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 of uniformly elliptic nonlinear equations with 1 < sigma < 2 (subcritical case) and to their subclass with 0 < sigma a parts per thousand currency sign 1. We show that still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Holder regularity, and C (1,alpha) -regularity of the solutions by obtaining decay estimates of their level sets in each cases.