Heat kernel estimates for general symmetric pure jump Dirichlet forms

In this paper we consider the following symmetric non-local Dirichlet forms of pure jump type on a metric measure space.(equation presented); where J.dx; dy/is a symmetric Radon measure on M × M n diag that may have different scalings for small jumps and large jumps. Under a general volume doubling condition on.M; d; μ/and some mild quantitative assumptions on J.dx; dy/that are allowed to have light tails of polynomial decay at infinity, we establish stability results for two-sided heat kernel estimates as well as heat kernel upper bound estimates in terms of jumping kernel bounds, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (respectively, the Poincaré inequalities). We also give stable characterizations of the corresponding parabolic Harnack inequalities.