$$L^p$$-Kato class measures for symmetric Markov processes under heat kernel estimates

Abstract

In this paper, we establish the coincidence of two classes of $$L^p$$ -Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $$L^p$$ -Kato class measures is defined by the pth power of positive order resolvent kernel, another is defined in terms of the pth power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that functions u such that $$\sup _{x\in E}\int _{B_1(x)}|u|^q\mathrm {d}\mathfrak {m} <\infty $$ are of $$L^p$$ -Kato class if q is greater than a constant related to p and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman–Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to $$L^p$$ -Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.