Schauder estimates for a class of non-local elliptic equations

Abstract

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.

final submitted version, 32 pages

References in corpus (2)

• On $L_p$-estimates for a class of non-local elliptic equations
• On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem

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