From the Pr\'ekopa-Leindler inequality to modified logarithmic Sobolev inequality

TL;DR

This paper introduces an improved method leveraging the Prékopa-Leindler inequality to establish a modified logarithmic Sobolev inequality applicable to various measures with convex potentials, enhancing existing inequalities.

Contribution

It develops a new approach that extends the modified logarithmic Sobolev inequality to all measures with strictly convex, super-linear potentials using the Prékopa-Leindler inequality.

Findings

Proves a modified logarithmic Sobolev inequality for all measures with convex potentials.

Shows the inequality implies classical Euclidean logarithmic Sobolev inequality.

Extends the applicability of Sobolev inequalities to broader measure classes.

Abstract

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux. Using the Pr\'ekopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on $\dR^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality for all uniform strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.