Sobolev spaces with respect to a weighted Gaussian measures in infinite dimensions

Abstract

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,\mu )$ and $\log w\in W^{1,t}(X,\mu )$ for some $s>1$ and $t>s^{\prime }$. In the present paper we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $\nu :=w\mu$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron--Martin space) and the trace of Sobolev functions on hypersurfaces $\{x\in X|G(x)=0\}$, where $G$ is a suitable version of a Sobolev function.

References in corpus (3)

• Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure
• Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case
• Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

Cited by in corpus (5)

• Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure
• Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces
• Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case
• On the domain of elliptic operators defined in subsets of Wiener spaces
• On functions of bounded variation on convex domains in Hilbert spaces