A new characterization of convexity in free Carnot groups

Bonfiglioli, Andrea; Lanconelli, Ermanno

doi:10.1090/S0002-9939-2012-11180-3

A new characterization of convexity in free Carnot groups
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by Andrea Bonfiglioli and Ermanno Lanconelli PDF

Proc. Amer. Math. Soc. 140 (2012), 3263-3273 Request permission

Abstract:

A characterization of convex functions in $\mathbb {R}^N$ states that an upper semicontinuous function $u$ is convex if and only if $u(Ax)$ is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix $A$. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups $\mathbb {G}$ when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps $x\mapsto Ax$ of the Euclidean case must be replaced by suitable group isomorphisms $x\mapsto T_A(x)$, whose differential preserves the first layer of the stratification of $\operatorname {Lie}(\mathbb {G})$.

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