Smooth Solutions to a Class of Nonlocal Fully Nonlinear Elliptic Equations

We show that a certain class of fully nonlinear nonlocal equations have smooth solutions as long as the right-hand side is nice and the boundary datum is bounded. To this end we follow the classical strategy. We first show that solutions are $C^{\sigma+\alpha}$ continuous, then develop a bootstrap argument robust enough for operators that may depend on the solution and its \sigma-derivatives.