A Liouville type theorem for Carnot groups

Details A Liouville type theorem for Carnot groups A Liouville type theorem for Carnot groups

2010-01-07

arxiv.org/abs/1001.1087v1

Mathematics

Analysis of PDEs

11 pages

Scientific paper

L. Capogna and M. Cowling showed that if $\phi$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $\phi$ 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 $\phi$ 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.

Affiliated with

Also associated with

No associations

Rating

A Liouville type theorem for Carnot groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Liouville type theorem for Carnot groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Liouville type theorem for Carnot groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-373744