Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Abstract

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general $F$-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures $\mu_\alpha (dx) = (Z_\alpha)^{-1} e^{-2|x|^\alpha} dx$, when $\alpha\in (1,2)$. As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdr\'e and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.