Spaces of Holomorphic Functions in the Unit Ball

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).

From the reviews:

"This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball … The book can be read comfortably by anyone familiar with single variable complex analysis … the proofs are originally constructed and considerably simpler than the existing ones in the literature … The book is essentially self contained …" (Eleonara A. Storozhenko, Zentralblatt MATH, 1067, 2005)

"This is a very good book for graduate students of mathematics and mathematicians. … it is very interesting and contains much of the research done in recent years in the area of holomorphic spaces. It is certainly very useful to all who want to learn about and do research in this field – young and old. It could also constitute the basic material for a graduate course or seminar." (Mihaela Poplicher, MathDL, September, 2005)

"In recent years there has been a considerably growing interest in properties of spaces of holomorphic functions … . The book under review concerns the case of the unit ball in Cⁿ. … This monograph addresses graduate students and expert mathematicians as well. … The book is well written, clear in its exposition and has an exhaustive bibliography. … This book is going to become a standard reference for mathematicians and students working on this subject." (Marco M. Peloso, Mathematical Reviews, Issue 2006 d)

"The book is concerned with the basic properties of the most well-known and widely used spaces in holomorphic functions in the open unit ball B_n of Cⁿ. The restriction to the unit ball of Cⁿ allows the author to present direct proofs of most of the results by straightforward formulas. … The book is well written and can be used as a textbook for advanced graduate courses in complex analysis and spaces of holomorphic functions." (Mirela Kohr, Studia Universitatis Babes-Bolyai Mathematica, Vol. 50 (2), 2005)

“Fora reader familiar with the univariate case, the book is essentially self-contained. … This book is basically written as an advanced course in complex analysis … it will also be of interest to researchers who get some survey of the scattered literature. … Also the unification of several of these spaces as Sobolev spaces has been avoided. This has the advantage that ideas can be explained in a simpler and clearer way for first reading.” (Adhemar Bultheel, Bulletin of the Belgian Mathematical Society, 2007)

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).