A blend of classical and modern techniques and viewpoints, this text examines harmonic and subharmonic functions, the basic structure of Hp functions, applications, conjugate functions, and mean growth and smoothness. Other subjects include Taylor coefficients, Hp as a linear space, interpolation theory, the corona theorem, and more. Information on Rademacher functions and maximal theorems appears in the appendixes. Essentially self-contained, with a list of exercises in each chapter, this text is appropriate for researchers or second- or third-year graduate students.1970 ed.
Author(s): Author Unknown
Series: Pure and Applied Mathematics
Publisher: Academic Press
Year: 1970
Language: English
Pages: 277
Theory of Hp Spaces, Volume 38......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 12
1.1. Harmonic Functions......Page 16
1.2. Boundary Behavior of Poisson-Stieltjes Integrals......Page 19
1.3. Subharmonic Functions......Page 22
1.4. Hardy's Convexity Theorem......Page 23
1.5. Subordination......Page 25
1.6. Maximal Theorems......Page 26
Exercises......Page 28
2.1. Boundary Values......Page 30
2.2. Zeros......Page 33
2.3. Mean Convergence to Boundary Values......Page 35
2.4. Canonical Factorization......Page 38
2.5. The Class N +......Page 40
2.6. Harmonic Majorants......Page 43
Exercises......Page 44
3.1. Poisson Integrals and H1......Page 48
3.2. Description of the Boundary Functions......Page 50
3.3. Cauchy and Cauchy-Stieltjes Integrals......Page 54
3.4. Analytic Functions Continuous in [ z ] ≤ 1......Page 57
3.5. Applications to Conformal Mapping......Page 58
3.6. Inequalities of Fejér-Riesz, Hilbert, and Hardy......Page 61
3.7. Schlicht Functions......Page 64
Exercises......Page 66
4.1. Theorem of M. Riesz......Page 68
4.2. Kolmogorov's Theorem......Page 71
4.3. Zygmund's Theorem......Page 73
4.4. Trigonometric Series......Page 76
4.5. The Conjugate of an h1 Function......Page 78
4.6. The Case p < 1: A Counterexample......Page 80
Exercises......Page 82
5.1. Smoothness Classes......Page 86
5.2. Smoothness of the Boundary Function......Page 89
5.3. Growth of a Function and its Derivative......Page 94
5.4. More on Conjugate Functions......Page 97
5.5. Comparative Growth of Means......Page 99
5.6. Functions with Hp Derivative......Page 103
Exercises......Page 105
6.1. Hausdorff–Young Inequalities......Page 108
6.2. Theorem of Hardy and Littlewood......Page 110
6.3. The Case p≤ 1......Page 113
6.4. Multipliers......Page 114
Exercises......Page 121
CHAPTER 7. Hp AS A LINEAR SPACE......Page 124
7.1. Quotient Spaces and Annihilators......Page 125
7.2. Representation of Linear Functionals......Page 127
7.3. Beurling's Approximation Theorem......Page 128
7.4. Linear Functionals on Hp, 0 < p < 1......Page 130
7.5. Failure of the Hahn-Banach Theorem......Page 133
7.6. Extreme Points......Page 138
Exercises......Page 141
8.1. The Extremal Problem and its Dual......Page 144
8.2. Uniqueness of Solutions......Page 147
8.3. Counterexamples in the Case p = 1......Page 149
8.4. Rational Kernels......Page 151
8.5. Examples......Page 154
Exercises......Page 158
9.1. Universal Interpolation Sequences......Page 162
9.2. Proof of the Main Theorem......Page 164
9.3. The Proof for p < 1......Page 168
9.4. Uniformly Separated Sequences......Page 169
9.5. A Theorem of Carleson......Page 171
Exercises......Page 179
10.1. Simply Connected Domains......Page 182
10.2. Jordan Domains with Rectifiable Boundary......Page 184
10.3. Smirnov Domains......Page 188
10.4. Domains not of Smirnov Type......Page 191
10.5. Multiply Connected Domains......Page 194
Exercises......Page 198
CHAPTER 11. Hp SPACES OVER A HALF-PLANE......Page 202
11.1. Subharmonic Functions......Page 203
11.2. Boundary Behavior......Page 204
11.3. Canonical Factorization......Page 207
11.4. Cauchy Integrals......Page 209
11.5. Fourier Transforms......Page 210
Exercises......Page 212
12.1. Maximal Ideals......Page 216
12.2. Interpolation and the Corona Theorem......Page 218
12.3. Harmonic Measures......Page 222
12.4. Construction of the Contour Γ......Page 226
12.5. Arclength of Γ......Page 230
Exercises......Page 233
Appendix A. Rademacher Functions......Page 236
Appendix B. Maximal Theorems......Page 246
References......Page 252
Author Index......Page 268
Subject Index......Page 271
Pure and Applied Mathematics......Page 275
