On a Dirichlet Problem Involving an Ornstein–Uhlenbeck Operator

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of R ⁿ . We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Hölder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc. 327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.