Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Abstract

Let \({\mathbb{G}}\) be a Carnot group of step r and m generators and homogeneous dimension Q. Let \({\mathbb{F}_{m,r}}\) denote the free Lie group of step r and m generators. Let also \({\pi:\mathbb{F}_{m,r}\to\mathbb{G}}\) be a lifting map. We show that any horizontally convex function u on \({\mathbb{G}}\) lifts to a horizontally convex function \({u\circ \pi}\) on \({\mathbb{F}_{m,r}}\) (with respect to a suitable horizontal frame on \({\mathbb{F}_{m,r}}\)). One of the main aims of the paper is to exhibit an example of a sub-Laplacian \({\mathcal{L}=\sum_{j=1}^m X_j^2}\) on a Carnot group of step two such that the relevant \({\mathcal{L}}\)-gauge function d (i.e., d2-Q is the fundamental solution for \({\mathcal{L}}\)) is not h-convex with respect to the horizontal frame {X1, . . . , Xm}. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).