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

TL;DR

This paper develops Sobolev spaces on infinite-dimensional Banach spaces with weighted Gaussian measures, analyzing divergence operators and trace theorems for functions on hypersurfaces.

Contribution

It introduces and studies Sobolev spaces with respect to weighted Gaussian measures, including divergence and trace results in infinite dimensions.

Findings

Defined Sobolev spaces with weighted Gaussian measures.

Established divergence operator properties.

Proved trace theorems on hypersurfaces.

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 $t>s'$. 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.