Estimates of gradient of $\mathcal{L}$-harmonic functions for nonlocal operators with order $\alpha>1$

TL;DR

This paper derives Grönwall-type estimates for the gradient of harmonic functions associated with nonlocal Lévy operators of order greater than one, under minimal assumptions on the Lévy measure.

Contribution

It provides new gradient estimates for harmonic functions of nonlocal operators with order > 1, expanding understanding under minimal Lévy measure assumptions.

Findings

Gradient estimates for harmonic functions of Lévy operators with order > 1

Results hold under minimal Lévy measure assumptions

Extension of classical estimates to nonlocal operators

Abstract

We obtain Gr\"onwall type estimates for the gradient of the harmonic functions for a L\'evy operator with order strictly larger than 1 and minimal assumptions of its L\'evy measure.