The Cauchy Transform

Preface ix

Overview 1

Chapter 1. Preliminaries 11

1.1. Basic notation 11

1.2. Lebesgue spaces 11

1.3. Borel measures 14

1.4. Some elementary functional analysis 17

1.5. Some operator theory 20

1.6. Functional analysis on the space of measures 22

1.7. Non-tangential limits and angular derivatives 25

1.8. Poisson and conjugate Poisson integrals 30

1.9. The classical Hardy spaces 32

1.10. Weak-type spaces 35

1.11. Interpolation and Carleson’s theorem 36

1.12. Some integral estimates 39

Chapter 2. The Cauchy transform as a function 41

2.1. General properties of Cauchy integrals 41

2.2. Cauchy integrals and H1 46

2.3. Cauchy A-integrals 48

2.4. Fatou’s jump theorem 54

2.5. Plemelj’s formula 56

2.6. Tangential boundary behavior 58

2.7. Cauchy-Stieltjes integrals 59

Chapter 3. The Cauchy transform as an operator 61

3.1. An early theorem of Privalov 62

3.2. Riesz’s theorem 64

3.3. Bounded and vanishing mean oscillation 69

3.4. Kolmogorov’s theorem 73

3.5. Weighted spaces 76

3.6. The Cauchy transform and duality 77

3.7. Best constants 79

3.8. The Hilbert transform 81

Chapter 4. Topologies on the space of Cauchy transforms 83

4.1. The norm topology 83

4.2. The weak-∗ topology 91

4.3. The weak topology 94

v

vi CONTENTS

4.4. Schauder bases 95

Chapter 5. Which functions are Cauchy integrals? 99

5.1. General remarks 99

5.2. A theorem of Havin 99

5.3. A theorem of Tumarkin 100

5.4. Aleksandrov’s characterization 102

5.5. Other representation theorems 109

5.6. Some geometric conditions 110

Chapter 6. Multipliers and divisors 115

6.1. Multipliers and Toeplitz operators 115

6.2. Some necessary conditions 118

6.3. A theorem of Goluzina 120

6.4. Some sufficient conditions 122

6.5. The F-property 127

6.6. Multipliers and inner functions 129

Chapter 7. The distribution function for Cauchy transforms 163

7.1. The Hilbert transform of a measure 163

7.2. Boole’s theorem and its generalizations 164

7.3. A refinement of Boole’s theorem 169

7.4. Measures on the circle 170

7.5. A theorem of Stein and Weiss 176

Chapter 8. The backward shift on H2 179

8.1. Beurling’s theorem 179

8.2. A theorem of Douglas, Shapiro, and Shields 180

8.3. Spectral properties 184

8.4. Kernel functions 185

8.5. A density theorem 186

8.6. A theorem of Ahern and Clark 192

8.7. A basis for backward shift invariant subspaces 192

8.8. The compression of the shift 194

8.9. Rank-one unitary perturbations 196

Chapter 9. Clark measures 201

9.1. Some basic facts about Clark measures 201

9.2. Angular derivatives and point masses 208

9.3. Aleksandrov’s disintegration theorem 211

9.4. Extensions of the disintegration theorem 212

9.5. Clark’s theorem on perturbations 218

9.6. Some remarks on pure point spectra 221

9.7. Poltoratski’s distribution theorem 222

Chapter 10. The normalized Cauchy transform 227

10.1. Basic definition 227

10.2. Mapping properties of the normalized Cauchy transform 227

10.3. Function properties of the normalized Cauchy transform 230

10.4. A few remarks about the Borel transform 241

CONTENTS vii

10.5. A closer look at the F-property 243

Chapter 11. Other operators on the Cauchy transforms 249

11.1. Some classical operators 249

11.2. The forward shift 250

11.3. The backward shift 252

11.4. Toeplitz operators 252

11.5. Composition operators 253

11.6. The Ces`aro operator 253

List of Symbols 255

Bibliography 257

Index 267