Function Spaces and Potential Theory

  • David R. Adams
  • Lars Inge Hedberg

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 314)

Table of contents

  1. Front Matter
    Pages I-XI
  2. David R. Adams, Lars Inge Hedberg
    Pages 1-16
  3. David R. Adams, Lars Inge Hedberg
    Pages 17-51
  4. David R. Adams, Lars Inge Hedberg
    Pages 53-83
  5. David R. Adams, Lars Inge Hedberg
    Pages 85-127
  6. David R. Adams, Lars Inge Hedberg
    Pages 129-153
  7. David R. Adams, Lars Inge Hedberg
    Pages 155-186
  8. David R. Adams, Lars Inge Hedberg
    Pages 187-214
  9. David R. Adams, Lars Inge Hedberg
    Pages 215-231
  10. David R. Adams, Lars Inge Hedberg
    Pages 233-280
  11. David R. Adams, Lars Inge Hedberg
    Pages 281-303
  12. David R. Adams, Lars Inge Hedberg
    Pages 305-327
  13. Back Matter
    Pages 329-368

About this book

Introduction

Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE). Potential theory, which grew out of the theory of the electrostatic or gravita­ tional potential, the Laplace equation, the Dirichlet problem, etc. , had a fundamen­ tal role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More re­ cently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development. The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch. -J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940's. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L.

Keywords

Approximation Theory Capacity Distribution Fourier transform Function Spaces Hilbert space Potential theory Singular integral convolution

Authors and affiliations

  • David R. Adams
    • 1
  • Lars Inge Hedberg
    • 2
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Department of MathematicsLinköping UniversityLinköpingSweden

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-03282-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1996
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08172-9
  • Online ISBN 978-3-662-03282-4
  • Series Print ISSN 0072-7830
  • About this book
Industry Sectors
Telecommunications
Aerospace