Let $\Omega \subset \mathbb{R}^N$ a bounded domain and $f \in L^{p}(\Omega)$ for some $p \geq 1$. The Newstonian potential of $f$ is defined by $$w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy,$$ where $\Gamma$ is the fundamental solution of the equation $\Delta u =0$.
With this definition, is possible to introduce the calderón-zygmund estimate which proof can be found in the book: Elliptic Partial Differential Equations of Second Order by David Gilbarg and Neil.S Trudinger Theorem 9.9 page 230:
Let $f \in L^p({\Omega})$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$. Then $w \in W^{2,p}(\Omega)$, $\Delta w=f$ a.e and $$\left\lVert D^{2}w \right\rVert_p \leq C(N,p)\lVert f \rVert_p.$$
The problem i'm facing is that this theorem is stated and proved if $\Omega$ is supposed to be an bounded domain. After the proof is finished in the book, the authors says that the result remain valid for $\Omega=\mathbb{R}^N$ for example, provided $N \geq 3$, but the hint he gives to prove this fact i'm not finding very useful and given the complexity of the proof I'm finding very hard to modify the argument to include this special case $(\Omega=\mathbb{R}^N).$
My question is: Anyone know some reference where the proof for the unbounded case of this theorem can be found in relative good detail? To be especific, the case $\Omega=\mathbb{R}^N$ is the one that I need.
you can give a look at Stein's book Singular Integrals and Differentiability Properties of Functions at page 59 proposition 3. There the estimate is proved by means of Riesz Transforms and for compactly supported functions. I think you can complete the proof removing the assumption on the support. Th estimate you're gonna get will have an extra term of course.