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The Cauchy Transform Mathematical Surveys and Monographs http://dx.doi.org/10.1090/surv/125 The Cauchy Transform Mathematical Surveys and Monographs Volume 125 The Cauchy Transform Joseph A. Citna Alec L. Matheson William T. Ross AttEM^ American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 30E20, 30E10, 30H05, 32A35, 32A40, 32A37, 32A60, 47B35, 47B37, 46E27. For additional information and updates on this book, visit www.ams.org/bookpages/surv-125 Library of Congress Cataloging-in-Publication Data Cima, Joseph A., 1933- The Cauchy transform/ Joseph A. Cima, Alec L. Matheson, William T. Ross. p. cm. - (Mathematical surveys and monographs, ISSN 0076-5376; v. 125) Includes bibliographical references and index. ISBN 0-8218-3871-7 (acid-free paper) 1. Cauchy integrals. 2. Cauchy transform. 3. Functions of complex variables. 4. Holomorphic functions. 5. Operator theory. I. Matheson, Alec L., 1946- II. Ross, William T., 1964- III. Title. IV. Mathematical surveys and monographs; no. 125. QA331.7:C56 2006 515/.43-dc22 2005055587 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 Contents Preface ix Overview 1 Chapter 1. Preliminaries 11 1.1. Basic notation 11 1.2. Lebesgue spaces 11 1.3. Borel measures 14 1.4. Some elementary functional analysis 17 1.5. Some operator theory 20 1.6. Functional analysis on the space of measures 22 1.7. Non-tangential limits and angular derivatives 25 1.8. Poisson and conjugate Poisson integrals 30 1.9. The classical Hardy spaces 32 1.10. Weak-type spaces 35 1.11. Interpolation and Carleson's theorem 36 1.12. Some integral estimates 39 Chapter 2. The Cauchy transform as a function 41 2.1. General properties of Cauchy integrals 41 2.2. Cauchy integrals and H1 46 2.3. Cauchy yl-integrals 48 2.4. Fatou's jump theorem 54 2.5. Plemelj's formula 56 2.6. Tangential boundary behavior 58 2.7. Cauchy-Stieltjes integrals 59 Chapter 3. The Cauchy transform as an operator 61 3.1. An early theorem of Privalov 62 3.2. Riesz's theorem 64 3.3. Bounded and vanishing mean oscillation 69 3.4. Kolmogorov's theorem 73 3.5. Weighted spaces 76 3.6. The Cauchy transform and duality 77 3.7. Best constants 79 3.8. The Hilbert transform 81 Chapter 4. Topologies on the space of Cauchy transforms 83 4.1. The norm topology 83 4.2. The weak-* topology 91 4.3. The weak topology 94 CONTENTS 4.4. Schauder bases 95 Chapter 5. Which functions are Cauchy integrals? 99 5.1. General remarks 99 5.2. A theorem of Havin 99 5.3. A theorem of Tumarkin 100 5.4. Aleksandrov's characterization 102 5.5. Other representation theorems 109 5.6. Some geometric conditions 110 Chapter 6. Multipliers and divisors 115 6.1. Multipliers and Toeplitz operators 115 6.2. Some necessary conditions 118 6.3. A theorem of Goluzina 120 6.4. Some sufficient conditions 122 6.5. The ^-property 127 6.6. Multipliers and inner functions 129 Chapter 7. The distribution function for Cauchy transforms 163 7.1. The Hilbert transform of a measure 163 7.2. Boole's theorem and its generalizations 164 7.3. A refinement of Boole's theorem 169 7.4. Measures on the circle 170 7.5. A theorem of Stein and Weiss 176 Chapter 8. The backward shift on H2 179 8.1. Beurling's theorem 179 8.2. A theorem of Douglas, Shapiro, and Shields 180 8.3. Spectral properties 184 8.4. Kernel functions 185 8.5. A density theorem 186 8.6. A theorem of Ahern and Clark 192 8.7. A basis for backward shift invariant subspaces 192 8.8. The compression of the shift 194 8.9. Rank-one unitary perturbations 196 Chapter 9. Clark measures 201 9.1. Some basic facts about Clark measures 201 9.2. Angular derivatives and point masses 208 9.3. Aleksandrov's disintegration theorem 211 9.4. Extensions of the disintegration theorem 212 9.5. Clark's theorem on perturbations 218 9.6. Some remarks on pure point spectra 221 9.7. Poltoratski's distribution theorem 222 Chapter 10. The normalized Cauchy transform 227 10.1. Basic definition 227 10.2. Mapping properties of the normalized Cauchy transform 227 10.3. Function properties of the normalized Cauchy transform 230 10.4. A few remarks about the Borel transform 241 CONTENTS vii 10.5. A closer look at the ^-property 243 Chapter 11. Other operators on the Cauchy transforms 249 11.1. Some classical operators 249 11.2. The forward shift 250 11.3. The backward shift 252 11.4. Toeplitz operators 252 11.5. Composition operators 253 11.6. The Cesaro operator 253 List of Symbols 255 Bibliography 257 Index 267 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 [email protected] [email protected] [email protected] List of Symbols A (disk algebra) p. 91 A(f> (Aleksandrov measures associated with (p) p. 202 BMO, BMOA (bounded mean oscillation) p. 69 C (complex numbers) p. 11 C (Riemann sphere) CU{oo} p. 11 C+ (upper half plane) p. 81 C(T) (continuous functions on T) p. 14 Cji p. 54 C(E) (interpolation constant for a sequence E cTb) p. 38 5(E) (uniform separation constant for a sequence EcP) p. 37 D (unit disk) p. 1 De (extended exterior disk) p. 54 D\i (symmetric derivative of a measure \i) p. 15 Ea p. 206 /* (decreasing rearrangement of /) p. 13 F^ (Borel transform of a measure /i) p. 231 S(/) (Garcia norm of a function) p. 69 j(E) (Carleson constant for a sequence EcD) p. 37 J-Cfi (Hilbert transform of a measure /i) p. 163 HJJL (Herglotz integral of a measure fi) p. 30 Hp (Hardy space) p. 32 p H {pe) (Hardy space of the exterior disk) p. 54 HP(T) p. 33 Hi (the set of / e H1 such that /(0) = 0) p. 34 F1'00, H^°° (analytic weak L1) p. 35 % (space of Cauchy transforms) p. 41 %a (Cauchy transforms of fi <^ m) p. 88 %s (Cauchy transforms of [i _L m) p. 88 Kfi (Cauchy transform of a measure fi) p. 41 k\ (reproducing kernel for ^*(iJ2)) p. 186 ^ p. 15 Lp (Lebesgue spaces on T) p. 12 L1'00 (weak L1) p. 35 Xf (distribution function for /) p. 13 Aa (Lipschitz class) p. 62 m (Lebesgue measure on T) p. 12 mi (Lebesgue measure on R) p. 163 M (Borel measures on T) p. 14 M(R) (finite Borel measures on R) p. 163 255 256 LIST OF SYMBOLS M+ (resp.
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