Regularity for fully nonlinear equations driven by spatial-inhomogeneous nonlocal operators

In this paper we consider a large class of fully nonlinear integro-differential equations. The class of our nonlocal operators we consider is not spatial homogeneous and we put mild assumptions on its kernel near zero. We prove the H\"older regularity for such equation. In particular, our result covers the case that the kernel $K(x,y)$ is comparable to $|x-y|^{-d-\alpha} \ln (|x-y|^{-1})$ for $|x-y|<c$ where $0<\alpha<2$.