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 $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{ \infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{ i=1}^{ 2n} X_i \bigg((1+|\nabla_0 u|^2)^{ \frac{ p-2}{ 2}} X_i u\bigg).$$
Categories: math.AP