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
BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups
- Mathematics
- 2015
Abstract Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such…
Lipschitz and Bi-Lipschitz maps from PI spaces to Carnot groups
- MathematicsIndiana University Mathematics Journal
- 2020
This paper deals with the problem of finding bi-Lipschitz behavior in non-degenerate Lipschitz maps between metric measure spaces. Specifically, we study maps from (subsets of) Ahlfors regular PI…
Bi-Lipschitz Pieces between Manifolds
- Mathematics
- 2013
A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large…
Ahlfors‐regular distances on the Heisenberg group without biLipschitz pieces
- Mathematics
- 2015
We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in Fractured fractals and broken dreams by David and Semmes, or equivalently, Question 22 and hence also…
Lipschitz Maps in Metric Spaces
- Mathematics
- 2014
In this dissertation we study Lipschitz and bi-Lipschitz mappings on abstract, non-smooth metric measure spaces. The dissertation consists of two separate parts.The first part considers a well-known…
1 3 D ec 2 01 3 Bi-Lipschitz Pieces between Manifolds
- Mathematics
- 2018
A well-known class of questions asks the following: If X and Y are metric measure spaces and f : X → Y is a Lipschitz mapping whose image has positive measure, then must f have large pieces on which…
NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS
- MathematicsForum of Mathematics, Sigma
- 2017
We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$ , where $P$ is an operator of order 0 with geometric origin and $f$ a multiplication operator by a function. When…
Quantitative decompositions of Lipschitz mappings into metric spaces
- Mathematics
- 2020
We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which…
Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
- Mathematics
- 2011
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…