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We establish the generalized Evans--Krylov and Schauder type estimates fornonlocal fully nonlinear elliptic equations with rough kernels of variableorders. In contrast to the fractional Laplacian type operators having a fixedorder of differentiability $\sigma \in (0,2)$, the operators underconsideration have variable orders of differentiability. Since the order is notcharacterized by a single number, we consider a function $\varphi$ describingthe variable orders of differentiability, which is allowed to oscillate betweentwo 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 andSchauder type $C^{\sigma+\alpha}$ estimates.

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