Dirichlet spaces on \(H\)-convex sets in Wiener space. (English) Zbl 1235.31010
Bull. Sci. Math. 135, No. 6-7, 667-683 (2011); erratum ibid. 137, No. 5, 688-689 (2013).
Let \(U\) be a subset of the abstract Wiener space \((E,H,\mu)\). In this paper a number of \(H\)-notions are introduced (e.g. \(H\)-convex set, \(H\)-open set, \(H\)-continuous function, \(H\)-Lipschitz function, etc.). The relations between these notions and the corresponding quasi-notions are clarified. From this it is proved that the \((1,2)\)-Sobolev space \(W^{1,2}(U)\) on a subset \(U\) in \(E\) has a dense subset of smooth cylindrical functions if and only if \(U\) is \(H\)-convex and \(H\)-open.
Reviewer: Feng-Yu Wang (Swansea)
MSC:
| 31C25 | Dirichlet forms |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 60J60 | Diffusion processes |
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