Function Spaces and Potential Theory

1. Front Matter

   Pages I-XI

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2. Preliminaries

   • David R. Adams, Lars Inge Hedberg

   Pages 1-16

3. L^p-Capacities and Nonlinear Potentials

   • David R. Adams, Lars Inge Hedberg

   Pages 17-51

4. Estimates for Bessel and Riesz Potentials

   • David R. Adams, Lars Inge Hedberg

   Pages 53-83

5. Besov Spaces and Lizorkin-Triebel Spaces

   • David R. Adams, Lars Inge Hedberg

   Pages 85-127

6. Metric Properties of Capacities

   • David R. Adams, Lars Inge Hedberg

   Pages 129-153

7. Continuity Properties

   • David R. Adams, Lars Inge Hedberg

   Pages 155-186

8. Trace and Imbedding Theorems

   • David R. Adams, Lars Inge Hedberg

   Pages 187-214

9. Poincaré Type Inequalities

   • David R. Adams, Lars Inge Hedberg

   Pages 215-231

10. An Approximation Theorem

    • David R. Adams, Lars Inge Hedberg

    Pages 233-280

11. Two Theorems of Netrusov

    • David R. Adams, Lars Inge Hedberg

    Pages 281-303

12. Rational and Harmonic Approximation

    • David R. Adams, Lars Inge Hedberg

    Pages 305-327

13. Back Matter

    Pages 329-368

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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.

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

"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society

• Department of Mathematics, University of Kentucky, Lexington, USA

  David R. Adams

• Department of Mathematics, Linköping University, Linköping, Sweden

  Lars Inge Hedberg

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