A natural framework for studying semigroups associated with elliptic operators with unbounded coefficients is given by L^p spaces related to invariant measures. This is the case, for instance, of the classical Ornstein-Uhlenbeck semigroup (P(t)), which enjoys many nice properties in L^p(m), where m denotes the standard Gaussian measure that turns out to be the unique associated invariant measure. One of the most relevant properties of the Ornstein--Uhlenbeck semigroup, proved by Nelson, concerns hypercontractivity; that is, for any 1<p<q (not infinity) there exists t_0>0 such that

||P(t)f||_{L^q(m)} \le ||f||_{L^p(m)},

for all f \in L^p(m) and t > t_0.

The hypercontractivity of P(t) is strictly connected to the validity of the classical logarithmic Sobolev inequality. Moreover, the above estimat allows one to deduce the asymptotic behavior of P(t) as t tends to infinity.

The Ornstein-Uhlenbeck semigroup can be interpreted as a particular case of a generalized Mehler semigroup and, as is well known, in the general case hypercontractivity fails to hold for such semigroups.

In this talk we consider generalized Mehler semigroups on L^p spaces related to invariant measures and investigate their summability-improving properties. We identify natural subspaces of L^p where hypercontractivity-type estimates are satisfied, providing both examples and counterexamples. The results we prove extend and, in some cases, improve the existing theory. This is joint work with Luciana Angiuli (Università del Salento).