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 non-Autonomous elliptic operators with unbounded coefficients, in the space of bounded and continuous functions over Rd. 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 LP-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 LP-spaces related to the so-called tight evolution system of measures.