Schauder estimates for a class of non-local elliptic equations

TL;DR

This paper establishes Schauder estimates for a broad class of non-local elliptic operators with irregular kernels, demonstrating their isomorphic properties between specific function spaces and extending classical regularity results.

Contribution

It provides new Schauder estimates for non-local elliptic equations with non-symmetric, irregular kernels, and shows these operators are isomorphisms between Lipschitz--Zygmund spaces.

Findings

Proved Schauder estimates for non-local elliptic operators with general kernels.

Established isomorphism between Lipschitz--Zygmund spaces via these operators.

Extended estimates to operators with variable kernels K(x,y).

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 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 $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.