I am studying Calderón-Zygmund estimates ($L^p$ estimates) for Poisson equation. What I already know is:
Let $\Omega$ be a bounded domain, $f \in L^p(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonial Potential of $f$. Then $w \in W^{2,p}(\Omega)$, $\Delta w = f$ a.e. and $$ ||D^{2}w||_{p} \leq C(n,p) ||f||_{p} $$ and $$ \int_{\mathbb{R}^{n}} |D^2w|^2 = \int_{\Omega} f^2. $$
Moreover, I know that
Let $\Omega$ a domain in $\mathbb{R}^n$, $u \in W^{2,p}_{0}(\Omega)$, $1<p<\infty$. Then $$ ||D^2u||_p \leq C(n,p) ||\Delta u||_p $$ and if $p=2$, $$ ||D^2 u||^2_2 = ||\Delta u||^{2}_{2}. $$
My question is how to extend the first theorem to solutions of the Poisson equation in $W^{1,2}(\Omega)$, i.e., given $u \in W^{1,2}(\Omega)$, if $\Delta u \in L^{p}$, then $u \in W^{2,p}(\Omega)$?
I'm following the book by Gilbarg and Trudinger "Elliptic Partial DIfferential Equations of Second Order" but I cannot find there what I really want to.
