# Modified logarithmic Sobolev inequalities in null curvature

### Ivan Gentil

Université Claude Bernard Lyon 1, Villeurbanne, France### Arnaud Guillin

Université de Provence, Marseille, France### Laurent Miclo

Université de Provence, Marseille, France

## Abstract

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $R$ between the exponential and the Gaussian measure. More precisely, assume that $Φ$ is a symmetric convex function on $R$ satisfying $(1+ε)Φ(x)≤xΦ_{′}(x)≤(2−ε)Φ(x)$ for $x≥0$ large enough and with $ε∈]0,1/2]$. We prove that the probability measure on $R$ $μ_{Φ}(dx)=e_{−Φ(x)}/Z_{Φ}dx$ satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants $A,B,C>0$ such that for all smooth functions $f>0$,

$Ent_{μ_{Φ}}(f_{2})≤A∫H_{Φ}(ff_{′} )f_{2}dμ_{Φ},$

with

$H_{Φ}(x)={x_{2}if∣x∣<C,Φ_{∗}(Bx)if∣x∣≥C, $

where $Φ_{∗}$ is the Legendre-Fenchel transform of $Φ$.

## Cite this article

Ivan Gentil, Arnaud Guillin, Laurent Miclo, Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 235–258

DOI 10.4171/RMI/493