On the Characterization of p-Harmonic Functions on the Heisenberg Group by Mean Value Properties

TL;DR

This paper characterizes p-harmonic functions on the Heisenberg group using an asymptotic mean value property, extending Euclidean results to a subelliptic setting with a new geometric lemma.

Contribution

It introduces a geometric lemma that links maxima and minima directions to the horizontal gradient, enabling mean value characterizations in the Heisenberg group.

Findings

Established mean value characterization for p-harmonic functions in the Heisenberg group.

Developed a geometric lemma relating maxima/minima directions to the horizontal gradient.

Extended Euclidean p-harmonic theory to subelliptic settings.

Abstract

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 Manfredi et al. (2009) 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.