Nonlocal operators related to nonsymmetric forms I: H\"older estimates

TL;DR

This paper develops regularity theory for parabolic equations driven by nonsymmetric nonlocal operators, establishing H"older estimates and weak Harnack inequalities, and connecting nonlocal operators to classical local differential operators with drift.

Contribution

It extends nonlocal energy methods to nonsymmetric operators, linking nonlocal and local PDE theories, and provides convergence results to classical differential operators.

Findings

Established H"older regularity for nonsymmetric nonlocal operators

Proved weak Harnack inequalities in this context

Connected nonlocal operators with classical second order differential operators

Abstract

The aim of this article is to develop the regularity theory for parabolic equations driven by nonlocal operators associated with nonsymmetric forms. H\"older regularity and weak Harnack inequalities are proved using extensions of recently established nonlocal energy methods. We are able to connect the theory of nonsymmetric nonlocal operators with the important results of Aronson-Serrin in the local linear case. This connection is exemplified by nonlocal-to-local convergence results identifying the limiting class of operators as second order differential operators with drift terms.