Estimating the heat kernels of non-local operators or non-local Dirichlet forms on metric measure spaces has received many interest recently. Chen and Kumagai  (Stochastic Process. Appl. 108 (1) (2003) 27-62.) proved that, if the so called walk dimension is strictly smaller than two, then the stable-like heat kernel bounds can be characterized by the volume growth condition and jump kernel estimate. However, the same method does not work on the case when the walk dimension is greater than or equal to two.
In this paper, we prove necessary and sufficient conditions for stable-like estimates of the heat kernel for non-local Dirichlet forms on metric measure spaces by using purely analytic method. These conditions are given in terms of the volume growth function, jump kernel and a so called generalized capacity. Our proof works for all possible values of walk dimension (especially including the case when the walk dimension is greater than or equal to two.
 Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108(1) (2003) 27–62.