The Harnack inequality for a class of nonlocal parabolic equations

In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.