Dirichlet spaces on H-convex sets in Wiener space

TL;DR

This paper studies the structure of Sobolev spaces on H-convex subsets of Wiener space, establishing density of smooth cylindrical functions under certain convexity and openness conditions.

Contribution

It proves that the Sobolev space W^{1,2}(U) has smooth cylindrical functions dense when U is H-convex and H-open, linking H- and quasi-notions.

Findings

Density of smooth cylindrical functions in W^{1,2}(U) for H-convex, H-open sets

Relations between H- and quasi-notions in Wiener space

Enhanced understanding of Dirichlet spaces on convex subsets

Abstract

We consider the $(1,2)$-Sobolev space $W^{1,2}(U)$ on subsets $U$ in an abstract Wiener space, which is regarded as a canonical Dirichlet space on $U$. We prove that $W^{1,2}(U)$ has smooth cylindrical functions as a dense subset if $U$ is $H$-convex and $H$-open. For the proof, the relations between $H$-notions and quasi-notions are also studied.