Year: 2010
Type: preprint
Abstract: L. Capogna and M. Cowling showed that if $ϕ$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $ϕ$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $ϕ$ is in fact $C^\infty$. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.
Source: arXiv (Cornell University)
Authors: Ottazzi, Alessandro, Warhurst, Ben
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1469297