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

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 Hölder estimates for general elliptic equations with no regularity assumption on $x$, including for the first time operators like ${\sum }_{i=1}^n(-{\mathrm{\partial }}_{{\mathbf{\text{v}}}_i(x)}^2{)}^s$, provided that the coefficients have ``small oscillation’'.