Modified log-Sobolev inequalities and isoperimetry

Abstract

We find sufficient conditions for a probability measure $\mu$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d \mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d \mu + B \int_{\R^d} f^2 d \mu, $$ 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 $$ \int_{0}^{\epsilon} \Phi\Bigl(\delta c\Bigl[\frac{t F(\frac{1}{t})}{{\mathcal I}_{\mu}(t)} \Bigr] \Bigr) dt < \infty, $$ where ${\mathcal I}_{\mu}$ is the isoperimetric function of $\mu$ and $\Phi = (y F(y) -y)^{*}$. In a partial case $${\mathcal I}_{\mu}(t) \ge k t \phi ^{1-\frac{1}{\alpha}} (1/t), $$ where $\phi$ 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.