Author | Adams, David R. author |
---|---|
Title | Function Spaces and Potential Theory [electronic resource] / by David R. Adams, Lars Inge Hedberg |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-662-03282-4 |
Descript | XI, 368 p. online resource |
1. Preliminaries -- 1.1 Basics -- 1.2 Sobolev Spaces and Bessel Potentials -- 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. Poincarรฉ 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