Spaces of holomorphic functions in the unit ball.

*(English)*Zbl 1067.32005
Graduate Texts in Mathematics 226. New York, NY: Springer (ISBN 0-387-22036-4/hbk). x, 271 p. (2005).

This book discusses the most well-known and widely used spaces of holomorphic functions in the unit ball of \(\mathbb C_n\): the Bergman spaces, the Hardy spaces, the Bloch space, BMOA, the Dirichlet space, the Besov spaces, and the Lipschitz spaces. The theme of the book is very simple. For each scale of spaces, the author discusses integral representations, characterizations in terms of various derivatives, atomic decomposition, complex interpolation, and duality. Very few other properties are discussed. The unit ball is chosen as the setting because 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. Few of the results in the book are new, but most of the proofs are originally constructed and considerably simpler than the existing ones in the literature. In fact, most of the results presented in the book are based on the explicit form and elementary properties of the Bergman and Cauchy-Szëgo kernels, the Bergman metric, and the automorphism group.

There is some obvious overlap between this book and Walter Rudin’s classic “Function Theory in the Unit Ball of \(\mathbb C_n\)” (1980; Zbl 0495.32001). But the overlap is not substantial, and hopefully the two books will complement each other.

The book is essentially self-contained, with two exceptions worth mentioning. First, the existence of boundary values for functions in the Hardy spaces \(H^p\) is proven only for \(p\geq 1\); a full proof can be found in Rudin’s book. Second, the complex interpolation between the Hardy spaces \(H^1\) and \(H^p\) (or BMOA) is not proven; a full proof requires more real variable techniques.

The exercises at the end of each chapter vary greatly in the level of difficulty. Some of them are simple applications of the main theorems, some are obvious generalizations or variations, while others are difficult results that complement the main text. In the latter case, at least one reference is provided for the reader.

The book can be read comfortably by anyone familiar with single variable complex analysis; no prerequisite on several complex variables is required. Few of the results in the book are new, but most of the proofs are originally constructed and considerably simpler than the existing ones in the literature. In fact, most of the results presented in the book are based on the explicit form and elementary properties of the Bergman and Cauchy-Szëgo kernels, the Bergman metric, and the automorphism group.

There is some obvious overlap between this book and Walter Rudin’s classic “Function Theory in the Unit Ball of \(\mathbb C_n\)” (1980; Zbl 0495.32001). But the overlap is not substantial, and hopefully the two books will complement each other.

The book is essentially self-contained, with two exceptions worth mentioning. First, the existence of boundary values for functions in the Hardy spaces \(H^p\) is proven only for \(p\geq 1\); a full proof can be found in Rudin’s book. Second, the complex interpolation between the Hardy spaces \(H^1\) and \(H^p\) (or BMOA) is not proven; a full proof requires more real variable techniques.

The exercises at the end of each chapter vary greatly in the level of difficulty. Some of them are simple applications of the main theorems, some are obvious generalizations or variations, while others are difficult results that complement the main text. In the latter case, at least one reference is provided for the reader.

Reviewer: Eleonora A. Storozhenko (Odessa)

##### MSC:

32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |