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In this paper, we consider fully nonlinear integro-differential equationswith possibly nonsymmetric kernels. We are able to find different versions ofAlexandroff-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 largenumber of nonlinear operators as well as linear operators. And we show aHarnack inequality, H\"older regularity, and $C^{1,\alpha}$-regularity of thesolutions by obtaining decay estimates of their level sets in each cases.

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