Heat kernel estimates for general symmetric pure jump Dirichlet forms

TL;DR

This paper establishes stable heat kernel estimates and characterizations for symmetric pure jump Dirichlet forms on metric measure spaces with general jump kernels, volume doubling, and mild tail decay assumptions.

Contribution

It provides new stability results for heat kernel estimates under broad conditions, including variable jump behaviors and polynomial tail decay.

Findings

Two-sided heat kernel estimates established

Heat kernel upper bounds derived from jump kernel bounds

Stable characterizations of parabolic Harnack inequalities

Abstract

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.