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 $u_t=\text{I}u$, where $\text{I}$ is translation invariant and elliptic with respect to the class ${\mathcal{L}}_0(\sigma )$ of Caffarelli and Silvestre, $\sigma \in (0,2)$ being the order of $\text{I}$. We prove that if $u$ is a viscosity solution in $B_1\times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$,then$u$is$C^\beta$inspaceand$C^{\beta/\sigma}$intimein$\overline{B_{1/2}} \times [-1/2,0]$,forall$\beta< \min{\sigma, 1+\alpha}$,where$\alpha>0$. Our proof combines a Liouville type theorem —relaying on the nonlocal parabolic $C^{\alpha }$ estimate of Chang and D'avila— and a blow up and compactness argument.