Modified log-Sobolev inequalities and isoperimetry

Abstract

We find sufficient conditions for a probability measure $μ$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d μ} \Bigr) d μ\le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d μ+ B \int_{\R^d} f^2 d μ, $$ 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 $δ>0$ and $ε>0$ one has $$ \int_{0}^ε Φ\Bigl(δc\Bigl[\frac{t F(\frac{1}{t})}{{\mathcal I}_μ(t)} \Bigr] \Bigr) dt < \infty, $$ where ${\mathcal I}_μ$ is the isoperimetric function of $μ$ and $Φ= (y F(y) -y)^{*}$. In a partial case $${\mathcal I}_μ(t) \ge k t ϕ^{1-\frac{1}α} (1/t), $$ where $ϕ$ is a concave function growing not faster than $\log$, $k>0$, $1 < α\le 2$ 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.