Pansu derivative

In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989). A Carnot group [math]\displaystyle{ G }[/math] admits a one-parameter family of dilations, [math]\displaystyle{ \delta_s\colon G\to G }[/math]. If [math]\displaystyle{ G_1 }[/math] and [math]\displaystyle{ G_2 }[/math] are Carnot groups, then the Pansu derivative of a function [math]\displaystyle{ f\colon G_1\to G_2 }[/math] at a point [math]\displaystyle{ x\in G_1 }[/math] is the function [math]\displaystyle{ Df(x)\colon G_1\to G_2 }[/math] defined by

[math]\displaystyle{ Df(x)(y) = \lim_{s\to 0}\delta_{1/s} (f(x)^{-1}f(x\delta_sy))\, , }[/math]

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

• 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