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-Houdré and Talagrand, and
provides a scale of dimension free isoperimetric inequalities as
well as comparison theorems.