Contents
Preface ix
Numbering and notation xi
Chapter 1. Overview 1
Chapter 2. Classical boundary value results 9
2.1. Limits 9
2.2. Pseudocontinuations 13
Chapter 3. The Hardy space of the disk 17
3.1. Introduction 17
3.2.
Hp
and boundary values 17
3.3. Fourier analysis and
Hp
theory 21
3.4. The Cauchy transform 23
3.5. Duality 28
3.6. The Nevanlinna class 39
Chapter 4. The Hardy spaces of the upper-half plane 45
4.1. Motivation 45
4.2. Basic definitions 47
4.3. Poisson and conjugate Poisson integrals 49
4.4. Maximal functions 52
4.5. The Hilbert transform 54
4.6. Some examples 55
4.7. The harmonic Hardy space 60
4.8. Distributions 61
4.9. The atomic decomposition 72
4.10. Distributions and W 75
4.11. The space
HP{C\R)
76
Chapter 5. The backward shift on
Hp
for p G [1, oc) 81
5.1. The case p l 81
5.2. The first and most straightforward proof 82
5.3. The second proof - using Fatou's jump theorem 85
5.4. Application: Bergman spaces 87
5.5. Application: spectral properties 94
5.6. The third proof - using the Nevanlinna theory 97
5.7. Application: VMOA, BMOA, and L1/~H^ 99
5.8. The case p = 1 101
5.9. Cyclic vectors 105
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