I'd like a reference to a proof of the Davies-Gaffney estimate; which is an off-diagonal decay result. See for instance assumption H2 in the paper "Hardy Spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates" by Hofmann, Lu, Mitrea, Mitrea, and Yan. What I'd like to see more specifically is the proof of the estimate for the heat semigroup of a strongly elliptic operator, and under what conditions it is proven.
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I think what you need is Theorem 5.5.6 of the book Davies, E. B. Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92.
