Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces

We give necessary and sufficient conditions for sub-Gaussian estimates of the heat kernel of a strongly local regular Dirichlet form on a metric measure space. The conditions for two-sided estimates are given in terms of the generalized capacity inequality and the Poincaré inequality. The main difficulty lies in obtaining the elliptic Harnack inequality under these assumptions. The conditions for upper bound alone are given in terms of the generalized capacity inequality and the Faber–Krahn inequality.