  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.
