Pansu derivative

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

$Df(x)(y)=\underset{s\to 0}{\lim }{\delta }_{1/s}(f(x{)}^{-1}f(x{\delta }_{s}y)),$

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