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

Understand

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\&#34;older regularity, and $C^{1,\alpha}$-regularity of the solutions by obtaining decay estimates of their level sets in each cases.