On the Ornstein-Uhlenbeck Operator in Spaces of Continuous Functions

Abstract

We study the realization of the Ornstein-Uhlenbeck operator A in the space of the uniformly continuous and bounded functions in R n . We prove that it generates a semigroup which is neither strongly continuous nor analytic, but enjoys nice smoothing properties. Such properties are used to prove a Schauder-type theorem for the elliptic equation λ u − A u = f.