Spaces of holomorphic functions in the unit ball在单位中Holomorphic功能的空间呈球状
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作者： Kehe Zhu著
出 版 社：
出版时间： 2005-2-1字数：版次： 1页数： 271印刷时间： 2005/02/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387220369包装： 精装内容简介
There has been a flurry of activity in recent years in the loosely defined area of holomorphic spaces. This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball of C^n. Spaces discussed include the Bergman spaces, the Hardy spaces, the Bloch space, BMOA, the Dirichlet space, the Besov spaces, and the Lipschitz spaces. Most proofs in the book are new and simpler than the existing ones in the literature. The central idea in almost all these proofs is based on integral representations of holomorphic functions and elementary properties of the Bergman kernel, the Bergman metric, and the automorphism group.
The unit ball was chosen as the setting since most results can be achieved there using straightforward formulas without much fuss. The book can be read comfortably by anyone familiar with single variable complex analysis; no prerequisite on several complex variables is required. The author has included exercises at the end of each chapter that vary greatly in the level of difficulty.
目录
1Preliminaries
1.1 Holomorphic Functions
1.2 The Automorphism Group
1.3 Lebesgue Spaces
1.4 Several Notions of Differentiation
1.5 The Bergman Metric
1.6 The Invariant Green's Formula
1.7 Subharmonic Functions
1.8 Interpolation of Banach Spaces
Notes
Exercises
2Bergman Spaces
2.1 Bergman Spaces
2.2 Bergman Type Projections
2.3 Other Characterizations
2.4 Carleson Type Measures
2.5 Atomic Decomposition
2.6 Complex Interpolation
Notes
Exercises
3The Bloch Space
3.1 The Bloch space
3.2 The Little Bloch Space
3.3 Duality
3.4 Maximality
3.5 Pointwise Multipliers
3.6 Atomic Decomposition
3.7 Complex Interpolation
Notes
Exercises
4Hardy Spaces
4.1 The Poisson Transform
4.2 Hardy Spaces
4.3 The Cauchy-Szeg6Projection
4.4 Several Embedding Theorems
4.5 Duality
Notes
Exercises
5Functions of Bounded Mean Oscillation
5.1 BMOA
5.2 Carleson Measures
5.3 Vanishing Carleson Measures and VMOA
5.4 Duality
5.5 BMO in the Bergman Metric
5.6 Atomic Decomposition
Notes
Exercises
6Besov Spaces
6.1 The Spaces Bp
6.2 The Minimal M6bius Invariant Space
6.3 Mobius Invariance of Bp
6.4 The Dirichlet Space B2
6.5 Duality of Besov Spaces
6.6 Other Characterizations
Notes
Exercises
7Lipschitz Spaces
7.1 Ba Spaces
7.2 The Lipschitz Spaces Aα ror 0
7.3 The ZygmundClass4
7.4 The case α1
7.5 A Unified Treatment
7.6 Growth in Tangential Directions
7.7 Duality
Notes
Exercises
References
Index
