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

Abstract

We establish the generalized EvansKrylov 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 σ ∈ (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 ϕ describing the variable orders of differentiability, which is allowed to oscillate between two functions r σ 1 and r σ 2 for some 0 < σ1 ≤ σ2 < 2. By introducing the generalized Hölder spaces, we provide C ϕψ estimates that generalizes the standard EvansKrylov and Schauder type C σ+α estimates.