Spaces of Holomorphic Functions in the Unit Ball (Graduate Texts in Mathematics, Vol. 226) (Graduate Texts in Mathematics (226))

The book presents a modern theory of holomorphic function spaces in the open unit ball. 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 proofs 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. 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.--publisher's Website. 1.1 Holomorphic Functions 1 -- 1.2 The Automorphism Group 3 -- 1.3 Lebesgue Spaces 9 -- 1.4 Several Notions Of Differentiation 17 -- 1.5 The Bergman Metric 22 -- 1.6 The Invariant Green's Formula 28 -- 1.7 Subharmonic Functions 31 -- 1.8 Interpolation Of Banach Spaces 33 -- 2 Bergman Spaces 39 -- 2.1 Bergman Spaces 39 -- 2.2 Bergman Type Projections 43 -- 2.3 Other Characterizations 48 -- 2.4 Carleson Type Measures 56 -- 2.5 Atomic Decomposition 62 -- 2.6 Complex Interpolation 73 -- 3 The Bloch Space 79 -- 3.1 The Bloch Space 79 -- 3.2 The Little Bloch Space 89 -- 3.3 Duality 93 -- 3.4 Maximality 94 -- 3.5 Pointwise Multipliers 97 -- 3.6 Atomic Decomposition 98 -- 3.7 Complex Interpolation 103 -- 4 Hardy Spaces 109 -- 4.1 The Poisson Transform 109 -- 4.2 Hardy Spaces 122 -- 4.3 The Cauchy-szego Projection 131 -- 4.4 Several Embedding Theorems 141 -- 4.5 Duality 149 -- 5 Functions Of Bounded Mean Oscillation 157 -- 5.1 Bmoa 157 -- 5.2 Carleson Measures 162 -- 5.3 Vanishing Carleson Measures And Vmoa Kehe Zhu. Includes Bibliographical References (p. [263]-268) And Index. 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. Kehe Zhu is Professor of Mathematics at State University of New York at Albany. His previous books include Operator Theory in Function Spaces (Marcel Dekker 1990), Theory of Bergman Spaces, with H. Hedenmalm and B. Korenblum (Springer 2000), and An Introduction to Operator Algebras (CRC Press 1993). 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[superscript 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. "This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball of C[superscript 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."--Jacket Can be used as a graduate text Contains many exercises Contains new results