Pansu derivative

In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989). A Carnot group ${\displaystyle G}$ admits a one-parameter family of dilations, ${\displaystyle \delta _{s}\colon G\to G}$. If ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ are Carnot groups, then the Pansu derivative of a function ${\displaystyle f\colon G_{1}\to G_{2}}$ at a point ${\displaystyle x\in G_{1}}$ is the function ${\displaystyle Df(x)\colon G_{1}\to G_{2}}$ defined by

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

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, MR 0979599