Characterizations of Sobolev spaces on sublevel sets in abstract Wiener spaces

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 nat...

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Published inJournal of mathematical analysis and applications Vol. 524; no. 1; p. 127075
Main Authors Addona, Davide, Menegatti, Giorgio, Miranda, Michele
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.08.2023
Subjects
Abstract Wiener spaces
Malliavin derivative
Sobolev spaces
Sublevel sets
Trace operator
Malliavin derivative
Sublevel sets
Sobolev spaces
Abstract Wiener spaces
Trace operator
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Summary: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).
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2023.127075