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. M+(M)) (positive measures on T (resp. R)) p. 14 Ms (absolutely continuous measures) p. 16 Ms (singular measures) p. 16 M/Hl p. 83 m{%) (multipliers of DC) p. 115 M^ (multiplication by 0) p. 115 /la (Aleksandrov measure) p. 202 HE p. 37 N (natural numbers) {1, 2, 3, • • • } p. 11 No (natural numbers along with zero) {0,1, 2, • • • } p. 11 7V+ (Smirnov class) p. 35 Pji (Poisson integral of a measure \i on T) p. 30 Tfi (Poisson integral of a measure /i on M) p.232 P$ (orthogonal projection of H2 onto $H2) p. 185 Pz (Poisson kernel) p. 30 Q/J, (conjugate Poisson integral) p. 30 Qz (conjugate Poisson kernel) p. 30 Rf (representing measures for a Cauchy transform /) p. 42

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A-integral, 48 Beurling's theorem, 179, 251 absolutely continuous measure, 15 Blaschke condition, 27 Adams, D., 59 Blaschke product adjoint, 21 Caratheodory's theorem, 152 Ahern, P., 27, 30, 192 definition, 27 Ahlfors, L., 28, 103, 110 Frostman's theorem, 27 Aleksandrov multiplier, 130 measure, see also Clark measure Tumarkin's theorem, 152 disintegration theorem, 212, 216, 242 Bochner integral, 121 Aleksandrov, A., 1, 4, 6, 8, 36, 48, 49, 67, Boole's lemma, 165 102, 109, 183, 188, 215, 217, 228-230, Boole, G., 6, 164 244, 250-252 Borel Aleman, A., 179, 180, 185, 253 algebra, 12 algebra, 11 function, 12 cr-algebra, 11 measure, 14 Aliev, R., 54 sets, 12 analytic self-map, 28, 201 transform, 231, 241 Andersson, M., 36 bounded mean oscillation, 69 angular derivative, 28, 192, 208, 211, 216 bounded operator, 20 annihilator, 18 bounded type, 34 Aronszajn, N., 9, 241 Bourdon, P., Ill, 253 atoms (of a measure), 17 Bockarev, S., 96 Brennan, J., 9 backward shift, see also Clark measure Brown, L., 93 2 H Burkholder, D., 36 analytic continuation, 182 basis, 192 Calderon, A., 65, 67, 163 density theorem, 187 capacity, 59 Douglas-Shapiro-Shields theorem, 181 Caratheodory, C, 152 kernel function, 186, 192 Carleson pseudocontinuation, 181 interpolation theorem, 38 spectrum, 184 measure, 37, 133 HP, 183, 192 square, 37 X, 252 Carleson, L., 5, 38, 96 other spaces, 185 carrier (of a measure), 16, 232 Baernstein, A., 79, 80 Cauchy Bagemihl, F., 26, 43 A-integral formula, 49 balanced hull, 18 integral formula, 47 Banach-Alaoglu theorem, 19, 24 Stieltjes integral, 1, 59 Bary, N., 54 Cauchy transform basis, 95, 192 A-integral formula, 49 Bell, S., 9 Aleksandrov's characterization, 102, 127, Besicovich covering theorem, 233 190, 244 best constants, 79, 82 and C(T), 72

267 268 INDEX

and L\ 68 norm, 204 and L°°, 68, 69 normalized Cauchy transform, 227 and LP, 65 point mass, 208, 211, 216, 222, 230, 243 and duality, 78 Clark, D., 1, 6, 7, 27, 30, 192, 193, 197, 199, and weighted Lp, 76 201, 220 boundary behavior, 42, 58 closed graph theorem, 21 Cauchy integral formula, 47 Cohn, W., 192 Clark measure, 203 Collingwood, E., 26, 27 definition, 41 composition operator, 250, 253 distribution function, 172, 222 compression, see also forward shift F-property, 127, 243 conditional expectation operator, 215 Fatou's jump theorem, 55 conjugate geometric characterization, 111 Poisson integral, 30 Havin's characterization, 99 function, 32, 62, 65, 69, 72, 73, 80 Lipschitz classes, 62 continuous M. Riesz's theorem, 65 measure, 17 multiplier, 115 operator, 20 non-tangential limit, 44 convex norm, 83 balanced hull, 18 normalized Cauchy transform, 227 hull, 18 Plemelj formula, 56 Conway, J., ix, 9, 17, 20 pointwise estimate, 87 coset, 18 principal value integral, 56 Cowen, C, 28, 209, 250 representing measures, 42 cyclic, 21, 195, 200, 236 space of Cauchy transforms, 41 backward shift, 252 Davis, B., 80, 82 basis, 97 Day, M., 12 composition operator, 253 deBranges-Rovnyak space, 229 duality, 89, 91 decreasing rearrangement, 13, 49 forward shift, 250 del Rio, R., 243 Lebesgue decomposition, 88 Delbaen, F., 95 multiplier, 115 Denjoy, A., 48 reflexive, 90 derivative (of a measure), 15 separable, 89, 93 Diestel, J., 94-97, 121, 193 Toeplitz operator, 252 discrete measure, 17 weak topology, 95 disintegration theorem, see also Aleksandrov's weak-* topology, 91 disintegration theorem, 242 weakly sequentially complete, 95 disk algebra, 91, 117 Tumarkin's characterization, 101 distribution function, 13, see also decreasing Cauchy, A., 1, 46, 60 rearrangement Cesaro Boole's lemma, 165 operator, 250 Cauchy transform, 172, 222 sum, 24 conjugate function, 73, 80, 222 Choquet, G., 25 Herglotz integral, 170 Cima, J., 26, 67, 111, 112, 181-183, 185, 250, Hilbert transform, 163, 176 252, 253 Hruscev-Vinogradov theorem, 164, 170 Clark measure normalized Cauchy transform, 227 Aleksandrov's disintegration theorem, 212, Poltoratski's distribution theorem, 222 216, 242 Stein-Weiss theorem, 176 angular derivative, 208, 211, 216 Tsereteli's theorem, 169 carrier, 207 Donoghue, W., 9, 222, 241 Cauchy transform, 203 Doob, J., 84 composition operator, 253 Douglas, R., 181, 182 deBranges-Rovnyak space, 229 dual extremal problems, 84 definition, 202 duality Fourier coefficients, 204 A, 91 Herglotz integral, 202 H1, 78 Lebesgue decomposition, 205 HP, 78 INDEX 269

X, 91, 95 Hankel operator, 145

Xai 89 Hardy space, see also forward shift, back­ i?*(ifp), 183 ward shift, Toeplitz operator Duren, P., 27, 31, 32, 36, 41, 45, 65, 68, 84, classical operators, 249 94, 111, 179, 180, 250 definition, 32 Dyakonov, K., 187 Riesz factorization, 34 Smirnov class, 35 Enflo, P., 96 standard facts, 33 Evans, L., 11, 15, 16, 233 Hardy's inequality, 68 Hardy, G., 36, 57, 62, 76 F-property, 127, 129, 151, 157, 243 harmonic majorant, 103 F. and M. Riesz theorem, 34 Hausdorff, F., 214 factorization Havin, V., 95, 99, 109, 122 bounded analytic function, 27 Havinson, S. Ja., 84 functions of bounded type, 34 Hayman, W., 38, 103 Hardy space functions, 34 Hedberg, L., 59 Fatou's theorem Helson, H., 76 jump theorem, 55 Herglotz on non-tangential limits, 26 integral, 30, 170, 202 on Poisson integrals, 31 theorem, 32, 201 Fatou, P., 2, 26, 31, 55 Herglotz, G., 32 Fefferman, C, 79 Hewitt, E., 16, 17 Fefferman-Stein duality theorem, 79 Hilbert transform, see also distribution func­ Fejer, L., 24 tion, 163, 164, 169, 170 Fomin, S., 11 Hobson, E., 214 forward shift 2 Hoffman, K., 31, 32, 38, 68, 93, 252 H Hollenbeck, B., 3, 67, 79 Beurling's theorem, 179 Hruscev, S., 3, 5, 6, 110, 127-130, 137, 164, compression, 194 170, 190 X, 250, 251 Hunt, R, 76 perturbations, 196 Fourier coefficient, 24 inner function Frost man's theorem angular derivative, 29, 192 on angular derivatives, 29 Clark measure, 202, 216, 222 on radial limits, 27, 130 definition, 27 Frostman, O., 27, 29, 130, 160 kernel function, 192 Fuentes, S., 243 measure preserving, 171, 215 multiplier, 129 Gaier, D., 86 non-tangential limits, 27 Gamelin, T., 9, 80 spectrum, 182 Garcia, S., 2, 54, 199 interpolating sequence, 37, 133 Gariepy, R., 11, 15, 16 Garnett, J., 9, 32, 36, 44, 69, 70, 72, 76, 79, Jaksic, V., 231 84, 86, 95, 103, 109, 141, 153, 164, 176, Janson, S., 249, 253 180, 182 John-Nirenberg inequality, 70 Garsia norm, 69 Jordan decomposition theorem, 14 Gelfer domain, 112 Julia-Caratheodory theorem, 28, 209-211 Gelfer, S., 112 Goldstine, H., 20 Kahane, J., 42 Goluzin, G., 57, 62 Kakutani, S., 95 Goluzina, M., 120, 122, 124, 130, 245 Kalton, N., 35 Grafakos, L., 13 Katznelson, Y., 176 Gundy, R., 36 Kelley, J., 94 GurariT, V., 127 Kennedy, P., 103 kernel function, 185, 192, 199 Holder's inequality, 12 Khavinson, D., 187 Hahn-Banach Kisljakov, S., 95 extension theorem, 17 Kolmogorov, A., 3, 5, 11, 48, 73, 80, 163, separation theorem, 17 227 270 INDEX

Koosis, P., 32, 36, 69, 70, 73, 79, 95, 164, Morera, G., 1, 60 207 Muckenhoupt, B., 76 Korenblum, B., 180, 252 multiplier HP, 116 Landau, E., 86, 125 BMO, 117 Last, Y., 231 definition, 115 Lebesgue Dirichlet space, 116 decomposition theorem, 16 F-property, 127, 129, 151, 157 and space of Cauchy transforms, 88 Frostman condition, 130 differentiation theorem, 15 inner function, 129 measurable functions, 12 multiplier norm, 115 measure, 12 necessary conditions, 118 Lebesgue, H., 24 non-tangential limits, 119, 120 Lieb, E., 234 sufficient conditions, 122 Lindelof, E., 26 Toeplitz operator, 117 Lipschitz class, 62, 250 Muskhelishvili, N., 9 Littlewood subordination theorem, 79, 250 Littlewood, J., 26, 36, 41, 57, 62, 76, 79, 250 Naftalevic, A, 38 Livsic, M., 184 Nagel, A., 58 Lohwater, A., 26, 27, 43 Natanson, L, 11 Loomis, L., 163, 164 Nazarov, F., 77 Loss, M., 234 Nevanlinna class, 34 Lotto, B., 129 Nevanlinna, R., 208 Newman, D., 38, 68 MacCluer, B., 28, 250 Nikol'skii, N., 179, 181, 194, 195 MacGregor, T., 9, 112 non-tangential limit Markushevich, A., 101 p H functions, 33 Matheson, A., 132, 180, 217, 226, 252, 253 Cauchy transform, 44 Maurey, B., 96 definition, 25 maximal function, 36, 233 Fatou's theorem, 26 Mazur's theorem, 19 Frostman's theorem, 27 Maz'ya, V., 116 Lindelof's theorem, 26 McDonald, G., 138 multiplier, 119, 120 McKenna, P., 137 normalized Cauchy transform, 231 measure Privalov's uniqueness theorem, 26 absolutely continuous, 15 non-tangential maximal function, 36 atoms, 17 norm Banach-Alaoglu theorem, 24 LP, 12 Borel, 14 Cauchy transform, 83 carrier, 16, 232 operator, 20 Cesaro sum, 24 total variation, 14 continuous, 17 normalized Cauchy transform derivative, 15 definition, 227 discrete, 17 distribution function, 227 Fourier coefficients, 24 mapping properties, 228-230, 240 Jordan decomposition, 14 non-tangential limits, 231 Lebesgue, 12 Lebesgue decomposition, 16 operator positive, 14 adjoint, 21 Radon-Nikodym derivative, 15 bounded, 20 Riesz representation theorem, 15 norm, 20 singular, 15 spectral theorem, 22 support, 16 spectrum, 21 total variation, 14 oricyclic limit, 58 Megginson, R., 17, 96, 193 outer function, 27, 34 Minkowski's inequality, 12 Moeller, J., 184 Pajot, H., 9 Monotone class theorem, 213 Paley, R., 193 Mooney, M., 95 Parthasarathy, K., 23 INDEX 271

Peck, N., 35 Schauder basis, 95 Peetre, J., 249, 253 Schauder, J., 96 Peller, V., 125 second dual, 19 perturbations Seidel, W., 26, 43 Clark's theorem, 220 Seip, K., 36 of self-adjoint operators, 242 self-adjoint operator, 22 unitary, 196, 197, 199 spectral theorem, 22 Pelczyhski, A., 79, 96 self-map, 28, 201 Pichorides, S., 3, 80, 82, 103 Semmes, S., 249, 253 Piranian, G., 26, 43 separable, 20 Plemelj's formula, 56 space of Cauchy transforms, 89, 93 Plemelj, J., 1, 2, 56, 60 space of measures, 24 Poincare, EL, 43 separated, 37, 133 Poisson integral, 30, 232 Shapiro, H. S., 36, 84, 179, 181, 182, 185, Poisson-Stieltjes integral, 31 187 polar, 18 Shapiro, J., 28, 58, 209, 250 Poltoratski, A., 1, 3, 6, 8, 199, 222, 226, 231, Shaposhnikova, T., 116 240, 243, 244, 246 Shields, A., 36, 93, 94, 181, 182 Pommerenke, C, 209 shift operator, see also forward shift, back­ pre-polar, 18 ward shift principle of uniform boundedness, 17 Shimorin, S., 180 Privalov's theorem Shirokov, N., 127, 180 on Lipschitz classes, 62 Silverstein, M., 36 principle value of Cauchy integrals, 56 Simon, B., 9, 241 uniqueness theorem, 26 singular inner function, 27 Privalov, I., 1, 3, 9, 26, 56, 60, 62 singular measure, 15 pseudo-hyperbolic distance, 37 Siskakis, A., 250, 253 pseudocontinuation, 181, 244 Smirnov class, 35 pure point spectrum, 22, 222, 243 Smirnov, V., 2, 34, 35, 43, 45, 62 Putinar, M., 199 Smithies, F., 46 Sokhotski, Y., 1, 56, 60 quotient space, 18 Spanne, S., 69 radial spectral theorem, 22, 218, 236, 241 limit, 25 spectrum maximal function, 36 backward shift, 184 Radon-Nikodym compression, 196 derivative, 15 inner function, 182 theorem, 15 kernel function, 192 reflexive, 20 operator, 21 space of Cauchy transforms, 90 pure point spectrum, 22, 222, 243 representing measures, 42 restriction of backward shift, 184 Richter, S., 179, 180 spectral theorem, 22, 218, 236, 241 Riesz unitary perturbations, 222 projection, 65, 67 Stegenga, D., 116, 117, 249 representation theorem, 12, 15 Stein, E., 69, 79, 82, 164, 176, 233 Riesz, F., 20, 34, 193 Stein, P., 65 Riesz, M., 3, 29, 34, 65, 164, 210 Stessin, M., 226 Roberts, J., 35 Stoltz region, 25 Rogosinski, W., 84 Stromberg, K., 16, 17 Romberg, B., 94 Stroock, D., 23 Ross, W., 26, 67, 179, 181-183, 185, 250, 252 subharmonic function, 103 Rudin, W., ix, 11, 15-17, 20, 31, 58, 62, 91, Sundberg, C, 79, 138, 180 180, 233, 252 support (of a measure), 16 Rybkin, A., 54 symmetric derivative, 15 Ryff, J., 13 Sz.-Nagy, B., 20, 193 Sz.-Nagy-Foia§ functional model, 194 Sarason, D., 2, 54, 72, 129, 194, 218, 226, Szego's theorem, 22 229 Szego, G., 22, 76 272 INDEX INDEX 269

tangential boundary behavior, 58 X, 91, 95 Hankel operator, 145 Thomson, J., 9 Xa, 89 Hardy space, see also forward shift, back­ Titchmarsh, E., 48, 164 tf*(#P), 183 ward shift, Toeplitz operator Toeplitz operator, see also multiplier Duren, P., 27, 31, 32, 36, 41, 45, 65, 68, 84, classical operators, 249 A, 117 94, 111, 179, 180, 250 definition, 32 1 if , 117, 249 Dyakonov, K., 187 Riesz factorization, 34 H°°, 117 Smirnov class, 35 HP, 116, 250 Enflo, P., 96 standard facts, 33 X, 252 Evans, L., 11, 15, 16, 233 Hardy's inequality, 68 Tolsa, X., 9 Hardy, G., 36, 57, 62, 76 topology F-property, 127, 129, 151, 157, 243 harmonic major ant, 103 weak, 19, 95 F. and M. Riesz theorem, 34 Hausdorff, F., 214 weak-*, 19, 91 factorization Havin, V., 95, 99, 109, 122 total variation, 14 bounded analytic function, 27 Havinson, S. Ja., 84 Treil, S., 77 functions of bounded type, 34 Hayman, W., 38, 103 Tsereteli, O., 3, 5, 6, 76, 169 Hardy space functions, 34 Hedberg, L., 59 Tumarkin, G., 4, 101, 152, 154 Fatou's theorem Helson, H., 76 Twomey, J., 58, 59 jump theorem, 55 Herglotz on non-tangential limits, 26 integral, 30, 170, 202 Uhl, J., 121 on Poisson integrals, 31 theorem, 32, 201 Ul'yanov, P., 2, 48, 49, 54 Fatou, P., 2, 26, 31, 55 Herglotz, G., 32 uniform boundedness principle, 17 Fefferman, C., 79 Hewitt, E., 16, 17 uniformly separated, 37, 133 Fefferman-Stein duality theorem, 79 Hilbert transform, see also distribution func­ unitary operator, 21 Fejer, L., 24 tion, 163, 164, 169, 170 spectral theorem, 21 Fomin, S., 11 Hobson, E., 214 unitary perturbations, see also perturbations forward shift Hoffman, K., 31, 32, 38, 68, 93, 252 H2 vanishing mean oscillation, 72 Hollenbeck, B., 3, 67, 79 Beurling's theorem, 179 Vasjunin, V., 130 Hruscev, S., 3, 5, 6, 110, 127-130, 137, 164, compression, 194 Verbitsky, L, 3, 67, 79 170, 190 X, 250, 251 Vinogradov, S., 3, 5, 6, 117, 122, 127-130, Hunt, R, 76 perturbations, 196 137, 164, 170, 190, 249 Fourier coefficient, 24 inner function weak topology, 19, 94 Frostman's theorem angular derivative, 29, 192 weak-* Schauder basis, 96 on angular derivatives, 29 Clark measure, 202, 216, 222 weak-* topology, 19, 91 on radial limits, 27, 130 definition, 27 weak-L1, 35 Frostman, O., 27, 29, 130, 160 kernel function, 192 weakly sequentially complete, 94, 95 Fuentes, S., 243 measure preserving, 171, 215 Weiss, G., 176 multiplier, 129 Gaier, D., 86 Wheeden, R., 11, 13, 65, 76 non-tangential limits, 27 Gamelin, T., 9, 80 Wiener algebra, 127 spectrum, 182 Wiener, N., 193 Garcia, S., 2, 54, 199 interpolating sequence, 37, 133 Williams, D., 213 Gariepy, R., 11, 15, 16 Wojtaszczyk, P., 17, 94-96 Garnett, J., 9, 32, 36, 44, 69, 70, 72, 76, 79, Jaksic, V., 231 Wolff, T., 9, 241 84, 86, 95, 103, 109, 141, 153, 164, 176, Janson, S., 249, 253 180, 182 John-Nirenberg inequality, 70 Zhu, K., 62 Garsia norm, 69 Jordan decomposition theorem, 14 Zygmund, A., 11, 13, 32, 42, 62, 64, 65, 68, Gelfer domain, 112 Julia-Caratheodory theorem, 28, 209-211 123, 163 Gelfer, S., 112 Goldstine, H., 20 Kahane, J., 42 Goluzin, G., 57, 62 Kakutani, S., 95 Goluzina, M., 120, 122, 124, 130, 245 Kalton, N., 35 Grafakos, L., 13 Katznelson, Y., 176 Gundy, R., 36 Kelley, J., 94 Gurarii, V., 127 Kennedy, P., 103 kernel function, 185, 192, 199 Holder's inequality, 12 Khavinson, D., 187 Hahn-Banach Kisljakov, S., 95 extension theorem, 17 Kolmogorov, A., 3, 5, 11, 48, 73, 80, 163, separation theorem, 17 227 Titles in This Series

125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 123 Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck's FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups I. Structure of strongly quasithin K-groups, 2004 110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003 104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 TITLES IN THIS SERIES

92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, second edition, 2004 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999

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