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Aspects of Sobolev-type inequalities / Laurent Saloff-Coste.

By: Saloff-Coste, LMaterial type: TextTextSeries: London Mathematical Society lecture note series ; 289.Publication details: Cambridge ; New York : Cambridge University Press, 2002. Description: 1 online resource (x, 190 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9781107360747; 1107360749; 9780511549762; 0511549768; 9781107365650; 1107365651Subject(s): Sobolev spaces | Inequalities (Mathematics) | Sobolev, Espaces de | Inégalités (Mathématiques) | MATHEMATICS -- Functional Analysis | Inequalities (Mathematics) | Sobolev spaces | Sobolevsche Ungleichung | Sobolev ruimten | Vergelijkingen (wiskunde) | Desigualdades (análise matemática)Genre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Aspects of Sobolev-type inequalities.DDC classification: 515/.782 LOC classification: QA323 | .S35 2002ebOther classification: 31.46 Online resources: Click here to access online
Contents:
1 Sobolev inequalities in R[superscript n] 7 -- 1.1 Sobolev inequalities 7 -- 1.1.2 The proof due to Gagliardo and to Nirenberg 9 -- 1.1.3 p = 1 implies p [greater than or equal] 1 10 -- 1.2 Riesz potentials 11 -- 1.2.1 Another approach to Sobolev inequalities 11 -- 1.2.2 Marcinkiewicz interpolation theorem 13 -- 1.2.3 Proof of Sobolev Theorem 1.2.1 16 -- 1.3 Best constants 16 -- 1.3.1 The case p = 1: isoperimetry 16 -- 1.3.2 A complete proof with best constant for p = 1 18 -- 1.3.3 The case p> 1 20 -- 1.4 Some other Sobolev inequalities 21 -- 1.4.1 The case p> n 21 -- 1.4.2 The case p = n 24 -- 1.4.3 Higher derivatives 26 -- 1.5 Sobolev -- Poincare inequalities on balls 29 -- 1.5.1 The Neumann and Dirichlet eigenvalues 29 -- 1.5.2 Poincare inequalities on Euclidean balls 30 -- 1.5.3 Sobolev -- Poincare inequalities 31 -- 2 Moser's elliptic Harnack inequality 33 -- 2.1 Elliptic operators in divergence form 33 -- 2.1.1 Divergence form 33 -- 2.1.2 Uniform ellipticity 34 -- 2.1.3 A Sobolev-type inequality for Moser's iteration 37 -- 2.2 Subsolutions and supersolutions 38 -- 2.2.1 Subsolutions 38 -- 2.2.2 Supersolutions 43 -- 2.2.3 An abstract lemma 47 -- 2.3 Harnack inequalities and continuity 49 -- 2.3.1 Harnack inequalities 49 -- 2.3.2 Holder continuity 50 -- 3 Sobolev inequalities on manifolds 53 -- 3.1.1 Notation concerning Riemannian manifolds 53 -- 3.1.2 Isoperimetry 55 -- 3.1.3 Sobolev inequalities and volume growth 57 -- 3.2 Weak and strong Sobolev inequalities 60 -- 3.2.1 Examples of weak Sobolev inequalities 60 -- 3.2.2 (S[superscript [theta] subscript r, s])-inequalities: the parameters q and v 61 -- 3.2.3 The case 0 <q <[infinity] 63 -- 3.2.4 The case 1 = [infinity] 66 -- 3.2.5 The case -[infinity] <q <0 68 -- 3.2.6 Increasing p 70 -- 3.2.7 Local versions 72 -- 3.3.1 Pseudo-Poincare inequalities 73 -- 3.3.2 Pseudo-Poincare technique: local version 75 -- 3.3.3 Lie groups 77 -- 3.3.4 Pseudo-Poincare inequalities on Lie groups 79 -- 3.3.5 Ricci [greater than or equal] 0 and maximal volume growth 82 -- 3.3.6 Sobolev inequality in precompact regions 85 -- 4 Two applications 87 -- 4.1 Ultracontractivity 87 -- 4.1.1 Nash inequality implies ultracontractivity 87 -- 4.1.2 The converse 91 -- 4.2 Gaussian heat kernel estimates 93 -- 4.2.1 The Gaffney-Davies L[superscript 2] estimate 93 -- 4.2.2 Complex interpolation 95 -- 4.2.3 Pointwise Gaussian upper bounds 98 -- 4.2.4 On-diagonal lower bounds 99 -- 4.3 The Rozenblum-Lieb-Cwikel inequality 103 -- 4.3.1 The Schrodinger operator [Delta] -- V 103 -- 4.3.2 The operator T[subscript V] = [Delta superscript -1]V 105 -- 4.3.3 The Birman-Schwinger principle 109 -- 5 Parabolic Harnack inequalities 111 -- 5.1 Scale-invariant Harnack principle 111 -- 5.2 Local Sobolev inequalities 113 -- 5.2.1 Local Sobolev inequalities and volume growth 113 -- 5.2.2 Mean value inequalities for subsolutions 119 -- 5.2.3 Localized heat kernel upper bounds 122 -- 5.2.4 Time-derivative upper bounds 127 -- 5.2.5 Mean value inequalities for supersolutions 128 -- 5.3 Poincare inequalities 130 -- 5.3.1 Poincare inequality and Sobolev inequality 131 -- 5.3.2 Some weighted Poincare inequalities 133 -- 5.3.3 Whitney-type coverings 135 -- 5.3.4 A maximal inequality and an application 139 -- 5.3.5 End of the proof of Theorem 5.3.4 141 -- 5.4 Harnack inequalities and applications 143 -- 5.4.1 An inequality for log u 143 -- 5.4.2 Harnack inequality for positive supersolutions 145 -- 5.4.3 Harnack inequalities for positive solutions 146 -- 5.4.4 Holder continuity 149 -- 5.4.5 Liouville theorems 151 -- 5.4.6 Heat kernel lower bounds 152 -- 5.4.7 Two-sided heat kernel bounds 154 -- 5.5 The parabolic Harnack principle 155 -- 5.5.1 Poincare, doubling, and Harnack 157 -- 5.5.2 Stochastic completeness 161 -- 5.5.3 Local Sobolev inequalities and the heat equation 164 -- 5.5.4 Selected applications of Theorem 5.5.1 168 -- 5.6.1 Unimodular Lie groups 172 -- 5.6.2 Homogeneous spaces 175 -- 5.6.3 Manifolds with Ricci curvature bounded below 176.
Summary: Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers.
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Includes bibliographical references (pages 183-188) and index.

1 Sobolev inequalities in R[superscript n] 7 -- 1.1 Sobolev inequalities 7 -- 1.1.2 The proof due to Gagliardo and to Nirenberg 9 -- 1.1.3 p = 1 implies p [greater than or equal] 1 10 -- 1.2 Riesz potentials 11 -- 1.2.1 Another approach to Sobolev inequalities 11 -- 1.2.2 Marcinkiewicz interpolation theorem 13 -- 1.2.3 Proof of Sobolev Theorem 1.2.1 16 -- 1.3 Best constants 16 -- 1.3.1 The case p = 1: isoperimetry 16 -- 1.3.2 A complete proof with best constant for p = 1 18 -- 1.3.3 The case p> 1 20 -- 1.4 Some other Sobolev inequalities 21 -- 1.4.1 The case p> n 21 -- 1.4.2 The case p = n 24 -- 1.4.3 Higher derivatives 26 -- 1.5 Sobolev -- Poincare inequalities on balls 29 -- 1.5.1 The Neumann and Dirichlet eigenvalues 29 -- 1.5.2 Poincare inequalities on Euclidean balls 30 -- 1.5.3 Sobolev -- Poincare inequalities 31 -- 2 Moser's elliptic Harnack inequality 33 -- 2.1 Elliptic operators in divergence form 33 -- 2.1.1 Divergence form 33 -- 2.1.2 Uniform ellipticity 34 -- 2.1.3 A Sobolev-type inequality for Moser's iteration 37 -- 2.2 Subsolutions and supersolutions 38 -- 2.2.1 Subsolutions 38 -- 2.2.2 Supersolutions 43 -- 2.2.3 An abstract lemma 47 -- 2.3 Harnack inequalities and continuity 49 -- 2.3.1 Harnack inequalities 49 -- 2.3.2 Holder continuity 50 -- 3 Sobolev inequalities on manifolds 53 -- 3.1.1 Notation concerning Riemannian manifolds 53 -- 3.1.2 Isoperimetry 55 -- 3.1.3 Sobolev inequalities and volume growth 57 -- 3.2 Weak and strong Sobolev inequalities 60 -- 3.2.1 Examples of weak Sobolev inequalities 60 -- 3.2.2 (S[superscript [theta] subscript r, s])-inequalities: the parameters q and v 61 -- 3.2.3 The case 0 <q <[infinity] 63 -- 3.2.4 The case 1 = [infinity] 66 -- 3.2.5 The case -[infinity] <q <0 68 -- 3.2.6 Increasing p 70 -- 3.2.7 Local versions 72 -- 3.3.1 Pseudo-Poincare inequalities 73 -- 3.3.2 Pseudo-Poincare technique: local version 75 -- 3.3.3 Lie groups 77 -- 3.3.4 Pseudo-Poincare inequalities on Lie groups 79 -- 3.3.5 Ricci [greater than or equal] 0 and maximal volume growth 82 -- 3.3.6 Sobolev inequality in precompact regions 85 -- 4 Two applications 87 -- 4.1 Ultracontractivity 87 -- 4.1.1 Nash inequality implies ultracontractivity 87 -- 4.1.2 The converse 91 -- 4.2 Gaussian heat kernel estimates 93 -- 4.2.1 The Gaffney-Davies L[superscript 2] estimate 93 -- 4.2.2 Complex interpolation 95 -- 4.2.3 Pointwise Gaussian upper bounds 98 -- 4.2.4 On-diagonal lower bounds 99 -- 4.3 The Rozenblum-Lieb-Cwikel inequality 103 -- 4.3.1 The Schrodinger operator [Delta] -- V 103 -- 4.3.2 The operator T[subscript V] = [Delta superscript -1]V 105 -- 4.3.3 The Birman-Schwinger principle 109 -- 5 Parabolic Harnack inequalities 111 -- 5.1 Scale-invariant Harnack principle 111 -- 5.2 Local Sobolev inequalities 113 -- 5.2.1 Local Sobolev inequalities and volume growth 113 -- 5.2.2 Mean value inequalities for subsolutions 119 -- 5.2.3 Localized heat kernel upper bounds 122 -- 5.2.4 Time-derivative upper bounds 127 -- 5.2.5 Mean value inequalities for supersolutions 128 -- 5.3 Poincare inequalities 130 -- 5.3.1 Poincare inequality and Sobolev inequality 131 -- 5.3.2 Some weighted Poincare inequalities 133 -- 5.3.3 Whitney-type coverings 135 -- 5.3.4 A maximal inequality and an application 139 -- 5.3.5 End of the proof of Theorem 5.3.4 141 -- 5.4 Harnack inequalities and applications 143 -- 5.4.1 An inequality for log u 143 -- 5.4.2 Harnack inequality for positive supersolutions 145 -- 5.4.3 Harnack inequalities for positive solutions 146 -- 5.4.4 Holder continuity 149 -- 5.4.5 Liouville theorems 151 -- 5.4.6 Heat kernel lower bounds 152 -- 5.4.7 Two-sided heat kernel bounds 154 -- 5.5 The parabolic Harnack principle 155 -- 5.5.1 Poincare, doubling, and Harnack 157 -- 5.5.2 Stochastic completeness 161 -- 5.5.3 Local Sobolev inequalities and the heat equation 164 -- 5.5.4 Selected applications of Theorem 5.5.1 168 -- 5.6.1 Unimodular Lie groups 172 -- 5.6.2 Homogeneous spaces 175 -- 5.6.3 Manifolds with Ricci curvature bounded below 176.

Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers.

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