$W^{2,p}$ estimates for Newtonian potential

Question

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f \in L^p(\Omega)$ for some $1<p<\infty$. Let $$ w(x) = \int_{\Omega} \Gamma(x-y)f(y)dy $$ be the Newtonian potential of $f$, where $\Gamma$ is the fundamental solution of Laplace's equation. By Theorem 9.9 in the book of Gilbarg/Trudinger, we then have $u \in W^{2,p}(\Omega)$, $\Delta w=f$ almost everywhere in $\Omega$, moreover $w$ satisfies the estimate $$||D^2 w||_{L^p(\Omega)} \leq C ||f||_{L^p(\Omega)}.$$ I am interested in the question if $w$ also satisfies an estimate of the form $$ ||w||_{W^{2,p}(\Omega)} \leq C ||f||_{L^p(\Omega)}, $$ in particular if the $L^p$ norm of the first order derivatives of $w$ can be controlled by the $L^p$ norm of $f$. If this estimate fails in general, is it satisfied if we assume that $\Omega$ is a Lipschitz domain?