$L^p$-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 $p$-th power of positive order resolvent kernel, another is defined in terms of the $p$-th power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that $q$-th integrable functions on balls with radius $1$ having uniformity of its norm with respect to centers 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.