Small perturbation solutions for nonlocal elliptic equations

We present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in $C^{\sigma+\alpha}$ for any $\alpha\in (0,1)$ as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.