This is a very good book for graduate students of mathematics and mathematicians. Although it is mostly self-contained, it does require the reader to be familiar with real and complex analysis, at least in one variable.
The book has evolved with the teaching/research experience of Professor Zhu at the State University of New York at Albany. It is not comprehensive, but it includes the most well known spaces of holomorphic functions in the unit ball of Cn. After a chapter called Preliminaries, there are chapters devoted to each of these spaces of functions: Bergman Spaces, The Bloch Space, Hardy Spaces, Functions of Bounded Mean Oscillation, Besov Spaces, Lipschitz Spaces.
Many of the results included in the book are not new, but their treatment and the proofs are (many of the proofs are simpler and more elegant than the ones already known.)
There are a few important aspects discussed in each chapter (about each function space): integral representations, characterizations in terms of various derivatives, atomic decompositions, complex interpolation, duality, and a few other properties.
Each chapter ends with sections titled Notes and Exercises. While in the Notes many references to published papers are made, the Exercises vary greatly in the level of difficulty: some are simple applications of the theorems discussed in the text, while others are important theorems that complement and enrich the chapters.
The book is not a very easy read, but 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 is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.