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 $\cS^{\fL_0}$ of uniformly elliptic nonlinear equations with $1<\sigma<2$ (subcritical case) and to their subclass $\cS^{\fL_0}_{\eta}$ with $0<\sigma\leq 1$. We show that $\cS^{\fL_0}_{\eta}$ still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, H\"older regularity, and $C^{1,\alpha}$-regularity of the solutions by obtaining decay estimates of their level sets in each cases.