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., d ^2-Q is the fundamental solution for ${\mathcal{L}}$) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).