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Pré-Publication, Document De Travail Année : 2016

Pliability, or the Whitney Extension Theorem for Curves in Carnot Groups

Résumé

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to numerous settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to R^d for some d ≥ 1. We study here the case of maps from R to a general Carnot group and introduce for this purpose the notion of pliable Carnot group. We prove that this geometric condition, not far from the non-rigidity of curves as defined by Bryant and Hsu, is equivalent to the suitable statement for a horizontal Whitney extension theorem of class C^1. We provide examples of pliable and non-pliable Carnot groups and establish a relation with the Lusin approximation problem.
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Dates et versions

hal-01285215 , version 1 (08-03-2016)
hal-01285215 , version 2 (24-03-2016)
hal-01285215 , version 3 (19-04-2016)
hal-01285215 , version 4 (08-12-2016)

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Nicolas Juillet, Mario Sigalotti. Pliability, or the Whitney Extension Theorem for Curves in Carnot Groups. 2016. ⟨hal-01285215v2⟩
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