AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a
metric measure space, an off-diagonal stable-like upper bound of the heat
kernel is equivalent to the conjunction of the on-diagonal upper bound, a
cutoff inequality on any two concentric balls, and the jump kernel upper
bound, for any walk dimension. If in addition the jump kernel vanishes, that
is, if the Dirichlet form is strongly local, we obtain a
sub-Gaussian upper bound. This gives a unified approach to obtaining heat
kernel upper bounds for both the non-local and the local Dirichlet forms.
Heat kernel upper bounds on a complete non-compact manifold
