Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

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 $W^{1,p}(X,\gamma )$.