Theory of H[superscript p] spaces (eBook)

Front Cover 1
Theory of Hp Spaces, Volume 38 4
Copyright Page 5
Contents 8
Preface 12
CHAPTER 1. HARMONIC AND SUBHARMONIC FUNCTIONS 16
1.1. Harmonic Functions 16
1.2. Boundary Behavior of Poisson-Stieltjes Integrals 19
1.3. Subharmonic Functions 22
1.4. Hardy's Convexity Theorem 23
1.5. Subordination 25
1.6. Maximal Theorems 26
Exercises 28
CHAPTER 2. BASIC STRUCTURE OF HP FUNCTIONS 30
2.1. Boundary Values 30
2.2. Zeros 33
2.3. Mean Convergence to Boundary Values 35
2.4. Canonical Factorization 38
2.5. The Class N + 40
2.6. Harmonic Majorants 43
Exercises 44
CHAPTER 3. APPLICATIONS 48
3.1. Poisson Integrals and H1 48
3.2. Description of the Boundary Functions 50
3.3. Cauchy and Cauchy-Stieltjes Integrals 54
3.4. Analytic Functions Continuous in [ z ] ? 1 57
3.5. Applications to Conformal Mapping 58
3.6. Inequalities of Fejér-Riesz, Hilbert, and Hardy 61
3.7. Schlicht Functions 64
Exercises 66
CHAPTER 4. CONJUGATE FUNCTIONS 68
4.1. Theorem of M. Riesz 68
4.2. Kolmogorov's Theorem 71
4.3. Zygmund's Theorem 73
4.4. Trigonometric Series 76
4.5. The Conjugate of an h1 Function 78
4.6. The Case p < 1: A Counterexample
Exercises 82
CHAPTER 5. MEAN GROWTH AND SMOOTHNESS 86
5.1. Smoothness Classes 86
5.2. Smoothness of the Boundary Function 89
5.3. Growth of a Function and its Derivative 94
5.4. More on Conjugate Functions 97
5.5. Comparative Growth of Means 99
5.6. Functions with Hp Derivative 103
Exercises 105
CHAPTER 6. TAYLOR COEFFICIENTS 108
6.1. Hausdorff–Young Inequalities 108
6.2. Theorem of Hardy and Littlewood 110
6.3. The Case p? 1 113
6.4. Multipliers 114
Exercises 121
CHAPTER 7. Hp AS A LINEAR SPACE 124
7.1. Quotient Spaces and Annihilators 125
7.2. Representation of Linear Functionals 127
7.3. Beurling's Approximation Theorem 128
7.4. Linear Functionals on Hp, 0 < p <
7.5. Failure of the Hahn-Banach Theorem 133
7.6. Extreme Points 138
Exercises 141
CHAPTER 8. EXTREMAL PROBLEMS 144
8.1. The Extremal Problem and its Dual 144
8.2. Uniqueness of Solutions 147
8.3. Counterexamples in the Case p = 1 149
8.4. Rational Kernels 151
8.5. Examples 154
Exercises 158
CHAPTER 9. INTERPOLATION THEORY 162
9.1. Universal Interpolation Sequences 162
9.2. Proof of the Main Theorem 164
9.3. The Proof for p < 1
9.4. Uniformly Separated Sequences 169
9.5. A Theorem of Carleson 171
Exercises 179
CHAPTER 10. Hp SPACES OVER GENERAL DOMAINS 182
10.1. Simply Connected Domains 182
10.2. Jordan Domains with Rectifiable Boundary 184
10.3. Smirnov Domains 188
10.4. Domains not of Smirnov Type 191
10.5. Multiply Connected Domains 194
Exercises 198
CHAPTER 11. Hp SPACES OVER A HALF-PLANE 202
11.1. Subharmonic Functions 203
11.2. Boundary Behavior 204
11.3. Canonical Factorization 207
11.4. Cauchy Integrals 209
11.5. Fourier Transforms 210
Exercises 212
CHAPTER 12. THE CORONA THEOREM 216
12.1. Maximal Ideals 216
12.2. Interpolation and the Corona Theorem 218
12.3. Harmonic Measures 222
12.4. Construction of the Contour ? 226
12.5. Arclength of ? 230
Exercises 233
Appendix A. Rademacher Functions 236
Appendix B. Maximal Theorems 246
References 252
Author Index 268
Subject Index 271
Pure and Applied Mathematics 275