On the characterization of p-harmonic functions on the Heisenberg group by mean value properties

We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1<p<\infty$, following the scheme described in \cite{MPR} for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma \ref{lemma1} below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.