Pansu derivative

In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by. A Carnot group $$G$$ 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 by
 * $$Df(x)(y) = \lim_{s\to 0}\delta_{1/s} (f(x)^{-1}f(x\delta_sy))\, ,$$

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.