Pansu derivative explained

G admits a one-parameter family of dilations, \delta _{s\colon} G\to G . If G _{1} and G _{2} are Carnot groups, then the Pansu derivative of a function f\colon G _{1\to} G _{2} at a point x\in G _{1} is the function Df(x)\colon G _{1\to} G _{2} defined byDf(x)(y) = \lim _{s\to} \delta _{1/s} (f(x) ^{-1} f(x\delta _{sy))} , 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.