Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Let $\nu =e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and $U:X\to \mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type $$\lambda u-L_{\nu }u=f,$$ where $\lambda >0$, $f\in L^2(X,\nu)$and$L_\nu$ is the self-adjoint operator associated with the quadratic form $$(\psi ,\phi )\mapsto {\int }_X{⟨{\mathrm{\nabla }}_H\psi ,{\mathrm{\nabla }}_H\phi ⟩}_{H}d\nu \psi ,\phi \in W^{1,2}(X,\nu )\mathrm{.}$$