On the estimates of the derivatives of solutions to nonautonomous Kolmogorov equations and their consequences

We consider evolution operators $G(t,s)$ associated to a class of nonautonomous elliptic operators with unbounded coefficients, in the space of bounded and continuous functions over ${\mathbb{R}}^d$. We prove some new pointwise estimates for the spatial derivatives of the function $G(t,s)f$, when $f$ is bounded and continuous or much smoother. We then use these estimates to prove smoothing effects of the evolution operator in $L^p$-spaces. Finally, we show how pointwise gradient estimates have been used in the literature to study the asymptotic behaviour of the evolution operator and to prove summability improving results in the $L^p$-spaces related to the so-called tight evolution system of measures.