  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.
