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.

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