Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

Abstract

In this paper we consider an abstract Wiener space (X,γ,H) and an open subset O⊆X which satisfies suitable assumptions. For every p∈(1,+∞) we define the Sobolev space W01,p(O,γ) as the closure of Lipschitz continuous functions which have support with positive distance from ∂O with respect to the natural Sobolev norm, and we show that under the assumptions on O the space W01,p(O,γ) can be characterized as the space of functions in W1,p(O,γ) which have null trace at the boundary ∂O, or, equivalently, as the space of functions defined on O whose trivial extension belongs to W1,p(X,γ).