROPOSITION 4.1.5. For a given

I

E Ll, there is a unique 9 E HI with

PROOF. Our first step is to show that a best approximant 9 E HI exists. Let lif - gnlll --+ dist(f,HI) as n --+ 00. This means that the HI norms of gn are uniformly bounded and so from eq.(1.9.3), (gn)n~l is a normal family on D. Passing to a subsequence if necessary, we can assume that 9n converges to an analytic function 9 pointwise on D. Moreover, for any 0 < r < 1, (gn)n~l C HI with

!

Ig{r()1 dm«)

~ n~! 19n(r()1 dm«) ~ n~! 19n«)1 dm{()

(integral means increase in r)

~ sup IIgnlit n

=c for some constant c independent of 0 < r < 1. Now take a supremum in r to conclude that 9 E HI. If P(f - 9)(r() is the Poisson integral of f - 9 evaluated at

4.1. THE NORM TOPOLOGY

r(, note that P(gn)(r()

85

= gn(r() --+ g(r() as n --+ 00 and so

J IP(f - g)(r()1 dm«()

~ !~~ J

IP(f - gn)(r()1 dm«()

= !~ J

IJ

Pr,(~)(f(~) - gn(~)) dm(~) Idm«()

~ n~~J J Prd~)I!(~) - gn(~)1 dm(~) dm«()

n~~ J

=

=

(J

Prd~) dm«(») I!({) -

gn({)1 dm({)

lim IIf - gnlll

= dist(f, Hl).

Observe from Theorem 1.8.6 (Fatou's theorem) that IIf - gill

~ r~- J

IP(f - g) (r() I dm«()

and so, using the obvious inequality dist(f, Hl) ~ II! - gll1, we have II! - gll1 = dist(f,H l ).

(4.1.6)

Thus a best approximant 9 exists. We now argue that it is unique. By the F. and M. Riesz theorem (Theorem 1.9.7), the annihilator of Hl in Ll via the dual pairing

Jl9dm,

fELl,

gEL OO

is H(f. Thus, from Theorem 1.4.7, dist(f,Hl) = sup {IJ !Fdml : F E ball(HOO)}. Let Fn E ball(HIf) with

J fFn dm

--+

dist(j,H l ).

By the Banach-Alaoglu theorem, the sequence (Fn)n~l has a weak-* limit point F E ball(H(f). With 9 being a best approximant in eq.(4.1.6), note, since F E HOO, that

J gFdm=O and so dist(f,H l ) = J(f - g)Fdm

~ II! -

gllllWiloo

~ II! -

This string of inequalities says that

JU-9)Fdm= Jlf-9l dm and so (4.1.7)

U - g)F = I! - 91 a.e.

gilt = dist(j,H 1 ).

86

4.

TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

H f e HI, clearly the best approximant 9 must be equal to f and it is unique. Otherwise, the above equation says that F is the unique function in ball{H8") satisfying

!

fFdm = dist(f,H 1 ).

If 9 is any best approximant in eq.{4.1.6), we can use eq.(4.1.7) to say that

$;.)(gF)

= 9(fF)

and so $;.)(gF) is unique. But since gF e H8", this determines gF uniquely. Finally, IFI > 0 almost everywhere and so 9 is uniquely determined. 0 COROLLARY 4.1.8. For a given fELl, there is a unique 9

inf{IIf+hlll:

e HJ

so that

h E HJ} = IIf+gIl1.

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) = Co + C1Z + C2Z2 + ... + cnz n is an analytic polynomial, one can consider the extremal problem

L=

inf {lip - gill:

9 e HJ}.

We know from the previous corollary that this extremal problem has a unique solution go e 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

HJ.

=

n

Ln = ~)AjI2, j=O

where

(2j)!

Aj

= tV 0!)2 .

A estimate using Stirling's formula yields

(4.1.10)

Cl

log(n+ 2) ~ Ln ~ c210g(n+2),

n E N,

for some universal constants Cl, C2 > O. Putting this in another way, the Cauchy transform norm of 1 + z + Z2 + ... + zn can be estimated as

111 + z + z2 + ... + znll =

Ln

x log n.

See [79, p. 175] for a nice exposition of this. PROOF OF PROPOSITION 4.1.4. First note that for any ve Rf, (4.1.11)

IIfil = inf{IIdll + gdmll : 9 e HJ}

~

IIvll·

By the definition of the norm on X, there is a sequence (IIn )-n;;;':l of measures from. RI such that lllinil ~ Ilfil + lin. Since this sequence is uniformly bounded in total variation norm, it has a weak-* cluster point v e M (Banach-Alaoglu theorem). Hence, we can pass to a subsequence (Vn)n~l such that lim !9dVn = !gdV 't/g E C(1l').

n-+oo

4.1. THE NORM TOPOLOGY

Thus, for each

Z

E lIJ>,

f

= ~ dvn «() ~ 1- (z

fez)

f~ 1- (z

87

dv«(),

n

~ 00,

and so v E Rf. We now need to prove the equality IIvll = Ilfli. One direction (Ilfll : ; ; IIvll) comes from eq.(4.1.1l). For the other direction, observe from Proposition 1.6.2 that

For uniqueness, first notice that when v E Rf with

IIvll = IIfll,

I :~" + IIvsll = Ilvll = inf{lIv + hdmll : h E HH

= IIvs II + inf {II :~ + hill: hE HJ}

(by Corollary 1.3.10).

Thus

and by Corollary 4.1.8, this infimum is achieved precisely when h == O. Hence (4.1.12) If IL, v E Rf with

IIILil = IIvll = IIfll, then by Proposition 2.1.5,

(4.1.13)

ILs =

Vs

and consequently' (4.1.14) Since IL, v E Rf, Proposition 2.1.5 says that

dIL = dv +h, hE -HJ. 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 (4.1.15)

dIL

dv

dm = dm' By eq.(4.1.13), ILs measure is unique.

= Vs

and hence IL = v. Thus the norm attaining representing

o

REMARK 4.1.16. One should be careful as to not over interpret this theorem. It does not say, for a particular measure IL, that IIKILII = UILII, although this is the case when IL E M+ (Corollary 4.1.3). Instead, it says that there is some (perhaps other) measure v with KIL = Kv such that IIKILII = IIvll.

This next result relates the growth near the boundary with the norm.

88

TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

4.

PROPOSITION 4.1.17. If f E X, then

If(z)]

(4.1.18)

~ II~l~I'

z E ]D).

PROOF. H f = Kp., then for all z E ]D),

IfCz)1 =

If

1

I

~ (z dl-'C') ~ f 11 ~ (zl dll-'I(() ~ II~l~I'

The desired estimate follows by taking I-' to be the unique I-' E RJ with (Proposition 4.1.4).

11111 = 1Ip.1I 0

The previous proposition implies that if (fn)n~l C X converges in norm to f, then this sequence also converges to f uniformly on compact subsets of D (as is the case with all the well-known Banach spaces of analytic functions on ]D). Thus ball (X) forms a normal family of functions on D. By the Lebesgue decomposition theorem (Theorem 1.3.9),

M=Ma+M., where

Ma:={I-'EM:I-'«m}.

The

Ms:={I-'EM:p.1.m}.

+sign here means that M

= {VI + ZI:z : VI E Ma. ZI:z EMs}

and Ma

n Ms = {OJ.

Moreover (Corollary 1.3.10),

where Thus

M=MaEBMs. We can use the above decomposition to rewrite X as

X=Xa+X s • where

Xa:= {Kp.: p. E Ma}, Xs:= {KI-': I-' EMs}. We will now see that in fact X = Xa ED X B • PROPOSITION 4.1.19. If V EMs, then

IIKvlI = IIvlI.

PROOF.

9E HJ} = inf {1I1111 + I/g/ll : 9 E HJ}

IIKp.11 = inf {1I 11 + gdml\

:

(since

II

1. m)

= \111\1.

o 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 J.I. = J.l.a + J.l.s. Then

IIKJ.l.II = IIKJ.l.a;1I + IIKJ.l.sl1 = IIKJ.l.a;1I + IIJ.l.slI· In particular, Xa is isometrically isomorphic to L1 / HJ while Xs is isometrically isomorphic to Ms. . The above says that Xa and Xs are indeed closed subspaces of X and that

X=Xa$Xs' PROPOSITION 4.1.21. (1) Xa is separable. In lact, the analytic polynomials are dense in Xa. (2) Xs is not separable. PROOF. For any I E LI, note that the Cesaro sums uN(f) approximate the L1 norm (Theorem 1.6.5) and so, by the definition of the norm on X,

lIuN(f)+ -

1+11 ~ lIuN(f) -

1111

-+

0 as N

I

in

-+ 00

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. 0 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 ql L1 / HJ is isometrically isomorphic to HOO. A pairing between these two spaces is given by

([/l,g):= As a consequence, the dual

J

fgdm,

01 Xa

[/1 E L1/HJ, g E H oo •

can be identified with HOO by the pairing

(I+,g):=

J

fgdm.

Let us compute, as it will be used later, the X-norms of the functions 1 zl-+-1 _,

-az

aElIr.

When a E 1', then 1 -1 - = (KcSa)(z)

-az

and so, since

When

cSa

.1. m, we can use Proposition 4.1.20 to get

lal < 1,

1!az = (1 ~a() + (z) and so, using Proposition 4.1.20 again and the definition of the norm in L1/ HJ, we obtain

4.

90

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

inf

{Ill ~

a( -

9111 : 9 E HJ}

{If

= sup I«() 1 ~ a( dm«()1 : 1 E OO11(H';")} = sup {If(a)l : 1 E ball(HOO)} =1. Thus for a E D-, (4.1.23)

111

~azll = 1.

REMARK 4.1.24. One can avoid the above argument by noticing that for each

aED,

and so

1 -1 - =K(Padm)(z).

-az

But since Pa dm E M+. we can apply Corollary 4.1.3 to see that

111 ~ az II = IlPa dmll = IlPa III == 1. 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 / HJ and reflexivity is preserved under isometric isomorphisms, we just need to show that M / HJ is not reflexive. By Theorem 1.4.11 (subspaces of reflexive spaces must be reflexive), we reduce the problem to showing that L1 / HJ 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 Hoo is not reflexive. Since L1 / HJ is separable (L1 is separable), we can use Proposition 1.4.13 to reduce the problem to showing that Hoo is not separable. For this last detail, let

be a family of atomic inner functions. A computation shows that

o

and so by'Proposition 1.4.12, Hoo is not separable.

=

1Actually, if one is willing to work a hit harder, one can show that II" - !{lIoo 2. To see this, note that ft.. is analytic at ( with I!{«)\ = 1 while" has JI) 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* IS1. ~ S*, to the case where X = G(T), the continuous functions on T. By the Riesz representation theorem (Theorem 1.3.6), the mapping from M to G(T)* defined by J.1.~ Lf,.t, where Lf,.t(f) :=

J

fdJ.1.,

is a conjugate linear isometric isomorphism. If A, the disk algebra, denotes the functions f E G(T) which have continuous extensions to 0- which are also analytic on D, one notices that

A1. =

{J.£ EM:

J

fdJ.£

=

0 for all

I E

A} .

Letting I«) = (n for n E No in the previous equation, we see, by the F. and M. Riesz theorem (Theorem 1.9.7), that A 1. -- HIo' Thus from Theorem 1.4.6,

A* ~ MIHJ 2and the dual pairing between A and M I HJ is

(I, [Jt]) =

(4.2.1)

J

fdJ.1..

By the discussion in the previous section, the map KJ-L ~ [JLl is an isometric,isomorphism between X and M 1HJ. Put this all together to obtain the following. THEOREM

4.2.2. A * is conjugate linearly isomorphic to X with dual pairing

(f, KJL) =

J

fdJL.

A computation with power series shows that this pairing can be written in the more familiar Cauchy pairing (4.2.3)

Indeed,

r~xr- ~ f(n)jL(n)r n ~ r~- ~ f(n) (J (ndJ.£(») rn =

r~- J(~1 0 are given, we can find an h E B with III - hll oo < f. With this

l(f - h,gn - g} + (h,gn - g)1 ~ III - hlloollgn - gil + i(h,gn ~ f(C+ IIgll) + I(h,gn - g)l·

l(f,gn - g}1 =

g}1

Since I(h,gn - g)1 -+ 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 0=

f

1 ~ 1- (Zdv = ~znv(n)

Vz ED.

n=O

By the F. and M. Riesz theorem, this takes place precisely when v E HJ Thus B1. = A1. and, by the Hahn-Banach theorem, the proof is complete.

= A1. . 0

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 X, 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 Xs are each weak-* dense in X. PRoliF. When M and M / HJ are endowed with their respective weak-* topologies, eq.(4.2.1) shows that the natural map 1r :

M

-+

M / HJ ~ X

is weak-* continuous. The fact that Ma and Ms are weak-* dense in M (Proposi0 tion 1.6.7) finishes the proof. REMARK 4.2.9. (1) If 9 = KJL and O"n(g) is the n-th Cesaro mean of g, that is O"n(g) = O"n(JL), then O"n(g) -+ 9 weak.-* in X as n -+ 00. One can see this directly by the identities

(f,O"n(g)} = where

f

100n(J.L) dm =

f

O"n(J) dJ.L,

I E A [101, p. 20]. This last integral converges to

f

IdJL = (f,g).

(2) From the proof of Proposition 4.1.21, we saw that if IE L1, then O"n(f)+, the n-th Cesaro mean of i+, converges to 1+ in the norm of Xa. However, if JL .1 m and JL t= 0, then O"n(KJL), although it converges weak-* to K J.L, does not converge in norm to anything. The reason for this is that O"n(KJL) E Xa and Xa is a norm closed subspace of X (Proposition 4.1.20). Thus if O"n(KJ.L) has a norm limit F, it must belong to Xa. However, it is easy to check that O"n(KJL) converges pointwise to KJL and so KJL = F E Xa which is a contradiction to the fact that KJL E Xs.

94

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

4.3. The weak topology, A topological vector space ~ is complete if every Cauchy net (see [113, p. 190] for a definition) in 11 converges to an element of~. If ~ is a Banach space, it is complete by definition. On the other hand, if ~ is either (X, wk) or (X*, Wk4), that is, a Banach space X endowed with its weak/weak-* topology, then ~ is complete if and only if '0 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 (Xn)n;?;l in a Banach space X is weak Cauchy if the numerical sequence (i(Xn))n;?;l is a Cauchy sequence for every i E X*. A sequence (in )n;?;1 c X* is weak4 Cauchy if the numerical sequence (in (X))n;?;1 is a Cauchy sequence for each x E 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 BanachAlaoglu 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 < 00, are weakly sequentially complete since they are reflexive and so the ~eak and weak-* topologies coincide. For the same' reason, the Hardy spaces HP, 1 < p n has a unique decomposition as 4>n = tPn + Vn , where tPn E Xa and Vn E Xs. Because of the direct sum decomposition of X*, it is evident that each of the sequences (tPn)n-;;;'l and (Vn )n-;;;'1 is a weak Cauchy sequence. By Mooney's theorem, the first sequence converges weakly to some tP E Xa. On the other hand, a theorem of Kakutani says that Ms is isometrically isomorphic to Ll(n, lJ, 1-') for some abstract ,measure space (n, lJ, 1-') [110]. It follows that Ms is weakly sequentially complete since every such space Ll (n, E, 1-') is weakly sequentially complete [60]. The second theorem we wish to present on the weak topology in X is a deep result due independently to Delbaen [59J and Kisljakov [115]. For each f E X, recall from Proposition 4.1.4 that 1-'1 is the unique measure such that I = KI-'I and

11/11

=

111-'111·

THEOREM 4.3.3. Let W be a weakly compact set in X, and let

W = {I-'I:

fEW}.

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 M: W c M is weakly compact if and only if it is norm bounded and unifonnly absolutely continuous (cf. [60]). 4.4. Schauder bases A sequence (X n )n;;;'l in a Banach space X is called a Schauder basis for X if every x E X can be written uniquely as

4A EBl B = ({a, b) : a E A, bE B} with norm II (a, b) II = lIaliA

+ IIbIiB.

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

96

where c,.,. are complex numbers and the = sign means convergence in the norm of X, i.e., N

lim

N-oo

x - L..J '" c,.,.Xn = O. n=1

The sequence space lP, 1 ~ P < 00, has a Schauder basis (en)n~b where en(j) = 6n,j' The space of continuous functions e[o, 1] has the classical Schauder basis discovered by J. Schauder [189J (see also [142, p. 352]). Schauder [190J (see also [142, p. 361]) also proved that the LP spaces for 1 ~ P < 00 have a basis, the Haar basis. As a consequence of this, and the fact that HP is isomorphic to V, one can show that HP, 1 < p < 00, has a basis. Maurey [138] proved that HI has an unconditional basis. Carleson [39] explicitly constructed such a basis. Wojtaszczyk [231J provided further improvements. Boekarev [29] discovered a basis (bn)n~1 for the disk algebra A with the additional property that

J

bnbk dm = 6n,k'

(4.4.1)

We will make use of this in a. moment. The existence of a Scha.uder basis is not automatic since there are separable Banach spaces (even reflexive ones) without a basis [67J. We refer the reader to [60, 142, 157, 231] for more on bases. A sequence (.en)n~l C X* is called a weak-* Schauder basis if every I. EX· can be written uniquely as

n=1

where dn are complex numbers and = in the above equation means weak-* convergence, i.e.,

for each x E X. For a Schauder basis (Xn)n~I' there is a natural sequence linear functionals defined by

(X~)n~l

of continuous

00

x~(x)

= c,.,.,

where x

=L

CnXn

n=1

(remember that the expansion of x is unique and so x~ is well-defined). The fact that the functionals x~ are continuous is a deep theorem of Banach [60, p. 32]. PROPOSITION 4.4.2. If (Xn)n~1 is a Schauder basis for X, then weak-* Schauder basis for X· . PROOF.

For lEX·, let dn:= i(xn ) and for each N N

iN:=

LdnX~. n=1

E

N let

(X~)n~l

is a

4.4. SCHAUDER BASES

97

Then for any 00

X= LCnxn EX, n=1 N N iN(X) = ~dnx~(X) = ~dnCn Thus iN

-+

l weak-* as N

-+ 00

N (N) = ~i(Xn)Cn =£ ~XnCn

.

and so every l E X* can be written as 00

l= LdnX:. n=1 The uniqueness follows from the identities X:(Xk) = On,k' Hence (X:)n;)1 is a weak-* Schauder basis for X*.

o

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~1 is a Schauder basis for X, then {X:)n~1 is a 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 Boekarev basis (bn)n~l for A. To clarify notation, we let

Bn := (bn )+. Certainly Bn = bn as analytic functions on IDl. We use this notation to avoid confusing, bn , the element of the disk algebra A, with B n , the element of X (in fact Bn E Xa). The result here is the following. PROPOSITION

4.4.4.

(Bn)n~1

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

(j, KJ-L)

= / f dJ-L.

So if f E A is written in terms of its Schauder basis (the Boekarev basis)

n=1

then for each k E fIl, we can use the orthogonality in eq.{4.4.1) to see that

(j,Bk)

=/

(~Cnbn) bkdm = ~Cn /

bnbkdm = Ck·

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 bt: (f) = Ck. Applying Proposition 4.4.2 we have shown that (Bnk~l is a weak-* Schauder basis for X. In order to prove that (Bn)n~1 is a Schauder basis for X a, we notice that Bn = (bn )+ E 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~l is all of Xo. From Theorem 4.1.22, with Hoo by means of the dual pairing

(/+,g) =

f

Jgdm,

f

E

L1,

X: can be identified

9 E H oo .

Thus if 9 E Hoo and annihilates every B n , then 0= (Bmg)

= «bn )+, g) =

f

bnydm 'tin E N.

But since the closed linear span of the bn's is the disk algebra A, we have

f

(ny dm = 0 'tin E No

and so g(n) = 0 for all n E No. This me~ that 9 == O. An application of the Hahn-Banach separation theorem completes the proof.

o

CHAPTER 5

Which functions are Cauchy integrals? 5.1. General remarks Which analytic functions on ]])) belong to the space of Cauchy transforms X? 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 = KiJ. for some iJ. E M. Then

(1) f satisfies the growth condition

If(z}1

~ 1 ~iz"

z E ]])).

(2) f has finite non-tangential limits m-almost everywhere on T and

m(lf! > y} ~ C" y

y > O.

(3) f E HP for all 0 < p < 1 and

IIfllp = (4) If f

= En;ll:oanz n ,

0(1 ~p)' p-1-.

then (an)n;ll:O is a bounded sequence of complex num-

bers. None of the above conditions is sufficient. The above necessary conditions can only be used to determine which analytic functions on ]])) 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 f is analytic on ]])) with power series

fez} = ao + atZ + a2 z2 + ... and we want to determine whether or not

f=

K iJ. for some IL EM. Since

(KiJ.)(z) = j1(0) + jL(l)z + j1(2}Z2 ... , we would be trying to determine, by equating an with j1( n) for n E No, the measure IL 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 X [91], albeit difficult to apply. 99

5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

100

THEOREM

5.2.1 (Havin). Suppose 00

1= Lak zk k=O

is analytic on D. Then the lollowing statements are equivalent. (1) There is some constant C > 0, depending only on I, such that

I~Akak ~cmax{ ~Ak(k

(5.2.2)

:(ET}

lor any complex numbers Ao,· .. , An. (2) 1= Kp lor some p E M. PROOF.

Recall that if I = K p, then

I(z)

=

f ---dJ.L«() = 1

1- (z

00

Lzk

J

00

tdp«() = LI1(k)Zk

k=O

k=O

and so the Taylor coefficients of I are equal to the (non-negative) Fourier coefficients of p. Also recall from eq.(4.2.3) that the dual of A (the disk algebra) can be identified with X via the Cauchy pairing 00

gEA.

More specifically, 00

r~~- Ly(n)p(n)rn ~ Cpllgll oo •

(5.2.3)

k=O

To prove (2) ~ (1), let I = Kp and Ao, At.··· , An be given complex numbers. With g(z) = Ao + AIZ + ... + Anzn, the inequality in eq.(5.2.2) follows from the inequality in eq.{5.2.3). To prove (I) ~ (2), the hypothesis imply that the linear functional i, defined first on polynomials p{z) = Ao + AIZ + A2z2 + ... + Anzn by

10;=0

extends to a bounded linear functional on the disk algebra A. Hence i(P) = (p, f) for some I = K p. Thus

o

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 I on T is a Cauchy transform

0 with

dt

U(Zo + sed) 21r·

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 I is analytic on n and p > 0, then I/IP is subharmonic on n. If u is subharmonic on C and I is analytic on n, then u 0 f is subharmonic on n. For a subharmonic function u on n, we say that a function U on n is a harmomc ma)orant for u if U is harmonic and u ~ U. A harmonic function U on n is called the least harmomc ma)orant of u if U is a harmonic majorant and U ~ V for any other harmonic majorant V of u. 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):=

increase as r /

h

u(r() dm«)

1-. This next result is found in [79, p. 38]

LEMMA 5.4.6. A subharmomc functwn u on]D) has a harmomc ma)orant ~f and only ~f the zncreaszng zntegral means M (r; u) are bounded as r -+ 1-. The least harmomc ma)orant ~s then

U(z):= hm (pz«()u(r() dm«(). r-+l-

1T

With these preliminaries about subharmonic functions in place, we begin the proof of Aleksandrov's theorem by proving a few technical lemmas. Define, for o < p < 1, the following function Gp on C.

(5.4.7)

Gp(z)

:= {lzlP cos (p9(z))

o

if z =f:. 0, , if z = O.

where (5.4.8)

arctan(Y/lxl) { 9(z).= 1r/2 -1r /2

Note that 19(z)1 ~ 1r/2 and 0 < p

(5.4.9)

x =f:. 0,

x=O,Y>O, x = 0, Y

< o.

< 1 and so Gp

~

O.

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).

5 WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

104

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 E C, there is an r(z) > 0 such that

Gp(Z) ~ - 1 111" Gp(Z + re it ) dt, for all r E (0, r(z)). 211" -11"

(54.11) Since

G (z) =

Gp is harmonic on {iRz {iRz f. OJ. If Z = 0, then

{~(ZP)

for )arg(z») < 11"/2, for! arg( -z)! < 11"/2,

~(-z)P

p

f. O} and thus the inequality in eq.(5.4.11) holds on the set

. dt = 2 ( - 1 111"/2 r Pcos(pt) dt ) = -2rP sin( p7r - 1 111" Gp(re't) -2 ) ~ 0 211" -11" 211" -11" /2 p7r since 0 < p < 1. But Gp(O) = 0, by definition, and so the inequality in eq.(5.4.11) is satisfied. Letting Hp(z) = iRzP on {z : ! arg z! < 11", z f. OJ, we first notice that Gp -Hp ~ o on this set. To see this, observe that

Hp(z) = iR(zP) = Gp(z),

Z

!¢! ~

= rei"',

11"/2, r > O.

11"/2 < ¢ < 11", then Gp(z) - Hp(z) = r P cos (p (¢ - 11"» - r P cos(p¢)

If z = re i , where r > 0 and

= 2rP sin (p (¢ -

i) )Sin(p~

)

~o.

If z = rei , where r > 0 and -11" < ¢
1 and (hn)n~l c Lq with IIhnll q ~ C for all ~ h almost everywhere as n ~ 00, then h E Lq and hn ~ h weakly

5 4 ALEKSANDROV'S CHARACTERIZATION

105

By Fatou's lemma,

PROOF.

f

Ihl qdm

iT

= f lim Ihnl q dm ~ lim f Ihnl qdm ~ C q < 00

Jr n->oo

n->oo iT

andsohELq. Let f > 0 and 9 E LP be given (lip + 11q = 1) and note, by basic properties of the integral, there exists a 0 > 0 such that for any measurable set A c 1l' with meA) < 0, we have

(L IglPdm) By Egorov's theorem, there is a set E uniformly on E. With this in place,

f

l-

Then

fez)

=

f ~ + z dJl(()

Jr",-z

for some real measure Jl E M sat'tSfyzng

ilJlII

~

lI'fRfll£1

+ ~ p~~_ (1 - p)II/II~·

Moreover, tf lim (1 - p)lIfll~ = 0, p-+l-

then dJl =

~fdm.

PROOF. Notice that Gp 0 I ~ 0 on JI)) (since Gp ~ 0 - see eq.(5.4.9». Moreover, since Gp is subharmonic on C (Lemma 5.4.10) and I is analytic on JI)), Gp 0 f is subharmonic on JI)) •

106

5 WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

Assume for a moment that !RI = 0 almost everywhere on T. For q E (I,l/p) and r E (0,1), the Lq integral means of Gp 0 I satisfy

h

(Gp 0 f)(r()qdm«()

h : :; h

I/(r()lpq (cospO(f(r()))q dm«()

=

I/(r()lpqdm

=

Ilfll:g

which is finite since 0 < pq < 1. Thus the function Gp 0 I is subharmonic 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~

a r~~- h

r~~-l P

is the least harmonic majorant for G p

(Gp 0 f)(a) :::;

«() (Gp 0

0

f) (r() dm«()

f. Thus for all a E D, Pa«()(Gp 0 f)(r() dm«().

-+ f«() almost everywhere as r -+ 1- and Gp is continuous, (G p 0 (Gp 0 f)«() almost everywhere as r -+ 1-. This, together with the uniform boundedness of the Lq means of Gp 0 I, say, via Lemma 5.4.12, that (G p 0 f)(r·) -+ Gp 0 I weakly in Lq. Thus

Since I(r()

f)(r()

-+

f =f

(Gp 0 f)(a) :::;

Pa«()(Gp 0 f)«() dm«() Pa«()!f«()IP cos (P27r ) dm«(),

a E D.

In the last integral above, notice that we are assuming R/«() = 0 and so argf«() = ±7r/2. Since cos(p7r/2) ;:::: 1 - p, we can use our assumption in eq.(5.4.14) to observe that

p~~_

h

If«()I P cos

This means that for some sequence Pn /

(~) dm«() < 00. 1, the measures

dVPn := Ifl Pn cos

(P;7r) dm

satisfy lim IIvpn II

(5.4.15)

n---+oo

= lim (1/«()iP cos (P7r) dm«(). p---+l- iT

2

Since these measures are uniformly bounded in total variation norm, they have a weal O.

~ P ~ 00,

!T(HP,E) =F (0) 1,f and only 'tf

5

110

WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

PROOF. Suppose m(E) = 0 and there is a measure J.L bupported in E with KJ.L E HP. Note, since J.L is supported E and m(E) = 0, that J.L .L m. Since fdm, where I is a boundary function for K J.L, is also a representing measure for K J.L (Cauchy integral formula - Proposition 2.2.1), and all representing measures have the same singular part (Proposition 2.1.5), then J.L 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 9 on C\ E and, by subtracting off a constant, we may assume g( 00) = o. Define f m-almost everywhere on '][' by f(():= lim (g(r() - g«(Jr)) r-+l-

and note that the measure Idm is supported in E. A power series computation shows that K(fdm) = glD is non-trivial. 0 Though the analysis is more complicated, HruSCev [103J 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 9='(A, E) -=I- (0) il and only if m(E) > O. (2) II Aoo is the space 01 functions f which are analytic on D such that f(n) E A lor all n = 0,1,2,·· , then 9='{Aoo, E) -=I- (O) if and only il E contains a closed subset F 01 positive Lebesgue measure satisfying the Carleson condition 00

L m{In) logm(In) > -00, where

{In)n~l

n=l 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 \ 1!') 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 f{D) is contained in a halfplane, then f E X (Proposition 2.1.13). In this section, we provide some other geometric conditions. LEMMA 5.6.1. II I E X and ¢ : D - D is analytic, then f

0

¢ E X.

PROOF. If J.L E M+, then !R{KJ.L 0 ¢) > 0 on II} and so by Proposition 2.1.13, K J.L 0 ¢ E X. Write any J.L E M as J.L = (J.Ll - J.L2) + i(J.L3 - J.L4), J.Lj E M+, and apply the above argunIent four times to conclude that K J.L 0 ¢ E X for any J.L EM. 0 This simple lemma yields the following corollary. COROLLARY 5.6.2. Suppose that n is a proper simply connected subset of C and tf; : D - n is a Riemann map. If tf; E X (this may not always happen!), then any analytic map f : D _ n 1 belongs to X PROOF. If tf; = KJ.L, apply Lemma 5.6.1 to ¢ := tf;-l

f = K J.L 0 ¢ E X. INote that f{D} need not be equal to

n

0

f : D - D to see that 0

SOME GEOMETRIC CONDITIONS

56

111

The following [31] is a nice geometric cond1tion for membership in X. THEOREM 5.6.3 (Bourdon and Cima). Suppose f 1.S analyt1.c and f(lO) 'tS contamed m a regwn that om1.ts two opposztely pomted half-lmes (see Fzgure 1), then

fEX. PROOF. Assume feD) is contained in a region that omits two oppositely pointed half-lines and let n be the complement of those two oppositely oriented lines. In a moment, we will construct an invertible analytic 1/J : D -+ n and show that the harmonic function h := ~1/J satisfies sup

(5.6.4)

O E rol(X) , we define the operator norm of M", via eq.(1.5.1) by

IIM",II

:= sup{II4>lll : 11/11 ~

I}

and call it the multzplzer norm of 4>.1 Since the constant functions belong to X, we know that rol(X) c X. It turns out that the multipliers of X have a certain amount of 'extra' regularity near 'j[' and so rol(X) S;; X. PROPOSITION 6.1.3.

114> E rol(X), then 4> E Hoo and 114>1100

PROOF. For each A E lDl, let l>..(f) functional on X. For each n E N,

=

~

IIM",II.

I(A) be the (continuous) evaluation

Thus

o IHere 11111, Without any subscnpts on the norm, denotes the Ca.uchy transform norm of (see eq (4 11» 115

I

6 MULTIPLIERS AND DIVISORS

116

The multiplier theory goes well beyond X to general Banach spaces of analytic functions X on the unit disk for which the evaluation functional f 1-+ f(z) is continuous on X for each z E 11)). In this setting, is a 'multiplier' of X if X c X. With nearly the same proofs, one can show that m(X) c Hoo and that when E m(X), the multiplication operator MqJ = f is continuous on X. For certain spaces, such as the Hardy spaces HP, the multipliers are easy to describe Indeed, m(HP) = Hoo. To see this, note that whenever E HOC and f E HP,

i

1(r()f(r(W dm«()

~ IIII~

i

If(r(W dm((),

0 < r < 1,

implying lIfllp ~ 1IlIoollfll p · Hence H oo c m(HP) The other direction follows from Proposition 6.1.3 For other spaces of analytic functions such as the Dirichlet space 'D of analytic functions f on II)) which have finite Dirichlet integral

.!. (I1'I 2 dx dy, 7r

10

the multipliers are more complicated [205]. By Proposition 6.1.3, m('D) c Hoo but, unlike HP, 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 X, we would like to connect the multipliers of X with the co-analytic Toeplitz operators. For 1 < p < 00, the Riesz projection operator

f 1-+ f+ is a bounded operator from LP onto HP (Theorem 3.2.1). Thus for each E Hoo, the co-analytic Toeplitz opemtor T¢: HP

-+

HP,

T¢f:= C(fJf)+

is also bounded Recall that if 9 E £P has Fourier series

n=-oo

then

00

g+(z)

=

LY(n)zn. n=O

Using the identity l,kEZ 00

=

00

L

(n

n=-oo

00

L

¢(k - n)i(k)

k=-oo

00

=

L

(n

n=-oo

=-__

L ¢(k -

we observe that (6.1.4)

(T?iJ)(z)

=

n)i(k),

k=n 00

00

n=O

k=n

=-__

L zn L ¢(k - n)i(k).

61

MULTIPLIERS AND TOEPLITZ OPERATORS

117

When p = 1 or p = 00, the situation becomes more complicated. For example, the continuity of T"4) on HI, which IS not automatic since (L1)+ :2 HI, depends on whether or not 'R4> and '.S4> are multiplIers of BMO [204]. Since

(L OC )+ = BMOA:2 HOC

and

(C)+

= VMOA;;2 A,

(see Chapter 3) the continuity of T"4) on HOC or A is also not automatic. However, we can determine which symbols 4> give rise to continuous T"4) on HOC or A [227J. PROPOSITION 6.1.5. For 4> E Hoc, the lollowmg are equwalent. (1) 4> E rol(X). (2) 4> E rol(Xa ), z.e., IE Xa

=}

4>1 E Xa 2 •

(3) T"4): A ~ A zs bounded. (4) T"4): HOC ~ HOC 'tS bounded. Moreover, I/T"4): A

~

All =

IIMII·

'*

PROOF. (1) (2): For 4> E rol(X) and I E X a , note that 4>1 E X. We need to prove that 4>1 E Xa. By Proposition 4.1.21, there exist a sequence (fn)n~l of polynomIals with In - I m X-norm. But 4>ln E HI C Xa (Proposition 2.2.1) and, since 4> E rol(X), 4>ln - 4>1 in X-norm. Thus, since Xa is a closed subspace of X (Proposition 4.1.20), 4>1 E Xa. It follows now that 4> E rol(Xa).

'*

(2) (1): Assume 4> E rol(Xa). Given I E X, there IS a sequence (fn)n~l C Xa such that In - I weak-* in X (Remark 4.2.9). Then 4>ln E Xa (by assumptIOn) and moreover, since weak-* convergent sequences are norm bounded (Proposition 4.2.5), 4>ln is norm bounded. Passmg to a subsequence if necessary, 4>ln converges weak-* to some 9 E X (Banach-Alaoglu). Now use the fact that In - I and 4>ln - 9 pointwise in lIJ> (Proposition 4.2.5) to argue that 9 = 4>1 and so 4>1 E X. Thus

4> E VJt(X). (3) (1): By Theorem 4.2.2, the dual of A can be identified with X via the

'*

pairing

(I, g)

=

r~rp--

h

Ir«()g«() dm«(),

I

E X,

9 E A.

If T"4) : A - A is continuous, then by Proposition 1.5.4, T~ : X - X is continuous and llT~ll = IIT"4)II· Now notice that T~ = M.

'*

(2) (4): By Theorem 4.1 22, the dual of Xa can be identified WIth HOC VIa the pairing

(I,g)=r~rp-_ilr«()g«()dm«(),

IEXa ,

gEH OC

Thus, as before, if M : Xa - Xa is continuous, then so is M;

= T"4) : HOC _ HOC>.

(4) =} (3): For any 9 E A, let 9n be the Cesaro polynomials for 9 and note that gn - 9 umformly (Theorem 1.6.5). From eq.(6.1.4), we know that T"4)9n is also a polynomial and hence belongs to A. If we assume T"4) : HOC _ HOC is continuous, then T"4)9n - T"4)g uniformly and so T"4)9 E A. It follows that T"4) : A - A is continuous.

o 2Recall from Chapter 4 that Xa = {K tL . tL

«: m}

6 MULTIPLIERS AND DIVISORS

118

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 Y

I dm =

1

JI

dT/s.

We know from eq.(6.3.2) that v(

= KA(

for some A( EM. If I is continuous and vanishes near (, we can use the fact that is bounded on the support of f (since multiphers are bounded) to say that for large enough y, the set {lv(1 > y} nsupp(f)

v(

122

6 MULTIPLIERS AND DIVISORS

is empty and so

j

fd(Ad s = lim try f fdm = o. y--oo J1vd>y This implies that the singular part of A( is at most a point mass at (. Now we use the identity r lim __ 1- (1- r)(Kij)(r() = ij({(}), ij E M,

(see eq.(2 1.8)) to prove that

= rlim(l __ l

A« {(})

=

r)(KAd(r()

lim (1 - r) ¢(r() - ¢«() 1- (r(

r __ l-

= r lim __ l- ¢(r() -

¢«()

=0 since ¢(r() -+ ¢( () as r E Xa as desired.

-+

1- (Theorem 6.2.6) This means that A( E Ma and so

0

v(

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 ¢ E H oo • (1) If

00

L 14>(n)llog(n + 2)
E 9Jt(X) and there is an absolute constant c > 0 such that for each n EN,

~ c (~II¢>'II= + log(ne)II¢>II= )

IIM,p11 PROOF.

.

From Theorem 6.4.1, part (2), there is a set E

such that IIM,p11

~

II¢>IIoo

+ sup (EE

1'" /

C

1r of full measure

¢>((e it )

_'"

- ¢>(() / dt. e,t - 1 211"

Let us now estimate the second term in the above expression. Fix n E N and write the integral as

I:

=

i' 0,

(ii) seE) > 0 and I(E)
0

6.610 A Blaschke product whose zeros E satzsfy both ap(E)
0, the correspondmg Blaschke product B has the representatzon

() 6.6.13

1 B(z) __1__ '"" 1 -lal 2 1 - B(O) lal Ba(a) 1 - az

fEB

The above senes converges unzformly on compact subsets of D. Moreover, zf we also assume that ap(E) < 00, thzs representatzon holds pomtwzse for every (E T. 6Such E exist by Lemma 6 6 5 and Remark 6 6 9

134

6 MULTIPLIERS AND DIVISORS

PROOF. Let {En : n E N} be an increasing sequence of subsets of E with card(En} = n and such that 00

n=l

For each n, let B(n) be the Blaschke product with En as its zeros. If a E En, we let Bin) be the Blaschke product with En \ { a} as its zeros. By considering each B(n) as a rational function on
zEW.

+ sei9 , we have for s > 1/4, 11-zI 4 =

(~_SCOS{}+S2)2

~ (G -8)' + 28.;n (0/2»), 2

~ 4s 2 sin4 ({}/2)

~ ~ sin4 ({}/2). Use this estimate along with eq.(6.6.21) to see that sup

r27r Ihl (~ + sei9 ) IV d{} ~ C 10r27r (sin dO({}/2»v
0, the pseudo-hyperbolic disks K(a, s(E)/2), a E E, are disjoint, and their union is contained in the set B of points z E ]]]) whose pseudo-hyperbolic distance to T(Q) is less than 1/2. Each pseudohyperbolic disk K(a, s(E)/2) has Euclidean radius proportional to m(I)s(E), where I is the arc upon which Q sits. Hence

area (

U

K(a'S(E)/2»)

o.EEnT(Q)

is proportional to card(EnT(Q»m(I)2s(E)2. On the other hand, the area of B is proportional to m(I)2, and so

card(E n T(Q»s(E)2 ~ c

o

for some absolute constant c.

6.6.26. If a sequence E ~ sepamted and E c B(,) for some' unzformly sepamted, and 8(E) depends only on seE).

LEMMA

then E

~

E

T,

PROOF To prove that 8(E) > 0, it is enough to use Carleson's theorem (Theorem 1.11.5) and reduce the problem to showing that J.LE 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 B(,), there is an absolute constant C1 so that for each n, at most C1 of the sets T(Qj) intersect B(,) (see Figure 4).

6 MULTIPLIERS AND DIVISORS

142

FIGURE

4. At any high enough 'level', S«() intersects only two T(Q)'l

Since E is separated, we see from Lemma 6.6.25 that for any n, eNd

(En (,QT(Qj))) .; S(~)2

For a Carleson square Q, it follows from the facts that 1-

and for each n, only that

Cl

m(I)

n

lal ~ 2ft'

a E Qj'

of the boxes T(Qj) intersect S«(), and of course E ( 2"

00

JLE(Q) = L LJLE(T(Qj)) n=Oj=l

~

~

f::o

Cl

C m(I) s(E)2 ~

C

= S(E)2 m(I).

Thus the Carleson constant 'Y(E) satisfies the inequality C

'Y(E) ~ s(E)2· Using the inequalities exp (

-C3

~~~ ) ~ t5(E) ~ s(E),

from eq.(111.7), and the inequality 'Y(E) obtain (6.6.27) where

~2(s(E)) ~

~ ~l(s(E)),

5(E)

~

s(E),

where

~l(S) =

cJ

6 6 MULTIPLIERS AND INNER FUNCTIONS

143

is a non-negative increasing function on [0,00). Hence 8(E) depends only on s(E).

o

LEMMA

6.6.28. If uF(E)

< 00, then

card(E n S(()) ~ cUF(E) for each ( E 11.', where

C UI

an absolute constant.

PROOF. If Z E S(() and the segment from z to ( makes angle () with the diameter through (, then the chord of the circle 11.' passing through z and (z) '= " Xa_ ~ 1-az aEE

belongs to H OO •

(1) If sup (ET

L aE

E

1-la1 2 IXa11a11 (I < 00 1- a

then 1> E W'l(X) and (6.6.32)

(2) If 1> E W'l(X) and E zs mterpolatmg, wzth mterpolatmg constant C(E) from eq.(llLlj, then

1-la1 L IXailal11 (I ~ 2C(E)II M ",1I < aEE - a 2

(6.6.33)

sup (ET

00.

Thus 'tf E zs an mterpolatmg sequence, we have

1-la1 2: IXallalll _ a(1 < aEE 2

1> E W'l(X) sup (ET

00.

PROOF. We begin the proof with a few preliminary comments. Since 1> E H oo , the Toeplitz operator T(f is well-defined operator from Hoo to H2 and so the corresponding Hankel operator

is a well-defined operator from operator With

HOO

to HJ. Here we use P+ for the Riesz projectlOn

1-la1 2 1>(z) = "~ Xa 1 -az ' aEE we first claim that for f E Hoo, (6.6.34)

,,_

_1-laI 2

(H~f)(z) = ~ aXaf(a)z 1- az'

aEE

Z E lIJ).

146

6 MULTIPLIERS AND DIVISORS

To see this, observe that for z E ill>, ¢(z) = """ xa(l ~ aEE

1

-laI 2 )l-az -_00

x a{1-laI 2) Lakz k aEE k=O

=L

t

zk ( L xa(l - laI2)ak) k=O aEE Since we are assuming that ¢ E H oo , we know that the almost everywhere defined boundary function for ¢ belongs to LOO and has a Fourier expansion equal to =

00

¢ rv

L ¢(k)(k, k=O

where (6.6.35) aEE

For each n E No, we have

(nlfi '"

00

00

k=O

l=-n

L ¢(k)(k-n = L

¢(l + nfr!

and so l=1

Thus for each z E ill> and n E No, 00 =-__

(H¢(n)(z) =

L ¢(l + n)i l=1

= t z l ( L Xa(1-la I2 )al+n ) l=1 aEE

(byeq.(6.6.35))

00

= L

aEE

xa(l - lal 2 )an L

ali

1=1

= """ x a {1-laI 2 )an az _. ~ l-az aEE

Thus the formula in eq.(6.6.34) is valid whenever I is a monomial, and hence by linearity, any analytic polynomial. To show that the formula is valid for a general I E Hoo, approximate I in the weak-* topology of H OO with its Cesaro polynomials In = unl (Le., pointwise on ill> with uniformly bounded sup-norms). Argue that In ~ I weakly in HZ and so, since H¢ : H2 ~ H6 is continuous, H¢ln ~ H(j)1 weakly in H6 and hence pointwise in ill>. The that for each z E ill>, 1 - lal 2 lim """ aXaln(a)z 1 = n-oo ~ - az aEE

dominated convergence theorem says 1 - lal 2 """ axal(a)z--'----''-~ 1 - az aEE

6 6 MULTIPLIERS AND INNER FUNCTIONS

147

and thus the formula in eq.(6.6 34) is valid for all I E Hoo To prove (1), use the assumption that

'"

1-laj2

sup ~ IXallallI C;E'f aEE

- a

(I < 00,

along with the inequality 1 - ra(1 ~ 2 11a(

and the dominated convergence theorem, to see that for each ( E 1[',

(H?>f)«() = lim_ (H?>f) (r() r-.l

= lim

L aXal(a)r(_1-jaI 1 - are 2

(byeq.(6.6.34))

r-.l- aEE

-1-laI 2 . 1- a(

= L aXal(a)( aEE

Thus, for any I E Hoo,

IIH?>IIIoo ~ jlflloo sup

L laliXal111 -

C;E'f aEE

and so H?> . Hoo satisfies

---+

HO'

-

lal 2 (I < 00, a

and Its operator norm IIH?>1i .- IIH?> : Hoo

IIH?>II ~ sup

L

1

jaj2

11 -

a(1

lallxal -

C;E'f aEE

---+

HO'II

.

The estimate liT?>!! ~ 114>1100 + IIH?>II, along with Proposition 6 1.5 shows that 4> E 9Jt(X). The identity IIM1>1I = liT?>!! from Proposition 6.1.5 implies the estimate in eq.(6.6.32}. To prove part (2), we first observe, that for fixed Z E Il}, we can choose E = {Ea : a E E} with lEal = 1 for all a and such that

L

(6.6.36)

aEE

l-laJ2 - _1- jal 2 Z IXallallzl,) = '" 1-az ~ XaaEa 1 -az . aEE

interpolating and f E baU(£OO(E)), there is a function IE E HOC such that for each a E E, and 11/.1100 ~ G(E), where C(E) IS the interpolating constant from eq.(1.11.4). But for such an IE> we know from eq.(6.6.34) that Since E

IS

IE(a) =

Ea

H:dE(Z) 0/

_ _ 1-laI 2

= '" XaaEa Z 1- az ~

.

aEE

However, we are assuming that 4> E 9Jt(X), and hence H?>: HOO Hence Thus we have (6.6.37)

---+

Hoo is bounded.

6 MULTIPLIERS AND DIVISORS

148

and so for each, E 'll',

E Ix llal{I-laI a

aEE

II-a'i

2)

~

lim

E Ix llalr{I-laI a

2)

II-ar'l '" Ixa llallzl{1 -laI 2 ) ~ sup L...J .zElI) 11 - azl aEE r-+1- aE E

~

IIH4iIlC{E) (by eq.{6.6.36) and eq.{6.6.37)).

The estimate in eq.(6.6.33) now follows from the estimates

o LEMMA 6.6.38. Let 0 < € < 1/2, and suppose that B is a finite Blaschke product with n zeros, all lying in the disk {z E O· Izi < €}. Then there is an absolute constant c such that

1~ c

(ne + 1 +logn 11MB II) .

PROOF. We write our finite Blaschke product B as

From Proposition 6.4.5 and the triangle inequality, (6.6.39) Now use Proposition 6.4.9 to get, (6.640)

IIMB(1).zn-BIl

~ c (~lInB(I)Zn-1 -

B'lIoo

To estimate the first term on the right, note that

+ log(en)IIB{I)zn -

Blloo) .

6 6 MULTIPLIERS AND INNER FUNCTIONS

149

Hence

IInB(I)zn-1 - B'I/oo

~ nIlB(I)zn -

Blloo +

~ II ~a~ ~~:: - 1/100

~ (1 -l -l akl)2 kl 2 - 1 ~ nIlB(I)zn - Blloo + ~ a

11

nIlB(I)zn - Blloo

=

~ nIlB(I)zn ~

+

I

t

2l akl- 21 akl 2 k=l (1 -l akl)2

Blloo + (1 2~:)2

nIlB(I)zn - Blloo + 8n€ (since 1 -

€

If bk is the Blaschke factor corresponding to the zero ak, we have B so by the triangle inequality,

> 1/2).

= b1' . bk and

n

IIB(I)(n - Blloo ~

L IKbk(I) - bklloo.

k=l Now for each k = 1,···

,n,

I

l(bk(I) - bk(()1 = al k - 1 ( - Iak ~( - ak - ak

I

+ ak - fik + fik( - ak(1 (1 - ak)(1 - ak()

= l'ak,2(( - ()

6€ (1 - 10)2 24€.

~ -;-.,-----c:-=

~

Hence

IIB(I)zn - Blloo ~ 24nt. Combine the above inequalities and put them in eq.(6.6.40) (and re-adjust the umversal constant c) to get

11MB(l)zn-BII Now bring

ill

~

c(€ + mlogn)

eq (6.6.39) to see that c1logn ~ C2(t + m log n) + IIMBII

and so, again re-adjusting the universal constant, we get

I~c(m+

€+IIMBII) logn

~c(m+

I+II MB logn

II). o

LEMMA

6.6.41. Let a

E ll}

and let

,),(z) If 4> E 9Jt(X), then 4> 0

(6.6.42)

')'

a-z I-az

:= - _ - .

E 9Jt(X) and

1

2"M",1 ~ IIM",o-yli ~ 211 M",II·

150

6

MULTIPLIERS

DIVISORS

AND

PROOF. Notice that "I is an automorphism ofID> and so from Lemma 5.6.1 both go'Y and gO'Y- 1 belong to X whenever 9 E X (i.e., the composition operator is well defined on X). So let! E X and apply the previous line to see that! 0 "1-1 EX. But since ¢ is a multiplier, then ¢U 0"1- 1) EX. Compose with "I to conclude that (¢ 0 "I)! E X. Thus ¢ 0 "I is a multiplier. We now prove the string of estimates in eq.(6.6.42). For h E A,

=

(T-h)(z) q,

I

4>(()h(() dm«) 1 - (z

and so, from Proposition 6.1.5 and the definition of the operator norm,

IIMq,1I = IIT~II

~ sup = sup = sup

{II! ~h~;) {II ~h~~) I: {II ¢~(2~ I:

dm((t . hE ball(A)}

dm«)

Izl =

dm«)

Izl = l,h E balI(A)}

l,h E ball(A)}

Observe that by a change of variables,

I

¢«)h(() dm«) = 1-(z

I

(4) 0 "I) «(h'«)(h 0 'Y)«() dm«) 1-'Y«)z

and, via a computation with partial fractions, that 'Y'«) z - -z= 1-'Y«()z 1/0,-( u-('

z,(E'lI'

where

l+az o,+z

U=--.

Notice that

l/a E ID>e and u E 'lI' (since z E 'lI') (1-+ /

z

la-(

and so the functions

z

and (1-+--

u-(

are Cauchy transforms which are at most of unit norm (see eq.(4.1.23)). Thus (6643) From our earlier argument that 4> 0 "I is a multiplier, we can use eq.(6.6.43) to conclude that for each Izl = 1, (6.6.44)

«h'«) II '"& 211Mq,o"( II • II (4) 1 -"I)'Y( ()z 0

By the Cauchy dual pairing

rl!..~- h(KIL)(r()g«)dm«),

IL E M,

9 E A,

151

6 6 MULTIPLIERS AND INNER FUNCTIONS

between A and X, we see that for all Izl = 1 and h E ball(A),

dm«() if (c/J if c/J«()h((j (z =

0

1-

,)(()-y'(()(h 0 ,)«() dm«() 1 -,«()z

~

II (c/J 0 ,)«()-y'«() 1IIIh 0 ,1100 1 -,(()z ~ 211Mo')'lI (by eq (6.6.44)). Thus, again usmg the definitIOn of the operator norm of IIT¢II and the equality IIT¢II = II MII, it follows that

1

"2 II M II ~

II M o')'ll·

Finally, to prove the inequality

we recall the argument at the beginning of the proof that c/J apply the above estimates to get 1

"2 II M o,), II

0,

is a multiplier and

~ IIM(o,),)o,),-lll = IIM II

o In a moment, we will need the following technical lemma relating pseudohyperbolic dISks and Carleson boxes. See [19, p 299] for details. LEMMA 6.6.45. For a gwen f > 0, there zs an mteger M > 0 such that for any Carleson box Q, the box T(Q) can be covered by at most M pseudo-hyperbolzc dzsks of radzus f. LEMMA 6.6 46 Let B be a Blaschke product such that B E 9Jl(X). If c zs the unwersal constant from Lemma 6.6.38, let N be an mteger such that

N ~ exp(2c(1 + 211MB I!)) and let

1

1

c= 4cN = 4c exp( -2c(1

Then for any a E ofB. PROOF

][J),

+ 211MB I!))

the pseudo-hyperbolzc dzsk K(a, f) contams less than N zeros

The automorphism

a-z

l(Z):= - _ I - az

maps K(a,f) onto the EuclIdean disk ~(O,f). Thus B has precisely as many zeros m K(a, f) as B 0 1 has in ~(O, f), Suppose BO l has at least N zeros in ~(O, f). We now derive a contradiction as follows. let B* be a sub-product of BO l with preczsely N zeros in ~(O, f). By the!tproperty for multiphers (Theorem 6.5.1), 11MB-Ii ~ IIMBo')'ll. Apply Lemma 66.41 to see that liMBOI'll ~ 211MB II· Combining these two estimates YIelds

11MB-II ~ 2I1MBII·

6 MULTIPLIERS AND DIVISORS

152

Notice that N was chosen so that

cl+211MBil 10gN

~ ~ '" 2·

Now apply Lemma 6.6 38 to B* to see that l~c

( N€+ 1+IIMB*II) logN

~cN€+c

1+211MBII 10gN

1 2

~cN€+-

Hence 1 1 1 2"" 4cN 4 which is a contradiction. Thus B has less than N zeros in K(a, c). -~CNf=cN--=-

o

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 If a Blaschke product B is a multiplier, then B =B1···Bn , where for each j E {I,··. ,n}, B j is a Blaschke product and the corresponding zero sequence ZB j is separated. PROOF. We first note that if 0 < € < 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 ofradius € If each such disk contains at most N zeros of B, then each T(Q) contains at most M N 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 0 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 ~ denote the collection of Blaschke products. An old theorem of CaratModory [38J says that ~ is a weak-* dense subset of baU(Hoo). Our needed technical lemma is the following. LEMMA 6.6.49 (Tumarkin [222]). For each c > 0, the set ~c :=

{B E

~ :

(J'F(ZB) ~ c}

is a weak-* closed subset of HOO . The proof of this lemma requires a few facts. The first is an application of Jensen's formula. LEMMA 6.650. If B is a Blaschke product and n(p) is the number of non-zero zeros of B in the disk {Izl < p}, we have the formula (6.6.51)

[log IB(p()1 dm() = Alogp -

iT

where A is the order of the zero of B at the origin.

t

i

p

nCr} dr, r

153

6 6 MULTIPLIERS AND INNER FUNCTIONS

PROOF. For fixed 0

< p < 1, we can WrIte

(;f' B{z).

B 1 {z) =

Then B and Bl have the same modulus on the cIrcle {Izl = p} and B1(O) Jensen's formula (see [5, p. 208]) applied to Bl yields (6.6.52)

L

log IBI (0)1 = -

i

log C:' I) +

la,l";p

=1=

O.

log IB(p()1 dm(().

However, from the definitIOn of B 1 , 00

B 1 (O) = / '

II la 1 3

3=1

and so

00

logIB1 (0)1 = Alogp+ Llogla31. 3=1

Combine thIS last identity with eq.(6.6.52) to get (6.6.53)

Alogp+ EIOgla31= -

L

log C:'I) +

la,l";p

3=1

h

10gIB(p()ldm(().

With the observation that

L

L

log C:'I) = n(p) logp -

la,l";p

log la3 1,

la,l";p

eq.(6.6.53) becomes (6.6.54)

Alogp+n(p)logp+

L

logla31= 110gIB(p()ldm(().

la,l>p

'Il'

Now if PI ~ P2 ~ .. denote the successive moduli of the zeros of B which are greater than p, we observe that

n(P3) = n(p) + J, and so

1 1

p

n{r)

-dr= r

l p

P1

n(r)

~

- d r + L..,.; r 3=1

l

P'+l

P,

n(r)

-dr r

00

= n(p) log PI + L

p

3=1

n(P3) log P3+I P3

00

= n(p) log PI + L{n(p) + J) log P3+I

P

3=1

P3

00

= -n(p)logp+ Llogp3' 3=1

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. 0 We also need the following characterization of Blaschke products [79, p. 56].

MULTIPLIERS AND DIVISORS

6

154

PROPOSITION 6.6.55. A function U E HOC with 1!Ulloc if and only if (6656)

lim flog IU(r()1 dm(() rjl

JT

~

1 is a Blaschke product

= o.

Finally, to prove Lemma 6.6.49 we will need another technical result of Tumarkin [221]. Let (aj)j~1 be the zeros of the Blaschke product Bk and let nk(p) be the number of non-zero zeros of Bk in {Izj < pl. LEMMA 6657. Let (Bk) be a sequence of Blaschke products which converges uniformly on compact sets of]])) 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 {Izl < r} is uniformly bounded in k; (2) for every f > 0 there is an 0 < R < 1 such that

L

(1 -

lajl) < f

laJI>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 {/zl = r}, the principle of the argument (assuming B has no zeros on {I z I = r}) shows that for large enough k, B and Bk have the same number of zeros in the disk {Izl < r}. Condition (1) now follows. Since B is a Blaschke product it follows from eq.(6.6.56) that for every f > 0 there is an o < R < 1, such that

i

Since Bk that

for k

~

-+

log IB(R()I dm(()

i

log IBk(R()1 dm(() > -2f

K. From eq.(6.6 51)

1 -1 1

~

-f.

B uniformly on the circle {Izl = R}, there is a positive integer K such

nk(r) d - r< 2f

r K, since Ak log r is negative. Finally, R

for k

>

1

11

1 nk(r) - nk(R) d nk(r) d nk(R) d - - rr+ -- r R r R r R r 1

~ fl ndr) - nk(R) dr

JR

=

L

laJI>R

r 1

log

lajl·

Condition (2) now follows because

1 3 1 - x < log -; < "2(1 - x),

0 < x < 1-

Now suppose that conditions (1) and (2) hold. Because of (1), B is not the zero function. Let f > 0 and let R be given by condition (2). Notice how there is

6 6 MULTIPLIERS AND INNER FUNCTIONS

155

an RI with R < RI < 1 and such that 1 nk(R) log RI < € V k,

since nk(R) is bounded in k by condItion (1) Thus

r

I nk(r) dr = JRI r ~

r

r

I nk(r) - nk(R) dr + I nk(R) dr JRI r JRI r

1 1

R

nk(r) - nk(R) d (R) I 1 r+nk og-R

L

=

r

1

1

log

laJI>R

la~1

1 + nk(R) log RI

J

~2€+€

=3€ for all k. Choosmg R2

> Rl so that

it follows from eq.(6.6.51) that

Ii Ii

10gIBk(p()ldm«()1 < 4€

for R2 < P < 1. Since Bk

~

B uniformly on each cIrcle

for R2 < P < 1. Hence hm rj1

{Izl =

p}, we have

log IB(p()1 dm«()/ o.

In that same proof, one can also show that the separation constant s( E J ), where E J are the zeros of B], for one of the factors B J , satisfies

s(E]) ~ cexp( -C'IIMBJ II). However by the ~-property for multipliers, IIMBJ ~ IIMBII and so

s(EJ) ~ cexp(-c'\lMBII)· Now follow the proof of Claim 2 above to obtain the inequality

,,(EJ) ~ dl>3(S(EJ»11MB II· But since 4(IIMBII)iP 4 (IIMB II)

= iPs(IIMBII), where iPs is increasing. The last line of the proof of Claim 2 says that

CTF(Ej) ~ cC(Ej ) 11MB II ~ ciPs (II MB II) 11MB II = 4>6(IIMB

II),

where iP6 is increasing. Finally, n

CTF(ZB) ~ LCT(Ej ) j=l

~ n4>6(IIMBII)

~ cexp(c"IIMBII)iP 6 (IIMBII)

= iP7 (II M BII), where 4>7 is increasing.

Claim 4: 8 =

81-'

D

is not a multiplier unless JL

== O.

PROOF. Suppose that 8 is a non-constant multiplier. By a well-known theorem of Frostman [79, p. 79]10, there is a sequence (an)n~l C JI} with an ---t 0 such that

B _ 8-an n - 1- a n 8 is a Blaschke product. We can assume that lanlliMslI < 1/2 for all n. Observe that

8- B n

=

S- 8-an l-a n 8 8 l-a n 8

S - a;;:8 2

-

+ an

00

(an - a n8 2 ) LankSk. k=O

Since lan l1l811 < 1/2, the multiplier norm of 00

Lan k 8 k k=O

is bounded by 2 Also observe by Proposition 6.1.3 that

I M sll

~ 1181100 = 1

IOThe theorem here is the following' suppose J is a non-constant inner function Then for all w E

JI)),

except possibly for a set of logarithmic capacity zero, the function Jw(z) = J(z) -

is a Blaschke product

t

I-wi z)

6 6 MULTIPLIERS AND INNER FUNCTIONS

and so the multiplier norm of an - a n S 2 is bounded above by again using Proposition 6.1.3,

liS -

161

21an 11lMs112.

Thus,

Bnlloo ~ II MS-Bn II ~ 4lan lliMslI2.

It follows from here that II S - Bn II co

-+

0 as n

-+ 00

IIMBnll ~ IIMBn-sll + IIMsl1 ~ 41an iliMsil But since each Bn is a multiplier, Claim 3 says that

and that for every n EN,

+ IIMsl1

~

311 Msll·

O"F(ZBn) ~ y) ~

C" y

fELl,

1

m 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 zn k , for an appropriate lacunary sequence (nk)k~b are nearly stochastically independent. In thIS chapter, we examine thIS distribution function Y 1-+

m(lil > y)

more closely, as well as the corresponding one for the Cauchy transform, paying special attention as to how one can obtain mformation about the singular part of the measure just by knowing the behavior of m(IKf.l1 > y) as y -+ 00. To make some of our proofs easier, and since it is interesting in its own right, we first look at the correspondmg results for the Hilbert transform. Let M(IR) denote the finite, complex Borel measures on IR and M+(IR) the positIve ones. Let ml denote Lebesgue measure on IR 1. For t: > 0 and f.l E M(IR), let

(1{ef.l)(x):=

r

_1_ df.l(s)

J1x-sl>e x -

S

be the truncated Hilbert transform, WhICh is defined for every x E 1R, and let

(1{f.l)(X) := lim(1{ef.l)(x) 0

denote the Hzlbert transform 2 , whenever this limit eXIsts. The existence of thIS limit for ml-almost every x E 1R, as well as the basIC distributIOnal inequality, is well known [37, 128)3. THEOREM 7.1.1. For f.l E M(IR), the Hzlberl transform (J{f.l)(x) exzsts for mIalmost every x E lR. Moreover, there zs a C > 0 such that

(7.1 2)

lUSIng thIs notation, ml f/- M(IR) (since It IS not fimte) 21n many books, the HIlbert transform has an extra factor of 1/7r 3The LoomIS proof [128]18 for one varIable The well-known and often cIted paper of Calderon and Zygmund [31] looks at the HIlbert transform In hIgher dImenSIons 163

164

7

THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS

As is convention here, when d/L = I dml, we will use 'J{I in place of the more cumbersome 'J{fdml. 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 1 < p
y) = o(l/y).

It is well known that if /L is compactly supported, say in the interval [0,211"], then for ml-almost every x, the Hilbert transform ('J{tt)(x) is equal to

~PV 1211" cot (~) d/L(s) + ~ 1211" (_2_ 211" 0 2 211" 0 X - s

cot

(~)) d/L(s). 2

The first integral is the conjugate function (Q/L)(e ix ) (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 'J{/L and Q/L have similar types of singularities and mapping properties 7.2. Boole's theorem and its generalizations For a non-negative singular measure /L on JR, the distribution function

Y f---+ ml ('J{/L > y) of its Hilbert transform 'J{tt can be computed explicitly [104]. THEOREM 7.2.1 (HruScev and Vinogradov). If /L E M+(JR) and is singular, then lor every y > 0,

As a consequence,

ml(I'J{/L1 > y}

=

2M. y

To prove this, we begin with Boole's lemma [30] which was discovered in 1857 and computes the distribution function for

~

Ci

LJ x-a-' i=l

•

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

165

LEMMA 7.2.2 (Boole's Lemma) Let n

g(x) :=

L -~-, a, t=l X -

where

~

> 0 for z = 1, .. ,n, and a1 < a2 < ... < an. Then 1

and m1(g < -y)

n

= - L~ Y,=l

for every Y > O.

PROOF. Consider the graph of g(x) and assume that y > 0 is given (see Figure

1).

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, ad, (aI, a2),"" (an, 00). It follows that the equation g(x) = y has precisely n solutions a, < 8. < a.+1 for z = 1, .. ,n -1, and an < 8n . Moreover, n

m1(g

> Y)

=

L)s, - a.) ,=1

81,

•• , 8 n

,

with

166

7 THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS

The expression

g(X-ai)(l-g~»)

ai

when multiplied out (and redefined at in the obvious way), turns out to be a polynomial p(x) of degree n whose roots are S1, .. , Sn. In fact, n 1 n n p(x) = II(x - ai) - - LCi II(x - aj) i=1 y i=1 i=1 =

xn -

itfj

(t ai ~ t +

Ci )

x n- 1 + h(x),

where h(x) is a polynomial of degree less than n - 1. By Viete's theorem\

or equivalently 1

n

n

L(si - ai) = - LCi. Y

i=1

i=l

This completes the proof of the first assertion. The second follows in a similar

D

w~

PROOF OF THEOREM 7.2.1: First suppose that p, E M+(~), 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 '}{p, is a differentiable function on ~ \ F and

d

dx '}{p,(x) = -

J

dp,(s) (x _ S)2' x E ~ \ F.

Thus '}{p, is strictly decreasing on each interval complementary to F. It follows for y > 0 that the set {x E ~ \ F: '}{p,(x) = y} is countable and so has Lebesgue measure zero. Let a

= min{x E F} and b = max{x E F}.

Note that

00

[a, b] \ F =

U In, n=1

where In are disjoint open intervals. We now go through the following approximation argument to produce a sequence of discrete measures P,n with finite support that approximate p, weak-* and such that each P,n places mass only at the end points of the complimentary intervals (In)n~l of F.5 Indeed, let Fl,l =

[a,b]

4Viete's theorem if p(x) = xn + an_1Xn-1 + + ao is a monic polynomial with roots ,sn,thenS1+S2+ +Sn=-an-l 50ne can prove this using Proposition 1610 However, for the sake of completeness, we provide a direct construction

51,

7 2 BOOLE'S THEOREM AND ITS GENERALIZATIONS

167

and define where al,1 = a. Write

[a, bj = II U F2 ,1 U F2 ,2 as a disjoint union, where F2,1 and F2,2 are disjoint closed intervals, and define J.L2 = J.L(F2,1)8a2 ,1 + J.L(F2,2)8a2 2' where a',3 is the left-hand endpoint of the closed interval F.,,. Doing thIs agaIn, write [a, bJ = II U 12 U F3,1 U F3,2 U F3,3 as a disjoint union, where F 3 ,1, F3,2, F3,3 are disjoint closed intervals. Define J.L3 = J.L(F3,1)8a3 ,1 + J.L{F3,2)8a3 2 + J.L(F3,3)Oa3 3 In general, write

and define

n

J.Ln = LJ.L(Fn,3)8an3 · 3=1

By our constructlOn,

lIJ.Lnll = !iJ.L11 'In E N We will now show that

J.Ln --+ J.L weak-*, n --+ 00. To this end, let f be a continuous functlOn on lR and note that fl[a,bj is uniformly continuous and so there is a 8 > 0 such that whenever x, y E [a, bj with Ix - yj < 8, we have If(x) - f(y)1 < E. Since m1(F) = 0, there is an n so that m1

(u

Fn,3) < 8.

3=1

With this n,

II'dp. -I fdl'l

= = =

=

t, It t Idl'l t, kCGnJ)I'CFnJ) - /..., Id"l t, I/CGnJ)I'CFn.,) - L.., Idp,. -

(f -

t1

3=1

(f - f(a n,3»dJ.L

F",3

n

~

L EJL(F ,3) n

3=1

ICa..,) + ICGnJ))d,,1

168

7 THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS

Thus J-Ln --+ J-L weak-* as n --+ 00 Using the fact that J-Ln --+ J-L weak-*, it follows that '){J-Ln converges uniformly to '){J-L on any compact set that is disjoint from F. If Xn denotes the characteristic function of the set {'){J-Ln (x) > y} and X the characteristic function of the set f}{J-L(x) > y}, then Xn converges pointwise and boundedly to X off F, except possibly on the countable set {x: JeJ-L(x) = y}. Hence, since F has Lebesgue measure zero, m1('){J-L > y) = jXdm1 =

lim

n-+oo

jXndm1 =

lim !IIJ-Lnll. y

n--+oo

The last equality is from Boole's lemma (Lemma 7.2.2). However, our construction says that IIJ-Ln II = IIJ-LII for all n and so 1 (7.2.3) m1(JeJ-L > Y) = -IIJ-LII y for singular J.L E M+(JR.) with compact support Now let J-L E M+(R) be singular but not necessarily of compact support. For fixed f > 0, there are positive singular measures v and p such that J-L = v + p, v has compact support on a set of Lebesgue measure zero, and Ilpll < f211J-L11 6 Applying the estimate in eq.(7 2.3) to v and the estimate in eq.(7.1.2) to p, we have m1(JeJ-L> Y)

~

m1

(Jev

> (1 -

f)Y)

+ m1('){p > fY)

~

II vII + cllell (1 - f)Y fy ~ IIJ-LII (_1_ + Y

Let

f --+

I-t

Ct)

0, to get

(7.24) To prove the reverse inequality in eq (7.2.4), we start with the inequality (7.25)

m1(Jev>

l~f) ~m1(JeJ-L>y)+m1(JeP

-y-) = 1- f II vII ~ IIJ-LII (1 _ f)(1 _ f2).

1- f Y From eq.(7.1.2) (applied to p) we know that m1

(JeP

< -f 1Y- )

-f

Y

~ m1 (IJepl > f1-y ) ~ Cllplll - f ~ Cf211J-Llll - f f yf yf

6Indeed, since I' 1. mI, there is a set F,. so that I'(F,,) = 111'11 (i e , F,. is a carrier for 1') and mI(F,,) = 0 Since I' is regular there is a compact set F C F" so that I'(F,. \ F) ~ £2111'11 [182, p 48] Let v = I'IF and p .= I'I(F,. \ F)

73 A

REFINEMENT OF BOOLE'S THEOREM

169

Apply these two facts to the inequality in eq.(7.2.5) to get

IIIlIl (1- f)(I- f2) ~ ml(XJ.L > y) + C€(I- €) 111l1i. Now let

f

-+

y

°

y

to obtain

111l1l ~ y m l(XIL > y) Combine this inequality with the one in eq.(7.2.4) to complete the proof that ml (XIL > y) =

1I1l1i. y

The other equality

o

is proved in a similar way. 7.3. A refinement of Boole's theorem By the Lebesgue decomposition theorem, every Il E M(JR) can be written as J.L = Ila

+ Ils,

Ila

«: ml,

Ils..l mI·

THEOREM 7.3.1 (Tsereteli [218, 219]). For Il E M(lR), the followmg are equzvalent (1) ml(IXILI > y) = o(l/y) as y (2) J.L

-+ 00.

= Ila·

PROOF. The result stated here is true for complex measures but to avoid some technical details, we prove it only for positive ones. Suppose dll = gdml for some non-negative 9 E Ll(lR). Given € > 0, choose an infinitely dlfferentiable with compact support such that II - gll1 < f. For large enough a, ml(IX a) = 0, since X(¢) is a bounded function 7 . Thus ml(IXILI > y)

~

=

ml(IX(Il- ¢dml)1 > y/2) ml(IX(Il- ¢dml)1 > y/2)

+ ml(IX1 > y/2)

for large enough y. From eq (7.1.2) it follows that €

ml(IXILI > y) ~ c-. y From here one can show that

Conversely, suppose that m(IXILI

> y) = o{1/y). Start with the inequality

ml(IXlLsl > y) ~ ml(IXlLal > y/2)

+ ml(IXILI > y/2)

and use the hypothesis that ml(IXILI > y) = o(l/y) and the above fact that ml(1Xllal > y) = o(l/y), to conclude ml(lXllsl > y) = o(l/y). However, {IXJ.Lsl > y}

= {XlLs > y}U{XlLs < -y}

and these sets are disjoint. Thus ml(IXJ.Lsl > y) 7Recall that when dJL =

= ml(XlLs > y) + ml(XlLs < -y).

f dml, we use 'XI In place of the more cumbersome 'X(f dmI)

170

7

THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS

Apply Theorem 72.1 twice to obtain ml (I1fJLs I > y) = 2I1JLsll/Y· The fact that ml (l1fJLs I > y)

= o(l/y)

implies that JLs = 0 and so JL

= JLa

0

THEOREM 732 (Hruscev and Vinogradov). For JL E M+(IR), lim yml(I1fJLI > y)

y->oo

= 211JLsll.

PROOF. By Theorem 7.2 1, this formula is certainly true if JL = JLs. Also note that ml(l1fJLal > y) = o(l/y) as y ---+ 00. If 0'.,f3 > 0 with 0'.+ f3 = 1, we can apply Theorem 72.1 to get ym l(I1fJLI > y)

~

ym l(I1fJLal > yO'.)

+ yml (I1fJLs I > yf3)

yml (l1fJLa I > yo'.)

+ 211~ II

Combining this with the above observations, we conclude that lim yml(I1fJLI > y)

y->oo

But since 0

< f3 < 1 is arbitrary,

~ 2I1JLf3s ll.

we get

lim yml(I1fJLI > y) ~ 211JLsll.

y->oo

In a similar way,

yml(!1fJLs I > y) ~ yml(IJeJLI > yO'.)

+ yml (I1fJLaI > f3y).

Hence

and so

o

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 JL E M+ recall from eq.(1.8.3) the Herglotz integral

(HJL)(z) =

J~ ~;

dJL(().

We saw earlier that H JL E HP for all 0 < p < 1 and so has non-tangential boundary values m-almost everywhere. Also recall that SRH JL = PJL, the Poisson integral of JL, and for m-a.e. ( E 'll',

(PJL)(()

= r--+llim (PJL)(r() = dJL () dm

(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 74.1. For a szngular measure IL E M+,

m(IHILI

> Y) =

~ arctan e~")

PROOF. Let us assume first that 1i1L11 = 1. At the end we will see how to adjust the proof if this is not the case. Smce IL E M+, H IL maps the unit dIsk to the domam {~z > O} Since

z-1 z+1

z~-­

maps

{~z

> O} to D, the analytic function ¢ defined by

(7.4.2) maps the disk to Itself Moreover, observe that

H _1+¢ IL- 1 _¢

Since (HIL)(O) a.e. (E 'll',

= IIILII =

1, it follows that ¢(O)

= O.

Now observe that for m-

But since IL -.L m, th18 last quantity is zero m-almost everywhere. Thus the boundary values of HIL are purely imaginary and, by eq.{7.4 2), I¢I = 1 m-a.e. on 'll' Hence ¢ IS an inner function. We now claim that the map ( ~ ¢{ () IS measure preservmg on 'll' Indeed, since ¢(O) = 0, the mean value property for harmonIc functions says that (7.4.3)

r ¢n dm -_ {1,0,

iT

n = 0; n E Z \ {OJ.

Thus

whenever h 18 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 ( ~ ¢{ () 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(IHIlI

> y).

Indeed,

I~ ~ :~gI> y } ) = m ( { ( E 1r : I~ ~ ~ I> y } ), q) is meas. pres. = 2~ ml ( {O E (-7r, 7r) : I~ ~ ::: I> y } )

m( {( E 1r : I(H Il) «() I > y}) = m ( { ( E 1r :

1

= 27rm1({O E (-7r,7r): cot(O/2)

> y})

1 7r

= -ml({O E (0,7r): cot(O/2) > y}) 2 = - cot- 1 Y

7r 2

= - arctan(l/y). To prove the result when 1I1l11

f.

7r 1, observe that HIl

1+q) l-q)

M=

o

for some inner q) 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 Il EM, lim ym (I Kill

y->oo

> y)

= '!'lIllsll.

7r

REMARK 7.4.5. Our notation gets a bit easier if we work with a modified definition of the Cauchy transform

(K' 1l)(Z):=

J(~Z

dll«()·

Notice that K Il = K' «( dll) and so if we can show that lim ym(IK'1l1 > y)

y-+oo

= .!.IIJLslI, 7r

then lim ym(IKIlI

y-+oo

> y)

= lim ym(IK'«(dll)1 y-+oo

> y)

= .!.1I«(dll)sll = .!.lIllsll.

7r

7r

The proof of this theorem depends on the following two technical lemmas. LEMMA 7.4.6. Let Il E M. For each € such that

> 0, there are junctions q)b

y-+oo

lim y>'K'J.t(Y) ~ lim Y>'K'IJ.tI(Y)·

(7.4 10)

y-+oo

y-+oo

PROOF. For the proof of eq.(7 4.9) we let to produce a ¢ E Coo('IT') with 1I¢1100 ~ 1 and

€

IldJ.L - ¢dlJ.L11i ~

> 0 be given and use Lemma 7.4 6 €2.

On 'IT', write

K'J.L = ¢K' IJ.L I + 1 + g, where

I(f,) = K'(¢dlJ.LI)(f,) - ¢(f,) (K'IJ.LI)(f,) =

J¢«(~ =

:(f,) dlJ.LI«()

and 9 = K'(dJ.L - ¢dlJ.Ll)· Since ¢ E Coo ('IT'), we can apply Privalov's theorem (Theorem 3.1 1) to see that 1 is continuous on 'IT' and so lim Y>.j(Y) = O.

(7.4.11)

y-+oo

By Kolmogorov's theorem (Theorem 3.4.1),

C

C

>.g(Y) ~ -lIdJ.L - ¢dlJ.L11i ~ -I?~ y y

(7.4.12) Since 1I¢1100 ~ 1, we have

IK'J.LI ~ IK'IJ.LII

+ 111 + 191

almost everywhere on 'IT'. This implies that

>'K'J.t(Y)

~ >'K'IJ.tI«l- €)y) + >'j Gy) + >.g Gy)

and so lim y>'K'J.t(Y)

y-+oo

~ ~

lim y>'K'IJ.tI«l- €)y)

+ y-+oo lim y>'g (:..y) 2

-

+ -f f2

y-+oo

lim y>'K'IJ.tI«l- f)Y)

y-+oo

2C

(by eq. (7.4. 11))

(by eq.(7 4.12))

One obtains the inequality in eq.(7.4.9) by letting f - t O. To prove eq.(7.4.1O), let f > 0 be given and, via Lemma 7.4.6, choose ¢ E Coo('IT') with 11¢1I00 ~ 1 and such that IIdlJ.Ll- ¢dJ.L1I ~

f2.

Now, on 'Jr, write

K'IJ.LI = ¢K'J.L + 1 + g, where

1 = K'(¢dJ.L) and proceed as before

¢K'J.L, 9 = K'(dlJ.Ll- ¢dJ.L),

o

74

MEASURES

ON

THE CIRCLE

175

PROOF OF THEOREM 7.4.4. By Lemma 7.4.8, It suffices to show . 1 (7.413) hm YAK'IJLI(Y) = -lilLsll· y--+oo 7r Observe that IILI = IILal O,a + (3 = 1, we have

+ IlLs I and

K'IILI = K'IILal + K'IILsl·

Thus for a, (3 >

YAK'IJLcl(ay) + yAK'IJL. I«(3y) 1 1 (aY)AK'IJLa I (ay) + -p«(3Y)AK'IJLsl«(3Y)

~

YAK'IJLI(Y)

so

a

The first term goes to zero as Y ~

00

(Proposition 3.4.11) and so

-

1-

lim YAK'IJLI(Y) ~ -(3 y--+oo lim YAK'IJL s I(Y)

y--+oo

Usmg the mequahty IK'IILsll ~ IK'IILall argument, we get,

Lettmg (3

~

+ IK'IILII

and an analog of the previous

1, we obtam the two inequalities

hm YAK'IJLI(Y) ~ y-+oo lim yAK'IJL.I(Y),

y--+oo

lim YAK'IJLsl(Y) ~ hm yAK'IJLI(Y)·

y-+oo

y-+oo

In light of these inequalities, to prove eq.(7.4.13), it suffices to show .

1

hm YAK'IJLBI(Y) = -lilLsll· 7r

y--+oo

Recall (Theorem 7.4.1) that for a positive singular measure v, AHv(Y) =

~ tan- 1 (11~1i) .

Also recall the formula

(Hv)(z) = 1 + 2z(K'v)(z),

z E ID>.

This previous formula says that m-almost everywhere on the circle, we have the inequality IK'IILsll Hence for a

+ (3 =

~ ~ + ~IHIILsll.

1,

YAK'IJLBI(Y)

~

YA1/2(ay) + YA~HIJLBI«(3y)

=

yA!HIJL.I«(3y) (since Al/2(ay)

=

yAHIJL.I(2(3y)

=

2(3 (2(3Y)AHIJLBI(2(3y)

1

~(2(3y)~tan-l (1iILsli) 2(3

Now take a lim as y

°

= when ay > 1/2)

~ 00

2(3y

7r

(by Theorem 7.41)

to get 1 lim YAK'IJL I(Y) ~ -(3lilLsll. y-+oo

•

7r

176

THE DISTRIBUTION FUNCTION FOR CAUCHY TRANSFORMS

7

Letting {3

-+

1 we obtain

To show the corresponding lim inequality, we note that on '][',

IHIJ-Lsil :::;; 1 + 2IK'IJ-Lsll and so for a

+ (3 =

1,

yAHIIL.I(Y)

:::;; YA1(ay) + YA2K'IILa I({3y) yA2K'IIL.I({3y) (since Al(ay) = 0 when ay > 1) yAK'IIL.1 =

(~Y)

~ (~Y) AK'lILsl (~Y) .

Use Theorem 7.4.1 and the above inequalities to obtain

~IIJ-Lsli : :; ~ lim yAK'IILsl(Y). 7r fJy--+oo Let {3 -+ 1 to get

.!.1iJ-Lsli : :; 7r

lim YAK'lILal (y).

y--+oo

Combine this with the above lim inequality to obtain the result.

o

1.5. 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 :J{XE (or QXE) does not depend on the geometry of the set E, just the its measure. THEOREM 7.5.1 (Stein - Weiss). (1) If E C R is measurable with ml(E) ml (I:J{XE I > y) = (2) If A

< 00, then

~ sinh- 1 (2m~(E») .

c '][' is measurable, then m ( IQ XA I > Y)

=4

tan

-1

(sin(m(A)/2»)

. n7ry h()·

SI

PROOF. We follow [79, p. 115] and only prove that for fixed y > 0, the quantity m(IQXAI > y) depends only on m(A), and not actually compute the function. The reader can get the exact formula by computing m(IQXA) > y) when A is an arc of the circle. Fix Y > 0 and let h be the harmonic function on the strip {O boundary function is XIO'zl>y, Le.,

+ it) = 0, It I : :; y, h(it) = h(l + it) = 1, It I > y. h(it) = h(l

< !Rz < I} whose

7 5 A THEOREM OF STEIN AND WEISS

The Herglotz integral H XA propertIes·

177

= PXA + ZQXA is analytic on ][) and has the following

0< rRHXA(re,8) < 1,

~HXA(e·8)

= XA(e,8),

HXA(O) = m(A).

The bounded harmonic function h 0 H XA on ][) has the property that

(h 0 HXA)(e,8) = XIQxAI>y(e,8) almost everywhere. Hence m(IQXAI > y)

=

i

hoHXAdm

= h(HXA(O))

= h(m(A)). D

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 = f - f(O)

z on the classical Hardy space H2 In order to place the next two chapters m better context, we spend some time here outlimng 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 ~ 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 {) is an mner function, then lI{)fll = IIfll for all f E H2 and so {)H2 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 (Beurhng). If {) zs an znner junctzon, the set {)H 2 zs an Smvanant subspace oj H2. Conversely, zJ M C H2, M =I- {O}, zs an S -tnvanant subspace, then M = {) H2 for some znner junctwn {). PROOF. The proof that {)H 2 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 8M "# M. If this were not the case, then f / z E M whenever f E M. Applying this k times we conclude that

f

k EM VkEN.

z

But thls would mean, since f / zk must be analytic on ]I), that to the assumption that M"# {OJ. Second, since 8M "# M, one observes that

f == 0, a contradlctlOn

M n (SM).l =I- {OJ and so M n (SM).l contains a non-trivial function {). We now argue that on a set of full measure in 'Jr. Indeed,

i 1{)((W,n

dm(() = ({),sn{)) = 0 Vn E N. 179

I{)I =

c

8 THE BACKWARD SHIFT ON H2

180

Taking complex conjugates of both sides of the above equation, we also see that

h

IfJ((W(ndm(() = 0 'In E N

This means that the Fourier coefficients of IfJI 2 all vanish except for n = 0 and so IfJI 2 = c almost everywhere on 1l'. Without loss of generality, we can assume that IfJl = 1 almost everywhere on 1l' and so fJ is an inner function. Third, let [fJ] denote the closed linear span of the functions

fJ, SfJ, S2fJ,· .. and observe that

[fJ] = fJH 2 To see this, notice that clearly [fJ] C fJH2. For the other containment, let g = fJG E fJH2 and let GN be the N-th partial sum of the Taylor series of G. Notice that fJG N E [fJ] since GN is a polynomial. From Parseval's theorem, GN ---* G in H2 and so, since fJ is a bounded function, fJG N converges to fJG in H2. Finally, observe that [fJ] =M. Indeed, fJ E M and so [fJ] c M. Now suppose that I E M and I ..L [fJ]. Since I ..L [fJ],

i

1(()fJ(()(n dm(()

= (f, snfJ) = 0

'In

E

No·

But since fJ ..L SM, we also know that

i

1(()fJ(()(n dm(() = (sn I, fJ) = 0 'In E N.

The previous two equations say that all of the Fourier coefficients of 119 vanish and so 119 = 0 almost everywhere on 1l'. But we have already shown that IfJl = 1 almost everywhere on 1l' and so I == o. 0 8.1 2. (1) The key to proving Beurling's theorem is the fact that the invariant subspace generated by M n (SM)!. 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 S-invariant subspaces of some other Banach spaces of analytic functions. (2) There is a Beurling theorem for the HP spaces [65, 19]: suppose 0 < p < 00 and M is a non-zero subspace of HP. Then M is S-invariant if and only if M = fJ HP for some inner function fJ.

REMARK

8.2. A theorem of Douglas, Shapiro, and Shields Using the inner product

(f,g)

= [ Igdm =

JT

f

j(n)g(n),

n=O

and the definitions of the forward and backward shifts on H2, one can quickly check that (S/, g) = (f, Bg) '11,g E H2,

82

A THEOREM OF DOUGLAS, SHAPIRO, AND SHIELDS

181

and so S*, the Hilbert space adjoint of S, is equal to B. Usmg basic properties of annihilators and adjoints, one can also check that

SM

c M {:} BM..L

C

M..L

for any subspace M c H2. Thus, via Beurling's theorem (Theorem 8.1.1), every S*-invariant subspace of H2 takes the form ('I9H2)..L for some inner function '19. The following theorem of Douglas, Shapiro, and Shields [64J describes ('I9H2)..L. THEOREM 82.1 (Douglas-Shapiro-Shields). For f E H2, the follounng are equwalent (1) f E ('I9H2)..L. (2) The boundary functwn ( 1--+ fee) belongs to H2('JI') n 'I9H8('JI') , ~.e., there zs agE H2 such that f = 'I9(g m-a.e. on 'JI'. (3) The meromorphzc functwn f /'19 on D has a pseudocontznuatwn1 to a functwn f,o E H2(De) unth f,o(oo) = O. PROOF We will first prove the statement

('I9H2)..L = H2 n'l9H8. Indeed If f E H2 n'l9H6, then f = {iii almost everywhere on 'JI' for some h E H6. Thus for every 9 E H2,

(j,'I9g) = j('I9h)(Dg)dm = j hgdm = hg(O) = 0 smce gh E HJ and '19 is an inner function. Hence f E ('I9H2)..L. Conversely, If f belongs to

(V {z 'l9: kENo})..L,

('I9H2)..L = then

k

j f1J(

-k

This implies that

dm = 0 Vk E No.

/19 E H6 and so f

E 1JH6. Hence

(1JH2)..L = H2 n 1JH6 To finish, we must show that f E H2 n '19 H6 if and only If f j '19 has a pseudocontinuation belonging to H2(De) whIch vanishes at infinity. If f E H2 n'l9HJ then f = 'I9h almost everywhere on 'JI' for some h E H5 Note that h(Ijz) E H2(De) and vanishes at infimty. Thus If

f,n(z)

:=

h(I/z),

Z E

De,

then (J /'I9)(() = ftJ«() almost everywhere on 'JI' (the non-tangentIal limit functions are equal almost everywhere) and ff) E H2(De) and vanishes at infinity. Conversely if f N has a pseudocontinuation ff) E H2(De) with ftJ(oo) = 0, then the function

h(z) := ftJ(I/z) IThe meromorphlc functions 9 (on D) and G (on De) are pseudoconhnuatwns of each other If the non-tangential lImits of these two functiOns eXist and are equal almost everywhere on T See [44, 64, 153, 118, 119] for more on pseudocontmuatIons

8 THE BACKWARD SHIFT ON H2

182

belongs to

H6.

Hence

f

= fJh

almost everywhere on 'll' and this completes the proof.

D

REMARK 8.2.2. Using a slightly different proof [44, p. 87], one can even find an integral formula for the pseudocontinuation fn of f IfJ. Indeed

/iJ(>') = ->.

f

J.r

>. E IDle.

f«()fJ«() dm«(), z->.

We leave it as a simple exercise to the reader to show the following. PROPOSITION

8.2.3.

For our inner function fJ, let

u(fJ) :=

{>. E IDl- : lim IfJ(z)1

=

Z--+A

o}

be the spectrum of fJ. If fJ = bsp., where b is a Blaschke product with zeros and sp. is the singular inner factor with positive singular measure J.L, then

(an)n~l

u(fJ) = (an)~~l U supp(J.L). We can define fJ to be meromorphic on IDle by -

1

fJ(z) := fJ(llz) '

z E IDle,

-

-

and notice that fJ has poles at the reflected zeros of fJ. Moreover, fJ and fJ are pseudo continuations of each other Using a version of Morera's theorem [79, p. 95], J is an analytic continuation of fJ across 'll' \ u(fJ) as long as u(fJ) does not contain all of'll'. It turns out that all functions in (fJH2)~ enjoy this same property [64] or [44, p. 84]. THEOREM 8.2.4. If f E (fJH2)~, then f has an analytic continuation across 'll' \ u( fJ) to a function analytic on the set

1

{z E IDle: liz Ii u(fJ)}. Furthermore, for>. in this set, the linear functional f !--+ le>.) is continuous on (fJH2)~. Moreover, if Z E IDle with liz E u(fJ), then has at pole at z whose order does not exceed the order of the pole of fJ at z.

1

For any 0 < P < 00, the backward shift operator B on HP is continuous. When 1 < p < 00, we can use duality to characterize the B-invariant subspaces of HP. Indeed, by Corollary 3.6.3, the dual of HP can be identified with Hq, where lip + 11q = 1, via the Cauchy pairing

h

f9dm .

8 2 A THEOREM OF DOUGLAS, SHAPIRO, AND SHIELDS

183

Thus ifM is a B-invariant subspace of HP, then its annihilator M..lIS an S-invariant subspace of Hq and, VIa Beurling's theorem, takes the form fJHq. Using the HahnBanach separatIOn theorem we conclude that

M = (fJHq)..l

=

{f E HP '!rfgfJdm = O\:lg E Hq}.

The following theorem is the Douglas-Shapiro-Shields theorem in the HP settmg. THEOREM 8.25 (Douglas-Shapiro-Shields) For 1 < p < 00 and f E HP, the followmg are equwalent: (1) f E (fJHq)..l (2) The boundary functwn ( ~ f(C) belongs to HP(,][,) n fJHC(']['), z e., there zs agE HP such that f = fJ(g m-a.e. on 1['. (3) The meromorph~c functwn f IfJ on D has a pseudocontmuatwn to a functzon ff) E HP(D e ) wzth ff)(oo) = O. REMARK 8.26. To avoid having to make reference to the conJugate index q, we will use the notation

fJ*(HP) to denote (fJHq)..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 B-invanant subspace of HI takes the form fJ*(HI) := HI n fJ HJ 2but the proof is more complicated due to some technical difficulties when dealing with the functIOns of bounded mean oscillation and the fact that HI is not reflexive. The second fact to mention is that when 0 < p < 1, not all B-invariant subspaces take the form HPnfJHC. A complete description of the B-mvariant 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.

8.2.7. If 1 ~ p < 00 and f E fJ*(HP), then f has an analytzc contmuatzon across 1[' \ O"(fJ) to a functzon analytzc on the set THEOREM

1

{z

E

De . liz ~ O"(fJ)}.

Furthermore, for>. m th~ set, the lmear functzonal f ~ 1.) zs contmuous on fJ*(HP). Moreover, zf Z E De wzth liz E O"(fJ), then has at pole at z whose order does not exceed the order of the pole of fJ at z

1

Finally, there is the following duality result [44, p 109]. PROPOSITION

8.28. For 1 < p
. E JI)), one can show by direct computation that (I - >'8*)-1 exists and is given by the formula (I - >'8*)-1 f

=

zf - >.f(>.) .

Z-A

Thus u(8*), the spectrum3 of 8* is contained in

JI))-.

Furthermore, since, for any

>. E JI)), 8* _1_

= >._1_

1->'z

1->.z'

then ker(>.I - 8*)

"# {O}.

Hence u(8*) = JI))-.

If fJ is an inner function and T(fJ) := 8*lfJ*(H2),

what is u(T( fJ))? First notice that if >. E u( fJ) n JI)), that is to say, fJ(X) (1

~ >.z' zkfJ) = >.kfJ(X) = 0

= 0, then

Vk E No·

But since the closed linear span of {zkfJ : kENo} is equal to fJH 2 (Theorem 8.1.1), we conclude that

However, from above, T(fJ)_I_ = 8*_1_ = >._1_ 1 - >.z 1 - >.Z 1 - >.Z

and thus u(fJ) c u(T(fJ)).

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 fJ and T(fJ) := 8*lfJ*(H2), we have u(T( fJ))

= u( fJ)

Moreover,

(1) >. E u(T(fJ)) n JI)) if and only if (1 - >.z)-l E fJ*(H2). (2) If ( E 'll', then ( ¢ u(T(fJ)) if and only if there is an open neighborhood U of ( such that every f E fJ*(H2) has an analytic continuation to U. 3See 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 ¢ T, and (I - AT('I?))-l exists, then

(I - AT('I?))-l f

zf - AC).(f), f Z-A for some complex number c).(f). If AE ][), then c).(f) then =

E

'I? * (H2),

=

f(A) as above. If IAI > 1,

This says that

c)'(f) =

\Z~A''I?) / \Z~A''I?) = \Z~A''I?) /

-19(1/1).

It turns out that the function A 1---+ c)'(f) on ][)e (minus some poles) is a pseudocontmuation of f. We refer the reader to [17, 44, 178, 179] where thls technique was used to explore the spectral properties of the backward shift on various Banach spaces of analytic functions on ][).4 8.4. Kernel functions Let Pi} denote the orthogonal projection of H2 onto 'I?*(H2) and let 9

1---+

g+ = K(gdm)

be the Riesz projection operator from L2 onto H2. PROPOSITION 8.4.1.

For f E H2,

Pi}f = f - 'I?(19f)+. PROOF.

Let Wf := f - 'I?(19f)+ and observe that Wf E 'I?*(H2). Indeed, for

any n E No,

(f - 'I?(19f)+, zn'l?} = (f, zn'l?) - ('I?(19f) + , zn'l?) = (f,zn'l?) - (19f)+,zn) = (f, zn'l?) - (19 f, zn) = (f, zn'l?) -

(I, zn'l?)

=0 and so Wf..l 'l?H2 (since {zn'l?: n E No} has dense linear span m 'l?H2). Clearly

f - Wf = 'I?(19f)+

E

'l?H2 = ('I?*(H2)).L

and so (see [49, p 9]) W must be the desired orthogonal projectlOn PfJ

0

For each A E ][), define 1

C),(z) = --_-, z E][), 1- AZ and note that by the Cauchy mtegral formula, f(A)

=

(I, C),} "If

E

H2.

Hence, the Cauchy kernels {C). . A E ][)} are called the reproducmg kernels for H2 4rhe c).{f) that anse from pseudocontmuatlOn

variOUS

Banach spaces of analytic functIons mayor may not be a

186

8 THE BACKWARD SHIFT

ON

H2

Since PiJ is the orthogonal projection onto the S* -invariant subspace '19* (H2), we have

1(>") = (PiJI, C>.) = (I, PiJC>.) for every I E 'I9*(H2) and>" E

Thus the functions

JD).

k>.(z) .= (PiJC>.)lz) are the reproducing kernels for 'I9*(H 2). Here is an explicit fornmla for k>.. PROPOSITION 8.4.2.

~'I9(z)

k>.(z) = 1 -

1- >..z

PROOF. Using Proposition 8.4.1, it suffices to show that

(z) = 'I9(~ (19~) l->"z + l->"z Indeed,

(z) = r 'I9(~ ~dm«() (19~) 1 - >..z + J.r 1 - >..( 1 - (z =

rf

= fxn n=O

=

'19«(1

JT n=O 1- (z

xn(ndm«()

1

'I9«()(n ftzkdm«()

l'

fxnfzk n=O

k=O

00

00

k=O

1

'I9«()(n- k dm«()

l'

L:xnL:zkJ'(n-k)

=

n=O

k=n

'I9(>"} 1-Xz D

8.5. A density theorem Recall that A, the disk algebra, is the space of continuous functions on JD)which are analytic on JD). The norm on this space is, via the maximum modulus theorem,

11/1100

=

sup{I/«()1 : ( E '1l'}

and we can identify A, by means of its boundary values and the F. and M. Riesz theorem, with the space

{f

E

C('1l'): fc-n) = OVn E N}

In several of his papers, Aleksandrov makes use of the following interesting density theorem for '19* (H2)

8 5 A DENSITY THEOREM THEOREM

8.5.1. If {}

'tS

znner and {}*(A)

then {}*(A)

'tS

187

.= An {}*(H 2 ),

a dense subset of {}*(H2)

8.52 (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 54.5 (Aleksandrov's characterization of Cauchy transforms), Theorem 6.5.1 (the !7-property for Cauchy transforms), and Theorem 7.4.4 (the distribution theorem for Cauchy transforms). (2) When {} has no singular inner factor (i.e., {} is a Blaschke product), then

REMARK

:,x E JD),{}(,x) = o} c {}*(A), {~ 1- ,xz

(8.5.3)

since (1- XZ)-1 = k,\(z) E {}*(H 2 ), and one can use duality to check that this set of kernels has dense hnear span in {}*(H2). (3) When {} has a singular inner factor, the set contamment m eq (8.5 3) is still valId although the kernels do not have dense hnear span in {}*(H2). What makes Theorem 85.1 remarkable 18 that {}*(A) is still dense in {}*(H2). (4) Even more remarkable is the case where {} 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 Hoc n{}*(H2) contaIns {s*n{} . n E N}

which has dense linear span in D*(H2) (PropositIOn 8.2.3). (6) The papers [114, 192] have further informatIOn about whether or not {}*(H2) contains functions from varIOUS smoothness classes Recall from Chapter 4 the space of Cauchy transforms X = {f = K J.L . J.L E M} with norm Ilfll = inf{IIJ.L1i : J.L E Rf}, where Rf = {J.L EM: f = KJ.L} is the set of measures that represent a particular Cauchy transform. By identIfying the dual of A with X via the Cauchy pairing

(g,KJ.L)

= j9dJ.L = r->llim ~g(n)fi(n)rn, L...J

9 E A,

J.L E M,

n=O

recall that X can be endowed with a weak-* topology. For an inner function {}, let

D(X) := {f EX: f /D E N+} and recall the !7-property from Theorem 65.1: if moreover, II! /DII ~ II!II· PROPOSITION

8.5 4. The space D(X)

'tS

f

E

D(X), then! /D E X and

weak-* closed zn X.

8

188

THE BACKWARD SHIFT ON

H2

PROOF. As a consequence of the Krein-Smulian theorem [49, p 165], it suffices to show that 'I?(X) is weak-* sequentially closed, i.e., if (fn)n~l is a sequence in 'I?(X) such that In ---+ I weak-*, then I E 'I?(X). By Proposition 4.2.5, In ---+ I weak-* if and only if II/nll ~ L for all n and In ---+ I pointwise on ID>. Notice from the !f-property that Il/n/'l?lI ~ L for all n. Furthermore, In/'I? ---+ I/'I? pointwise on ID> and so I/'I? E X (Banach-Alaoglu). Hence I E 'I?(X) as desired. D For an inner function 'I? , let

An := 'I?(Xh = {f E A: (f,g) = 0 Vg E 'I?(X)} be the pre-annihilator of 'I?(X). PROPOSITION 8.5.5. With the above notation, we have the lollowing: (1) An is a norm closed subspace 01 A.

(2) An C 'I?*(A). PROOF. The first item follows from the fact that pre-annihilators are norm closed subspaces. Since H2 C X, we know that 'l?H2 C 'I?(X) and so

An = 'I?(Xh c ('I?H 2h = 'I?*(H2) n A = 'I?*(A). D

This proves the second item. The Hahn-Banach separation theorem says that (A'!9)1. = 'I?(X) from which

'I?* (A)1. c (An)1. = 'I?(X). This yields the following corollary. COROLLARY 8.5.6. II>' E M such that K>' E 'I? * (A)1. , then K>./'I? E N+. Let

'I?*(Hoo)

:=

'I? * (H2)

n H oo

closed5 subspace of Hoo.

note that 'I?* (Hoo) is a weak-* To see this, it suffices, via the Krein-Smulian theorem [49, p. 165], to show that 'I?*(Hoo) is weak-* sequentially closed Thus if (fn)n~l C 'I?*(Hoo) with In ---+ I weak-* in Hoo, then In ---+ I weakly in H2 But 'I? * (H2) is norm closed and hence weakly closed [49, p. 129] and so I E 'I?*(H2). Hence I E H oo n 'I?*(H2) = 'I?*(Hoo) This next theorem is the key to proving Theorem 8.5.1. THEOREM 8.5.7 (Aleksandrov [15]). 'I?*(A) is weak-* dense in 'I?*(Hoo). PROOF. For U C L oo , recall from Chapter 1 the pre-polar

°U =

{g E L1: If fgdml ~ 1 VI E U}

and for V C Ll, the polar

Vo

=

{I

E

L OO

:

If fgdml ~

1

Vg E V}.

If U1 C U2 C L oo , then

°U2

C

°U1

and (OU)O is the weak-* closure of the convex balanced hull of U 5Here the weak-* topology on Hoo is the restriction of the weak-* topology on Loo, given by the pairing f fgdm,f E LOO,g E L1

8 5 A DENSITY THEOREM

189

Since It suffices to show that

Hence it suffices to show that if 9 E Ll and satisfies

If

(8.5.8)

~1

jgdml

'ilf E ball (19* (A)) ,

then (8.5.9) For such a 9 satisfying eq.(8.5 8), we can use the Hahn-Banach extension theorem to produce a I-" E M with 111-"11 ~ 1 such that

f

jgdm =

f

jdl-" 'If E ball(19*(A))

6

Let .x E M be defined by d.x = dl-"- gdm

(8.5 10)

and note that K.x E (ball(19*(A)))l. and so by linearity, K.x E 19*(Ah. By Corollary 8.5 6, we know that K.xj19 E N+. For f E ball(19*(Hoo)), note that f = 19(h on 1I' for some h E ball(Hoo) and moreover, IIflloo = IIhlloo We let

(C.x)(z) :=

f ~d.x«(), l-(z

Izl =f 1

and define

G(z)

:= {

zh(z) 19tz) (C.x)(z),

Izl < 1;

f(l/z) (C.x)(z),

Izl > 1.

Smce K.xj19 E N+, then G is analytIc on ][) and, for 0 norm estimate

y} = {IK>"llfl > y} c {IK>"lllflloo > y} = {IK>"I > Yllfll~l}

and so

ym(IKvl > y)

~ Ilflloollf~oo m (IK>..I > lI~oo)'

Now we use the HruScev-Vinogradov asymptotic formula in Theorem 7.4.4, that is lim 7rtm(IK111 > t) =

t~oo

lI11sll, 11 E M,

to show that

However, by the definition of the measure>.. in eq.(8 5.10), we see that (8.5.13)

and so (8.5.14)

85 A

191

DENSITY THEOREM

It now follows that

117g

dm l = = = = =

111 17 dJ.£ -

dAj (byeq.(8.5.1O})

II f f I II 11~ -I :~ I II 1:~ -I :~ + I dvl 111:~ + I dvsj -dJ.£ dm - l-dA dm

(

(byeq.(8.5.13))

(byeq.(8.5.12» (by eq (8.5.11)

~ IIfilexo 1I:~1I1 + /lvsll

~ 11:~1I1 + IIvsll ~ II :~ 111 + IIJ.£s II

(by eq.(8.5.14))

~1

Thus we have shown eq.(8.5 9) and so the proof is now complete.

o

PROOF OF THEOREM 8.5.1. From ProposItion 84.2, the kernel functions for

1J*(H2) are k),(z) = 1 -

~1J(z) 1-Az

and moreover, k), E 1J*(HOO). If f E H2 annihilates 1J*(A), we can use the fact that 1J*(A) is weak-* dense in 1J*{HOO) (Theorem 8 5.7) to conclude that

However,

1 f,~) -1J(A) 1 f'~) \

1- Az

\

1- Az

f(A) -1J(A) 119f,~) \

1- Az

f()..) -1J(A)(19f)+(A). Smce (19f)+ E H2, then f E 1JH2. = (1J*{H2».i. An applicatIOn of the HahnBanach separatIOn theorem says that

The reverse inclusion is ObVIOUS.

o

192

8 THE BACKWARD SHIFT ON H2

8.6. A theorem of Ahern and Clark When ( E 'Jr \ 0"('19), Theorem 8.2.4 says that every f E 'I9*(H2) has an analytic continuation across ( and the functional f f-+ f«) is continuous on 'I9*(H2). Moreover, since '19 is analytic near (, we can use the identity

Iwl > 1 , =1 ¢ 0"('19), w

'I9(w) 1 - 'I9(ljw) ' to conclude that the kernel function

kd z ) = 1 -1?(()'I9(z) 1- (z belongs to H2. An application of the dominated convergence theorem will show that kr and a singular inner function 8"" then, for ( E 'll', the kernel kt; belongs to t?*(H2) if and only if (87.1)

(8.7.2)

~

l-Ian l

~ I( -

an l

(

1

2+ iT I( _ eI2dl-t(e) < 00.

Note that for kt; to be even defined, t?() (the non-tangential hmit of t? at () must actually exist The condition m eq.(8.7.2) guarantees this. For the second questIOn, Clark produces a unitary operator Uo : t?*(H 2) -+ t?*(H2), where a E 'll', such that whenever ( E 'll' satisfies the condition in eq (8 7.2) and t9(() = a, then kt; is an eigenvector for Uo 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 Uo has pure point spectrum (in that the spectrum of Uo consists solely of the eigenvalues of Uo , or equivalently, the spectral measure for Uo 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~1 must satisfy in order to span t?*(H2) (see [46] for details). 7It mIght be the case that l' C 0-(19), for example, when accumulate on all of l'

f}

IS a Blaschke product whose zeros

8 THE BACKWARD SHIFT

194

ON

H2

This kernel function approximation problem was also studied by Sarason [186] in the special case where -0 is the atomic inner function

-o(z) =exp ( -~~;). Here, by means of a certain unitary operator between L2 and -o*(H2), the kernel functions k>"n correspond to the functions ei "fn9 , where "In

.1

+ An

= z-----=.

1- An Sarason then uses the original Paley-Wiener theorem in eq.(8.7I) 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 -o*(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 -o*(H2). The 'Clark' measure aa E M will be the spectral measure for Ua in that Ua is unitarily equivalent to the operator 'multiplication by (' on L2(aa). The next two sections set up the compression as well as its unitary perturbation. The next chapter covers the Clark measure aa:. 8.8. The compression of the shift To avoid unnecessary technical details, we shall assume that

-0(0) = 0 for the rest of this chapter. With this technical assumption in force, let (8.8.1)

N:= -o*(H2)

eC~

= -0* (H2)

n ( c~) l.

Recall that -o*(H2) .= (-oH2)l.. L)!:MMA 8.8.2. A function 9 E H2 belongs to :N if and only if both 9 and zg belonf, to -o*(H 2).

PROOF. Suppose g, zg E -o*(H2). Then

0= (zg,-o) =

\g,~)

and so 9 1. C-o / z. Hence 9 E:N. Conversely, suppose that gEN. Then 9 1. zn-o for all n E No and 9 .1 -0/ z. This means that zg 1. zn-o for all n E No and it follows that zg.l -oH2 0 DEFINITION 8.8.3. Define the operator So on -0* (H2) to be the compression of S to -o*(H2), that is 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 (IITI! ~ 1) on a Hilbert space such that lim T m =0 n-+oo

88

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 umtarily equivalent to S{} for some inner function {}. See [153J for more. The followmg precise formula for S{}I will be Important. PROPOSITION 8.8.4. If {}(O)

= 0,

then

S{}I = z (I - (/,~)~) PROOF. Wnte

'if IE fJ*(H2).

I E {}* (H2) as

(8.8.5) Notice that h E 'N, since the operator I ~ (I, {} / z){} / z is the orthogonal projection of {}*(H 2) onto C{}/z, while h E C{}/z. Since zh E {}H2, then p(}(zh) = 0 By Lemma 8.8.2, zh E {}*(H2) and so p{}(zh) = zh and the proof is complete. 0 The model operator S{} turns out to be a cyclic operator wIth an easily identifiable cychc vector. We discuss this m the followmg two propositions. PROPOSITION 8.8.6. For each n E N,

s:; =

PDS n l{}*(H 2 ).

PROOF. Let /,g E {}*(H 2). Then, smce S*kg E {}*(H2) for all kEN, (S;/,g)

= «P-oS) (p-os)n-1 /,g) = (S(P-oSt-1/,g) = (p{}st-1/, S*g)

= (I,s*ng) = (sn I, g) = (p{}sn)/,g)·

o

The proposition now follows.

that

PROPOSITION 8.8 7. The constant functton X == 1 zs a cyclzc vecto.,s for S{}, tS to say, V{s:;x: n E No} = {}*(H 2 ).

PROOF. Suppose n E No,

I

E {}*(H2) and

I

1. S;X for all n E No. Then for each

0= (I,S;X> =

(I, p{)sn X } (by PropositIOn 8.8.6) (I, p-ozn)

=

(I, zn) (since I E {}*(H2»

=

This says that

I == 0 which establishes the proposition.

o

8There 18 a general result here f E 1J* (s') IS cyclic for S6 If and only If the Inner part of relatively prime to 1J [153, p 82J Also notice, smce 1J(O) = 0, that X E 1J*(H2) Indeed, for any n E No, (zn1J, X) = (z n1J)(O) = 0

f

IS

196

8

THE BACKWARD SHIFT ON

H2

Theorem 8.3.1 computes the spectrum of S*I19*(H2). By using the well-known fact that cr(A*) = cr(A) for a bounded operator A on a Hilbert space, as well as the observation that S~ = S*I19*(H2), we can compute the spectrum of Sf}. COROLLARY 8.8.8 For an inner function 19,

cr(SfJ) = cr(19). Moreover, if ( E 'll', then ( rt cr(Sf}) if and only if there is an open neighborhood U of ( such that every f E 19*(H2) has an analytic continuation to U. 8.9. Rank-one unitary perturbations DEFINITION 891. Recall from eq.(8.8 1) that

19*(H2) =:N EEl C~ z and so, for each a E 'll', we can define the linear operator Ua : 19*(H2) ~ 19*(H2) by Uag = zg for g E :N, 19 Ua(-)=a z Notice from Lemma 8.82 that zg E 19*(H2) whenever 9 E:N. The fact that constant function a belongs to 19* (H2) follows from our standing assumption that 19(0) = O. Using the decomposition in eq.(8.8.5) and Proposition 8.8.4, we observe that (8.9.2) where

Xaf := a \ f,

~) .

Thus Ua is a rank-one perturbation of Sf}. One can easily show that

X~f = 7i~(f, 1) z and so, since S~ = S*I19*(H 2),

U~f = S* f + 7i~(f, 1). z THEOREM 8.9.3. For each a E 'll', Ua is unitary. PROOF. Notice that

Ua = Sf}

+ Xa

and

U~

= S* + X~

Thus U~Ua

= S* S{} + X~S{} + S* Xa + X~Xa

Routine computations with the definitions of these operators show that for each f E 19*(H2),

S* S{}f = f f'\ \ f,

~) ~,

X:S,J! = S* Xaf = 0,

8 9 RANK-ONE UNITARY PERTURBATIONS

197

X~Xal = (I,~)~. Hence

U:XUa =

I. For the other dIrection,

S{}SJ + XaSJ + S{}X~ + XaX~ Again, str8J.ghtforward computations show that for each I E 19*(H2), UaU~ =

S{}SJI = I - f(O),

and so

o

It follows that Ua is unitary.

This next result says that {Ua : a E 'f} are all of the unitary rank one perturbations of S{} THEOREM 8.9.4 (Clark [46]). If 19(0) = 0 and U zs a rank-one perturbatwn of S{} that zs also unztary, then U = Ua for some a E 'f. PROOF. Before beginning the formal proof, we make two observations. The first is that (8.9.5) To see this, recall from Proposition 8.8.4 that

and so if

Ilfll = IIS{}III, then 111112 = IIS{}III 2 = =

=

Ik - (I,~) ~112 1(1, ~)( - 2iR(l, (I,~)~) 111112 -1(1, ~)r

111112 +

Thus 1.1 19/z. The second preliminary observation IS that

IISJIII

(8.9.6) Indeed, for

I

=

IIII!

=}

I .11.

E 19*(H2),

SJI = S* I = f - 1(0) . z

8 THE BACKWARD SHIFT ON H2

198

IISu/1l = 11111, then 11/112 = 118Ufll2 = III - 1{0}11 2 = 11/112 + I/(OW = 11/112 -1f(OW. Hence 0 = I(O} = (1,1).

Thus if

2SR(I, 1(0))

We now proceed to the formal proof. If U is a unitary rank one perturbation of S'I9, then there are vectors h, k so that

UI = 81)1 + (I,h)k and

IIUIII = 11111 = IIU* III for all I E fJ*(H2). The goal here is to show that h = fJ/z and k is a unimodular constant function. If 11.. h, then UI = 8'191 and so

11/11 = IIUIII = 118'19/11· Now bring in eq.(8.9.5) to conclude that (8.9.7)

1 ..L fJ/z. From this, we observe that

(Ch}-L C (

C~)

-L ,

where ..L is the annihilator in fJ*(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= Suppose that

I

fJ C1-' Z

..L k. Then U* I =

Sui + (I, k)h =

8~1

from which

11I11 = IIU* III = IISull1 and so, from eq (8.9.6), I ..L 1. An argument similar to the one used in eq.(8 9.7) shows that k = C21. Since fJ h = Cl - and k = c21, z then for all I E fJ*(H2),

UI=S1)I+\I,~)C for some complex number c. To finish, we need to show that Icl = 1 and thus U = Uc . One sees this by first observing that fJ(O) = 0 and so 1 E fJ*(H2). Hence U*1 = S*1

+ c~(1, 1) = c~. z

z

RANK-ONE UNITARY PERTURBATIONS

89

Fmally, since U

IS

199

umtary, we know that

1= 11111 = IIU*lll =

IIc~ I = 14 D

REMARK 8.9.8. (1) Poltoratski [164] examined the finite rank perturbations of Sa. (2) There is a generalization of the operators Sa and Uo: m [75, 76]. For ( E 11', Theorem 8.6 1 says that

ke;

E

{}*(H2) ~ W(()I
:= {J-ter : 0: E 'll'} denote the family of measures associated with the function 4>. We will call Act> the family of Clark measures of 4> when 4> is an inner function. When 4> is a general analytic self-map of the disk, we will call the family Act> the family Aleksandrov measures of 4>. 2 REMARK

9.1.4. If J-t is any measure in M+, the Herglotz transform of J-t,

(HJ-t)(z)

=

h~ ~;

dJ-t«(),

satisfies 1RHJ-t > O. One can use this, along with the fact that w-l WJ-+ - w+l maps {1Rw > O} onto [J), to verify that the function

(91.5)

4>(z)

:=

(HJ-t)(z) -1 (HJ-t)(z) + 1

is an analytic self map of the disk. A little algebra shows that

(HJ-t)(z) = 1 + 4>(z) 1 - 4>(z) and hence

(PJ-t)(z) = lR(HJ-t)(z) =

l-I4>(zW 11 - 4>(z)l2·

Thus J-t is an Aleksandrov measure for 4>, that is to say, J-t = J-t1 E Act>. If J-t is not only a positive measure but also singular with respect to m, we can use Theorem 1.8.6 to conclude that . . 1 -14>(r()1 2 dJ-t hm I 4>( ()12 = hm (PJ-t)(r() = -d r---+l- 1r r---+1m «() = 0 m-a.e. But since lim 11 - 4>(r() I > 0 r---+l-

for almost every ( E 'll' (Theorem 1.9.4), it must be the case that lim (1 -

r---+l-

14>(r()1) = 0 a.e.,

that is to say, 4> 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. If /Ler

E

A, then

{ ( + z d/Ler«() = 0: + 4>(z) + 2i

iT (- Z

0: - 4>(z)

~(a4>(O» .

10: - 4>(0)1 2

2The literature has a variety of notation for this Some call A<j> Clark measures (even if t/J is not inner) while others call A<j> Aleksandrov measures (even if t/J is inner) Still others compromise and call A<j> 'AC' measures

203

9 1 SOME BASIC FACTS ABOUT CLARK MEASURES

PROOF. For

f3 E T and w f3+w f3-w

E lIJ), the identity

!R (f3 + w)

f3-w

f3-w

2. ~(7Jw)

1 _JwJ2 =

+ i~ (f3 + w)

Jf3 - WJ2 + 'lJf3 - WJ2

yields the equations

a + ¢>(z) = 1-J¢>(z)J2 a - ¢>(z) Ja - ¢>(z)J2

+ 2i

~(a¢>(z)) Ja - ¢>(z)J2

and

f ~ + z dJ.ta«() = (PJ.ta)(Z) + i(QJ.ta)(Z).

iT" -z But since

1-J¢>(z)J2

(PJ.ta)(z) = Ja _ ¢>(z)J2' the functions (QJ.ta)(z) and

2 ~(a¢>(z)) Ja - ¢>(Z)J2 are harmonic conjugates of the same function and so ~(a¢>(z))

(QJ.ta)(z) = 2 Ja _ ¢>(z)J2

+ c.

Now plug in z = 0 to see that

c = -2 ~(a¢>(O)) Ja - ¢>(O) J2 and so

a + ¢>(z) a - ¢>(z)

!R (a + ¢>(Z)) + i~ (a + ¢>(Z)) a-¢>(z) a-¢>(z) . _ ~(a¢>(O)) (PJ.ta)(z) + z(QJ.ta)(z) - 2z Ja _ ¢>(0)J 2

f ( + z dJ.ta«() _ 2i

iT ( - z

~(a¢>(O)) .

Ja - ¢>(0)J 2

o There is also a nice formula for the Cauchy transform of an Aleksandrov measure. COROLLARY 9.1.7. If J.ta E A"" then

f ~ dJ.ta«() =

iT 1 -

(z

1 1 - a¢>(z)

+ JJJ.tall- 1 + i ~(a¢>~o)) 2

.

Ja - ¢> 0)12

PROOF. Use the identity

f3 + w = 2 1 f3-w 1-f3w and Proposition 9.1.6.

_ 1,

f3 E T, WE lIJ),

o

204

9 CLARK MEASURES

PROPOSITION

9.18. If J.Lo: E A4>, then

lIJ.Lo:lI

=

l-Iq,(OW la - q,(0) 12 .

Hence each J.Lo: is a probability measure whenever q,(0) = O. PROOF.

Since Uo: = PJ.Lo:, we have

uo:(O) =

f

Po dJ.Lo: =

f

dJ.Lo: = lIJ.Lo:lI·

On the other hand, by the definition of Uo: from eq.(9.1.3),

uo:(O)

=

1- 1q,(0)1 2 la - q,(O)l2·

o

Combine these two identities to prove the result. We can also compute the Fourier coefficients of an Aleksandrov measure. PROPOSITION

9.1.9. If q,(0)

= 0 and J.La

ta f k

E

A, then

q,«)k( dm«), n

~ 1;

k=l

1, PROOF.

n=O.

Proposition 9.1.6 says that whenever q,(0) = 0 and J.Lo: E A4>, we have 1 + ~q,(z) =

(9.1.10)

l-aq,(z)

f (+

Z dJ.La«). (-z

For zED and ( E 'lI.", observe that

(+z = 1 +2 (z_ (- z

1- (z

and so

Looking at the left-hand-side of eq.(9.1.lO), we start with the identity 1 + aq,(z) = 1 + 2 aq,(z) = 1 + 2 I:c?q,(z)n. 1 - aq,(z) 1 - aq,(z) n=l Bringing in our computation with the right-hand-side of eq.(9.1.lO), we get

f

dJ.La«() + 2

~ zn (f ( dJ.Lo:«)) =

1+2

~ c?q,(z)n

91

SOME BASIC FACTS ABOUT CLARK MEASURES

205

By Proposition 9.1.8, IIJLall = 1 and so

~zn (f(d JLa ((») = ~anq>(Z)n.

(9.1.11)

Applying the Cauchy integral formula to both sides of eq.(9.1.11), we get, for each 0 < r < 1, (9.1.12)

f(

dJLa(()

=

r

~akq>(w)k dw.

1111w 1~ - r k=l

=

27rt

wn+l

~ak

~ k=l

r

111'w 1_ _r

q>(w)k dw... w n + 1 27rt

Since q>(0) = 0, the function

q>(W)k w n +1

w~-­

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

f

(dJLa(() = t ak

hq>~~k

dm((),

n E N.

Take complex conjugates of both sides of this identity to handle the case when

0

n(()1 2

h a (() =

Ill! -

q>(()12

PROOF. Use Fatou's theorem (Theorem 1.8.6) to see that for m-a.e. ( E T,

.

1 - jq>(r()j2 q>( r ()12 lim (PJLa)(r() .

lIm ua(r() = r-+llIm r-+l-

IIl! -

= r-+l= (DJLa)(() = ha (().

o We can use the fact that (Dv)(() = 0 m-almost everywhere implies v ..L m, along with Proposition 9.1.14, to prove the following corollary. COROLLARY 9.1.15. If q> is an inner /unction, then JLa 1. m for every JLa E A.p. From Proposition 1.3.11 we deduce the following. PROPOSITION 9.1.16. The Borel set {I!.JLa = oo} is a carrier for O"a.

9 CLARK MEASURES

206

It will be important for what follows to consider sets of the form ¢-1 (B), where B is a Borel subset of T. Since ¢«) is defined by means of its radial boundary function ¢(():= lim ¢( r(), r-+1for m-a.e. ( E T (and not necessarily every ( E T), we need to be more specific about what we mean by ¢-l(B). Of course, we can say that ¢-l(B) is defined to be the set of ( E T for which ¢«) exists and belongs to B. However, since we will also be considering Borel measures on such sets ¢-l(B), we want to guarantee that ¢-l(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 ¢ : II)) which lim ¢(r() r-+lexists is an Fuo set.

--+ II)),

the set of points ( E T for

PROOF. Let (En)n~l be a sequence of positive numbers with En --+ 0 and be a sequence in (0,1) with rn --+ 1. For each k, n E N let F(Ek' rn) be the set of points ( E T for which (rn)n~l

1¢(Sl() -

I

¢(S2() =:::; Ek

for all

Sl

>

S2

> rn

and notice by continuity that F( Ek, r n) is a closed set. The set for which the radial limit of ¢ exists is the Fuo set

nU 00

00

F(Ek,rn).

k=ln=l

o Let G be the set of points ( E T for which lim ¢(rc;,) r-+lexists and observe from Proposition 9.1.17 that G is a Borel set. With ¢r«) := ¢(r(), the functions Xa¢r are certainly Borel functions and so the limit function

¢*«):= lim Xa«)¢r«) r-+l-

is a Borel function. The above argument proves the following result. PROPOSITION 9.1.18. If B is a Borel subset oflI))-, the set (¢*)-l(B) is a Borel subset of T

By the above proposition, the sets (9.1.19) are Borel subsets of T and moreover, for a Borel set BeT, (9.1.20)

(¢*)-l(B) =

U Eo oEB

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 {¢> = a} for Eo; and ¢> -1 (G) for (¢>*) -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

l-I¢>(z)12 uo;(z) = 100 _ ¢>(z)12 PROPOSITION

9 1.22. The set of (

E

1l' for which

lim uo;(r()

= 00

r--+ 1-

is a Borel set. PROOF.

For MEN and (rn)n;;'1 C (0,1) with rn ~ 1, let F(M,r n ):= {( E 1l'. uo;(rn() ~ M}

and notice this set is closed. The set of ( E 1l' for which the radial limit is equal to infinity is

nun F(M, rn). 00

00

00

M=1 k=1 n=k

The sets {uo;

=

o

oo} and Eo; (from eq.(9.1.19)) are Borel sets and

{uo;=OO} cEo;.

(9.123)

For f-lo; E Aq" we see from eq.(1.8.8) that {Df-lo;

= oo} C {uo; = oo}.

However, these sets need not be equal [118, p. 13]. In summary, we have the following string of containments

{I2po; = oo} C {uo; = oo}

c Eo;

that yield the following useful corollary. 9.1.24. (1) For f-lo; E Aq" the Borel sets

COROLLARY

{I2pa

= oo}

C {un:

= oo}

C Ea

are all carriers for a a' (2) aa .1 af3 for a =I- (3. (3) For aa-a.e. (E 1l', ¢>(() = oo.

An obvious observation from this corollary is that if ¢> has no radial limits of unit modulus (e.g., ¢> maps ~ to a compact subset of ~3), then f-ln: «m. Let us end this section with a comment about the absolutely continuous part of f-la, that is,

l-I¢>(()1 2

ha dm

= la _ ¢>(()12 dm.

3With somewhat more work, one can also construct an analytic self-map such that (lJ}) = lJ} but has no radial limits if modulus one

9 CLARK MEASURES

208

Since h;I({O}) = 14>I-I({I}) m-a.e., we know that for any ex, (3 E 'll', the measures hadm and h{3dm are mutually absolutely continuous. 9.2. Angular derivatives and point masses

In this section, we examine the points of 'll' where an Aleksandrov measure JLa 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 JLa E At/> and (E 'll', then JLo:({(}) if and only if L lim 4>(z} = ex and 14>'«(}1 < 00.

>0

z---+(

Furthermore, in this case, 1 JLo:({(}) = 14>'«()I'

PROOF. We will make the simplifying assumption that 4>(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

For fixed ( E

J J

ex + 4>(z) = ~ + z dJLa(~). ex - 4>(z) ~- z 'll', multiply both sides of this formula by « - z) to get

(9.2.2)

(-z

(ex + 4>(z)) ex _ 4>(z)

Notice how for any

~ E

=

(-z

(~+ z) ~ _ z dJ-La(~)'

'll' and for any z in a fixed Stoltz domain with vertex at (,

~ K=.:.l ~ c /(-z/ ~ - z "" l-lzl ~ for some c > 0 which only depends on the opening of the Stoltz domain. Since

Llim(~+z)(-z={ 2(, if~=~; ~-

z--.(

z

0,

otherWIse,

we can apply the Lebesgue dominated convergence theorem to obtain L lim

z---+(

J(~ + z);.. -- zz dJLa(~) =

2(J-Lo:( {(}).

Now apply eq.(9.2.2) to see that

(~~) =2(J-Lo:({(}). ex - z We are now ready for the proof. If JLo:({(}) > 0, we use eq.(9.2.3) to see that

(9.2.3)

Llim(ex+4>(z)) z---+(

L lim 4>(z) = ex z---+(

and 4>'«()

= L lim 4>(z) - ex = z---+(

Hence 14>'«()I
'«()1
'«()

=f. O. If we

L lim 4>(z) = a,

z--+,

we can use eq.(9.2.3) again to conclude that 1

o i- 4>'«()

a(/La({(})'

=

o 9.2.4. (1) If 4> is any self map of the disk, we know that (/La)d, the discrete part4 of /La must take the form

EXAMPLE

I:

(/La)d =

c,8,.

p.a({W>o

(9.2.5)

By what was said above in Theorem 9.2.1, 1

(/La)d

I:

=

W«()1 8"

'EEa\E

where E = {WI = co}. Notice, since the measure is discrete, how the sum in eq.(9 2.5) is at most a countable one. This implies the following

4> of Jl} and any a a, 14>'«()! < co}

interesting fact: for any analytic self map

Ea \ E = {(

E

4>«() =

'lI"

E

'lI', the set

is at most countable. This was observed before in [52, Thm. 8.1] and [194, p. 385]. (2) If 4> is a finite Blaschke product, then 4> is inner and so /La = era' Moreover, a carrier for era is Ea = {4> = o:} which must be a finite set {(t, ... ,(n} since 4> has an analytic continuation across 'lI'. Thus, from eq.(9.2.5) and the fact that angular derivative at (j is equal to the ordinary derivative at (j, we have n

/La =

1

?: 14>'«(_)1 8,;-

3=1

3

(3) If 4> is the atomic inner function

4>( z) = exp (~ ~ ;) , then /La = era' Moreover, since 4> has an analytic continuation across 'lI' \ {I}, the set Ea = {4> = o:} is countable and clusters only at the point ( = 1. Thus the measure /La = era is discrete and, since the angular derivative is the ordinary derivative at all points of 'lI' \ {I}, we observe that /La ({(})= 1(~112, (EEa. Thus

_ L...,; '" I( -2 112 8,.

/La -

'EE",

4See Theorem 1 3 13 for precisely what we mean by the discrete part of a measure

210

MEASURES

9 CLARK

We end this section with a result of M. Riesz [175] that relates Aleksandrov measures with the existence of angular derivatives. ~ l\)

THEOREM 9.2.6 (M. Riesz). Let 4> : l\) (1) If there is a f3 E 1l' such that

J _ (1

d~,8(~)

I~

then 14>'«()I (2) If

< 00,

2

< 00. and 14>'«()I
(z) = 0:

z-+
'«()I
(r(W.

(92.7) Since

be analytic and ( E 1l'.

is an Aleksandrov measure for 4>, we have 1 -14>(r()1 2 1 - r2 10: _ 4>(r() 12 = Ie _ r(12 d~o:(e)·

J

f3 E 1l' \ {o:}

J

and observe that

d~i1(e)

.

I~ _ (12 ::;; r~~.

=

r~~

J

d~,8(e) , Ie - r(12 (Fatou s lemma)

1 1 - 14>(r() 12 1 - r 2 1f3 _ 4>(r()12 (from eq.(9 2.8)

14>'«()1

= 1f3 _ 0:12


(r()1 2

1f3 _ 4>(r() 12

2 ::;; C(I- r ).

The above inequality says that 4>( ()

=

lim 4>(r()

r-+l-

Ie

(12

d~,8(e)

9 3 ALEKSANDROV'S DISINTEGRATION THEOREM

211

exists and 14>«() 1= 1. Since lim 1 - 14>(Z)1 ~ lim 1 - 14>(r()! , 1 - Izl """ r-+l1- r

z-+,

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 1-14>(r()11 + 14>(r() I 1 + r 1-r 1+14>(r()11+r

1-14>(r()1 1-r

~

21 - 14>(r()1 2 1-r2

= 21 - 14>(r()1 2 1/1- 4>(r()1 2

1/1- 4>(r() 12 2 ~ 21 -14>(r()1 (1 + 14>(r()J)2 """ 1 - r2 1/1- 4>(r()1 2 1- r2

~ 81 - 14>(r()!2 1 """ 1 - r2 1/1- 4>(r()1 2 ~ C by eq.(9.2 9).

Now take a lim as r

->

1 to see that 14>'«()1
(z) = 0

and /

ti~~~


fh

H2 .

a computation reveals that

Thus

o COROLLARY 9.5.2. II t?(0) = 0 and A E D, then

V;C a

A-I _

1

at?(A)

k

A·

PROOF. Recall that

(Vag)(z) = (1 - at?(z»K(gd.. to a constant multiple of k>.. Moreover, if n

g= LcJC>"J J=1 IS

a finite linear combination of the C>.. 's, then

Iiglii2(O"a) =

(g, g) £2(0"",) n

= L

cJc/(C>"J,C>..,h2(O"a)

J,I=1

=

t

cJCi (

k>"J

,

k>..,

)

(Lemma 9 5.1)

1 - a:D(AJ ) 1 - a:D(AI) H2

J,I=l n

= L

cJCi{VaC>"J' VaC>../)H2 (Corollary 9.52)

J,I=l

= {Vag, Vag} H2 =

IlVagIi1f2·

Thus Va is an isometry on the linear manifold of L2(O'a) generated by the C>.. to the linear manifold of D*(H2) generated by the kernels k>... The latter manifold is dense in D*(H2). LEMMA 9.5.3. The lmear mamlold generated by {C>.. : A E D} zs dense m L2(O'a). PROOF. Suppose that 9 E L2(O'a) and (g, C>..) £2 (O"a) = 0 VA E D. Working out the inner product says that the Cauchy transform K(gdO'a) 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 0 by {C>.. : A E D} is dense in L2(O'oJ We leave it to the reader to fill in the details to prove the followmg. PROPOSITION 9.5.4. II D(O) = 0, the operator Va L2(O'a) onto D*(H2).

~s

a umtary operator from

We are now ready for Clark's theorem. Recall the bounded multiplication operator Z : L2(O'a) --+ L2(O'a), (Zg)«() = (g() and notice that (Z*g)() = (g«). THEOREM 9.5.5 (Clark). When D(O) = 0, VaZ = UaVa. PROOF. It suffices to prove that

221

9 6 SOME REMARKS ON PURE POINT SPECTRA

For an analytic function h on ID> (it need not belong to any special class of functions), let Bh := h - h(O) .

z Recall that S* is the adjoint of the forward shift (Sf)(z)

= zf(z)

on H2 and that

S* = BIH2. For 9 E Ll(aoJ, let us use the shorthand Ko.g.= K(gdao.). A routine computation shows that

B(hh) = hBh + h(O)Bh

(9.5.6) and (9.5.7)

Also remember that t?( 0) = 0 and so

Bt?

=~. z

Thus for 9 E L2(ao.),

(Vo.Z*g)(z)

= (1- at?(z))(Ko.Z*g)(z) = (1- at?(z))(BKo.g)(z) (eq.(9.5.7))

= B«1 -

at?(z))Ko.g)(z) + aBt?(g, l)p(uQ) (eq.(9.5.7))

= (S*Vo.g)(z) + a(g, Ih2(uQ)~ = (S*Vo.g)(z)

+ a(Vo.g, Vo.l)H2~

= (S*Vo.g)(z)

+ a(Vo.g, I}H2~z

z

(Proposition 9.54)

(note Vo.l = 1).

Now use the function 9 = V'; f for an f E t?*(H2) in the above identity to see that

Vo.Z*V'; f =

s" f + aU, 1) H2 ~z = U~f·

o A nice identity relating Clark measures, unitary perturbations, and Cauchy transforms is the following formula. COROLLARY

9.5.8. Suppose t?(O) = O. Then for z E ID> and a E 'll',

(1 - zU~)-11, I)

=

(KIJo.)(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~1 c ID> with IAnl -+ 1, does the sequence of kernel functions (k'>'n)n~1 have dense linear span in t?*(H2)? As dIscussed earlier, the approach one takes to this problem is to observe from Theorem 8.6.1 that for (E'll',

222

9 CLARK MEASURES

Moreover, from Theorem 8.9.9,

{k,: 1'!9'«)I < 00,'!9«)

=

a}

are all the eigenvectors for Uo. But since UOI is unitary, we know that

k,

1-

ke, (# {,

From our discussions in Chapter 8, we saw that if

{k, : 1'!9'«)I < 00, '!9«() = a} has dense linear span in '!9* (H2), we could apply a Paley-Wiener type approximation theorem to our onginal problem By Theorem 9.5.5, the Clark measure 0'0. is a spectral measure for U0.. If 0'0. happens to be discrete, the spectral theorem says that Uo. would have pure point spectrum (see Definition 1.5.8) in that the eIgenvectors k, form a spanning set for '!9*(H2) 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 '!9(0) = 0 and the set {( E '][':

1'!9'«)I = oo}

has Lebesgue measure zero. Then for m-a.e. a E '][', the Clark measure 0'0. zs dzscrete and so U0. has pure pomt spectrum. There are examples of families of Clark measures AD for which measure but 0'0. is singular continuous for all a E 'lr \ {I} [62].

0'1

is a discrete

9.7. Poltoratski's distribution theorem

We know from Theorem 7.4.4 that for any I-' EM, lim

y-+oo

7rym(IKI-'I > y) = III-'sll·

In this section we use Aleksandrov measures to prove Poltoratski's generalization of this result [163]. THEOREM 97.1 (Poltoratski). If J.L E M, then 7rYXIKI-'I>y . m -+ II-'sl weak-* as y -+ 00

The key to provmg this is the following THEOREM 9.7.2. If I-' E M+, then 7rYXQI-'>Y . m -+ J.Ls 7rYXQI-'y m -+ 21-'8

weak-* as y

-+

00

Before getting to the proof of Theorem 9.7.2, let us show how one can prove Theorem 9.7.1 from Theorem 9.72 at least for I-' E M+ 7. We already know that

2(KJ.L)(z) = (PJ.L)(z)

+ z(QJ.L)(z) + 1,

z E Jl))

7To aVOId some techmcal detaIls, we wIll only prove PoltoratskI's theorem for POSItive

223

97 POLTORATSKI'S DISTRIBUTION THEOREM

Taking non-tangential limits we get

2(K/L)«() = g«() + 1,(Q/L) «(),

m-a.e. ( E 'll',

where

g«()

d/L

= dm «()

+ l.

Since /L E M+, 9 and Q/L are real-valued and so

IQ/LI ::::; 21K/L1 ::::; Igl + IQ/LI m-a.e. Thus for y > 0 and

I

I ;;:: 0, we get

E C('ll') with

rry f

Idm::::; 2rr'#.. f

Idm.

2 J1KI-'I>'Y/2 By assummg the conclusion of Theorem 9.72, we know that J1QI-'I>'Y

lim 'Try f

J1QI-'I>Y

y-+oo

Idm = 2j l d/LS,

and so lim 'TrU f I dm. j l d/Ls::::; u--+oo J1K,..I>u

(9.7.3) For any

f

>0 {2I K/L1 > y} c {igi > fY}

U{IQ/LI > (1 - f)Y}

I E C('ll'), I;;:: 0,

and so for any

~f

J2IKI-'I>Y

I~::::;~f

J1g1>ey

f

= 'Try

I~+~f

J1QI-'I>(1-e)y

I

J1g1>EY

I~

dm + _l_'Try(1 - f) f I dm. 1- f J/Q,..I>(l-e)y

Since

m(lgl > fY)

= o(l/y),

8it follows that

2 lim 'TrU f u-+oo

and hence, letting

f ~

J1KI-'I>u

Idm::::; - 21 jld/Ls -

f

0, lim 'Try y--+oo

f

J1K,..I>y

Idm::::; jld/Ls,

Combming this inequality with the one in eq.(9.7.3) yields (9.74)

lim 'Try f

y--+oo

J1K,..I>y

Idm=jld/LS,

Now write any complex-valued

I

1= «lRI)+ 8Indeed, If At = {Igl basIC measure theory,

tm(At)

~r.. Igldm

> t},

1;;::0.

E C('ll') as

(lRl)_)

+ 1,«CSI)+ - (CSI)_)

then meAt) -

0 as t -

0 as t -

[149, p

r Igldm -

JAt

IE C(1l') ,

00

00

(Proposition 124) Moreover, by

148)

Fmally, tXAt

~ IglXAt

and so

9 CLARK MEASURES

224

and apply eq.(9.7.4) four times to get lim

'Try ( iIKIJI>Y

y_oo

fdm = jfdf-Ls,

f E G('If),

that is to say, lim

y-oo

'TrYXIKIJI>y • m

= f-Ls

weak-*.

The proof Poltoratski's theorem (Theorem 9.7.2), requires a few preliminaries. Suppose that f-L is a positive, singular, probability measure. By Remark 9.1.4, f-L = C71 E At? for some inner function '19 with '19(0) = O. Moreover, each C7a E At? is also a probability measure. Let x+z w : 1R - 'If, w(x):= - - . X-'l The following technical lemma is a version of Aleksandrov's disintegration theorem (Theorem 9.3.2). LEMMA 9.7.5. For 9 E G('If) and Y

(

g() dm()

> 0,

1. (iT(

g«) dC7a

=

iQul>Y

w(y,oo)

«(») dm(a).

PROOF By Corollary 9.1.24, each measure C7a E At? is carried by the Borel set Ea = {'19 = a}. Notice that 1 + 'I9(z) 1 _ 'I9(z) = (HC7d(z) = (PC71)(Z) + Z(Q C7 1)(Z), zED. Thus, since C71 .1. m, lim (PC71)(r()

r ..... l-

= 0 m-a.e. (

E 'If.

Using this and taking radial limits above, we see that 1 + '19«() 1 _ '19«() = 'l(Q C7 1)«), m-a.e. ( E 'If.

Let us make the observation that if

.1+'I9«) -1 Y = (Q C71)«) = -1,1_ '19«() = w ('19«» , then

Y+'l Y-'l

'19«() = -

= w(y).

Hence, the two sets {( E 'If: (QC71)«()

> y} and {( E 1r '19«) E w(y, oo)}

are equal m-a.e. and so (9.7.6)

{QC7I > y}

=

U

Ea m-a.e.

aEw(y,oo)

Let h be the characteristic function of the Borel set9

U

Ea

aEw(y,oo)

9Byeq (9120), thIS set

IS

equal to 'I9- 1 (w(y,oo)) and hence

IS

a Borel set

225

97 POLTORATSKI'S DISTRIBUTION THEOREM

Then for any 9 E C(T),

f

JQul>Y

g( () dm( ()

f

=

JT = i

g( ()h( () dm( () (by eq.(9.7 6» ( i 9«()h«()dUa

f

=

Jw(y,oo)

«(») dm(o:)

( f 9(()dUa ((») dm(o:) JT

(by Theorem 9.4.5) (by Corollary 9.1.24)

o We now proceed to the proof of Poltoratski's theorem. PROOF OF THEOREM

9.7.2. We will prove, for any f..L

E

M+ with 1If..L1I = 1,

that

'TrYXQp,>y m

-+

Y -+ 00,

f..Ls,

weak-*. Let us first prove this when f..L is also a singular measure. From Remark 9.1.4 and eq.(9.1.5), f..L = Ul E A19 for some inner function {) with {)(O) = O. From Proposition 9.4 2 it follows that UO/. -+

UI, weak-* as 0: -+ 1.

Routine estimates show that 1 - - - : - - - rv

m(w(y,oo»

'Try

and so for any 9 E C(T), lim 'Try

y....oo

1

QUl>y

gdm

11

lim

=

y-+oo m(w(y, 00»

QU1>y

gdm

when these limits actually exist (we will show they do). Now recall from Proposition 9.4.2 that

(Gg)(o:) := i9«()dUa (() is continuous. Use Lemma 9.7.5 to see that

f

1 f (Gg)(o:) dm(o:) m(w(y,oo» Jw(y,oo) However the last integral is the average of (Gg)(o:) over the arc w(y,oo) and this arc approaches the point w(oo) = 1. Thus 1

m(w(y, 00» JQul>y

lim

gdm =

(/

y-+oo m w y,oo

»JW(1I,00) f (Gg) (0:) dm(o:)

= lim(Gg)(o:) 0/.-+1

= (Gg)(l) = h9«()dUI«()

since Gg is continuous. Hence we have shown that

'TrYXQUl >11 . m weak-* as y

-+

00.

-+

UI

9 CLARK MEASURES

226

To prove that weak-* as Y -+

00

7rYXQp.>y • m -+ f.1.s for any positive measure f.1., note that if

f.1.=f.1.a+f.1.s is the usual Lebesgue decomposition, then Qf.1. = Qf.1.a m(IQf.1.al

+ Qf.1.s,

> y)

=

o(l/y),

and lim

y-oo

For any

f

7rYXQ/J-.>y • m

= J.Ls weak-*.

> 0, observe that {Qf.1. > y} C {Qf.1.a > €y} U {Qf.1.s > (1- fly}·

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 9 E G (T), lim 7ry y-+oo

f

1Q p.>y

gdm

~

{ gdf.1.s.

1T

For the other direction, notice that {Qf.1.s

> y}

C {Qp

> (1- fly} U {Qf.1.a < -fY}

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 9 E G(T),

f gdf.1.s ~

1T

The result follows as before.

lim 7ry y-oo

f

1Qp.>y

gdm.

o

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 {}(O) = 0 and a E 'll', we know from Theorem 9.5 5 that the operator Va . L2«(1a) - 4 {}*(H2) defined by

VaJ

K(fd(1a) K(1a is a unitary operator that intertwines the rank-one unitary perturbatIOn Ua on {}*(H2) with the multiplication operator (Zg)«() = (g«() on L2«(1a). In this chapter, we examme a generalization of Va. (10.11)

=\

7

For WED notice that

and so for JL E M+,

(10.1.2)

~(KJL)(z) =

r 1~«(z) dJL«() ~ M 11 - (z12 2

Vz

J.Jr

E D.

Thus for each J E Ll(JL), we can define the function

(V J)(z) J1.

.= K(fdJL)(z) •

(KJL)(z)'

In\

Z E

llY.

By eq.(10.1.2) this function, called the normalzzed Cauchy transform, is analytic on Jl) and, since V,. is the quotient of two HP functions (Theorem 2.1.10), has m-a.e. defined boundary values given by

(V,.J)«():= lim (V,.J)(r(). r-+l-

When {} is inner, {}(O) = 0, and JL =

(1a

E

A", then

Vu", = Va· Thus V,. 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 34.1). PROPOSITION

f

E

10.2.1. For JL

E

M+ there zs constant C > 0 such that for any

L1(JL),

227

228

10

THE NORMALIZED CAUCHY TRANSFORM

PROOF. Since IKJLI ~ 111L1i/2 on T, we know that for any y > 0,

{IV,../I > y} = {IKldlLl > IKJLIY} C {IKld lLl

>

11~lIy}

Thus, by the Kolmogorov weak-type estlmate in Theorem 3.4.1,

m(IVI'/I > y)

~ m (IK1dlLl > 1I~lIy) 2

~ C 1IILily IlldJLl1

~

CI'II/II£1(I'). y

o REMARK 10.2.2. 1(he previous result says that VI'L1{1L) C H 1 ,00. One can see this by observing that VI'L 1 (1L) c N+ (since it is the quotient of an HP function and an outer function) and has L 1 ,00 boundary values. Now use Theorem 1.10 4. One concludes from tills and Proposition 1 10 1 that

VI'L 1 (1L)

c

n

HP.

O.(z) = 1 -

~iJ(z) 1- AZ

10

230

THE NORMALIZED CAUCHY TRANSFORM

belong to 'I9*(HP)l and have dense linear span = 0, we have the formula

From Corollary 95.2, assuming

'19(0)

v;C a

1 k 1 _ a'l9(A) A'

A -

From these facts, together with the continuity of Va . £P(ua } --> HP when 1 < p (Corollary 1O.2.5), we obtain the following corollary ~s

COROLLARY 10.2.7. Suppose '19 Ua

mner and '19(0) = O. If 1

< p

~

~

2

2 and

E A1?, then

For p ~ 2, we look at Va-Ion '19* (HP). Recall that Va is a unitary operator from L2(ua ) onto '19* (H2). But since p ~ 2, we have, via Holder's mequality, the obvious contamment 'I9*(HP) C '19* (H2) and so Va- 1 f E L2(Ua ) whenever f E '19* (HP). COROLLARY 10.2.8. Suppose '19 ~s mner and '19(0) then

= O. If p

~

2 and Ua EAt?,

Va- 1 (t?*(HP» C LP(ua ). PROOF. Since the dual of '19* (HP) can be identIfied with '19* (Hq), where lip + 1/q = 1, via the pamng

h

tYdm ,

(see Proposition 8.2.8) and Va : U(u a ) --> 'I9*(Hq) (since 1 < q ~ 2), the result follows by noting that the adjoint of Va : Lq(ua ) --> 'I9*(Hq) is Va-I: 'I9*(HP) -->

LP(uac ).

0

We end this section by mentIoning some related results. When 1 < p ~ 2 and JL E M+, the operator VI-' maps LP(JL) continuously to HP. Moreover, when JL = Ua EAt? ('19 inner with '19(0) = 0), then VI-' = Va and Va£P(u a ) C 'I9*(HP). Is this map ever onto? When p = 2, the statement that Va L2(O'a) = 'I9*(H2) is part of Clark's theorem. For other values of p, there is this result of Aleksandrov [14]. 2

THEOREM 10 2 9 (Aleksandrov). If 1 < p < 2 and VaLP(O'a) = 'I9*(HP) or zf and Va- 1 ('I9*(HP» = LP(ua }, then O'a ~ a d~crete measure.

< p < 00

Can the continuity of VI-' . LP (JL)

-->

HP for 1 < p

~

2 be extended to p > 2?

THEOREM 10.2.10 (Aleksandrov). Suppose JL E M+ and ~ smgular. If p > 2 and VI-'C('][') C HP, then JL ~s a dzscrete measure. Thus zf VI-' zs a contmuous operator from £P(JL) to HP, then JL zs d~crete.

10.3. Function properties of the normalized Cauchy transform If

lRecail the defimtlon of 1?*(HP) from Remark 826

10 3

FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM

231

is a discrete measure in M and fELl (J.l), observe, using the dominated convergence theorem, the two formulas lim (1 - r)(KJ.l)(r(J) = cJ'

r----+l-

hm (1- r)(KfdJ.l)(r(J) = cJf«J}. r-+lThese two formulas say that . K(f dJ.l)(r(} hm (VJ-Lf)(r() = hm (K)( () = f«(,) J.l-a e. r-+lr-+lJ.l r A theorem of Poltoratski [162] extends this observation to general measures. THEOREM 10.3.1 (Poltoratski). If J.l E M and f E L 1 (J.l), then

where J.ls

~

lim (VJ-Lf}(r() = f«} J.ls-a.e., r-+lthe smgular part of J.l wzth respect to Lebesgue measure m.

REMARK 10.3.2. Notice In the theorem that J.l need not be a positIve measure and so VJ-Lf is a meromorphzc function on D. However, since VJ-Lf IS the quotient of two Cauchy transforms, it is a functIOn of bounded type and as such has nontangential limits m-a.e The significance of PoltoratskI's result is that VJ-Lf has non-tangential limits J.l-a.e and that /Ls-a e., these limits equal f. 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(JR) denotes the finite Borel measures on JR, M+(JR) denotes the posItIve ones, and ml denotes Lebesgue measure on JR. For J.l E M(JR), let

J

_1_ dJ.l(t), Z E C \ JR, t-z be the Borel transform of J.l and note that FJ-L is analytic on C \ JR. For convenience of notation, let

FJ-L(z) =

FfJ-L(Z} = whenever f E following

L 1 (J.l).

J

-1_f(t) dJ.l(t)

t-

Z

The Jaksic and Last verSIon of Poltoratski's theorem IS the

THEOREM 10.3.3. For J.l E M(JR) and f E L 1 (J.l), lim FfJ-L(x + zy) = f(x) y-+O+ FJ-L(x + zy) for J.ls-a.e. x E JR, where J.ls zs the smgular part of J.l wzth respect to Lebesgue measure ml Before we get to the proof of this, let us see how Theorem 10.3.3 proves Theorem 103.1. We only provide an outline and leave some of the details to the reader. The function 1jJ(z) = n + 1

zz -1

maps the upper half plane {~z > O} onto D. For simplicity, we assume that the point (, = 1 is not in the support of J.l E M and so the measure v := J.l 0 1jJ belongs to M(JR) and has compact support. With the substitutions,

( = 1jJ(x)

E 11'

and w

= 1jJ(z) ED, x

E

JR,

~z

> 0,

232

10 THE NORMALIZED CAUCHY TRANSFORM

the Cauchy transform

! ~d~«() l-(w becomes (zz -1)

!

(-~x

-l)dv(x) x-z

= (zz -l)F(_lx_l)II(Z).

Hence (V/..J)(w) = F(-.x-l)(Jo,p)II(z) F(-.x-l)II(Z)

and the result will follow. To prove Theorem 10.3.3, we need a few preliminaries. Let 1 PAt) := ~--, z E C+, t E R t-z be the Poisson kernel for the upper half-plane C+ := {~z > O} and notIce that y Px+.y(t) = ( t)2 x+y2 > 0 Vy > 0, VX,t E R.

(P~)(z) :=

!

Pz(t) d~(t)

will denote the Poisson integral of~. A version of Fatou's theorem (Theorem 1 8.6) says that whenever ~ E M+(R) and (Dp,)(x) exists (and we include the pOSSIbility that (Dp,)(x) = 00), then hm (P~)(x + zy) = 7r(Dp,)(x).

y-+O+

An upper-half plane analog of Proposition 1.3.11 says that {x . 0 < (l2J.t)( x) < oo} is a carrier for ~a and {x : (l2J.t)( x) = oo} 18 a carrier for P,s. Here ~ = P,a

+ ~8'

is the Lebesgue decomposition

of~.

~a« mb

P,s..L

ml,

Thus, on a set of full J.L-measure we have

0< lim (P~)(x + ~y) ~ 00.

(10.3.4)

y-+O+

When p, E M+(R) and

f

E

£1(p,) is real-valued, ~FfJ.L(z) = (Pf~)(z).

We begin with some technical lemmas. LEMMA

10.3.5. For p,

E

M+(R),

~s

zs earned by

lim 1FJ.L(x + zy)! = { x: y-+o+

co} .

PROOF. Observe that Thus lim IFJL(x + ~y)! { x: y-+o+

=

oo}

:)

{ X'

lim

y-+o+

(P~)(x + zy) =

:) {x: (Dp,)(x)

=

co}.

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 carner for JLs. 0 For JL E M+(JR) and h E LI(JL), let

(Mh)(X)

:= sup

(

1

€>o JL x - c, x

)

+c

r+€ Ih(t)1 dJL(t).

1x-€

This next inequahty is a generalization of the famous Hardy-Littlewood inequality for maximal functions [207, p. 5] (see also [182, p. 137J for a related result). LEMMA 10.3.6. There'tS a constant N zndependent of any hE L 1 (JL) and y > 0 so that for any bounded znterval [a, b], N JL ([a, b] n {Mh > y}) ~ -llhll£l(I')· Y

PROOF. For fixed y > 0, each point of the set {x: (Mh)(X) > y} is the center of a closed interval Ix for which

( Ihl dJL > YJL(Ix )

11",

For a compact interval [a, bj, consider the family of intervals

1':= {Ix: x E [a,bJ n {Mh > y}}. Apply the BesicovlCh covering theorem2 to extract a countable covering of [a, bJ n {Mh > y} by intervals of the form N

00

UUIJ,k, k=lJ=l

where N IS some universal integeI (a dImensionality constant) and for each k 1,2,· .. ,N, the intervals IJ,k, J EN belong to l' and are disjoint. Finally, we have N

JL ([a, bj n {Mh > y}) ~

=

00

L: L: JL(IJ,k) k=lJ=l

o LEMMA 10.3.7. Let JL E M+(JR) and h E L1(JL) Then for each x E JR,

(Mh)(X) ~ (PlhIJL)(x + zy) (PJL)(x + zy)

vy> O.

2BesicoVICh covering theorem [68, p 30J There IS a Universal constant N WIth the follOWing property If:1' IS any collection of non-degenerate closed intervals 1D It With sup{ dlam( I) I E :J'} < 00 and If A is the set of centers of the intervals in :1', then there exISts GI, G N E 3" such that each G. IS a countable collectIOn of dISJOint intervals from 3" and A C Ui=l UIEG. I

234

10

THE NORMALIZED CAUCHY TRANSFORM

PROOF. For v E M+(JR) and non-negative f E LI(v), there is the distributional equality [123, p. 26] (see also ProposItIon 1.2.4)

!

(10.3.8)

f(t)dv(t) =

1

00

v(f > s)ds.

We will apply this to the function

1 1 f(t) = -Px+ty(t) = ( )2 2· Y t-x +y

A computation shows that whenever y > 0 and s E (0,1/y2), { tEJR: (t-x

~2 +y

2

q=q(s,Y)=J~_y2. s

>s}=(X-q,x+ q),

Using the distributIOnal equalIty in eq.(10.3.8), we get, for any v E M+(JR),

1 -(Pv)(x + ty) =

(lO 3.9)

y

10r

l/y2

v (x - q(s, y), x + q(s, y)) ds.

Let I(a, r) = (a - r, a + r) and notice from the definition of (Mh)(x), that for any s and y

(Mh)(X) ~

1

1 Ihl df.L f.L(I(x, q(s, y») I(x,q(s,y»

and so

f.L(I(x,q(s,y)))(Mh)(x)

~

r

1I(x,q(s,y»

Ihldf.L

Now integrate to get

(Mh)(X) t/y2 f.L(I(x, q(s, y)))ds

10

Using eq.(10.3.9) twice, once with v

~ r 10

l y2 / (

r

1I(x,q(s,y»

Ih(t)1 df.L(t») ds.

= f.L and again with dv = Ihl df.L, we get

(Mh)(X) ~ (Plhlf.L)(x + iy) ~ (Pf.L)(x + ty)

V y > O.

o PROPOSITION 1O.3.lO. For f.L E M+(JR) and real-valued f E L1(f.L), lim (Pff.L)(x + iy) y->O+ (Pf.L) (x + ty)

=

f(x)

for f.L-a.e. x E JR. PROOF. To prove the result, it suffices to show that lim I(P(f - f(x»f.L)(x

y-O+

(Pf.L) (x + zy)

+ zy)1 = 0

f.L- a .e .

To do this, let (9n)n~1 be a sequence of continuous functions that approximate f the norm of LI(f.L) and let hn = f - 9n. Thus IIh n ll£1(I') --4 O. Without loss of generalIty, we can also assume, by passing to a subsequence, that hn --40 f.L-a.e. The continuity of the 9n'S imply that for every x E JR, III

lim (P(9n - 9n(X))f.L)(x + zy) =

y->O+

o.

10 3 FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM

235

From eq.(10.3.4), lim (:1'J..£) (x + ty) > 0 J..£-a.e.

y-+O+

and thus hm

(10.3.11)

y-+o+

I(:1'(gn - gn(x»J..£)(x + zy)1 (:1'J..£)(x + zy)

=

0

J..£-a e.

Hence for J..£-a.e. x E lR and every n E N, lim 1(:1'(f - f(x»J..£)(x + zy)1 :::;; lim (:1'lhn IJ..£) (x + zy) + Ihn(x)l.

(:1'J..£)(x+zy)

y-+o+

y-+O+

(:1'J..£)(x+zy)

Now use Lemma 10 3.7 to see that this last quantity

IS

bounded above by

(10.3.12) Let

W(x + zy)

f(x»J..£)(x + zy)l. (:1'J..£)(x + zy)

= 1(:1'(f -

To establish J..£ ( {x:

y~~+ W(x + zy) > 0}) = 0,

it suffices to show that for each fixed t > 0 and fixed compact interval [a, b],

J..£([a,b)n{x:

y~+ W(x+zy) >t}) =0.

For each fixed t > 0, it follows from eq (10 3.12) that for any n E N (10.3.13)

lim W(x + iy) > t} { y-+O+

c {Mh n > t/2} U{Ihnl > t/2}

J..£-a.e.

Let € > 0 be given. Since h n - 0 J..£-a e. we can use Egorov's theorem 3 , to produce a set A C lR with J..£(lR \ A) < €/2 and such that hn - 0 umformly on A. Thus for n ~ K, Ihnl < t/2 on A and so

J..£(lhnl > t/2) < €/2 for all n

(10.314) Since Ilh n llL1(p)

-

0, we can, by making IIhnIlLl(p) :::;;

€

K

~

K

even larger, assume that

t

4" N for all n

~ K,

where N is the constant from Lemma 10.3.6 By Lemma 10.3.6, (10.315) for n

~

K. From eq.(10.3.13), eq.(10.3.14), and eq.(10.3.15), J..£ ([a, b)

n t~+ W(x+zy) >

and the result follows. 3Note that p.

IS

a fimte measure

t}) : :;

€/2+€/2 = €

o

236

10

THE NORMALIZED CAUCHY TRANSFORM

COROLLARY 10.3.16. For any JL,v E M+(lR),

lor almost every x E v.

lim (Pv)(x + zy) = 0 y--->o+ (PJL)(x + zy) IR W'tth respect to the part 01 JL that

M

smgular wzth respect to

PROOF. Let

JL = JL~ + JL~, JL~« v, JL~ 1. v, be the decomposition of JL with respect to v (see Remark 1.3.12). Since have disjoint carriers, there is a function I with

1=1

(10.3.17)

JL~-a.e

and

JL~

and v

1=0 v-a.e.

By Proposition 10.3.10,

PI(v + JL~) _ " ) - 1 JLs-a.e. v + JL~ (For typesetting purposes, we are suppressing the x + zy.) Hence by eq.(10.3.17), hm P(

y--->o+

lim

PJL~

= lim PI(v + JL~) = 1

y--->o+ P(v + JL~)

y--->o+ P(v + JL~)

JL~-a e

Consequently, the identIty

Implies that

. ;nPv = lim . (P(v+ v hm P JL~») - 1 = 0 JLs-a.e y--->o+ .rJL~ y---+O+ JL~ Finally, observe that SInce all measures involved here belong to M+(IR), we have Pv Pv O~-~­ "" PJL "" PJL~ from which

· m:::;; Pv I·Im;nPv O :::;; 11m y--->o+ .r JL y--->o+.r JL~

=0

" JLs-a.e.

o One of the keys to proving the main theorem (Theorem 1O.3.3) is a certain resolvent formula. For this we need the spectral theorem for self-adjoint operators on a Hllbert space (Theorem 157). For a JL E M+(IR) with compact support let

(f,g)

=

f

I(x)g(x) dJL(x)

be the usual Hilbert space inner product on L 2 (JL). Let

A . L 2 (JL)

-+

L 2 (JL), (Ag)(x)

= xg(x)

and notice that the constant function

X(x)=l, xElR. is a cychc vector for A in the sense that

V{ATlx: n

E

No}

= L 2 (JL).

10 3 FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM

237

One can see this last fact by observing that An X = xn and so the linear span of {An X : n E No} contains the polynomials. Now use the Stone-Weierstrass theorem and the density of the continuous functions with compact support in the space L 2(JL). Notice also that (A*g)(x) = xg(x) and so A is self-adjoint. Define a rank one perturbatIOn Al of A by Al : L 2(JL) -+ L 2(JL), Al = A + (', X)X and notice that Al is self-adjoint. Also notice that

A~X =

x,

Ah =

x + CI,OX, 2 A~X = x + C2,IX + C2,OX,

and so

V{Afx: n E No}:::> V{Anx: n E No} =

L 2 (JL). Thus Al 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 JLl E M(lR.) and a unitary operator with

Ux = X and UAIU* = M x , where (Mxf)(x) = xl(x) on L2(JLI)' Furthermore, Ux n = UAnX = UAU*Ux n- 1 = U(AI - (" X}x)U·Ux n -

1

= (UAIU* - U( ,X}xU*)Uxn = MxUx n- 1

xUx n- 1

=

-

-

U(',X)xx n -

1

1

(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, U I is a real-valued element of L2(JLd whenever I E L 2(JL) is real-valued. We will also need the following formulas that relate the Borel transform to certain resolvent operators:

Fp(z)

=!

Ffp.(z) =

J

_1_ dJL(t) = «A - zI)-lX,X) ,

t-z

_1_/(t) dJL(t) = «A - zI)-l I, X) , t-z

F(Uf)l-'l (z) =

J

_1_(U f)(t) dJLl(t) t-z = «Mx - zI)-lUI,x)p(,..t> =

(U*(Mx - zI)-IUI,U*x)

=

«AI - zI)-l I,x).

238

10

THE NORMALIZED CAUCHY TRANSFORM

From the basic operator IdentIty

A-I _ B- 1 = A-l(B _ A)B- I one shows that for any h, 9 E L 2(J.L)

«({A - zI)-l - (AI - zI)-l) h,g)

= «A - zI)-1 «AI - zI) - (A - zI)) (AI - zI)-lh,g) =

«A - zI)-I(A I - A)(AI - zI)-lh,g)

= «A-zI)-l( ,X)x(A1-zI)-lh,g) =

«A - zI)-I(A I - zI)-lh,X)X,g)

=

«AI - zI)-lh,X) «A - zI)-lX,g).

ThIS yields the identity (10.318)

(AI - zI)-lh,g) = (A - zI)-lh,g) - (A - zI)-IX,g) (AI - zI)-lh, 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 /1 E M+(JR.) with compact support and real-valued f E L 2 (/1). Setting 9 = h = 1 in eq.(10.3.18), we have

F,,(z) Fl'l (z) = 1 + F,,(z)"

(10.3.19)

Let XI' :=

{x:

lim 1F,,(x + zy)1 = oo}

y-+O+

and recall from Lemma 10.3 5 that X" IS a carrier for J.Ls (the singular part of J.L with respect to m1). If x E X"' we can use eq.(10.3.19), to see that IF"l (x + zy)1 ~ 1 and so x tJ. X"l. In a similar way, If x E X"l then x tJ. Xw Thus, by Lemma 10.3.5, J.Ls and J.L1,s (the singular part of /11 wIth respect to ml) have disjoint carriers and so /1s 1- /1l,s· Take imaginary parts of both sides of eq.(10.3.19) and use the Identity ~Fv(z) = (Pv)(z), for v real, to get (10.320) Using eq.{10.3.18) again, this time with 9

(10.3.21)

= X and h = f,

we get

Ffl'(z) = (1 + F,,(z))F(Uf)"l (z).

Take imaglllary parts of both sides of eq.(10.3.21) and apply the identity ~(ab) = (~a)(~b)

to get (10.3.22) where (10323)

+ (~b)(~b),

a,b E C,

10 3 FUNCTION PROPERTIES OF THE NORMALIZED CAUCHY TRANSFORM

239

In the above computations, we are using the fact that U f IS real valued whenever f is real valued and so S(FUf~J = P(Uf)J.ll. To estimate P(Uf)J.ll we use the Cauchy-Schwartz inequalIty (in a clever way) to get

I(P(Uf)J.ll)(X+zy)1

)/Y(Uf)(t)dJ.ll(t)1 (t - x)2 + y2

=

(10.3.24) Smce J.ls 1.. J.ll, then J.ls 1.. (Uf) 2J.ll. Thus by Corollary 10.3.16,

(P(Uf) 2J.ll)(X+ZY) (PJ.l)(x + zy)

h

() 10.325

y-.:g+

0 =

J.ls-a.e.

Now observe from eq.(10.3.20) that (10.3.26) and so eq.(10.3.24) and eq.(10.3.25) imply that

+ zy)1 =

lim I(P(U f)J.ll) (x

(10.3.27)

y-+o+

0 J.ls-a.e

By using eq.(l0 3.24) and the equality in eq (10.3.24) to estimate I(P(Uf)J.lr)(z)1 in eq.(10 3.23), we have

IL(x + zy)1

~

+ F~(x + zy»1 I(PJ.l) (x + zy)1 I(PJ.l) (x + zy)1 11 + F~(x + zy)1

1~{1

(P(Uf)2J.lr)(X + zy) (PJ.l) (x + zy)

(P(U f) 2J.ll) (x + zy) (PJ.l)(x + zy)

=

Hence from eq.(10.3.25), lim IL(x + zy)1 = 0 J.ls-a.e

(10.3.28)

y-+o+

Combining eq.(10.3.22), eq.(10.3.27), eq.(10.3 28) along with ProposItion 10.3.10 implies that

.

hm F(Uf)~l (x + zy)

y-+o+

.

(PfJ.l)(x+zy)

= y-+o+ hm (P)( ) J.l x + zy

= f(x) J.ls-a.e.

Divide both sides of the identity in eq.(10.3.21) by Ftt(z) and apply Lemma 10.3.5 (which will say that IFtt(x + zy)1 --t 00 J.ls-a.e.) to see that (10.329)

.

Fftt(x + zy) ) /Jo x + zy

hm F (

y-+o+

=

.

lim

y-+o+

(1+ zy + )F(Uf)/Jol (X + zy) f (x ) F (

/Jo X

)

1

=

J.ls-a.e.

This completes the proof of Poltoratski's theorem in the special case where J.l E M+(IR) with compact support and f is real-valued and belongs to £2(J.l). One can remove the assumption that J.l E M+(IR) has compact support by making some

240

10

THE NORMALIZED CAUCHY TRANSFORM

technical adjustments coming from the fact that the operators A and Al used to prove the resolvent formula become unbounded. To finish, we follow Poltoratski's original proof in [162]: Let 1 E £1(J.L) be positIve and J.L E M+(R). Set 1

g:= 1 + 1

and

v = (1 + f)J.L.

Observe that 9 E L2(v) and the measures J.Ls and Vs have the samE' carner. Apply eq.(l0 3.29) to obtam the identities lim Y-'o+

FI'(x + zy) F(1+f)I'(x + zy)

=

hm Fgv(x + zy) = g(x) Fl/(x + zy)

Y-'o+

=

1 J.Ls-a e 1 + I(x)

and

· Ffl'(x+'lY) 11m = 1·1m F(1+f)I'(x+zy) - 1 = I() x J.Ls-a e FI'(x + 'ty) Y-'O+ FI'(x + zy)

Y-'O+

This proves the result for J.L E M+(R) and positive 1 E Ll(J.L). Write a complexvalued 1 EL I (J.L) as a complex linear combmation of four positive functions and use the linearity of the Borel transform to get the general result (still for J.L E M+(R» For the general case where J.L is a complex measure, we note that J.L = glJ.Ll, where 9 E Ll(IJ.Li) with 191 = 1 (recall the definition of the total variation measure from eq.(1.3.4)) Hence

1. Ffgll'l(x + 'tY) . Ffl'(x + zy) 11m = 1m FI'(x+zy) Y-'o+ Fgll'l(x+zy)

Y-'O+

=

.

hm Y-'o+

Ffgll'l(x + zy) FIl'l (x + zy) FII'I(x+zy) Fgll'l(x+'ty)

= I(x) /Ls- a.e

and the proof is now complete.

o

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 mentiomng (without proofs) a few related results of Poltoratski. The main theorem of this section shows that for J.L E M+ and 1 E L1(/L), (VI'f)«().= L lim (VI'f)(z)

z-.,

exists J.L-a.e. and is equal to 1 J.Ls-a.e. Since VI'I is now J.L-measurable, one can ask the natural question as to whether or not VI'I belongs to any LP(J.L) class. The following result of Poltoratskl [165] sheds some light on this. THEOREM 10.331 (Poltoratskl). For J.L E M+ and 1 < p zs contmuous from LP(J.L) to V(J.L).

~

2, the operator VI'

REMARK 10.3.32. This result is no longer true when p = 1 or when p > 2. In fact, if J.L = m + 0', where 0' 18 a non-discrete singular measure, then for any p > 2, there IS an 1 E LP(J.L) for which V/'I (j. V(J.L).

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 Uc., the unitary rank-one perturbations of the model operator Sf). We have discussed the spectral measures Uc< for Uc. = L...J A2 9(x) 8x • xEe).

As we discussed earlier with the operators Uo:, we wish to know when the operators A>. have pure point spectrum (see Definition 1.5.8).

10.4.8. The followzng are equwalent. (1) For ml-a.e. A E JR, A>. has pure pomt spectrum. (2) For ml-a.e. x E JR, 9(x) < 00.

THEOREM

Tills result has been generalized in several directions by Poltoratski [57, 58, 164]. 10.5. A closer look at the :J'-property Recall the :J'-property for the space of Cauchy transforms (Theorem 6.5.1)' if {} is inner and KJ1./{} E HP for some p > 0, then

KJ1.=Kv {}

for some v E M What is the relationship between the measures 1', 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

-e

'tS

tnner and

Kf-L=Kv

-e

Jor some f-L, v EM. Then -e(() = lim -e(r() r ..... l-

eX'tsts J.t-a.e. Moreover, the measure v can be chosen to satzsfy dv = 19dJ.t. The proof of this theorem reqUIres a few preliminarIes To avoid confusion, we will use the notation K J.t to denote the Cauchy transform regarded as an analytic function on the disk, while C J.t will denote the Cauchy transform regarded as an analytic function on C\ T. If -e is an inner function, recall that -e is defined as an analytic functIOn on

C\ {liz: where

z

E

u(-e)},

{Z E l!)- : .>.lim !-e(A)1 = o} . ..... z

O'(-e) =

Recall from Chapter 8 that if 9 is a meromorphic function on l!) and G is meromorphic on l!)e, then 9 and G are pseudocontznuatwns of each other if the non-tangential limits of 9 and G exist and are equal almost everywhere. When 9 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 pseudocontinuatIOn. Throughout thIS section we will consIder an inner function -e as an analytic function on l!) and a meromorphIc function on l!)e' From the identity,

-e(z)

1

=

19(1/z) ,

Z

E l!)e \ {liz,

Z

E O'(-e)},

one concludes that the functions 9 = -e!l!) and G = -ell!)e are pseudocontinuations of each other. Furthermore if Z E l!) is a zero of -e of order n, then liz, the reflection of Z across T, will be a pole of -e of order n. Also notice that IN E HCXl(l!)e) and is 'inner' on l!)e (Le., 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 -e be znner and 9 E HCXl(l!)e) W'tth g(oo) = O. Suppose J E HCXl(l!) has a pseudocontznuatwn equal to -eg and that Jor some f-L EM, -eCf-L E HP(l!)e) Jor some p > O. Then there 'tS a v E M such that

JCf-L=Cv

onC\ T. PROOF. Let

F(z) _ { J(z) (KJ.t) (z), z E l!); -e(z)g(z)(CJ.t)(z), z E l!)e' One can quickly check, using Fatou's jump theorem (Corollary 2.4.2), that F is an analytic function on C \ T with J(oo) = 0 which satisfies the hypothesis of Aleksandrov's representation theorem (Theorem 5.4.5) Thus F = Cv for some vE M 0

10 5

A CLOSER LOOK AT THE 3'-PROPERTY

245

COROLLARY 10 5.3. Under the assumptwns of Theorem 10.5.2,

dV=fdJ.L. dm

dm

PROOF. If

= r-.llim (h(r() - h«/r»

(Jh)«)

is the jump function for an h analytic on iC \ 'Jr, it follows from Fatou's jump theorem (Corollary 2 4.2) and the fact that f has a pseudo continuation, which we also denote by f, that dJ.L dv f dm = J(fCJ.L) = J(Cv) = dm'

o COROLLARY 10.5.4 (Goluzina, [83]) Under the assumptzons of Theorem 10.5.2,

v« J.L. PROOF. For a measurable function F on 'Jr and a Borel subset E

L(F,E)

=

lim 7rym({( E E:

y--->oo

1J.LI(lh)

=

Iv l(8"()

'Jr, let

IF«) I > y})

whenever this limit exists From Theorem 7.4.4, L(KJ.L, 'Jr) Let "( be any arc of 'Jr for which (10.5.5)

c

= lIJ.Lsll.

= O.

If

m, Poltoratski's distnbution theorem (Theorem 9.7.1) says that a y := 7rYX{IKJLI>y} .

a y --+

From the fact that ay(E) sition 16.1), we see that

--+

lJ.Lsl weak-* as y

--+ 00.

lJ.Lsl(E) for every Borel set with lJ.Lsl(8E) = 0 (Propo-

(10.5.6)

L(KJ.L,,,(} = lJ.Lslh) and L(Kv,,,(} Since fKJ.L = Kv, we also have

= Ivslh)·

(10.57) A calculation with the definition of L yields the estimate

L(fKJ.L,,,()

~

sup{lf«} I : (E "(}L(KJ.L,,,()

which by eq.(10.5.6) and eq.(10.5.7) gives us Ivslh) ~ IIflloolJ.Lslh}· Combine thIS with Corollary 10.5.3 to get (10.5.8) for any arc "( with 1J.L1(8"() = Ivl(8"() = O. Using the fact that the atoms of Ivl and lJ.Ll are at most countable, along with a limit argument using eq.(10.5.8), one can show that Ivl({(}} ~ IlflloolJ.LI({(}) V( E 'Jr. From here it follows that eq.(10.5.8) is valid for all arcs "( C 'Jr. It now follws from 0 basic measure theory that Ivl « IJ.LI and hence, by Remark 1.3 12, v « J.L.

10

246

THE NORMALIZED CAUCHY TRANSFORM

COROLLARY 10.5.9 (Poltoratski, [162]). Assummg the hypothests of Theorem 10.5.2, we have the followmg. (1) lim f(r() = f«() r-+1-

ensts for p,-a e (E ']['. (2) In the tdenttty fGp, = Gv on C\ '][', the measure v can be chosen to sattsfy dv = f dp" that tS to say, fGp, = GUp,)·

PROOF. By Corollary 10 5.3, we know that dv = f dP,. dm dm By Corollary 10.5 4, dv = 9 dp, and so 9 = f m-a e. To finish, we need to show that f has radial limIts ILs-a.e. and that these limits are equal to the boundary values of 9 p,s-a.e. Observe that for z E J[} (minus some poles), f( ) = (Kv)(z) = (Kgp,)(z) = (V )() z (Kp,)(z) (Kp,)(z) I!g z

and so from Theorem 10 31, f has radial limits p,-a e. that are equal to the boundary 0 values of 9 p,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 ~ l/z. THEOREM 10.5.10. Let 1/J E HOO(J[}e) be mner and 9 E HOO(J[}) wtth g(O) = O. Suppose that f E HOO(J[}e) has a pseudocontznuatwn equal to 1/Jg and that for some a E M, 1/JKa E HP(J[}) for some p > O. Then there ts an TJ EM such that fGa = GTJ on

C\ T.

Moreover, for a-a.e. ( E '][',

lim f«(/r)

r-+1-

eX'tsts and m the tdenttty fGa

= GTJ,

the measure TJ can be chosen to sattsfy

dTJ = fda. PROOF OF THEOREM 105.1. Suppose Kp,

= Kv

'19

on

JI)).

In Theorem 10 5.10, let 1

f:=

0'

1

1/J:= zf)'

9 = z.

Note that f E HOO(JI))e), 1/J is an inner function on Jl))e, and ation equal to 1/Jg. Let jJ, := -j.t( -l)m + (JL and observe that zKp, = Kfl,

f has a pseudocontmu-

10 5 A CLOSER LOOK AT THE !r-PROPERTY

and

'1f;KiJ,

247

= ~KiJ, = ~ZK/-L = Kp, = Kv E HP(]f). Z{} Z{} {}

Thus by the above theorem, there is a III E M with

~GiJ, =

Gillon

C\ 'll'

Furthermore, .

1

r~- -;a((jr) exists iJ, almost everywhere. Usmg the identity

{}(w) 1 Iwl > 1 - {}(ljw) ' , we conclude that lim {}(r()

r-+l-

exists iJ,-a.e. By standard HP theory, this limit already exists m-a e Moreover, (iJ,)s = (P,s and so the above limit exists /-Ls-a.e. To finish the proof, we need to show that in the identity

K/-L

=

{}

KII

on ]f), we can choose II to be II = {}/-L. To this end, use Theorem 10.5.10 to show that m the identity

~GiJ, =

Gillon

C\ 'll',

III can be chosen to be (10 5.11) On]f), we have that (10.5.12) where

Kill

= ~KiJ, = ~ZK/-L = zKv = Kv,

v = -v(-I)m + (II.

For U E M, let [u] denote the coset in Mj HJ represented by u. From eq.(10.5.11), eq.(10.5.12), and Proposition 2.1.5, we get

[19iJ,] From this and the definitions of iJ, and

=

[v].

v, we get

19 (ii(-I)m + (/-L) = ¢m + v(-l)m + (II,

4> E HJ,

and so

({}ii( -1)m + 19/-L = (4)m + (v( -1)m + II. Hence

[19/-L] = [II]. A final applicatlOn of Proposition 2.1.5 YIelds K(19p,) = KII.

D

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)

f(z) - f(O) (backward shift) z (Tq,f)(z) = (f)+(z) (Toeplitz operator) (C",f)(z) = f((z» (composition operator) (Cf)(z)

=

=!

r

f(t) dt (Cesaro operator).

z 10 1- t

In this bnef chapter, we mentIOn how these operators act on the space of Cauchy transforms X 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 E LOC), define the Toephtz operator T", : H2

T",f

=

-+

H2 by

(f)+

and note that T", is bounded. By Riesz's theorem (Theorem 3.2.1), Tq, is also bounded on HP whenever E LOO and 1 < p < 00. When p = 1, Riesz's theorem is no longer available to us and, as to be expected, Tq, is not always bounded. For example, a result of Stegenga [204] says that whenever t/J E HOC), the co-analytic Toeplitz operator T~ : HI -+ HI is bounded if and only if ~t/J and ~t/J are multipliers of BMO. For a general symbol E Loo, there is a definitive result of Janson, Peetre, and Semmes [108] which says that Tq, : HI -+ HI is bounded if and only if - + is of 'logarithmic mean oscillation'. Here a function 9 E L1 on T is of 'logarithmic mean oscillation' if the supremum of log

(~U») m~I)

1

1g -

gIl dm,

over all arcs leT, is finite. l Compare this to the definition of bounded mean oscillation in Definition 3.3.1. When p = 00, we know from Proposition 6.1.5 that the co-analytic Toeplitz operator T~ : HOC) -+ HOC) is bounded if and only if t/J is a multiplier of X. 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 = (f)+ does not make sense since the 1Note that gl =

m~I)

f

9 dm

IS

the mean of 9 on I 249

11 OTHER OPERATORS ON THE CAUCHY TRANSFORMS

250

product on H2 can be extended to be bounded on HP. Aleksandrov [10] (see also [44, p. 189]) showed that whenever t/J E Hoo and 0 < p < 1, the co-analytic Toeplitz operator T1j) can be extended to a bounded operator on HP if and only if t/J 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

can be extended to a bounded operator on HP if and only if

(X, II . II)

zs

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.(1l.2.2)) of the forward shift 8 on A. Since 118 : A --> All = 1, then liB : X --> XII = 1 (Proposition 1.5.4). 0 This next theorem of Aleksandrov [11] (see also [44, p. 99]) is the DouglasShapiro-Shields theorem (Theorem 8.2.1) for Xa' THEOREM 11.3.2 (Aleksandrov). If J'v( zs a norm closed B-mvarzant subspace of X a , then there zs an znner functwn fJ such that f E J'v( zf and only zf there zs a G E N+(JI])e)4 W'tth G(oo) = 0 and such that

r~xr- ~(r() = r~xr- G«(jr) for m-almost every ( E ']J.". For a weak-* closed B-invariant subspace N of X, the dual pairing eq.(11.2.2) tells us that NJ. (the pre-annihilator of N) is an 8-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 B-invariant subspaces of (X, II . II)· 11.4. Toeplitz operators

For what E L CXJ can we define a meaningful Toeplitz operator T,p : X --> X? Certainly when 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 t/J is an analytic polynomial, then T~Kp, =

K(t/J*dp,),

253

11 6 THE CEsARO OPERATOR

where 1/J*{z) = 1/J{z). Now approximate any function 1/J in the disk algebra A uniformly on JD)- by its Cesaro polynomials (Theorem 1.6.5) to show that T;pKJL = K(1/J*dJL) is bounded on X. Are these the only co-analytic Toeplitz operators on X? Is there a definitIve characterization, like the one in [108J, of the symbols that yield continuous Toeplitz operators on X? 11.5. Composition operators

For an analytic map : JD) ----+ JD), we know from Lemma 5.6.1 that the composition operator CIP : f 1--+ f 0 maps X to itself. Without too much difficulty, one can show that CIP has closed graph and so CIP is bounded on X. Bourdon and Cima [31] proved that 2 + 2V2 IICIP !! ~ 1 - I{O) I which was improved to IIG II ~ 1 + 21 {O)I '" 1 - I{O) I by Cima and Matheson [42]. Moreover, equality is attained for certain linear fractional maps . This same paper also discusses compactness properties of C. In fact, CIP : X ----+ X is compact if and only if for every a E 11', the Aleksandrov measure JLoo corresponding to , satisfies JLa « 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

(f,g}o:=

r~xr- h9{()f(r()dm{(),

f EX,

9 E A.

A short computation yields (Cf,g}o = (f,Lg}o where L is the operator acting on A by

(Lg)(z):=

11 + 1g{tz

t)dt

=

~ (~k ~

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

f

11

1--+ -

z

z

f(w)g'(w) dw,

0

on HP considered in [16, 19] bounded on X?

9 E BMOA,

List of Symbols A (disk algebra) .. ............ .......... . ... ....... .... ..... ... p. 91 .4. (Aleksandrov measures assoCIated with .. (reproducIng kernel for 19*(H2)) .. .. .. .. .. .. .... ........ .. ............ p. 186 .eJL ••.••..•• ••..•. •.•••••• •••••••• • • . . • • • • . . • • . • • • • • • • • • • • •• • •• p. 15 £p (Lebesgue spaces on 1r) .............. .. . .. . ........... p 12 L 1 ,co (weak LI) ........ ....... ....... .. . . . .. . . . . . . . . ....... , ..... p. 35 >..f (distribution function for f) .. .. .. .. .. . .... ............. .. .. .. ... p. 13 Au (Lipschitz class) ....... ....... .. ........ ....... ......... .... . .. p. 62 m (Lebesgue measure on 1r). . . .. ....... . . . . . . . . . . . .. ............... . p. 12 ml (Lebesgue measure on JR) ....... ..... .. ....... ....... . ........ p. 163 M (Borel measures on 1r) . . . . . . . . . . . .. ........ ....... ........ . . . .. . p. 14 M(JR) (finite Borel measures on JR) .. , ........ ....... ....... .. ...... p. 163 255

256

LIST OF SYMBOLS

M+ (resp. M+(lR)) (positive measures on 'll' (resp. JR)) ................... p. 14 Ms (absolutely continuous measures) ..................... ... .. ......... p. 16 Ms (singular measures).............. ... .. ...................... . p. 16 M/HJ ... ........................................................... p. 83 9Jl(X) (multipliers of X)........... ... .. ................... .... . p. 115 M", (multiplication by