Lifting of convex functions on Carnot groups and lack of convexity for a gauge function (Q1038681)

Let \((\mathbb G,*)\) be a Carnot group, i.e., a connected and simply connected Lie group whose Lie algebra \({\mathbf g}\) admits a stratification \({\mathbf g}=V_{1}\oplus \cdot \cdot \cdot \oplus V_{r}\) with \( [V_{1},V_{i}]=V_{i+1}\) for \(i\leq r-1\) and \([V_{1},V_{r}]=\{0\}.\) A function \(u:\mathbb G\to\mathbb R\) is called horizontally convex if the function \(\mathbb R\ni t\mapsto u(x*\operatorname{Exp}(tX))\) is convex for every \(X\in V_{1}\) and every \(x\in\mathbb G\). Let \(m=\dim (V_{1})\) and denote by \({\mathbf f}_{m,r}\) the nilpotent Lie algebra of step \(r\) generated by \(m\) of its elements \(F_{1},\dots,F_{m},\) such that for every nilpotent Lie algebra \({\mathbf n}\) of step \(r\) and for every map \(L:\{F_{1},\dots,F_{m}\} \to{\mathbf n}\) there exists a Lie algebra morphism from \({\mathbf f}_{m,r}\) to \({\mathbf n}\) extending \(L.\) Consider the stratification \({\mathbf f}_{m,r}=\widetilde{V}_{1}\oplus \dots \oplus \widetilde{V}_{r}\). Let \(\mathbb F_{m,r}\) be a connected and simply connected Lie group whose Lie algebra is isomorphic to \({\mathbf f}_{m,r}\), and \(\pi :\mathbb F_{m,r}\to\mathbb G\) be a surjective Lie group morphism such that the restriction of \(d\pi :{\mathbf f} _{m,r}\to{\mathbf g}\) to \(\widetilde{V}_{1}\) is a bijection onto \(V_{1}\). The main result in this paper states that if \(u:\mathbb G\to\mathbb R\) is horizontally convex (with \(V_{1}\) as horizontal layer), then \(u\circ \pi \) is horizontally convex (with \( \widetilde{V}_{1}\) as horizontal layer). The author also provides an example showing that Theorem 6.8 in [\textit{D. Danielli, N. Garofalo} and \textit{D.-M. Nhieu}, Commun. Anal. Geom. 11, No. 2, 263--341 (2003; Zbl 1077.22007)] does not extend to Carnot groups with \(r=2\).