On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties
Fausto Ferrari, Qing Huo Liu, Juan J. Manfredi · Discrete and Continuous Dynamical Systems · 2013
We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1 < p <\infty$, following the schemedescribed in [16] for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 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.