Mean value property and harmonicity on Carnot-Carath\'eodory groups

TL;DR

This paper investigates strongly harmonic functions in Carnot-Carathéodory groups, establishing their regularity, connection to the sub-Laplace equation, and illustrating these properties with examples.

Contribution

It demonstrates that strongly harmonic functions are Sobolev regular, satisfy the sub-Laplace equation, and provides explicit examples including spherical harmonic polynomials.

Findings

Strongly harmonic functions are Sobolev regular and smooth.

Such functions satisfy the sub-Laplace equation with respect to the gauge norm.

Spherical harmonic polynomials in  are both strongly harmonic and satisfy the sub-Laplace equation.

Abstract

We study strongly harmonic functions in Carnot-Carath\'eodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that spherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.