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

Abstract We study integro‐differential elliptic equations (of order ) 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 , including for the first‐time operators like , provided that the coefficients have “small oscillation.”