Modified logarithmic Sobolev inequalities in null curvature

Abstract

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $\mathbb{R}$ between the exponential and the Gaussian measure. More precisely, assume that $\Phi$ is a symmetric convex function on $\mathbb{R}$ satisfying $(1+\epsilon )\Phi (x)\le x{\Phi }^{\prime }(x)\le (2-\epsilon )\Phi (x)$ for $x\ge 0$ large enough and with $\epsilon \in ]0,1/2]$. We prove that the probability measure on $\mathbb{R}$ ${\mu }_{\Phi }(dx)=e^{-\Phi (x)}/Z_{\Phi }dx$ satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants $A,B,C>0$ such that for all smooth functions $f>0$,

with

where ${\Phi }^{\ast }$ is the Legendre–Fenchel transform of $\Phi$.