Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

Abstract

Abstract In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L 2 {L^{2}} -norm estimate of the survival function 1 - P t B ⁢ 1 B {1-P_{t}^{B}1_{B}} for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L 2 {L^{2}} mean-value inequality is borrowed to use.