HAMILTON–JACOBI SEMIGROUP ON LENGTH SPACES AND APPLICATIONS
13 pages

HAMILTON–JACOBI SEMIGROUP ON LENGTH SPACES AND APPLICATIONS

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HAMILTON–JACOBI SEMIGROUP ON LENGTH SPACES AND APPLICATIONS JOHN LOTT AND CEDRIC VILLANI Abstract. We define a Hamilton–Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length space implies a global Poincare inequality and (2) if the space satisfies a doubling con- dition, a local Poincare inequality and a log Sobolev inequality then it also satisfies a Talagrand inequality. Keywords: metric-measure spaces, Hamilton–Jacobi semigroup, Talagrand inequality, logarithmic Sobolev inequality, Poincare inequality, Ricci curvature. Resume. Nous definissons un semigroupe de Hamilton–Jacobi agissant sur les fonctions continues definies sur un espace de longueurs compact. Nous utilisons les proprietes de ce semigroupe pour etudier certaines inegalites geometriques liees au phenomene de concen- tration de la mesure, selon une strategie initiee par Bobkov, Gentil et Ledoux. Nos prin- cipaux resultats stipulent que (1) une inegalite de Talagrand sur un espace de longueurs mesure implique une inegalite de Poincare globale, et (2) si l'espace verifie en outre une condition de doublement, une inegalite de Poincare locale et une inegalite de Sobolev lo- garithmique, alors il admet aussi une inegalite de Talagrand. Mots-cles : espaces metriques mesures, semigroupe de Hamilton–Jacobi, inegalite de Talagrand, inegalite de Sobolev logarithmique, inegalite de Poincare, courbure de Ricci.

  • espace de longueur compact

  • inegalite de talagrand

  • espaces metriques

  • sobolev inequality

  • poincare inequality

  • hamilton–jacobi semigroup

  • local poincare inequality

  • implies


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HAMILTON–JACOBI SEMIGROUP ON LENGTH SPACES AND APPLICATIONS ´ JOHN LOTT AND CEDRIC VILLANI
Abstract. We define a Hamilton–Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length spaceimpliesaglobalPoincare´inequalityand(2)ifthespacesatisesadoublingcon-dition,alocalPoincar´einequalityandalogSobolevinequalitythenitalsosatisesa Talagrand inequality. Keywords: metric-measure spaces, Hamilton–Jacobi semigroup, Talagrand inequality, logarithmicSobolevinequality,Poincare´inequality,Riccicurvature.
R´ ´. Nousd´enissonsunsemigroupedeHamiltonJacobiagissantsurlesfonctions esume continuesd´eniessurunespacedelongueurscompact.Nousutilisonslespropri´ete´sdece semigroupepoure´tudiercertainesin´egalit´esge´ometriqueslie´esauphe´nome`nedeconcen-´ trationdelamesure,selonunestrate´gieinitie´eparBobkov,GentiletLedoux.Nosprin-cipauxre´sultatsstipulentque(1)unein´egalite´deTalagrandsurunespacedelongueurs mesure´impliqueuneine´galit´edePoincar´eglobale,et(2)silespaceve´rieenoutreune conditiondedoublement,unein´egalite´dePoincar´elocaleetuneine´galite´deSobolevlo-garithmique,alorsiladmetaussiunein´egalite´deTalagrand. Mots-cle´s: espacesme´triquesmesur´es,semigroupedeHamiltonJacobi,in´egalite´de Talagrand,in´egalit´edeSobolevlogarithmique,in´egalite´dePoincare´,courburedeRicci. Links between concentration of measure, log Sobolev inequalities, Talagrand inequalities andPoincar´einequalitieshavebeenstudiedinthesettingofRiemannianmanifolds[1, 2, 3, 8, 9, 12]. The main result in the paper of Otto and Villani [12] can be informally stated as follows : on a Riemannian manifold, a log Sobolev inequality implies a Talagrand inequality,whichinturnimpliesaPoincare´(orspectralgap)inequality,allofthisbeing without any degradation of the constants. On the other hand, there has been intense recent activity to develop a theory of Ricci curvature bounds, log Sobolev inequalities and related inequalities in the more general setting of metric-measure length spaces satisfying minimal regularity assumptions [10, 11, 14, 15, 16]. Date : March 28, 2007. The research of the first author was supported by NSF grant DMS-0604829. The second author is a member of the Institut Universitaire de France. 1
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