Gradient contractivity of a rescaled resolvent on domains in Wiener spaces

Abstract

Given an abstract Wiener space $(X,\gamma ,H)$, we consider an open set $O\subseteq X$ which satisfies certain smoothness and mean-curvature conditions. We prove that the rescaled resolvent operator associated to the Ornstein-Uhlenbeck operator with homogeneous Dirichlet boundary conditions on $O$ is gradient contractive in $L^p(X,\gamma )$ for every $p\in (1,\infty )$. This is the Gaussian counterpart of an analogous result for the rescaled resolvent operator associated to the Laplace operator $\Delta$ in $L^p$ with respect to the Lebesgue measure, $p\in [1,\infty )$, with homogeneous Dirichlet boundary conditions on a bounded convex open set $O\subseteq {\mathbb{R}}^n$.