Mean Value Property and Harmonicity on Carnot-Carathéodory Groups

We study strongly harmonic functions in Carnot–Carathéodory 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 appropriate spherical harmonic polynomials in ℍ1 are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.