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 (M, d, mu): sigma(f, g) = integral(M x M) (f(x) - f(y))(g(x) - g(y)) J(dx, dy), where J(dx, dy) is a symmetric Radon measure on M x M \ diag that may have different scalings for small jumps and large jumps. Under a general volume doubling condition on (M, d, mu) 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 Poincare inequalities). We also give stable characterizations of the corresponding parabolic Harnack inequalities.