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Lifting of convex functions on Carnot groups and lack of convexity for a gauge function. (English) Zbl 1181.22013

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 [D. Danielli, N. Garofalo and D.-M. Nhieu, Commun. Anal. Geom. 11, No. 2, 263–341 (2003; Zbl 1077.22007)] does not extend to Carnot groups with \(r=2\).

MSC:

22E25 Nilpotent and solvable Lie groups
26B25 Convexity of real functions of several variables, generalizations
43A80 Analysis on other specific Lie groups
35A30 Geometric theory, characteristics, transformations in context of PDEs

Citations:

Zbl 1077.22007
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References:

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