Real-Variable Methods in Harmonic Analysis

Alberto Torchinsky, Samuel Eilenberg, Hyman Bass

• Publisher: Academic Press
• Year: 2016
• ISBN:
  9781483268880
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Real-Variable Methods in Harmonic Analysis deals with the unity of several areas in harmonic analysis, with emphasis on real-variable methods. Active areas of research in this field are discussed, from the Calderón-Zygmund theory of singular integral operators to the Muckenhoupt theory of Ap weights and the Burkholder-Gundy theory of good? inequalities. The Calderón theory of commutators is also considered.

Comprised of 17 chapters, this volume begins with an introduction to the pointwise convergence of Fourier series of functions, followed by an analysis of Cesàro summability. The discussion then turns to norm convergence; the basic working principles of harmonic analysis, centered around the Calderón-Zygmund decomposition of locally integrable functions; and fractional integration. Subsequent chapters deal with harmonic and subharmonic functions; oscillation of functions; the Muckenhoupt theory of Ap weights; and elliptic equations in divergence form. The book also explores the essentials of the Calderón-Zygmund theory of singular integral operators; the good? inequalities of Burkholder-Gundy; the Fefferman-Stein theory of Hardy spaces of several real variables; Carleson measures; and Cauchy integrals on Lipschitz curves. The final chapter presents the solution to the Dirichlet and Neumann problems on C1-domains by means of the layer potential methods.

This monograph is intended for graduate students with varied backgrounds and interests, ranging from operator theory to partial differential equations.

Table of contents

• Front Cover
• Real-Variable Methods in Harmonic Analysis
• Copyright Page
• Table of Contents
• Dedication
• Preface
• CHAPTER I. Fourier Series
• CHAPTER II. Cesàro Summabìlity
• CHAPTER III. Norm Convergence of Fourier Series
• CHAPTER IV. The Basic Principles
• CHAPTER V. The Hilbert Transform and Multipliers
• CHAPTER VI. Paley's Theorem and Fractional Integration
• CHAPTER VII. Harmonic and Subharmonic Functions
• CHAPTER VIII. Oscillation of Functions
• CHAPTER IX. Ap Weights
• CHAPTER X. More about Rn
• CHAPTER XI. Calderón–Zygmund Singular Integral Operators
• CHAPTER XII. The Littlewood-Paley Theory
• CHAPTER XIII. The Good λ Principle
• CHAPTER XIV. Hardy Spaces of Several Real Variables
• CHAPTER XV. Carleson Measures
• CHAPTER XVI. Cauchy Integrals on Lipschitz Curves
• CHAPTER XVII. Boundary Value Problems on C1-Domains
• Bibliography
• Index