Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group

We extend to the parabolic setting some of the ideas originated with Xiao Zhong’s proof in \cite{Zhong} of the H"older regularity of $p-$harmonic functions in the Heisenberg group $\text{Hn}$. Given a number $p\ge 2$, in this paper we establish the $C^{\mathrm{\infty }}$ smoothness of weak solutions of quasilinear pde’s in $\text{Hn}$ modelled on the equation $${\text{p}}_{t}u=\sum _{i=1}^{2n}{X}_i((1+\mathrm{\mid }{\mathrm{\nabla }}_{0}u{\mathrm{\mid }}^2{)}^{\frac{p-2}{2}}{X}_{i}u)\mathrm{.}$$