Modified log-Sobolev inequalities and isoperimetry

Abstract

We find sufficient conditions for a probability measure $\mu$ to satisfy an inequality of the type $${\int }_{{\mathbb{R}}^d}{f}^{2}F\left(\frac{f^2}{{\int }_{{\mathbb{R}}^d}{f}^{2}d\mu }\right)d\mu \le C{\int }_{{\mathbb{R}}^d}{f}^2{c}^{\ast }\left(\frac{|\nabla f|}{|f|}\right)d\mu +B{\int }_{{\mathbb{R}}^d}{f}^{2}d\mu ,$$ where $F$ is concave and $c$ (a cost function) is convex. We show that under broad assumptions on $c$ and $F$ the above inequality holds if for some $\delta >0$ and $\epsilon >0$ one has $${\int }_0^{\epsilon }\Phi \left(\delta c\right[\frac{tF(\frac{1}{t})}{{\mathcal{I}}_{\mu }(t)}\left]\right)dt<\infty ,$$ where ${\mathcal{I}}_{\mu }$ is the isoperimetric function of $\mu$ and $\Phi =(yF(y)-y{)}^{\ast }$. In a partial case $${\mathcal{I}}_{\mu }(t)\ge kt{\varphi }^{1-\frac{1}{\alpha }}(1/t),$$ where $\varphi$ is a concave function growing not faster than $\log$, $k>0$, $1<\alpha \le 2$ and $t\le 1/2$, we establish a family of tight inequalities interpolating between the $F$-Sobolev and modified inequalities of log-Sobolev type. A basic example is given by convex measures satisfying certain integrability assumptions.

26 pages

References in corpus (1)

• Characterization of Talagrand's Like Transportation-Cost Inequalities on the Real Line