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Pansu derivative

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In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989). A Carnot group admits a one-parameter family of dilations, . If and are Carnot groups, then the Pansu derivative of a function at a point is the function defined by

provided that this limit exists.

A key theorem in this area is the Pansu–Rademacher theorem, a generalization of Rademacher's theorem, which can be stated as follows: Lipschitz continuous functions between (measurable subsets of) Carnot groups are Pansu differentiable almost everywhere.

References

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  • Pansu, Pierre (1989), "Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un", Annals of Mathematics, Second Series, 129 (1): 1–60, doi:10.2307/1971484, ISSN 0003-486X, JSTOR 1971484, MR 0979599