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Aspects of Sobolev-type inequalities

Author: L Saloff-Coste
Publisher: Cambridge ; New York : Cambridge University Press, 2002.
Series: London Mathematical Society lecture note series, 289.
Edition/Format:   Print book : EnglishView all editions and formats
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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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: L Saloff-Coste
ISBN: 0521006074 9780521006071
OCLC Number: 46810848
Description: x, 190 pages : illustrations ; 23 cm.
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.
Series Title: London Mathematical Society lecture note series, 289.
Responsibility: Laurent Saloff-Coste.
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Abstract:

This book, first published in 2001, is a concise introduction to analysis on manifolds, covering functional inequalities and their applications, for graduates and researchers.  Read more...

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