Heat Kernels and Green Functions on Metric Measure Spaces

Grigoryan A, Hu J. Heat Kernels and Green Functions on Metric Measure Spaces. Canadian Journal of Mathematics. 2014;66(3):641-699.We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball