Nonlocal operators with singular anisotropic kernels

TL;DR

This paper investigates anisotropic nonlocal operators with singular kernels, establishing regularity results and inequalities for solutions, which broadens understanding of jump processes with direction-dependent behaviors.

Contribution

It introduces a general framework for anisotropic nonlocal operators with measurable coefficients and proves key regularity properties and inequalities for their solutions.

Findings

Proved a weak Harnack inequality for solutions.

Established Hölder regularity results.

Analyzed operators with anisotropic jump processes.

Abstract

We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. We prove a weak Harnack inequality and H\"older regularity results for solutions to corresponding integro-differential equations.