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

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