Theory of Hp Spaces

276 Pages·2000·3.726 MB·English

by  Peter L. Duren

THEORY OF HP SPACES
This is Volume 3 8 in
PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Editors: PAUL A. SMITH AND SAMUEL ElLENBERG
A complete list of titles in this series appears at the end of this volume
THEORY OF HP SPACES
Peter L. Duren
Department of Mathematics
University of Michigan
Ann Arbor, Michigan
Academic Press
New York and London
1970
COPYRIGHT© 1970, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY
OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM
THE PUBLISHERS.
ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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LmRARY OF CoNGRESs CATALOG CARD NUMBER: 74-117092
PRINTED IN THE UNITED STATES OF AMERICA
TO MY FATHER
William L. Duren
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CONTENTS
Preface, xi
1. HARMONIC AND SUBHARMONIC FUNCTIONS
1.1. Harmonic Functions, 1
1.2. Boundary Behavior of Poisson-Stieltjes Integrals, 4
1.3. Subharmonic Functions, 7
1.4. Hardy's Convexity Theorem, 8
1.5. Subordination, 1 0
1.6. Maximal Theorems, 11
Exercises, 13
2. BASIC STRUCTURE OF HP FUNCTIONS
2.1. Boundary Values, 15
2.2. Zeros, 18
2.3. Mean Convergence to Boundary Values, 20
2.4. Canonical Factorization, 23
2.5. The Class N+. 25
2.6. Harmonic Majorants, 28
Exercises, 29
3. APPLICATIONS
3.1. Poisson Integrals and H'. 33
3.2. Description of the Boundary Functions, 35
3.3. Cauchy and Cauchy-Stieltjes Integrals, 39
3.4. Analytic Functions Continuous in [z[ ~ 1, 42
3.5. Applications to Conformal Mapping, 43
3.6. Inequalities of Fejer-Riesz, Hilbert, and Hardy, 46
3.7. Schlicht Functions, 49
Exercises, 51
4. CONJUGATE FUNCTIONS
4.1. Theorem of M. Riesz, 53
4.2. Kolmogorov's Theorem, 56
vii
viii CONTENTS
4.3. Zygmund's Theorem, 58
4.4. Trigonometric Series, 61
4.5. The Conjugate of an h' Function, 63
4.6. The Case p < 1 : A Counterexample. 65
Exercises, 67
5. MEAN GROWTH AND SMOOTHNESS
5.1. Smoothness Classes, 71
5.2. Smoothness of the Boundary Function, 74
5.3. Growth of a Function and its Derivative, 79
5.4. More on Conjugate Functions, 82
5.5. Comparative Growth of Means, 84
5.6. Functions with HP Derivative, 88
Exercises. 90
6. TAYLOR COEFFICIENTS
6.1. Hausdorff-Young Inequalities, 93
6.2. Theorem of Hardy and Littlewood, 95
6.3. The Case p:::::: 1. 98
6.4. Multipliers, 99
Exercises. 106
7. HP AS A LINEAR SPACE
7.1. Quotient Spaces and Annihilators, 110
7.2. Representation of Linear Functionals, 112
7.3. Beurling's Approximation Theorem, 113
7 .4. Linear Functionals on HP, 0 < p < 1. 11 5
7.5. Failure of the Hahn-Banach Theorem, 118
7.6. Extreme Points, 123
Exercises. 126
8. EXTREMAL PROBLEMS
8.1. The Extremal Probl.em and its Dual, 129
8.2. Uniqueness of Solutions, 132
8.3. Counterexamples in the Case p = 1, 134
8.4. Rational Kernels, 136
8.5. Examples, 139
Exercises, 143
9. INTERPOLATION THEORY
9.1. Universal Interpolation Sequences, 147
9.2. Proof of the Main Theorem, 149
CONTENTS lx
9.3. The Proof for p < 1, 153
9.4. Uniformly Separated Sequences, 154
9.5. A Theorem of Carleson, 156
Exercises, 164
10, HP SPACES OVER GENERAL DOMAINS
1 0.1. Simply Connected Domains, 167
1 0.2. Jordan Domains with Rectifiable Boundary, 169
10 .3. Smirnov Domains, 173
10 .4. Domains not of Smirnov Type, 176
10 .5. Multiply Connected Domains, 179
Exercises. 183
11. HP SPACES OVER A HALF-PLANE
11.1. Subharmonic Functions, 188
11.2. Boundary Behavior, 189
11.3. Canonical Factorization, 192
11.4. Cauchy Integrals, 194
11.5. Fourier Transforms, 195
Exercises. 197
12. THE CORONA THEOREM
12.1. Maximal Ideals, 201
12.2. Interpolation and the Corona Theorem, 203
12.3. Harmonic Measures, 207
12.4. Construction of the Contour r, 211
r.
12.5. Arclength of 215
Exercises. 218
Appendix A. Rademacher Functions, 221
Appendix B. Maximal Theorems, 231
References, 237
Author Index, 253
Subject Index. 256
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