Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

TL;DR

This paper establishes sharp derivative estimates and Schauder regularity results for nonautonomous Ornstein-Uhlenbeck operators in weighted continuous function spaces, also proving the compactness of the evolution operator.

Contribution

It provides new uniform derivative estimates and optimal Schauder estimates for nonautonomous Ornstein-Uhlenbeck operators in weighted spaces, along with compactness results for the evolution operator.

Findings

Sharp uniform estimates for spatial derivatives of the evolution operator

Optimal Schauder estimates for solutions to related parabolic problems

Compactness of the evolution operator in weighted spaces

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.