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 δ s : G → G . If G 1 and G 2 are Carnot groups, then the Pansu derivative of a function f : G 1 → G 2 at a point x ∈ G 1 is the function D f ( x ) : G 1 → G 2 defined by

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.