Generalized Evans-Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders

Abstract

We establish the generalized Evans-Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability sigma is an element of (0, 2), the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function phi describing the variable orders of differentiability, which is allowed to oscillate between two functions r(sigma 1) and r(sigma 2) for some 0 < sigma(1) <= sigma(2) < 2. By introducing the generalized Holder spaces, we provide C-phi psi estimates that generalizes the standard Evans-Krylov and Schauder type C sigma+alpha estimates.