John Garnett 
Bounded Analytic Functions [PDF ebook] 

"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, Math Sci Net"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