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The purpose of this paper is to study and classify singular solutions of thePoisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned}{\mathcal L}^s_\mu u = f \quad\ {\rm in}\ \, \Omega\setminus \{0\},\\ u =0\quad\ {\rm in}\ \, {\mathbb R}^N \setminus \Omega\ %\\ %\liminf_{x \to0}\:|u(x)| /\Phi_\mu(x) = k. \end{aligned} \right. $$ for the fractional Hardyoperator ${\mathcal L}_\mu^s u= (-\Delta)^s u +\frac{\mu}{|x|^{2s}}u$ in abounded domain $\Omega \subset {\mathbb R}^N$ ($N \ge 2$) containing theorigin. Here $(-\Delta)^s$, $s\in(0,1)$, is the fractional Laplacian of order$2s$, and $\mu \ge \mu_0$, where $\mu_0 =-2^{2s}\frac{\Gamma^2(\frac{N+2s}4)}{\Gamma^2(\frac{N-2s}{4})}<0$ is the bestconstant in the fractional Hardy inequality. The analysis requires a thoroughstudy of fundamental solutions and associated distributional identities.Special attention will be given to the critical case $\mu= \mu_0$ whichrequires more subtle estimates than the case $\mu>\mu_0$.

The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions