Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels

TL;DR

This paper establishes the most general Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with variable coefficients and singular kernels, including operators with minimal regularity assumptions.

Contribution

It extends Schauder and Cordes-Nirenberg estimates to a broad class of nonlocal elliptic equations with singular kernels and minimal regularity on coefficients.

Findings

Proved Schauder-type estimates for nonlocal elliptic equations with variable coefficients.

Established H"older estimates without regularity assumptions on the coefficients.

Included operators like sum of fractional Laplacians with variable directions.

Abstract

We study integro-differential elliptic equations (of order $2s$) with variable coefficients, and prove the natural and most general Schauder-type estimates that can hold in this setting, both in divergence and non-divergence form. Furthermore, we also establish H\"older estimates for general elliptic equations with no regularity assumption on $x$, including for the first time operators like $\sum_{i=1}^n(-\partial^2_{\textbf{v}_i(x)})^s$, provided that the coefficients have ``small oscillation''.