Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in R^N.We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator P_{s,t} , which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t > s, the evolution operator P_{s,t} is compact in the previous weighted spaces.