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

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

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.

This book, first published in 2001, focuses on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds. Applications covered include the ultracontractivity of the heat diffusion semigroup, Gaussian heat kernel bounds, the Rozenblum-Lieb-Cwikel inequality and elliptic and parabolic Harnack inequalities. Emphasis is placed on the role of families of local Poincaré and Sobolev inequalities. The text provides the first self contained account of the equivalence between the uniform parabolic Harnack inequality, on the one hand, and the conjunction of the doubling volume property and Poincaré's inequality on the other. It is suitable to be used as an advanced graduate textbook and will also be a useful source of information for graduate students and researchers in analysis on manifolds, geometric differential equations, Brownian motion and diffusion on manifolds, as well as other related areas.