Bounded Analytic Functions (eBook)

Front Cover 1
Bounded Analytic Functions 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 14
List of Symbols 16
CHAPTER I. PRELIMINARIES 20
1. Schwarz’s Lemma 20
2. Pick’s Theorem 25
3. Poisson Integrals 30
4. Hardy–Littlewood Maximal Function 39
5. Nontangential Maximal Function and Fatou’s Theorem 47
6. Subharmonic Functions 52
Notes 58
Exercises and Further Results 60
CHAPTER ll. HP SPACES 69
1. Definitions 69
2. Blaschke Products 72
3. Maximal Functions and Boundary Values 76
4. (1/p) f (log|ƒ(t)|/(1 + t2))dt > – 8
5. The Nevanlinna Class 87
6. Inner Functions 94
7. Beurling’s Theorem 101
Notes 106
Exercises and Further Results 106
CHAPTER lll. CONJUGATE FUNCTIONS 121
1. Preliminaries 121
2. The LP Theorems 129
3. Conjugate Functions and Maximal Functions 134
Notes 142
Exercises and Further Results 144
CHAPTER IV. SOME EXTREMAL PROBLEMS 151
1. Dual Extremal Problems 152
2. The Carleson–Jacobs Theorem 158
3. The Helson–Szegö Theorem 163
4. Interpolating Functions of Constant Modulus 169
5. Parametrization of K 176
6. Nevanlinna’s Proof 184
Notes 193
Exercises and Further Results 194
CHAPTER V. SOME UNIFORM ALGEBRA 201
1. Maximal Ideal Spaces 201
2. Inner Functions 211
3. Analytic Discs in Fibers 216
4. Representing Measures and Orthogonal Measures 219
5. The Space L1/H10 224
Notes 231
Exercises and Further Results 232
CHAPTER VI. BOUNDED MEAN OSCILLATION 241
1. Preliminaries 241
2. The John–Nirenberg Theorem 249
3. Littlewood–Paley Integrals and Carleson Measures 254
4. Fefferman’s Duality Theorem 260
5. Vanishing Mean Oscillation 269
6. Weighted Norm Inequalities for Maximal Functions and Conjugate Functions 272
Notes 286
Exercises and Further Results 288
CHAPTER VII. INTERPOLATING SEQUENCES 303
1. Carleson’s Interpolation Theorem 303
2. The Linear Operator of Interpolation 313
3. Generations 318
4. Harmonic Interpolation 320
5. Earl’s Elementary Proof 327
Notes 332
Exercises and Further Results 333
CHAPTER VIII. THE CORONA CONSTRUCTION 337
1. Inhomogeneous Cauchy–Riemann Equations 337
2. The Corona Theorem 342
3. Two Theorems on Minimum Modulus 351
4. Interpolating Blaschke Products 355
5. Carleson's Construction 361
6. Gradients of Bounded Harmonic Functions 366
7. A Constructive Solution of ab/af = µ 377
Appendix: The Koszul Complex 383
Notes 385
Exercises and Further Results 386
CHAPTER IX. DOUGLAS ALGEBRAS 392
1. The Douglas Problem 392
2. H8 + C 395
3. The Chang–Marshall Theorem 397
4. The Structure of Douglas Algebras 404
5. The Local Fatou Theorem and an Application 408
Notes 413
Exercises and Further Results 414
CHAPTER X. INTERPOLATING SEQUENCES AND MAXIMAL IDEALS 419
1. Analytic Discs in m 419
2. Hoffman's Theorem 429
3. Approximate Dependence between Kernels 434
4. Interpolating Sequences and Harmonic Separation 441
5. A Constructive Douglas–Rudin Theorem 447
Notes 457
Exercises and Further Results 457
Bibliography 463
Index 480
Pure and Applied Mathematics 487