Function theory on planar domains

Front Cover......Page 1
Title Page......Page 4
Copyright Information......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 10
Notation and Numbering......Page 14
1.2. The Poisson Formula and Some Preliminaries......Page 18
1.3. Subharmonic Functions......Page 23
1.4. Solution of the Dirichlet Problem......Page 28
1.5. The Green\'s Function of a Domain......Page 33
1.6. Harmonic Measure......Page 36
1.7. Logarithmic Capacity......Page 41
Exercises......Page 52
2.2. The Uniformization Theorem......Page 55
2.3. Conditional Expectation and the Space N......Page 60
2.4. Harmonic Measure and L¹/K......Page 65
Exercises......Page 66
3.2. Basic Properties of H^{p}(Ω)......Page 68
3.3. Hp on the Unit Disc......Page 72
3.4. H^{p}/K and H^{p}(Ω)......Page 79
3.5. Null Sets and Essential Boundary Points for H^{p}(Ω)......Page 81
3.6. H^{∞}(Ω) Determines Ω*......Page 83
3.7. Weak Peak Points for H^{∞}(Ω)......Page 87
Additional Readings and Notes......Page 90
Exercises......Page 91
4.1. Introduction......Page 94
4.2. The Defect of ReR(Ω) in C_{r}(Γ)......Page 95
4.3. Measures Orthogonal to R(Ω)......Page 98
4.4. H^{p}(Ω)......Page 102
4.5. N Again......Page 110
4.6. Functions with Periods......Page 114
4.7. The Factorization of H^{p}(Ω) Functions......Page 120
Additional Readings and Notes......Page 123
Exercises......Page 124
5.1. The Ahlfors Function......Page 126
5.2. Blaschke Products......Page 133
5.3. Approximation by Inner Functions......Page 137
5.4. Pick-Nevanhnna Interpolation......Page 147
5.5. Interpolation Sequences......Page 158
5.6. The Maximum Principle for Multiple-Valued Bounded Analytic Functions......Page 168
Additional Readings and Notes......Page 177
Exercises......Page 178
6.2. Peak Points and Parts......Page 182
6.3. The Fibers of M(Ω)......Page 188
6.4. Distinguished Homomorphisms......Page 194
6.5. The Shilov Boundary of H^{∞}(Ω)......Page 207
6.6. The Corona Theorem......Page 212
Additional Readings and Notes......Page 219
Exercises......Page 220
7.1. The Isometries of H^{p}(Ω)......Page 223
7.2. Self-Mappings of a Domain......Page 234
7.3. General Properties of Composition Operators......Page 245
7.4. Compact Composition Operators on H^{∞}(Ω)......Page 249
7.5. Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 1. The H^{∞} Case......Page 254
7.6. Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 2. The H² Case......Page 268
Exercises......Page 275
Bibliography......Page 279
Index......Page 284