  Let $X$ be a separable Banach space endowed with a non-degenerate centered
Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$.
Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$
and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and
$\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the domain of
the the self-adjoint operator associated with the quadratic form \begin{gather}
(\psi,\varphi)\mapsto
\int_\Omega\langle\nabla_H\psi,\nabla_H\varphi\rangle_Hd\nu\qquad\psi,\varphi\in
W^{1,2}(\Omega,\nu).\qquad\qquad (\star) \end{gather} In particular we obtain a
complete characterization of the Ornstein-Uhlenbeck operator on half-spaces,
namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the
operator defined via $(\star)$ is the space \[\{u\in W^{2,2}(\Omega,\mu)\,|\,
\langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }\rho\text{-a.e. }x\in
G^{-1}(0)\},\] where $\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface
measure.
