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

Abstract

We study integro-differential elliptic equations (of order 2s$2s$) 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 & ouml;lder estimates for general elliptic equations with no regularity assumption on x$x$, including for the first-time operators like & sum;i=1n(-partial derivative vi(x)2)s$\sum _{i=1}<^>n(-\partial <^>2_{{\bf v}_i(x)})<^>s$, provided that the coefficients have small oscillation.