On the gradient estimates for evolution operators associated to Kolmogorov operators

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a class of operators which satisfy our conditions

On improvement of summability properties in nonautonomous Kolmogorov equations

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided

Non autonomous parabolic problems with unbounded coefficients in unbounded domains

Given a class of nonautonomous elliptic operators \A(t) with unbounded coefficients, defined in \overline{I \times \Om} (where $I$ is a right-halfline or $I=\R$ and \Om\subset \Rd is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to \A(t) in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved