From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality

Gentil, Ivan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 2, pp. 291-308.

Dans cet article nous améliorons la méthode exposée par S. Bobkov et M. Ledoux dans [BL00]. En utilisant l’inégalité de Prékopa-Leindler, nous prouvons une inégalité de Sobolev logarithmique modifiée, adaptée à toutes les mesures sur ${ℝ}^n$ possédant un potentiel strictement convexe et super-linéaire. Cette inégalité implique en particulier une inégalité de Sobolev logarithmique modifiée, développée dans [GGM05, GGM07], pour les mesures ayant un potentiel uniformément strictement convexe mais aussi une inégalité de Sobolev logarithmique de type euclidien.

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on ${ℝ}^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.

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