This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.
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Janna Lierl, Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 687–738DOI 10.4171/RMI/1001