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The theory of HP spaces has its origins in discoveries made forty or fifty years ago by such mathematicians as G. H. Hardy, J. E. Littlewood, I. I. Privalov, F. and M. Riesz, V. Sm… Read more
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The theory of HP spaces has its origins in discoveries made forty or fifty years ago by such mathematicians as G. H. Hardy, J. E. Littlewood, I. I. Privalov, F. and M. Riesz, V. Smirnov, and G. Szego. Most of this early work is concerned with the properties of individual functions of class HP, and is classical in spirit. In recent years, the development of functional analysis has stimulated new interest in the HP classes as linear spaces. This point of viewhas suggested a variety of natural problems and has provided new methods of attack, leading to important advances in the theory.
This book is an account of both aspects of the subject, the classical and the modern. It is intended to provide a convenient source for the older parts of the theory (the work of Hardy and Littlewood, for example), as well as to give a self-contained exposition of more recent developments such as Beurling’s theorem on invariant subspaces, the Macintyre-RogosinskiShapiro-Havinson theory of extremal problems, interpolation theory, the dual space structure of HP with p < 1, HP spaces over general domains, and Carleson’s proof of the corona theorem. Some of the older results are proved by modern methods. In fact, the dominant theme of the book is the interplay of “ hard” and “ soft” analysis, the blending of classical and modern techniques and viewpoints.
Dedication
Preface
Chapter 1: Harmonic and Subharmonic Functions
1.1. Harmonic Functions
1.2. Boundary Behavior of Poisson–Stieltjes Integrals
1.3. Subharmonic Functions
1.4. Hardy’s Convexity Theorem
1.5. Subordination
1.6. Maximal Theorems
Exercises
Notes
Chapter 2: Basic Structure of Hp Functions
2.1. Boundary Values
2.2. Zeros
2.3. Mean Convergence to Boundary Values
2.4. Canonical Factorization
2.5. The Class N+
2.6. Harmonic Majorants
Exercises
Notes
Chapter 3: Applications
3.1. Poisson Integrals and H1
3.2. Description of the Boundary Functions
3.3. Cauchy and Cauchy–Stieltjes Integrals
3.4. Analytic Functions Continuous in
PD