Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

Abstract: 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 $\OU$, 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.