Hypercontractivity type property for generalized Mehler semigroups

We investigate the hypercontractivity property of generalized Mehler semigroups on the $L^p$-scale with respect to invariant measures. This property is first obtained in the purely theoretical setting of skew operators and, subsequently, deduced for generalized Mehler semigroups arising from linear stochastic differential equations perturbed by L\'evy noise. When the associated invariant measure $\mu$ lacks a purely Gaussian structure, jump components may prevent the validity of Nelson's classical $L^p$-$L^q$ estimates. However, a summability-improving property can be obtained in the setting of mixed-norm spaces $\mathcal{X}_{p,q}(E;\gamma,\pi)$ related to the factorization of the invariant measure $\mu = \gamma * \pi$ into a Gaussian part $\gamma$ and an infinitely divisible non-Gaussian part $\pi$. As in the classical Gaussian case, some modified logarithmic Sobolev inequalities with respect to invariant measures can be inferred.