Bergman spaces Ap are subspaces of holomorphic Lp functions on the unit disc. In many respects, they resemble the Hardy spaces Hp, which were intensively studied from the 30s till the 50s. However, a majority of problems turned out to be much more difficult in the Bergman space setting and for decades they remained essentially intractable. This situation has changed radically in the 90s. Many significant advances have taken place, attracting in turn a lot of research activity in the area. The present book gives a systematic overview of the current state of this exciting subject.

The first two chapters present a crash course on the classical theory of Hardy spaces, the Bergman kernel function, hyperbolic geometry, biharmonic Green functions and a lot of other prerequisites, thus making the book very self-contained and accessible to anyone with basic knowledge of complex function theory and functional analysis. Chapter 3 deals with properties of individual functions in Bergman spaces (growth and boundary behaviour, Taylor coefficients, etc.). Chapters 4 and 5 develop properties of zero-sets of Ap functions and of the Hedenmalm canonical zero-divisors (analogues of Blaschke products), respectively. Chapters 6 and 7 contain an exposition of Seip's beautiful theory of interpolation and sampling in Ap spaces. Finally, Chapters 8 and 9 are devoted to the structure of invariant subspaces of Ap spaces, including the study of cyclic elements and the proofs of what may be the most profound result in the area, the Aleman-Richter-Sundberg analogue of Beurling's theorem. The exposition is on a masterly level, neatly and tightly organized, and yet highly readable.

So is, by the way, an earlier book on the subject, Theory of Bergman spaces by Hedenmalm, Korenblum and Zhu (Springer, 2000); there is, of course, a lot of overlap between the two books, but the current one contains more of the prerequisites (especially on Hp spaces), discusses some material not covered by the other book and treats some material in a different way. Similarly, the HKZ book contains several topics barely dealt with in the present volume, such as the invertible noncyclic functions or the log-subharmonic weights. It is extremely likely that both books are going to become standard references on the subject and should not be missing on the shelf of anyone seriously interested in this area.

Reviewer:

men