# Lipschitz and bilipschitz maps on Carnot groups

@article{Meyerson2010LipschitzAB, title={Lipschitz and bilipschitz maps on Carnot groups}, author={William Meyerson}, journal={Pacific Journal of Mathematics}, year={2010}, volume={263}, pages={143-170} }

Suppose A is an open subset of a Carnot group G and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is bilipschitz on a subset of A of positive Hausdorff measure. We also construct Lipschitz maps from open sets in Carnot groups to Euclidean space that do not decrease dimension. Finally, we discuss two counterexamples to explain why Carnot group structure is necessary for these results.

## 10 Citations

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We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be…

## References

SHOWING 1-10 OF 32 REFERENCES

Lipschitz extensions into Jet space Carnot groups

- Mathematics
- 2009

The aim of this article is to prove a Lipschitz extension theorem for partially defined Lipschitz maps to jet spaces endowed with a left-invariant sub-Riemannian Carnot-Carath\'eodory distance. The…

Lipschitz Non-extension Theorems into Jet Space Carnot Groups

- Mathematics
- 2009

We prove non-extendability results for Lipschitz maps with target space being jet spaces equipped with a left-invariant Riemannian distance, as well as jet spaces equipped with a left-invariant…

A Tour of Subriemannian Geometries, Their Geodesics and Applications

- Mathematics
- 2006

Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of…

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

- Mathematics
- 2007

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of…

Contact equations, Lipschitz extensions and isoperimetric inequalities

- Mathematics
- 2007

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined…

Quantitative rectifiability and Lipschitz mappings

- Mathematics
- 1993

The classical notion of rectifiability of sets in R n is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with L p estimates…

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

- Mathematics
- 2007

The Isoperimetric Problem in Euclidean Space.- The Heisenberg Group and Sub-Riemannian Geometry.- Applications of Heisenberg Geometry.- Horizontal Geometry of Submanifolds.- Sobolev and BV Spaces.-…

Métriques de Carnot-Carthéodory et quasiisométries des espaces symétriques de rang un

- Mathematics
- 1989

We exhibit a rigidity property of the simple groups Sp(n, 1) and F7-20 which implies Mostow rigidity. This property does not extend to O(n, 1) and U(n, 1). The proof relies on quasiconformal theory…

Measure-preserving quality within mappings

- Mathematics
- 2000

In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on…

Rectifiability and Lipschitz extensions into the Heisenberg group

- Mathematics
- 2009

Denote by $${\mathbb{H}^n}$$ the 2n + 1 dimensional Heisenberg group. We show that the pairs $${(\mathbb{R}^k ,\mathbb{H}^n)}$$ and $${(\mathbb{H}^k ,\mathbb{H}^n)}$$ do not have the Lipschitz…