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Bounded Analytic Functions Book

Bounded Analytic Functions
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  • Bounded Analytic Functions
  • Written by author John B. Garnett
  • Published by Springer-Verlag New York, LLC, October 2006
  • "This book is an account of the theory of Hardy spaces in one dimension, with emphasis on some of the exciting developments of the past two decades or so. The last seven of the ten chapters are devoted in the main to these recent developments. The motif o
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Preface     vii
List of Symbols     xiii
Preliminaries
Schwarz's Lemma     1
Pick's Theorem     5
Poisson Integrals     10
Hardy-Littlewood Maximal Function     19
Nontangential Maximal Function and Fatou's Theorem     27
Subharmonic Functions     32
Notes     38
Exercises and Further Results     39
H[superscript P] Spaces
Definitions     48
Blaschke Products     51
Maximal Functions and Boundary Values     55
(1/[Pi]) [Integral] (log | f(t)|/(1+t2))dt [greater than] - [Infin]     61
The Nevanlinna Class     66
Inner Functions     71
Beurling's Theorem     78
Notes     83
Exercises and Further Results     84
Conjugate Functions
Preliminaries     98
The L[superscript p] Theorems     106
Conjugate Functions and Maximal Functions     111
Notes     118
Exercises and Further Results     120
Some Extremal Problems
Dual Extremal Problems     128
The Carleson-Jacobs Theorem     134
The Helson-Szego Theorem     139
Interpolating Functions ofConstant Modulus     145
Parametrization of K     151
Nevanlinna's Proof     159
Notes     168
Exercises and Further Results     168
Some Uniform Algebra
Maximal Ideal Spaces     176
Inner Functions     186
Analytic Discs in Fibers     191
Representing Measures and Orthogonal Measures     193
The Space L[superscript 1]/H[subscript 0][superscript 1]     198
Notes     205
Exercises and Further Results     206
Bounded Mean Oscillation
Preliminaries     215
The John-Nirenberg Theorem     223
Littlewood-Paley Integrals and Carleson Measures     228
Fefferman's Duality Theorem     234
Vanishing Mean Oscillation     242
Weighted Norm Inequalities     245
Notes     259
Exercises and Further Results     261
Interpolating Sequences
Carleson's Interpolation Theorem     275
The Linear Operator of Interpolation     285
Generations     289
Harmonic Interpolation     292
Earl's Elementary Proof     299
Notes     304
Exercises and Further Results     305
The Corona Construction
Inhomogeneous Cauchy-Riemann Equations     309
The Corona Theorem     314
Two Theorems on Minimum Modulus     323
Interpolating Blaschke Products     327
Carleson's Construction     333
Gradients of Bounded Harmonic Functions     338
A Constructive Solution of ab/az = [Mu]     349
Notes     357
Exercises and Further Results     358
Douglas Algebras
The Douglas Problem     364
H[superscript Infin] + C     367
The Chang-Marshall Theorem     369
The Structure of Douglas Algebras     375
The Local Fatou Theorem and an Application     379
Notes     384
Exercises and Further Results     385
Interpolating Sequences and Maximal Ideals
Analytic Discs in M     391
Hoffman's Theorem     401
Approximate Dependence between Kernels     406
Interpolating Sequences and Harmonic Separation     412
A Constructive Douglas-Rudin Theorem     418
Notes     428
Exercises and Further Results     428
Bibliography     434
Index     453


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