Lifting of convex functions on Carnot groups and lack of convexity for a gauge function
type: Item
instance of: scholarly article
title: Lifting of convex functions on Carnot groups and lack of convexity for a gauge function
zbMATH Open document ID: 1181.22013
published in: Archiv der Mathematik
publication date: 2009-11-20T00:00:00Z
review text: 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 \(r=2\).
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 5636262
zbMATH Keywords: convex functions
zbMATH Keywords: Carnot groups
zbMATH Keywords: gauge functions
author: Andrea Bonfiglioli
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
full work available at URL: s00013-009-0033-4
OpenAlex ID: W1979655769
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DOI: 10.1007/S00013-009-0033-4
description: scientific article; zbMATH DE number 5636262