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 $(1+\varepsilon)\Phi(x)\leq {x}\Phi'(x)\leq(2-\varepsilon)\Phi(x)$ for $x\geq 0$ large enough and with $\varepsilon\in ]0,1/2]$. 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 $f>0$, $$ \mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi, $$ with $$ H_{\Phi}(x)= \left\{ \begin{array}{l} x^2 \text{ if }\left|x\right|< C,\\ \Phi^*\left(Bx\right) \text{ if }\left|x\right|\geq C, \end{array} \right. $$ where $\Phi^*$ is the Legendre-Fenchel transform of $\Phi$.