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