Lipschitz extensions of maps between Heisenberg groups
Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1653-1665.

Let n be the Heisenberg group of topological dimension 2n+1. We prove that if n is odd, the pair of metric spaces ( n , n ) does not have the Lipschitz extension property.

Soit n le groupe d’Heisenberg de dimension topologique 2n+1. On montre que si n est impair, le couple d’espaces métriques ( n , n ) ne possède pas la propriété d’extension lipschitzienne.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3046
Classification: 43A80
Keywords: Heisenberg group, Lipschitz extension property
Mot clés : Groupe d’Heisenberg, propriété d’extension lipschitzienne

Balogh, Zoltán M. 1; Lang, Urs 2; Pansu, Pierre 3

1 Department of Mathematics University of Bern Sidlerstrasse 5 CH-3012 Bern, Switzerland
2 Department of Mathematics ETH Zürich Rämistrasse 101 CH-8092 Zürich, Switzerland
3 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS, Univ. Paris-Saclay Bâtiment 405 F-91405 Orsay France
@article{AIF_2016__66_4_1653_0,
     author = {Balogh, Zolt\'an M. and Lang, Urs and Pansu, Pierre},
     title = {Lipschitz extensions of maps between {Heisenberg} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1653--1665},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     doi = {10.5802/aif.3046},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3046/}
}
TY  - JOUR
AU  - Balogh, Zoltán M.
AU  - Lang, Urs
AU  - Pansu, Pierre
TI  - Lipschitz extensions of maps between Heisenberg groups
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1653
EP  - 1665
VL  - 66
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3046/
DO  - 10.5802/aif.3046
LA  - en
ID  - AIF_2016__66_4_1653_0
ER  - 
%0 Journal Article
%A Balogh, Zoltán M.
%A Lang, Urs
%A Pansu, Pierre
%T Lipschitz extensions of maps between Heisenberg groups
%J Annales de l'Institut Fourier
%D 2016
%P 1653-1665
%V 66
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3046/
%R 10.5802/aif.3046
%G en
%F AIF_2016__66_4_1653_0
Balogh, Zoltán M.; Lang, Urs; Pansu, Pierre. Lipschitz extensions of maps between Heisenberg groups. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1653-1665. doi : 10.5802/aif.3046. https://aif.centre-mersenne.org/articles/10.5802/aif.3046/

[1] Ambrosio, L.; Rigot, S. Optimal mass transportation in the Heisenberg group, J. Funct. Anal., Volume 208 (2004) no. 2, pp. 261-301 | DOI

[2] Balogh, Zoltán M.; Fässler, Katrin S. Rectifiability and Lipschitz extensions into the Heisenberg group, Math. Z., Volume 263 (2009) no. 3, pp. 673-683 | DOI

[3] Capogna, Luca; Danielli, Donatella; Pauls, Scott D.; Tyson, Jeremy T. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, 259, Birkhäuser Verlag, Basel, 2007, xvi+223 pages

[4] Fonseca, Irene; Gangbo, Wilfrid Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications, 2, The Clarendon Press, Oxford University Press, New York, 1995, viii+211 pages (Oxford Science Publications)

[5] Kirszbraun, M. D. Über die zusammenziehende und Lipschitzsche Transformationen., Fundam. Math., Volume 22 (1934), pp. 77-108

[6] Lang, U.; Pavlović, B.; Schroeder, V. Extensions of Lipschitz maps into Hadamard spaces, Geom. Funct. Anal., Volume 10 (2000) no. 6, pp. 1527-1553 | DOI

[7] Lang, U.; Schroeder, V. Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 535-560 | DOI

[8] Lang, Urs; Schlichenmaier, Thilo Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. (2005) no. 58, pp. 3625-3655 | DOI

[9] Lloyd, N. G. Degree theory, Cambridge University Press, Cambridge-New York-Melbourne, 1978, vi+172 pages (Cambridge Tracts in Mathematics, No. 73)

[10] Magnani, Valentino The coarea formula for real-valued Lipschitz maps on stratified groups, Math. Nachr., Volume 278 (2005) no. 14, pp. 1689-1705 | DOI

[11] McShane, E. J. Extension of range of functions, Bull. Amer. Math. Soc., Volume 40 (1934) no. 12, pp. 837-842 | DOI

[12] Monti, Roberto Some properties of Carnot-Carathéodory balls in the Heisenberg group, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., Volume 11 (2000) no. 3, p. 155-167 (2001)

[13] Pansu, Pierre Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2), Volume 129 (1989) no. 1, pp. 1-60 | DOI

[14] Rigot, Séverine; Wenger, Stefan Lipschitz non-extension theorems into jet space Carnot groups, Int. Math. Res. Not. IMRN (2010) no. 18, pp. 3633-3648 | DOI

[15] Valentine, F. A. A Lipschitz condition preserving extension for a vector function, Amer. J. Math., Volume 67 (1945), pp. 83-93

[16] Wenger, Stefan; Young, Robert Lipschitz extensions into jet space Carnot groups, Math. Res. Lett., Volume 17 (2010) no. 6, pp. 1137-1149 | DOI

Cited by Sources: