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

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, H?lder regularity, and C ^1,α -regularity of the solutions by obtaining decay estimates of their level sets in each cases.