Heat kernel bounds for nonlocal operators with singular kernels

We prove sharp two-sided bounds of the fundamental solution for integro-differential operators of order alpha is an element of (0, 2) that generate 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. (C) 2022 Published by Elsevier Masson SAS.

Automated summary

We prove sharp two-sided bounds of the fundamental solution for integro-differential operators of order alpha is an element of (0, 2) that generate 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.