Cordes-Nirenberg type results for nonlocal equations with deforming kernels

Abstract: We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Amp`ere equation. Under a natural integrability assumption on the Monge-Amp`ere solution, we prove a stability lemma allowing the ellipticity class to vary. Using a compactness method, we then derive H\"older regularity estimates for the gradient of the solutions.