Abstract
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.
Highlights
The main subject of our studies are harmonic functions on Carnot-Caratheodory groups with emphasis on the setting of Heisenberg groups since in this case the pseudodistance induced by the fundamental solution of the sub-Laplacian is a metric
By using the convolution and scaling techniques available in Carnot groups, we show in Theorem 4.2 the smoothness of harmonic functions
Let us mention that a counterpart of Theorem 4.3 in more general metric spaces is not known and is a subject of an ongoing investigation to determine the relation between strongly harmonic functions and the p-harmonic functions defined as local minima of the p-Dirichlet energy with respect to weak upper gradients
Summary
The main subject of our studies are harmonic functions on Carnot-Caratheodory groups with emphasis on the setting of Heisenberg groups since in this case the pseudodistance induced by the fundamental solution of the sub-Laplacian is a metric (see below for relevant definitions). It turns out that for the proof of smoothness, one needs (1) to hold only for balls defined by a pseudodistance, i.e. the triangle inequality for d in Eq 1 can be relaxed Another topic we are especially interested in, is the interplay between harmonic functions and solutions to the subelliptic Laplace equation on a Carnot group (called the L-harmonic equation). Let us mention that a counterpart of Theorem 4.3 in more general metric spaces is not known and is a subject of an ongoing investigation to determine the relation between strongly harmonic functions and the p-harmonic functions defined as local minima of the p-Dirichlet energy with respect to weak upper gradients Another aspect of harmonicity studied in our work relates to the fact that the subelliptic harmonic functions are known to satisfy the kernel-type mean value property, see Formula (25) in Theorem 4.4 below and Appendix for its proof. Our presentation is largely self contained and for the readers convenience in Sections 2 and 3 we recall the necessary definitions and observations regarding Carnot-Caratheodory groups, pseudonorms, subelliptic Laplacians and their fundamental solutions
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