How Singular Integrals Theory is applied on Partial Differential Equations Currently I'm interested in Singular Integrals Theory (I'm a beginner). I have read that this theory has deep relations with PDE's. For that reason I would like to know if there exists some web page, guide, essay or book which explain how Singular Integrals Theory are applied in PDE's or how it is used nowadays.
Thanks for your help!

Take for example Poisson’s equation: $-\Delta u = f \text{ on }\mathbb{R}^3$, where $f\in L^2(\mathbb{R}^3)$ is a compactly supported function. Then a solution to this partial differential equation is given by:
$$
u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}\,{\mathrm d}y,
$$
where $|x-y|$ is the Euclidean distance between $x$ and $y$. It can be shown that this solution is the only solution to Poisson’s equation that fulfils $u(x)\to 0$ as $|x|\to\infty$.
You see that $u$ is given by a singular integral. The study of singular integrals now allows us to conclude properties of $u$. For example, one can show that $u\in H^2_{\mathrm{loc}}(\mathbb{R}^3)$, that is $u$ is twice weakly-differentiable. Similar results also hold for $f\in L^p(\mathbb{R}^3)$, $1<p<\infty$.
