$C^{1,\alpha }$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations With Rough Kernels

We prove a $C^{1,\alpha }$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.