Regularity theory for nonlocal equations with VMO coefficients

We prove higher regularity for nonlinear nonlocal equations with possibly discontinu-ous coefficients of VMO type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher Holder regularity for such nonlocal equations.