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Schauder estimates for a class of non-local elliptic equations

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  • We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder 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 $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
    Mathematics Subject Classification: Primary: 45K05, 35B65; Secondary: 60J75.


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