Aspects of Sobolev-type inequalitiesLaurent Saloff-Coste.
- Cambridge ; New York : Cambridge University Press, (c)2002.
- 1 online resource (x, 190 pages) : illustrations.
- London Mathematical Society lecture note series ; 289 .
Includes bibliographies and index.
Sobolev inequalities in R[superscript n] -- Sobolev inequalities -- The proof due to Gagliardo and to Nirenberg -- p = 1 implies p [greater than or equal] 1 -- Riesz potentials -- Another approach to Sobolev inequalities -- Marcinkiewicz interpolation theorem -- Proof of Sobolev Theorem 1.2.1 -- Best constants -- The case p = 1: isoperimetry -- A complete proof with best constant for p = 1 -- The case p > 1 -- Some other Sobolev inequalities -- The case p > n -- The case p = n -- Higher derivatives -- Sobolev -- Poincare inequalities on balls -- The Neumann and Dirichlet eigenvalues -- Poincare inequalities on Euclidean balls -- Sobolev -- Poincare inequalities -- Moser's elliptic Harnack inequality -- Elliptic operators in divergence form -- Divergence form -- Uniform ellipticity -- A Sobolev-type inequality for Moser's iteration -- Subsolutions and supersolutions -- Subsolutions -- Supersolutions -- An abstract lemma -- Harnack inequalities and continuity -- Harnack inequalities -- Holder continuity -- Sobolev inequalities on manifolds -- Notation concerning Riemannian manifolds -- Isoperimetry -- Sobolev inequalities and volume growth -- Weak and strong Sobolev inequalities -- Examples of weak Sobolev inequalities -- (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v -- The case 0 < q < [infinity] -- The case 1 = [infinity] -- The case -[infinity] < q < 0 -- Increasing p -- Local versions -- Pseudo-Poincare inequalities -- Pseudo-Poincare technique: local version -- Lie groups -- Pseudo-Poincare inequalities on Lie groups -- Ricci [greater than or equal] 0 and maximal volume growth -- Sobolev inequality in precompact regions -- Two applications -- Ultracontractivity -- Nash inequality implies ultracontractivity -- The converse -- Gaussian heat kernel estimates -- The Gaffney-Davies L[superscript 2. estimate -- Complex interpolation -- Pointwise Gaussian upper bounds -- On-diagonal lower bounds -- The Rozenblum-Lieb-Cwikel inequality -- The Schrodinger operator [Delta] -- V -- The operator T[subscript V] = [Delta superscript -1.V -- The Birman-Schwinger principle -- Parabolic Harnack inequalities -- Scale-invariant Harnack principle -- Local Sobolev inequalities -- Local Sobolev inequalities and volume growth -- Mean value inequalities for subsolutions -- Localized heat kernel upper bounds -- Time-derivative upper bounds -- Mean value inequalities for supersolutions -- Poincare inequalities -- Poincare inequality and Sobolev inequality -- Some weighted Poincare inequalities -- Whitney-type coverings -- A maximal inequality and an application -- End of the proof of Theorem 5.3.4 -- Harnack inequalities and applications -- An inequality for log u -- Harnack inequality for positive supersolutions -- Harnack inequalities for positive solutions -- Holder continuity -- Liouville theorems -- Heat kernel lower bounds -- Two-sided heat kernel bounds -- The parabolic Harnack principle -- Poincare, doubling, and Harnack -- Stochastic completeness -- Local Sobolev inequalities and the heat equation -- Selected applications of Theorem 5.5.1 -- Unimodular Lie groups -- Homogeneous spaces -- Manifolds with Ricci curvature bounded below
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.