Heat kernel bounds for nonlocal operators with singular kernels

Abstract

We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $\alpha \in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$ independent copies of one-dimensional jump processes, i.e., the jumping measure is singular with respect to the $d$-dimensional Lebesgue measure.

Changes from version 1 to version 2 are minor. They mainly concern Lemma 4.5 and the proof of Lemma 5.1