Conformality and Q-harmonicity in Carnot groups

Abstract

We show that if $f$ is a $1$-quasiconformal map defined on an open subset of a Carnot group $G$, then composition with $f$ preserves $Q$-harmonic functions. We combine this with a regularity theorem for $Q$-harmonic functions and an algebraic regularity theorem for maps between Carnot groups to show that $f$ is smooth. We give some applications to the study of rigidity