Bounded Analytic Functions

Review:

The best function-space reference.

The study of spaces of holomorphic functions brings together techniques of complex function theory, real analysis, harmonic analysis, and functional analysis. Anyone engaged in this study should have a copy of this book, which does an excellent job of laying out the subject from all of these perspectives. The main body of the text serves as an excellent reference, and the exercises add much more. As far as this reviewer can tell, the book is cited more often in the literature than any other book on the subject. The only problem is that the book is out of print. The reviewer's (faint) hope is that this review will help convince a publisher to republish the book

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John B. Garnett, «Bounded Analytic Functions (Graduate Texts in Mathematics)»

Springer | ISBN: 0387336214 | 1 edition (October 17, 2006) | 466 pages | PDF | 2.3 Mb

"This book is an account of the theory of Hardy spaces in one dimension, with emphasis on some of the exciting developments of the past two decades or so. The last seven of the ten chapters are devoted in the main to these recent developments. The motif of the theory of Hardy spaces is the interplay between real, complex, and abstract analysis. While paying proper attention to each of the three aspects, the author has underscored the effectiveness of the methods coming from real analysis, many of them developed as part of a program to extend the theory to Euclidean spaces, where the complex methods are not available...Each chapter ends with a section called Notes and another called Exercises and further results. The former sections contain brief historical comments and direct the reader to the original sources for the material in the text."

-Donald Sarason, MathSciNet

"The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs."

-From the citation for the AMS Steele Prize for exposition