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 $\sigma \in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $\varphi$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{\sigma_1}$ and $r^{\sigma_2}$ for some $0 < \sigma_1 \leq \sigma_2 < 2$. By introducing the generalized H\"older spaces, we provide $C^{\varphi\psi}$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{\sigma+\alpha}$ estimates.