Hölder estimates for viscosity solutions of nonlocal equations with variable-order fractional Laplace term

We consider a class of linear integro-differential equations with variable-order fractional Laplacian. Under sharp assumptions on the variable-order α(x,y), we prove the local Hölder continuity of bounded viscosity solutions. In particular, for the continuous right-hand side f, we show that weak solutions are viscosity solutions. Furthermore, we prove a comparison principle for viscosity solutions when the function f is strictly away from zero.