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

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 $σ\in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $φ$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{σ_1}$ and $r^{σ_2}$ for some $0 < σ_1 \leq σ_2 < 2$. By introducing the generalized Hölder spaces, we provide $C^{φψ}$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{σ+α}$ estimates.

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