Fractional $p$-caloric functions are Lipschitz

We study the parabolic fractional $p-$Laplace equation $${\text{p}}_{t}u+(-{\mathrm{\Delta }}_p{)}^{s}u=0$$ in the degenerate range $2\le p<2\mathrm{/}(1-s)$. We show that weak solutions are Lipschitz continuous in space and, if $p>1\mathrm{/}(1-s)$, also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.