Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

TL;DR

This paper characterizes Sobolev spaces on sublevel sets within abstract Wiener spaces, establishing boundary trace properties and extension criteria for functions in these spaces under certain conditions.

Contribution

It provides a new characterization of Sobolev spaces on open subsets of abstract Wiener spaces, linking boundary trace and extension properties.

Findings

Sobolev space $W_{0}^{1,p}(O, ext{gamma})$ equals functions with null boundary trace.

Functions with trivial extension belong to $W^{1,p}(X, ext{gamma})$.

Characterization holds under specific assumptions on the open set $O$.

Abstract

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$.