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
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 ...
admits a one-parameter family of dilations,
. If
and
are Carnot groups, then the Pansu derivative of a function
at a point
is the function
defined by
:
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
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