In mathematics, the Pansu derivative is a derivative on a

Carnot group In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigens ...

, introduced by . A

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_\delta_ (f(x)^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 In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' di ...

, which can be stated as follows:

Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...

functions between (measurable subsets of) Carnot groups are Pansu differentiable almost everywhere.

References

* Lie groups {{mathanalysis-stub