Schauder estimates for a class of non-local elliptic equations

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)\mathrm{/}\mathrm{\mid }y{\mathrm{\mid }}^{d+\sigma }$ and either Dini or H"older continuous data. Here $0<\sigma <2$ 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 ${\mathrm{\Lambda }}^{\alpha +\sigma }$ and ${\mathrm{\Lambda }}^{\alpha }$ for any $\alpha >0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.