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)=\lim _{s\to 0}\delta _{1/s}f(x\delta _{s}y)

provided that this limit exists.

A key theorem in this area is the Pansu–Rademacher theorem, the following generalization of Rademacher's theorem: Lipschitz continuous functions between (measurable subsets of) Carnot groups are Pansu differentiable a.e.

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 979599

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