Regularity estimates for elliptic nonlocal operators

TL;DR

This paper investigates regularity properties of solutions to elliptic nonlocal equations with measurable kernels, providing robust estimates and a general method for deriving H"older continuity from weak Harnack inequalities.

Contribution

It introduces new regularity results for nonlocal operators with singular kernels and develops a versatile tool for obtaining H"older estimates from weak Harnack inequalities.

Findings

Regularity results robust to differentiability order

A general tool for H"older estimates from weak Harnack inequality

Comparability results for nonlocal quadratic forms

Abstract

We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general, possibly singular, measurable kernels. We obtain regularity results which are robust with respect to the differentiability order of the equation. Furthermore, we provide a general tool for the derivation of H\"{o}lder a-priori estimates from the weak Harnack inequality. This tool is applicable for several local and nonlocal, linear and nonlinear problems on metric spaces. Another aim of this work is to provide comparability results for nonlocal quadratic forms.