Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations u(t) = Iu, where is translation invariant and elliptic with respect to the class L-0(sigma) of Caffarelli and Silvestre, sigma is an element of (0, 2) being the order of I. We prove that if is a viscosity solution in B-1 x (-1, 0] which is merely bounded in R-n x (-1, 0], then u is C-beta in space and C-beta/sigma in time in (B-1/2) over bar x [-1/2, 0], for all beta < min{sigma, 1 + alpha}, where alpha > 0. Our proof combines a Liouville type theorem-relaying on the nonlocal parabolic C-alpha estimate of Chang and Davila-and a blow up and compactness argument.