Elliptic Regularity with Gibbs Measure Satisfying Bakry-Emery Condition

For the estimate of the Hessian you might use Bochner's formula: on such weighted $R^d$ for any smooth compactly supported function $u$ it holds $$ \nabla^*\nabla\frac{|\nabla u|^2}2=|D^2u|^2+\langle\nabla u,\nabla\nabla^*\nabla u\rangle+D^2V(\nabla u,\nabla u) $$ Now using the assumption $D^2V\geq \lambda Id$, $\nabla^*\nabla u=f$ and integrating (the left hand side integrates to 0), after rearrangement we get $$ \int |D^2u|^2\leq \int |f|^2-\lambda |\nabla u|^2 $$ In particular, if $\lambda\geq 0$ then your desired inequality holds with $C=1$. Once you got this for smooth and compactly supported $u$'s, the general case follows by approximation.