Estimates for Poisson's equation

I am studying Calderón-Zygmund estimates ($$L^p$$ estimates) for Poisson equation. What I already know is:

Let $$\Omega$$ be a bounded domain, $$f \in L^p(\Omega)$$, $$1, and let $$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 $$\mathbb{R}^n$$, $$u \in W^{2,p}_{0}(\Omega)$$, $$1. Then $$||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.