Interpolated inequalities between exponential and Gaussian, \nOrlicz hypercontractivity and isoperimetry

Abstract

We introduce and study a notion of Orlicz hypercontractive\nsemigroups. We analyze their relations with general $F$-Sobolev\ninequalities, thus extending Gross hypercontractivity theory. We\nprovide criteria for these Sobolev type inequalities and for\nrelated properties. In particular, we implement in the context of\nprobability measures the ideas of Maz'ja's capacity theory, and\npresent equivalent forms relating the capacity of sets to their\nmeasure. Orlicz hypercontractivity efficiently describes the\nintegrability improving properties of the Heat semigroup associated\nto the Boltzmann measures $\\mu_{\\alpha}(dx) = (Z_{\\alpha})^{-1}\ne^{-2|x|^{\\alpha}} dx$, when $\\alpha\\in (1,2)$. As an application\nwe derive accurate isoperimetric inequalities for their products.\nThis completes earlier works by Bobkov-Houdré and Talagrand, and\nprovides a scale of dimension free isoperimetric inequalities as\nwell as comparison theorems.