The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$\{\begin{array}{rcll}(-\mathrm{\Delta }{)}^{s}u-\lambda {\displaystyle \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^{2s}}}& =& f(x,u)& \text{ in }\mathrm{\Omega },\\ u& =& 0& \text{ in }{\mathbb{R}}^N\setminus \mathrm{\Omega },\\ u& >& 0& \text{ in }\mathrm{\Omega },\end{array}$$ where $(-\mathrm{\Delta }{)}^s$, $s\in (0,1)$, is the fractional laplacian operator, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary such that $0\in \mathrm{\Omega }$ and $N>2s$. We will mainly consider the solvability in two cases:

1. The linear problem, that is, $f(x,t)=f(x)$, where according to the summability of the datum $f$ and the parameter $\lambda$ we give the summability of the solution $u$.
2. The problem with a nonlinear term $f(x,t)=\frac{h(x)}{t^{\sigma }}$ for $t>0$. In this case, existence and regularity will depend on the value of $\sigma$ and on the summability of $h$. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with \emph{singular coefficients} that seems to be new.