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 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''.