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

Keywords:

AMSC: 35R11, 35B65

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