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

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