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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 , who proved (with modem rigor supplied almost a century later by O. Frostman ) 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  in the 1940's. In the thesis of J. Deny , and in the subsequent work of J. Deny and J. L.
Content Level »Research
Keywords »Approximation Theory - Capacity - Distribution - Fourier transform - Function Spaces - Hilbert space - Potential theory - Singular integral - convolution
1. Preliminaries.- 1.1 Basics.- 1.1.1 Convention.- 1.1.2 Notation.- 1.1.3 Spaces of Functions and Their Duals.- 1.1.4 Maximal Functions.- 1.1.5 Integral Inequalities.- 1.1.6 Distributions.- 1.1.7 The Fourier Transform.- 1.1.8 The Riesz Transform and Singular Integrals.- 1.2 Sobolev Spaces and Bessel Potentials.- 1.2.1 Sobolev Spaces.- 1.2.2 Riesz Potentials.- 1.2.3 Bessel Potentials.- 1.2.4 Bessel Kernels.- 1.2.5 Some Classical Formulas for Bessel Functions.- 1.2.6 Bessel Potential Spaces.- 1.2.7 The Sobolev Imbedding Theorem.- 1.3 Banach Spaces.- 1.4 Two Covering Lemmas.- 2. Lp-Capacities and Nonlinear Potentials.- 2.1 Introduction.- 2.2 A First Version of (?, p)-Capacity.- 2.3 A General Theory for LP-Capacities.- 2.4 The Minimax Theorem.- 2.5 The Dual Definition of Capacity.- 2.6 Radially Decreasing Convolution Kernels.- 2.7 An Alternative Definition of Capacity and Removability of Singularities.- 2.8 Further Results.- 2.9 Notes.- 3. Estimates for Bessel and Riesz Potentials.- 3.1 Pointwise and Integral Estimates.- 3.2 A Sharp Exponential Estimate.- 3.3 Operations on Potentials.- 3.4 One-Sided Approximation.- 3.5 Operations on Potentials with Fractional Index.- 3.6 Potentials and Maximal Functions.- 3.7 Further Results.- 3.8 Notes.- 4. Besov Spaces and Lizorkin-Triebel Spaces.- 4.1 Besov Spaces.- 4.2 Lizorkin-Triebel Spaces.- 4.3 Lizorkin-Triebel Spaces, Continued.- 4.4 More Nonlinear Potentials.- 4.5 An Inequality of Wolff.- 4.6 An Atomic Decomposition.- 4.7 Atomic Nonlinear Potentials.- 4.8 A Characterization of L?,P.- 4.9 Notes.- 5. Metric Properties of Capacities.- 5.1 Comparison Theorems.- 5.2 Lipschitz Mappings and Capacities.- 5.3 The Capacity of Cantor Sets.- 5.4 Sharpness of Comparison Theorems.- 5.5 Relations Between Different Capacities.- 5.6 Further Results.- 5.7 Notes.- 6. Continuity Properties.- 6.1 Quasicontinuity.- 6.2 Lebesgue Points.- 6.3 Thin Sets.- 6.4 Fine Continuity.- 6.5 Further Results.- 6.6 Notes.- 7. Trace and Imbedding Theorems.- 7.1 A Capacitary Strong Type Inequality.- 7.2 Imbedding of Potentials.- 7.3 Compactness of the Imbedding.- 7.4 A Space of Quasicontinuous Functions.- 7.5 A Capacitary Strong Type Inequality. Another Approach.- 7.6 Further Results.- 7.7 Notes.- 8. Poincaré Type Inequalities.- 8.1 Some Basic Inequalities.- 8.2 Inequalities Depending on Capacities.- 8.3 An Abstract Approach.- 8.4 Notes.- 9. An Approximation Theorem.- 9.1 Statement of Results.- 9.2 The Case m = 1.- 9.3 The General Case. Outline.- 9.4 The Uniformly (1, p)-Thick Case.- 9.5 The General Thick Case.- 9.6 Proof of Lemma 9.5.2 for m = 1.- 9.7 Proof of Lemma 9.5.2.- 9.8 Estimates for Nonlinear Potentials.- 9.9 The Case Cm p(K) = 0.- 9.10 The Case Ck,p(K) = 0, 1 ? k < m.- 9.11 Conclusion of the Proof.- 9.12 Further Results.- 9.13 Notes.- 10. Two Theorems of Netrusov.- 10.1 An Approximation Theorem, Another Approach.- 10.2 A Generalization of a Theorem of Whitney.- 10.3 Further Results.- 10.4 Notes.- 11. Rational and Harmonic Approximation.- 11.1 Approximation and Stability.- 11.2 Approximation by Harmonic Functions in Gradient Norm.- 11.3 Stability of Sets Without Interior.- 11.4 Stability of Sets with Interior.- 11.5 Approximation by Harmonic Functions and Higher Order Stability.- 11.6 Further Results.- 11.7 Notes.- References.- List of Symbols.