On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11.40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.