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The Cauchy Transform
http://dx.doi.org/10.1090/surv/125
Mathematical Surveys
and Monographs
Volume 125
At tEM^
The Cauchy Transform
Joseph A. Citna Alec L. Matheson Wil l iam T. Ross
Amer ican Mathemat ica l Society
EDITORIAL COMMITTEE Jer ry L. Bona Peter S. Landweber Michael G. Eas twood Michael P. Loss
J. T. Stafford, Chair
2000 Mathematics Subject Classification. P r imary 30E20, 30E10, 30H05, 32A35, 32A40, 32A37, 32A60, 47B35, 47B37, 46E27.
For addi t ional information and upda te s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 1 2 5
Library of Congress Cataloging-in-Publicat ion D a t a Cima, Joseph A., 1933-
The Cauchy transform/ Joseph A. Cima, Alec L. Matheson, William T. Ross. p. cm. - (Mathematical surveys and monographs, ISSN 0076-5376; v. 125)
Includes bibliographical references and index. ISBN 0-8218-3871-7 (acid-free paper) 1. Cauchy integrals. 2. Cauchy transform. 3. Functions of complex variables. 4. Holomorphic
functions. 5. Operator theory. I. Matheson, Alec L., 1946- II. Ross, William T., 1964- III. Title. IV. Mathematical surveys and monographs; no. 125.
QA331.7:C56 2006 515/.43-dc22 2005055587
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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.
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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
Chapter 5. Which functions are Cauchy integrals? 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
General remarks A theorem of Havin A theorem of Tumarkin Aleksandrov's characterization Other representation theorems Some geometric conditions
Chapter 6. Multipliers and divisors 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.
Multipliers and Toeplitz operators Some necessary conditions A theorem of Goluzina Some sufficient conditions The ^-property Multipliers and inner functions
Chapter 7. The distribution function for Cauchy transforms 7.1. 7.2. 7.3. 7.4. 7.5.
The Hilbert transform of a measure Boole's theorem and its generalizations A refinement of Boole's theorem Measures on the circle A theorem of Stein and Weiss
Chapter 8. The backward shift on H2
8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.
Beurling's theorem A theorem of Douglas, Shapiro, and Shields Spectral properties Kernel functions A density theorem A theorem of Ahern and Clark A basis for backward shift invariant subspaces The compression of the shift Rank-one unitary perturbations
Chapter 9. Clark measures 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.
Some basic facts about Clark measures Angular derivatives and point masses Aleksandrov's disintegration theorem Extensions of the disintegration theorem Clark's theorem on perturbations Some remarks on pure point spectra Poltoratski's distribution theorem
Chapter 10. The normalized Cauchy transform 10.1. 10.2. 10.3. 10.4.
Basic definition Mapping properties of the normalized Cauchy transform Function properties of the normalized Cauchy transform A few remarks about the Borel transform
95
99 99 99
100 102 109 110
115 115 118 120 122 127 129
163 163 164 169 170 176
179 179 180 184 185 186 192 192 194 196
201 201 208 211 212 218 221 222
227 227 227 230 241
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 E c P ) 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 E c D ) 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 Hp{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 F 1 ' 0 0 , 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 O F 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 <Tjv(/x) (iV-th Cesaro sum of a measure /i) p. 24 aa (singular part of an Aleksandrov measure) p. 205 <JF(E) (Frostman constant for a sequence ^ C O ) p. 130 s(E) (separation constant for a sequence E c D ) p. 37 S (shift on H2) p. 179 S# (compression of the shift) p. 194 T (unit circle) p. 1 T-T (co-analytic Toeplitz operator) p. 116 ua p. 201 Va p. 218 VM (normalized Cauchy transform) p. 227 VMO, VMOA (functions of vanishing mean oscillation) p. 72 tf*(iJP) p. 183 /+ (Riesz projection of / ) p. 61 H/IIP (LP (or HP) norm) p. 34 j2(n) (n-th Fourier coefficient of a measure /i) p. 24 f(ri) (n-th Fourier coefficient of an L1 function / ) p. 24 \\/JL\\ (total variation norm of a measure //) p. 14 Ji p. 80 Z (non-tangential limit) p. 33 Z (integers) p. 11
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Index
A-integral, 48 absolutely continuous measure, 15 Adams, D., 59 adjoint, 21 Ahern, P., 27, 30, 192 Ahlfors, L., 28, 103, 110 Aleksandrov
measure, see also Clark measure disintegration theorem, 212, 216, 242
Aleksandrov, A., 1, 4, 6, 8, 36, 48, 49, 67, 102, 109, 183, 188, 215, 217, 228-230, 244, 250-252
Aleman, A., 179, 180, 185, 253 algebra, 11
cr-algebra, 11 Aliev, R., 54 analytic self-map, 28, 201 Andersson, M., 36 angular derivative, 28, 192, 208, 211, 216 annihilator, 18 Aronszajn, N., 9, 241 atoms (of a measure), 17
backward shift, see also Clark measure H2
analytic continuation, 182 basis, 192 density theorem, 187 Douglas-Shapiro-Shields theorem, 181 kernel function, 186, 192 pseudocontinuation, 181 spectrum, 184
HP, 183, 192 X, 252 other spaces, 185
Baernstein, A., 79, 80 Bagemihl, F., 26, 43 balanced hull, 18 Banach-Alaoglu theorem, 19, 24 Bary, N., 54 basis, 95, 192 Bell, S., 9 Besicovich covering theorem, 233 best constants, 79, 82
Beurling's theorem, 179, 251 Blaschke condition, 27 Blaschke product
Caratheodory's theorem, 152 definition, 27 Frostman's theorem, 27 multiplier, 130 Tumarkin's theorem, 152
Bochner integral, 121 Boole's lemma, 165 Boole, G., 6, 164 Borel
algebra, 12 function, 12 measure, 14 sets, 12 transform, 231, 241
bounded mean oscillation, 69 bounded operator, 20 bounded type, 34 Bourdon, P., I l l , 253 Bockarev, S., 96 Brennan, J., 9 Brown, L., 93 Burkholder, D., 36
Calderon, A., 65, 67, 163 capacity, 59 Caratheodory, C , 152 Carleson
interpolation theorem, 38 measure, 37, 133 square, 37
Carleson, L., 5, 38, 96 carrier (of a measure), 16, 232 Cauchy
A-integral formula, 49 integral formula, 47 Stieltjes integral, 1, 59
Cauchy transform A-integral formula, 49 Aleksandrov's characterization, 102, 127,
190, 244 and C(T), 72
267
268 INDEX
and L \ 68 and L°°, 68, 69 and LP, 65 and duality, 78 and weighted Lp, 76 boundary behavior, 42, 58 Cauchy integral formula, 47 Clark measure, 203 definition, 41 distribution function, 172, 222 F-property, 127, 243 Fatou's jump theorem, 55 geometric characterization, 111 Havin's characterization, 99 Lipschitz classes, 62 M. Riesz's theorem, 65 multiplier, 115 non-tangential limit, 44 norm, 83 normalized Cauchy transform, 227 Plemelj formula, 56 pointwise estimate, 87 principal value integral, 56 representing measures, 42 space of Cauchy transforms, 41
backward shift, 252 basis, 97 composition operator, 253 duality, 89, 91 forward shift, 250 Lebesgue decomposition, 88 multiplier, 115 reflexive, 90 separable, 89, 93 Toeplitz operator, 252 weak topology, 95 weak-* topology, 91 weakly sequentially complete, 95
Tumarkin's characterization, 101 Cauchy, A., 1, 46, 60 Cesaro
operator, 250 sum, 24
Choquet, G., 25 Cima, J., 26, 67, 111, 112, 181-183, 185, 250,
252, 253 Clark measure
Aleksandrov's disintegration theorem, 212, 216, 242
angular derivative, 208, 211, 216 carrier, 207 Cauchy transform, 203 composition operator, 253 deBranges-Rovnyak space, 229 definition, 202 Fourier coefficients, 204 Herglotz integral, 202 Lebesgue decomposition, 205
norm, 204 normalized Cauchy transform, 227 point mass, 208, 211, 216, 222, 230, 243
Clark, D., 1, 6, 7, 27, 30, 192, 193, 197, 199, 201, 220
closed graph theorem, 21 Cohn, W., 192 Collingwood, E., 26, 27 composition operator, 250, 253 compression, see also forward shift conditional expectation operator, 215 conjugate
Poisson integral, 30 function, 32, 62, 65, 69, 72, 73, 80
continuous measure, 17 operator, 20
convex balanced hull, 18 hull, 18
Conway, J., ix, 9, 17, 20 coset, 18 Cowen, C , 28, 209, 250 cyclic, 21, 195, 200, 236
Davis, B., 80, 82 Day, M., 12 deBranges-Rovnyak space, 229 decreasing rearrangement, 13, 49 del Rio, R., 243 Delbaen, F., 95 Denjoy, A., 48 derivative (of a measure), 15 Diestel, J., 94-97, 121, 193 discrete measure, 17 disintegration theorem, see also Aleksandrov's
disintegration theorem, 242 disk algebra, 91, 117 distribution function, 13, see also decreasing
rearrangement Boole's lemma, 165 Cauchy transform, 172, 222 conjugate function, 73, 80, 222 Herglotz integral, 170 Hilbert transform, 163, 176 Hruscev-Vinogradov theorem, 164, 170 normalized Cauchy transform, 227 Poltoratski's distribution theorem, 222 Stein-Weiss theorem, 176 Tsereteli's theorem, 169
Donoghue, W., 9, 222, 241 Doob, J., 84 Douglas, R., 181, 182 dual extremal problems, 84 duality
A, 91 H1, 78 HP, 78
INDEX 269
X, 91, 95 Xai 89 i?*(ifp), 183
Duren, P., 27, 31, 32, 36, 41, 45, 65, 68, 84, 94, 111, 179, 180, 250
Dyakonov, K., 187
Enflo, P., 96 Evans, L., 11, 15, 16, 233
F-property, 127, 129, 151, 157, 243 F. and M. Riesz theorem, 34 factorization
bounded analytic function, 27 functions of bounded type, 34 Hardy space functions, 34
Fatou's theorem jump theorem, 55 on non-tangential limits, 26 on Poisson integrals, 31
Fatou, P., 2, 26, 31, 55 Fefferman, C , 79 Fefferman-Stein duality theorem, 79 Fejer, L., 24 Fomin, S., 11 forward shift
H2
Beurling's theorem, 179 compression, 194
X, 250, 251 perturbations, 196
Fourier coefficient, 24 Frost man's theorem
on angular derivatives, 29 on radial limits, 27, 130
Frostman, O., 27, 29, 130, 160 Fuentes, S., 243
Gaier, D., 86 Gamelin, T., 9, 80 Garcia, S., 2, 54, 199 Gariepy, R., 11, 15, 16 Garnett, J., 9, 32, 36, 44, 69, 70, 72, 76, 79,
84, 86, 95, 103, 109, 141, 153, 164, 176, 180, 182
Garsia norm, 69 Gelfer domain, 112 Gelfer, S., 112 Goldstine, H., 20 Goluzin, G., 57, 62 Goluzina, M., 120, 122, 124, 130, 245 Grafakos, L., 13 Gundy, R., 36 GurariT, V., 127
Holder's inequality, 12 Hahn-Banach
extension theorem, 17 separation theorem, 17
Hankel operator, 145 Hardy space, see also forward shift, back
ward shift, Toeplitz operator classical operators, 249 definition, 32 Riesz factorization, 34 Smirnov class, 35 standard facts, 33
Hardy's inequality, 68 Hardy, G., 36, 57, 62, 76 harmonic majorant, 103 Hausdorff, F., 214 Havin, V., 95, 99, 109, 122 Havinson, S. Ja., 84 Hayman, W., 38, 103 Hedberg, L., 59 Helson, H., 76 Herglotz
integral, 30, 170, 202 theorem, 32, 201
Herglotz, G., 32 Hewitt, E., 16, 17 Hilbert transform, see also distribution func
tion, 163, 164, 169, 170 Hobson, E., 214 Hoffman, K., 31, 32, 38, 68, 93, 252 Hollenbeck, B., 3, 67, 79 Hruscev, S., 3, 5, 6, 110, 127-130, 137, 164,
170, 190 Hunt, R, 76
inner function angular derivative, 29, 192 Clark measure, 202, 216, 222 definition, 27 kernel function, 192 measure preserving, 171, 215 multiplier, 129 non-tangential limits, 27 spectrum, 182
interpolating sequence, 37, 133
Jaksic, V., 231 Janson, S., 249, 253 John-Nirenberg inequality, 70 Jordan decomposition theorem, 14 Julia-Caratheodory theorem, 28, 209-211
Kahane, J., 42 Kakutani, S., 95 Kalton, N., 35 Katznelson, Y., 176 Kelley, J., 94 Kennedy, P., 103 kernel function, 185, 192, 199 Khavinson, D., 187 Kisljakov, S., 95 Kolmogorov, A., 3, 5, 11, 48, 73, 80, 163,
227
270 INDEX
Koosis, P., 32, 36, 69, 70, 73, 79, 95, 164, 207
Korenblum, B., 180, 252
Landau, E., 86, 125 Last, Y., 231 Lebesgue
decomposition theorem, 16 and space of Cauchy transforms, 88
differentiation theorem, 15 measurable functions, 12 measure, 12
Lebesgue, H., 24 Lieb, E., 234 Lindelof, E., 26 Lipschitz class, 62, 250 Littlewood subordination theorem, 79, 250 Littlewood, J., 26, 36, 41, 57, 62, 76, 79, 250 Livsic, M., 184 Lohwater, A., 26, 27, 43 Loomis, L., 163, 164 Loss, M., 234 Lotto, B., 129
MacCluer, B., 28, 250 MacGregor, T., 9, 112 Markushevich, A., 101 Matheson, A., 132, 180, 217, 226, 252, 253 Maurey, B., 96 maximal function, 36, 233 Mazur's theorem, 19 Maz'ya, V., 116 McDonald, G., 138 McKenna, P., 137 measure
absolutely continuous, 15 atoms, 17 Banach-Alaoglu theorem, 24 Borel, 14 carrier, 16, 232 Cesaro sum, 24 continuous, 17 derivative, 15 discrete, 17 Fourier coefficients, 24 Jordan decomposition, 14 Lebesgue, 12 Lebesgue decomposition, 16 positive, 14 Radon-Nikodym derivative, 15 Riesz representation theorem, 15 singular, 15 support, 16 total variation, 14
Megginson, R., 17, 96, 193 Minkowski's inequality, 12 Moeller, J., 184 Monotone class theorem, 213 Mooney, M., 95
Morera, G., 1, 60 Muckenhoupt, B., 76 multiplier
HP, 116 BMO, 117 definition, 115 Dirichlet space, 116 F-property, 127, 129, 151, 157 Frostman condition, 130 inner function, 129 multiplier norm, 115 necessary conditions, 118 non-tangential limits, 119, 120 sufficient conditions, 122 Toeplitz operator, 117
Muskhelishvili, N., 9
Naftalevic, A, 38 Nagel, A., 58 Natanson, L, 11 Nazarov, F., 77 Nevanlinna class, 34 Nevanlinna, R., 208 Newman, D., 38, 68 Nikol'skii, N., 179, 181, 194, 195 non-tangential limit
Hp functions, 33 Cauchy transform, 44 definition, 25 Fatou's theorem, 26 Frostman's theorem, 27 Lindelof's theorem, 26 multiplier, 119, 120 normalized Cauchy transform, 231 Privalov's uniqueness theorem, 26
non-tangential maximal function, 36 norm
LP, 12 Cauchy transform, 83 operator, 20 total variation, 14
normalized Cauchy transform definition, 227 distribution function, 227 mapping properties, 228-230, 240 non-tangential limits, 231
operator adjoint, 21 bounded, 20 norm, 20 spectral theorem, 22 spectrum, 21
oricyclic limit, 58 outer function, 27, 34
Pajot, H., 9 Paley, R., 193 Parthasarathy, K., 23
INDEX 271
Peck, N., 35 Peetre, J., 249, 253 Peller, V., 125 perturbations
Clark's theorem, 220 of self-adjoint operators, 242 unitary, 196, 197, 199
Pelczyhski, A., 79, 96 Pichorides, S., 3, 80, 82, 103 Piranian, G., 26, 43 Plemelj's formula, 56 Plemelj, J., 1, 2, 56, 60 Poincare, EL, 43 Poisson integral, 30, 232 Poisson-Stieltjes integral, 31 polar, 18 Poltoratski, A., 1, 3, 6, 8, 199, 222, 226, 231,
240, 243, 244, 246 Pommerenke, C , 209 pre-polar, 18 principle of uniform boundedness, 17 Privalov's theorem
on Lipschitz classes, 62 principle value of Cauchy integrals, 56 uniqueness theorem, 26
Privalov, I., 1, 3, 9, 26, 56, 60, 62 pseudo-hyperbolic distance, 37 pseudocontinuation, 181, 244 pure point spectrum, 22, 222, 243 Putinar, M., 199
quotient space, 18
radial limit, 25 maximal function, 36
Radon-Nikodym derivative, 15 theorem, 15
reflexive, 20 space of Cauchy transforms, 90
representing measures, 42 Richter, S., 179, 180 Riesz
projection, 65, 67 representation theorem, 12, 15
Riesz, F., 20, 34, 193 Riesz, M., 3, 29, 34, 65, 164, 210 Roberts, J., 35 Rogosinski, W., 84 Romberg, B., 94 Ross, W., 26, 67, 179, 181-183, 185, 250, 252 Rudin, W., ix, 11, 15-17, 20, 31, 58, 62, 91,
180, 233, 252 Rybkin, A., 54 Ryff, J., 13
Sarason, D., 2, 54, 72, 129, 194, 218, 226, 229
Schauder basis, 95 Schauder, J., 96 second dual, 19 Seidel, W., 26, 43 Seip, K., 36 self-adjoint operator, 22
spectral theorem, 22 self-map, 28, 201 Semmes, S., 249, 253 separable, 20
space of Cauchy transforms, 89, 93 space of measures, 24
separated, 37, 133 Shapiro, H. S., 36, 84, 179, 181, 182, 185,
187 Shapiro, J., 28, 58, 209, 250 Shaposhnikova, T., 116 Shields, A., 36, 93, 94, 181, 182 shift operator, see also forward shift, back
ward shift Shimorin, S., 180 Shirokov, N., 127, 180 Silverstein, M., 36 Simon, B., 9, 241 singular inner function, 27 singular measure, 15 Siskakis, A., 250, 253 Smirnov class, 35 Smirnov, V., 2, 34, 35, 43, 45, 62 Smithies, F., 46 Sokhotski, Y., 1, 56, 60 Spanne, S., 69 spectral theorem, 22, 218, 236, 241 spectrum
backward shift, 184 compression, 196 inner function, 182 kernel function, 192 operator, 21 pure point spectrum, 22, 222, 243 restriction of backward shift, 184 spectral theorem, 22, 218, 236, 241 unitary perturbations, 222
Stegenga, D., 116, 117, 249 Stein, E., 69, 79, 82, 164, 176, 233 Stein, P., 65 Stessin, M., 226 Stoltz region, 25 Stromberg, K., 16, 17 Stroock, D., 23 subharmonic function, 103 Sundberg, C , 79, 138, 180 support (of a measure), 16 symmetric derivative, 15 Sz.-Nagy, B., 20, 193 Sz.-Nagy-Foia§ functional model, 194 Szego's theorem, 22 Szego, G., 22, 76
272 INDEX
tangential boundary behavior, 58 Thomson, J., 9 Titchmarsh, E., 48, 164 Toeplitz operator, see also multiplier
A, 117 if1, 117, 249 H°°, 117 HP, 116, 250 X, 252
Tolsa, X., 9 topology
weak, 19, 95 weak-*, 19, 91
total variation, 14 Treil, S., 77 Tsereteli, O., 3, 5, 6, 76, 169 Tumarkin, G., 4, 101, 152, 154 Twomey, J., 58, 59
Uhl, J., 121 Ul'yanov, P., 2, 48, 49, 54 uniform boundedness principle, 17 uniformly separated, 37, 133 unitary operator, 21
spectral theorem, 21 unitary perturbations, see also perturbations
vanishing mean oscillation, 72 Vasjunin, V., 130 Verbitsky, L, 3, 67, 79 Vinogradov, S., 3, 5, 6, 117, 122, 127-130,
137, 164, 170, 190, 249
weak topology, 19, 94 weak-* Schauder basis, 96 weak-* topology, 19, 91 weak-L1, 35 weakly sequentially complete, 94, 95 Weiss, G., 176 Wheeden, R., 11, 13, 65, 76 Wiener algebra, 127 Wiener, N., 193 Williams, D., 213 Wojtaszczyk, P., 17, 94-96 Wolff, T., 9, 241
Zhu, K., 62 Zygmund, A., 11, 13, 32, 42, 62, 64, 65, 68,
123, 163
INDEX 269
X, 91, 95 Xa, 89 tf*(#P), 183
Duren, P., 27, 31, 32, 36, 41, 45, 65, 68, 84, 94, 111, 179, 180, 250
Dyakonov, K., 187
Enflo, P., 96 Evans, L., 11, 15, 16, 233
F-property, 127, 129, 151, 157, 243 F. and M. Riesz theorem, 34 factorization
bounded analytic function, 27 functions of bounded type, 34 Hardy space functions, 34
Fatou's theorem jump theorem, 55 on non-tangential limits, 26 on Poisson integrals, 31
Fatou, P., 2, 26, 31, 55 Fefferman, C., 79 Fefferman-Stein duality theorem, 79 Fejer, L., 24 Fomin, S., 11 forward shift
H2
Beurling's theorem, 179 compression, 194
X, 250, 251 perturbations, 196
Fourier coefficient, 24 Frostman's theorem
on angular derivatives, 29 on radial limits, 27, 130
Frostman, O., 27, 29, 130, 160 Fuentes, S., 243
Gaier, D., 86 Gamelin, T., 9, 80 Garcia, S., 2, 54, 199 Gariepy, R., 11, 15, 16 Garnett, J., 9, 32, 36, 44, 69, 70, 72, 76, 79,
84, 86, 95, 103, 109, 141, 153, 164, 176, 180, 182
Garsia norm, 69 Gelfer domain, 112 Gelfer, S., 112 Goldstine, H., 20 Goluzin, G., 57, 62 Goluzina, M., 120, 122, 124, 130, 245 Grafakos, L., 13 Gundy, R., 36 Gurarii, V., 127
Holder's inequality, 12 Hahn-Banach
extension theorem, 17 separation theorem, 17
Hankel operator, 145 Hardy space, see also forward shift, back
ward shift, Toeplitz operator classical operators, 249 definition, 32 Riesz factorization, 34 Smirnov class, 35 standard facts, 33
Hardy's inequality, 68 Hardy, G., 36, 57, 62, 76 harmonic major ant, 103 Hausdorff, F., 214 Havin, V., 95, 99, 109, 122 Havinson, S. Ja., 84 Hayman, W., 38, 103 Hedberg, L., 59 Helson, H., 76 Herglotz
integral, 30, 170, 202 theorem, 32, 201
Herglotz, G., 32 Hewitt, E., 16, 17 Hilbert transform, see also distribution func
tion, 163, 164, 169, 170 Hobson, E., 214 Hoffman, K., 31, 32, 38, 68, 93, 252 Hollenbeck, B., 3, 67, 79 Hruscev, S., 3, 5, 6, 110, 127-130, 137, 164,
170, 190 Hunt, R, 76
inner function angular derivative, 29, 192 Clark measure, 202, 216, 222 definition, 27 kernel function, 192 measure preserving, 171, 215 multiplier, 129 non-tangential limits, 27 spectrum, 182
interpolating sequence, 37, 133
Jaksic, V., 231 Janson, S., 249, 253 John-Nirenberg inequality, 70 Jordan decomposition theorem, 14 Julia-Caratheodory theorem, 28, 209-211
Kahane, J., 42 Kakutani, S., 95 Kalton, N., 35 Katznelson, Y., 176 Kelley, J., 94 Kennedy, P., 103 kernel function, 185, 192, 199 Khavinson, D., 187 Kisljakov, S., 95 Kolmogorov, A., 3, 5, 11, 48, 73, 80, 163,
227
Titles in This Series
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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 M a and Shouhong Wang, Geometric theory of incompressible flows with
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physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l and Larry Smith , Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:
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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
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1999 64 Rene A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential
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