In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989). A Carnot group G {\displaystyle G} admits a one-parameter family of dilations, δ s : G → G {\displaystyle \delta _{s}\colon G\to G} . If G 1 {\displaystyle G_{1}} and G 2 {\displaystyle G_{2}} are Carnot groups, then the Pansu derivative of a function f : G 1 → G 2 {\displaystyle f\colon G_{1}\to G_{2}} at a point x ∈ G 1 {\displaystyle x\in G_{1}} is the function D f ( x ) : G 1 → G 2 {\displaystyle Df(x)\colon G_{1}\to G_{2}} defined by
D f ( x ) ( y ) = lim s → 0 δ 1 / s ( f ( x ) − 1 f ( x δ s y ) ) , {\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, JSTOR 1971484, MR 0979599