Schauder estimates for a class of non-local elliptic equations

We prove Schauder estimates for a class of non-local elliptic operators with kernel K(y) = a(y)/|y|^d+σ and either Dini or Hölder continuous data. Here 0 < σ < 2 is a constant and a is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz-Zygmund spaces Λ^α+σ and Λ^α for any α > 0. Several local estimates and an extension to operators with kernels K(x, y) are also discussed.