Nonlocal operators related to nonsymmetric forms I: Hölder estimates

Abstract The aim of this article is to develop the regularity theory for parabolic equations driven by nonlocal operators associated with nonsymmetric forms. Hölder 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.