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In this paper we study the influence of the Hardy potential in the fractionalheat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{\begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ cf\mbox{ in } \Omega\times (0,T),\\ u(x,t)&>&0\inn \Omega\times (0,T),\\u(x,t)&=&0\inn (\ren\setminus\Omega)\times[ 0,T),\\ u(x,0)&=&u_0(x) \mbox{ if}x\in\O, \end{array} \right. $$ where $N> 2s$, $0<s<1$, $(-\Delta)^s$ is thefractional Laplacian of order $2s$, $p>1$, $c,\l>0$, $u_0\ge 0$, $f\ge 0$ arein a suitable class of functions and $\theta=\{0,1\}$. Notice that $(P_0)$ is alinear problem, while $(P_1)$ is a semilinear problem.  The main features in the article are: \begin{enumerate} \item Optimal resultsabout \emph{existence} and \emph{instantaneous and complete blow up} in thelinear problem $(P_0)$, where the best constant $\Lambda_{N,s}$ in thefractional Hardy inequality provides the threshold between existence andnonexistence. Similar results in the local heat equation were obtained by Barasand Goldstein in \cite{BaGo}. However, in the fractional setting the argumentsare much more involved and they require the proof of a weak Harnack inequalityfor a weighted operator that appear in a natural way. Once this Harnackinequality is obtained, the optimal results follow as a simpler consequencethan in the classical case. \item The existence of a critical power$p_+(s,\lambda)$ in the semilinear problem $(P_1)$ such that: \begin{enumerate}\item If $p> p_+(s,\lambda)$, the problem has no weak positive supersolutionsand a phenomenon of \emph{complete and instantaneous blow up} happens. \item If$p< p_+(s,\lambda)$, there exists a positive solution for a suitable class ofnonnegative data. \end{enumerate} \end{enumerate}

Optimal results for the fractional heat equation involving the Hardy potential