# Multidimensional Singular Integrals and Integral Equations

## 1st Edition

**Authors:**S. G. Mikhlin

**Editors:**I. N. Sneddon M. Stark S. Ulam

**eBook ISBN:**9781483164496

**Imprint:**Pergamon

**Published Date:**1st January 1965

**Page Count:**172

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Preface

I. Introduction

§ 1. Review of Previous Work

§ 2. Some Theorems on Linear Equations in Banach Spaces

§ 3. Stereographic Projection

§ 4. Completely Continuous Operators

II. Simplest Properties of Multidimensional Singular Integrals

§ 5. Basic Concepts

§ 6. Lipschitz Conditions

§ 7. Order of Singular Integrals at Infinity

§ 8. Differentiation of Integrals with a Weak Singularity

III. Compounding of Singular Integrals

§ 9. Compounding of Singular and Ordinary Integrals

§ 10. Compounding of Double Singular Integrals

§ 11. The Concept of a Singular Operator

§ 12. Compounding of Double Singular Integrals. The Symbol

§ 13. Compounding of Multidimensional Singular Integrals

§ 14. Formulae for Reference

§ 15. Product of the Operators A1 and An

§ 16. Product of the Operators A2 and An

§ 17. Calculation of x1,m

§ 18. Symbol of a Multidimensional Singular Integral

IV. Properties of the Symbol

§ 19. Fourier Transform of a Singular Kernel

§ 20. Fourier Transform of a Kernel and the Symbol of a Singular Operator

§ 21. Transformation of the Symbol Under Change of Variables

§ 22. Differentiability of the Symbol

§ 23. The Conditions for the Continuity of the Symbol

V. Singular Integrals in Lp Spaces

§ 24. The Simplest Corollaries from the Fourier Transform. First Theorem on Boundedness in L2 Space

§ 25. Symbol Dependent on the Pole. Second Theorem on Boundedness in L2 Space

§ 26. On the Boundedness of a Singular Integral Operator in Lp Space

§ 27. Integrals Taken Over any Manifold

§ 28. Differential Properties of Singular Integrals

VI. Further Investigation of the Symbol

§ 29. More About the Differentiation of Integrals with a Weak Singularity

§ 30. Polyharmonic Potentials

§ 31. Series of Spherical Functions

§ 32. Differential Properties of the Symbol and the Characteristic

§ 33. Rule for the Multiplication of the Symbols in the General Case

§ 34. Conjugate Singular Operator

VII. Singular Integral Equations

§ 35. The Case Where the Symbol is Independent of the Pole

§ 36. The Case Where the Symbol is Dependent on pole. Regularization and Domains of Constancy of the Index

§ 37. Equivalent Regularization. Index Theorem

§ 38. Equations with an Integral Taken Over a Closed Manifold

§ 39. Extension by Means of the Parameter

§ 40. Systems of Singular Integral Equations

§ 41. Singular Integral Equations in Classes of Lipschitz Functions

VIII. Miscellaneous Applications

§ 42. Leading Derivatives of Volume Potential

§ 43. Problem of the Oblique Derivative

§ 44. Inequality Involving the Tangential and Normal Components of the Gradient of a Harmonic Function

§ 45. Equilibrium of an Isotropic Elastic Body

§ 46. Diffraction of Stationary Elastic Waves

Appendix. Multipliers of Fourier Integrals

Bibliography

Index

Other Titles in the Series

## Description

Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals.

This volume is comprised of eight chapters and begins with an overview of some theorems on linear equations in Banach spaces, followed by a discussion on the simplest properties of multidimensional singular integrals. Subsequent chapters deal with compounding of singular integrals; properties of the symbol, with particular reference to Fourier transform of a kernel and the symbol of a singular operator; singular integrals in Lp spaces; and singular integral equations. The differentiation of integrals with a weak singularity is also considered, along with the rule for the multiplication of the symbols in the general case. The final chapter describes several applications of multidimensional singular integral equations to boundary problems in mathematical physics.

This book will be of interest to mathematicians and students of mathematics.

## Details

- No. of pages:
- 172

- Language:
- English

- Copyright:
- © Pergamon 1965

- Published:
- 1st January 1965

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483164496