Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

We introduce and study a notion of Orlicz hypercontractive semigroups. Weanalyze their relations with general $F$-Sobolev inequalities, thus extendingGross hypercontractivity theory. We provide criteria for these Sobolev typeinequalities and for related properties. In particular, we implement in thecontext of probability measures the ideas of Maz'ja's capacity theory, andpresent equivalent forms relating the capacity of sets to their measure. Orliczhypercontractivity efficiently describes the integrability improving propertiesof 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 applicationwe derive accurate isoperimetric inequalities for their products. Thiscompletes earlier works by Bobkov-Houdr\'e and Talagrand, and provides a scaleof dimension free isoperimetric inequalities as well as comparison theorems.