Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

Abstract

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class \({\mathcal {S}^{\mathfrak {L}_0}}\) of uniformly elliptic nonlinear equations with 1 < σ < 2 (subcritical case) and to their subclass \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) with 0 < σ ≤ 1. We show that \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and C 1,α-regularity of the solutions by obtaining decay estimates of their level sets in each cases.