Continuity of solutions to equations with weakly singular nonlocal operators

We prove regularity estimates for weak solutions to linear nonlocal equations. The nonlocal operators under consideration are integro-differential operators with almost no order of differentiability. Under rather weak assumptions, which allow the operator to be anisotropic and weakly singular, we show that locally bounded weak solutions are continuous. We prove uniform a-priori estimates for the modulus of continuity. Furthermore, we provide sufficient conditions for the boundedness of weak solutions.