Regularity results for fully nonlinear integro-differential equations

• For general fully nonlinear integro-differential equations, interior $C^{1,\alpha}$ estimates can be proved in a variety of situations. The simplest assumption would be for a translation invariant uniformly elliptic equations with respect to the class of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 1 . There are several other $C^{1,\alpha}$ estimates for variations of this situation (smooth coefficients, kernels close to the smoothness class, etc...) .

• A nonlocal version of Evans-Krylov theorem says that for the Bellman equation, for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$ . This is enough regularity for the solutions to be classical.