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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A new characterization of convexity in free Carnot groups


Authors: Andrea Bonfiglioli and Ermanno Lanconelli
Journal: Proc. Amer. Math. Soc. 140 (2012), 3263-3273
MSC (2000): Primary 31C05, 26B25, 43A80; Secondary 35J70
Published electronically: January 30, 2012
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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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Additional Information

Andrea Bonfiglioli
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
Email: bonfigli@dm.unibo.it

Ermanno Lanconelli
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
Email: lanconel@dm.unibo.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11180-3
PII: S 0002-9939(2012)11180-3
Received by editor(s): October 13, 2010
Received by editor(s) in revised form: March 30, 2011
Published electronically: January 30, 2012
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2012 American Mathematical Society