Bounded Analytic Functions

By: John B. Garnett

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Hardcover | 23 November 2006 | Edition Number 2

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Hardcover
476 Pages

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23.39 x 15.6 x 2.69

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"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 of the theory of Hardy spaces is the interplay between real, complex, and abstract analysis. While paying proper attention to each of the three aspects, the author has underscored the effectiveness of the methods coming from real analysis, many of them developed as part of a program to extend the theory to Euclidean spaces, where the complex methods are not available...Each chapter ends with a section called Notes and another called Exercises and further results. The former sections contain brief historical comments and direct the reader to the original sources for the material in the text."

Donald Sarason, MathSciNet

"The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs."

From the citation for the AMS Steele Prize for exposition

Industry Reviews

From the reviews:

"After having been out of print for many years, the long-awaited revised first edition of this wonderful book has finally reappeared, this time with Springer Verlag. In 2003, Garnett was awarded the AMS Steel prize for this book; one of the highest distinctions for mathematical exposition. ... To sum up, Garnett's book is as useful now as it was in 1981 and is highly recommended to everyone interested in function theory." (Raymond Mortini, Zentralblatt MATH, Vol. 1106 (8), 2007)

"The book contains information about Hardy spaces, conjugate functions, bounded mean oscillation and duality, interpolating sequences, corona construction, Douglas algebras and many other topics. ... The book ... is a gold mine for researchers and experts." (EMS Newsletter, September, 2008)

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

Bounded Analytic Functions

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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