# Real-Variable Methods in Harmonic Analysis

## 1st Edition

**Authors:**Alberto Torchinsky

**Editors:**Samuel Eilenberg Hyman Bass

**eBook ISBN:**9781483268880

**Imprint:**Academic Press

**Published Date:**6th November 1986

**Page Count:**474

## Description

*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

Preface

Chapter I Fourier Series

1. Fourier Series of Functions

2. Fourier Series of Continuous Functions

3. Elementary Properties of Fourier Series

4. Fourier Series of Functionals

5. Notes; Further Results and Problems

Chapter II Cesàro Summability

1. (C, 1) Summability

2. Fejér's Kernel

3. Characterization of Fourier Series of Functions and Measures

4. A.E. Convergence of (C, 1) Means of Summable Functions

5. Notes; Further Results and Problems

Chapter III Norm Convergence of Fourier Series

1. The Case L2(T); Hilbert Space

2. Norm Convergence in Lp(T), 1 ≤ p ≤ ∞

3. The Conjugate Mapping

4. More on Integrable Functions

5. Integral Representation of the Conjugate Operator

6. The Truncated Hilbert Transform

7. Notes; Further Results and Problems

Chapter IV The Basic Principles

1. The Calderón-Zygmund Interval Decomposition

2. The Hardy-Littlewood Maximal Function

3. The Calderón-Zygmund Decomposition

4. The Marcinkiewicz Interpolation Theorem

5. Extrapolation and the Zygmund L in L Class

6. The Banach Continuity Principle and a.e. Convergence

7. Notes; Further Results and Problems

Chapter V The Hilbert Transform and Multipliers

1. Existence of the Hilbert Transform of Integrable Functions

2. The Hilbert Transform in Lp(T), 1 ≤ p ≤ ∞

3. Limiting Results

4. Multipliers

5. Notes; Further Results and Problems

Chapter VI Paley's Theorem and Fractional Integration

1. Paley's Theorem

2. Fractional Integration

3. Multipliers

4. Notes; Further Results and Problems

Chapter VII Harmonic and Subharmonic Functions

1. Abel Summability, Nontangential Convergence

2. The Poisson and Conjugate Poisson Kernels

3. Harmonic Functions

4. Further Properties of Harmonic Functions and Subharmonic Functions

5. Harnack's and Mean Value Inequalities

6. Notes; Further Results and Problems

Chapter VIII Oscillation of Functions

1. Mean Oscillation of Functions

2. The Maximal Operator and BMO

3. The Conjugate of Bounded and BMO Functions

4. Wk-Lp and Kf. Interpolation

5. Lipschitz and Morrey Spaces

6. Notes; Further Results and Problems

Chapter IX Ap Weights

1. The Hardy-Littlewood Maximal Theorem for Regular Measures

2. Ap Weights and the Hardy-Littlewood Maximal Function

3. A1 Weights

4. Ap Weights, p > 1

5. Factorization of Ap Weights

6. Ap and BMO

7. An Extrapolation Result

8. Notes; Further Results and Problems

Chapter X More about Rn

1. Distributions. Fourier Transforms

2. Translation Invariant Operators. Multipliers

3. The Hilbert and Riesz Transforms

4. Sobolev and Poincare Inequalities

Chapter XI Calderón-Zygmund Singular Integral Operators

1. The Benedek-Calderón-Panzone Principle

2. A Theorem of Zó

3. Convolution Operators

4. Cotlar's Lemma

5. Calderón-Zygmund Singular Integral Operators

6. Maximal Calderón-Zygmund Singular Integral Operators

7. Singular Integral Operators in L∞ (Rn)

8. Notes; Further Results and Problems

Chapter XII The Littlewood-Paley Theory

1. Vector-Valued Inequalities

2. Vector-Valued Singular Integral Operators

3. The Littlewood-Paley g Function

4. The Lusin Area Function and the Littlewood-Paley gλ Function

5. Hormander's Multiplier Theorem

6. Notes; Further Results and Problems

Chapter XIII The Good ƒλ Principle

1. Good λ Inequalities

2. Weighted Norm Inequalities for Maximal CZ Singular Integral Operators

3. Weighted Weak-Type (1,1) Estimates for CZ Singular Integral Operators

4. Notes; Further Results and Problems

Chapter XIV Hardy Spaces of Several Real Variables

1. Atomic Decomposition

2. Maximal Function Characterization of Hardy Spaces

3. Systems of Conjugate Functions

4. Multipliers

5. Interpolation

6. Notes; Further Results and Problems

Chapter XV Carleson Measures

1. Carleson Measures

2. Duals of Hardy Spaces

3. Tent Spaces

4. Notes; Further Results and Problems

Chapter XVI Cauchy Integrals on Lipschitz Curves

1. Cauchy Integrals on Lipschitz Curves

2. Related Operators

3. The T1 Theorem

4. Notes; Further Results and Problems

Chapter XVII Boundary Value Problems on C1-Domains

1. The Double and Single Layer Potentials on a C1 -Domain

2. The Dirichlet and Neumann Problems

3. Notes

Bibliography

Index

## Details

- No. of pages:
- 474

- Language:
- English

- Copyright:
- © Academic Press 1986

- Published:
- 6th November 1986

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483268880

## About the Author

### Alberto Torchinsky

### Affiliations and Expertise

Department of Mathematics,Indiana University, Bloornington, Indiana

## About the Editor

### Samuel Eilenberg

### Affiliations and Expertise

Columbia University

### Hyman Bass

### Affiliations and Expertise

Department of Mathematics, Columbia University, New York, New York