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 metric measure space $(M,d,\mu)$: $$\mathcal{E}(f,g)=\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $J(dx,dy)$ is a symmetric Radon measure on $M\times M\setminus {\rm diag}$ that may have different scalings for small jumps and large jumps. Under 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 (resp.\ the Poincar\'e inequalities). We also give stable characterizations of the corresponding parabolic Harnack inequalities.