Stability of heat kernel estimates for symmetric non-local Dirichlet forms

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability two-sided heat kernel estimates upper bounds. We obtain their stable equivalent characterizations in terms the jumping kernels, variants cut-off Sobolev inequalities, Faber-Krahn inequalities. particular, for ? \alpha -stable-like even with alttext="alpha greater-than-or-equal-to 2"> ? 2 encoding="application/x-tex">\alpha \ge 2 when underlying have walk dimensions larger than alttext="2"> encoding="application/x-tex">2 , which has been one major open problems area.