Function Spaces and Potential Theory

Description

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. Front Matter....Pages I-XI Preliminaries....Pages 1-16 L p -Capacities and Nonlinear Potentials....Pages 17-51 Estimates for Bessel and Riesz Potentials....Pages 53-83 Besov Spaces and Lizorkin-Triebel Spaces....Pages 85-127 Metric Properties of Capacities....Pages 129-153 Continuity Properties....Pages 155-186 Trace and Imbedding Theorems....Pages 187-214 Poincaré Type Inequalities....Pages 215-231 An Approximation Theorem....Pages 233-280 Two Theorems of Netrusov....Pages 281-303 Rational and Harmonic Approximation....Pages 305-327 Back Matter....Pages 329-368