Regularity theory for singular nonlocal diffusion equations

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover H"older continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.