Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

Abstract

We are concerned with heat kernel estimates for a non-local Dirichlet form on an Ahlfors regular metric measure space. We use an analytic approach to obtain the full lower stable-like estimate of the heat kernel from the near diagonal lower estimate. Combining with other known results, we obtain certain equivalent conditions for two-sided stable-like estimates of the heat kernel. The results can be simplified in certain cases, for example, when the Dirichlet form admits an effective resistance.