Modified logarithmic Sobolev inequalities in null curvature

Abstract

We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that $\Phi$ is a symmetric convex function on $\dR$ satisfying $(1+\e)Φ(x)\leq {x}Φ'(x)\leq(2-\e)Φ(x)$ for $x\ge 0$ large enough and with $\e\in]0,1/2]$. We prove that the probability measure on $\dR$ ${\mu }_{\Phi }(dx)=e^{-\Phi (x)}/Z_{\Phi }dx$ satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant $A,B,D>0$ such that for all smooth $f>0$, \begin{equation*} \ent{μ_Φ}{f^2}\leq A\int H_Φ\PAR{{\frac{f'}{f}}}f^2dμ_Φ, \text{with} H_Φ(x)= {\begin{array}{rl} Φ^*\PAR{Bx} &\text{if }\ABS{x}\geq D, x^2 &\text{if}\ABS{x}\leq D. \end{array} . \end{equation*}