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The subject of this book is the interplay between function space theory and potential theory. The theory has reached a state of maturity and elegance by now that makes it natural to present it in book form. The book is accessible to graduate students with a good background in real analysis. No previous knowlegde of potential theory is assumed.
The subject of this book is the interplay between function space theory and potential theory. The theory has reached a state of maturity and elegance by now that makes it natural to present it in book form. The book is accessible to graduate students with a good background in real analysis. No previous knowlegde of potential theory is assumed.
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Inhaltsangabe
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. Poincaré TypeInequalities.- 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.
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. Poincaré TypeInequalities.- 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.
Rezensionen
"..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
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