Domains of elliptic operators on sets in Wiener space

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron-Martin space is denoted by H. Consider two sufficiently regular convex functions U: X → R and G: X →R We let ν = e-Uμ and ω = G-1(-∞, 0]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form {equation presented} and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein-Uhlenbeck operator in Hilbert space with ω = X and on half-spaces, namely if U 0 and G is an affine function, then the domain of the operator defined via (0.1) is the space u {equation presented}, where ρ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.