  We settle the open question concerning the Harnack inequality for globally
positive solutions to non-local in time diffusion equations by constructing a
counter-example for dimensions $d\ge\beta$, where $\beta\in(0,2]$ is the order
of the equation with respect to the spatial variable. The equation can be
non-local both in time and in space but for the counter-example it is important
that the equation has a fractional time derivative. In this case, the
fundamental solution is singular at the origin for all times $t>0$ in
dimensions $d\ge\beta$. This underlines the markedly different behavior of
time-fractional diffusion compared to the purely space-fractional case, where a
local Harnack inequality is known.
  The key observation is that the memory strongly affects the estimates. In
particular, if the initial data $u_0 \in L^q_{loc}$ for $q$ larger than the
critical value $\tfrac d\beta$ of the elliptic operator $(-\Delta)^{\beta/2}$,
a non-local version of the Harnack inequality is still valid as we show.
  We also observe the critical dimension phenomenon already known from other
contexts: the diffusion behavior is substantially different in higher
dimensions than $d=1$ provided $\beta>1$, since we prove that the local Harnack
inequality holds if $d<\beta$.
