Regularity results of nonlinear perturbed stable-like operators

TL;DR

This paper investigates the regularity properties of a class of fully nonlinear integro-differential operators combining stable-like and lower order Lévy measures, establishing key inequalities and boundary properties.

Contribution

It introduces new regularity results for nonlinear operators with mixed stable-like and lower order Lévy components, which lack global scaling.

Findings

Established Hölder regularity for solutions.

Proved Harnack inequality for the operators.

Demonstrated boundary Harnack property.

Abstract

We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of \textit{lower order} L\'evy measures. Such operators do not have a global scaling property. We establish H\"{o}lder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.