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We consider evolution operators $G(t,s)$ associated to a class ofnonautonomous elliptic operators with unbounded coefficients, in the space ofbounded and continuous functions over $\mathbb{R}^d$. We prove some newpointwise 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 toprove smoothing effects of the evolution operator in $L^p$-spaces. Finally, weshow how pointwise gradient estimates have been used in the literature to studythe asymptotic behaviour of the evolution operator and to prove summabilityimproving results in the $L^p$-spaces related to the so-called tight evolutionsystem of measures.

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