Lower Comparison Inequalities For Heat Kernel And Heat Kernel Estimate For A Family Of Non-local Operators

The heat kernel is a very useful tool in modern mathematics, which has importantapplications in many other fields. The estimate of heat kernel is studied in this paper,which mainly includes two parts.Firstly, consider a regular and strong local Dirichlet form on an abstract metric mea-sure space. We obtain the comparison inequalities for the non-negative weak supersolu-tion of heat equation. Furthermore, assuming the existence of the Dirichlet heat kernel,the lower comparison inequalities of the heat kernel are obtained. These inequalities canbe used to derive the lower bound of heat kernel in some examples. In the proofs, wemainly use the purely analytic method which is the positive maximal principle on theabstract metric measure space.Secondly, consider the existence and the sharp bounds of the heat kernel of theoperator+aα α/2+b· onRd, where is Laplace operator, α/2(0<α <2) is fractionalLaplacian, the constant a∈(0,M](M>0) and function b∈Kd,1. The Duhamel’s formulais used to obtain the existence of the heat kernel, and the sharp upper bound is obtainedsimultaneously. Then, we use chain argument and Le′vy system from the probabilisticview to obtain the lower bound of the heat kernel.