TY  - BOOK
AU  - Saloff-Coste,L.
TI  - Aspects of Sobolev-type inequalities
T2  - London Mathematical Society lecture note series
SN  - 9780511549762 (ebook)
AV  - QA323 .S35 2002
U1  - 515/.782 21
PY  - 2002///
CY  - Cambridge
PB  - Cambridge University Press
KW  - Sobolev spaces
KW  - Inequalities (Mathematics)
N1  - 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
N2  - 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
UR  - https://doi.org/10.1017/CBO9780511549762
ER  -