H\"older and Harnack estimates for integro-differential operators with kernels of measure

TL;DR

This paper proves H"older and Harnack estimates for solutions to nonlocal elliptic equations with measure-based kernels, extending previous work and using a De Giorgi-type approach.

Contribution

It introduces generalized estimates for nonlocal operators with measure kernels, broadening the scope of prior results in the field.

Findings

Established H"older continuity for solutions.

Proved Harnack inequalities for weak solutions.

Extended previous results to more general kernels.

Abstract

We establish H\"older and Harnack estimates for weak solutions of a class of elliptic nonlocal equations that are modeled on integro-differential operators with kernels of measure. The approach is of De Giorgi-type, as developed by DiBenedetto, Gianazza and Vespri in a local setting. Our results generalize the work by Dyda and Kassmann (2020).