Annales Mathématiques
Blaise Pascal
no. 1
Averages and the ℓ q,1 cohomology of Heisenberg groups
[Moyennes et cohomologie ℓ q,1 des groupes d’Heisenberg]
Pierre Pansu1 ; Francesca Tripaldi2
1 Laboratoire de Mathématiques d’Orsay Université Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay FRANCE
2 Department of Mathematics and Statistics University of Jyväskylä 40014, Jyväskylä FINLAND
Annales mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 81-100.

Les moyennes sont des invariants definis sur la cohomologie ℓ 1 des groupes de Lie. Nous démontrons que les moyennes dans les groupes abéliens et les groupes d’Heisenberg sont nulles. Ce résultat complète des travaux précédents et montre que la cohomologie ℓ 1 est nulle pour les groupes de Lie étudiés.

Averages are invariants defined on the ℓ 1 cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the ℓ 1 cohomology vanishes in these cases.

Publié le :
DOI : 10.5802/ambp.384
Classification : 35R03, 58A10, 43A80
Keywords: Heisenberg groups, Rumin complex, $\ell ^{p}$ cohomology, parabolicity
Mot clés : Heisenberg groups, Rumin complex, $\ell ^{p}$ cohomology, parabolicity

Pierre Pansu 1 ; Francesca Tripaldi 2

1 Laboratoire de Mathématiques d’Orsay Université Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay FRANCE
2 Department of Mathematics and Statistics University of Jyväskylä 40014, Jyväskylä FINLAND
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AMBP_2019__26_1_81_0,
     author = {Pierre Pansu and Francesca Tripaldi},
     title = {Averages and the $\ell ^{q,1}$ cohomology of {Heisenberg} groups},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {81--100},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {26},
     number = {1},
     year = {2019},
     doi = {10.5802/ambp.384},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/}
}
TY  - JOUR
AU  - Pierre Pansu
AU  - Francesca Tripaldi
TI  - Averages and the $\ell ^{q,1}$ cohomology of Heisenberg groups
JO  - Annales mathématiques Blaise Pascal
PY  - 2019
SP  - 81
EP  - 100
VL  - 26
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/
DO  - 10.5802/ambp.384
LA  - en
ID  - AMBP_2019__26_1_81_0
ER  - 
%0 Journal Article
%A Pierre Pansu
%A Francesca Tripaldi
%T Averages and the $\ell ^{q,1}$ cohomology of Heisenberg groups
%J Annales mathématiques Blaise Pascal
%D 2019
%P 81-100
%V 26
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/
%R 10.5802/ambp.384
%G en
%F AMBP_2019__26_1_81_0
Pierre Pansu; Francesca Tripaldi. Averages and the $\ell ^{q,1}$ cohomology of Heisenberg groups. Annales mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 81-100. doi : 10.5802/ambp.384. https://ambp.centre-mersenne.org/articles/10.5802/ambp.384/

[1] Annalisa Baldi; Bruno Franchi; Pierre Pansu Poincaré and Sobolev inequalities for differential forms in Heisenberg groups (2017) (https://arxiv.org/abs/1711.09786) | Zbl

[2] Annalisa Baldi; Bruno Franchi; Pierre Pansu L 1 -Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups (2019) (https://arxiv.org/abs/1902.04819) | Zbl

[3] Annalisa Baldi; Bruno Franchi; Maria C. Tesi Compensated compactness in the contact complex of Heisenberg groups, Indiana Univ. Math. J., Volume 57 (2008) no. 1, pp. 133-185 | DOI | MR | Zbl

[4] Jean Bourgain; Haïm Brezis New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., Volume 009 (2007) no. 2, pp. 277-315 | DOI | MR | Zbl

[5] Sagun Chanillo; Jean Van Schaftingen Subelliptic Bourgain–Brezis estimates on groups, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 487-501 | DOI | MR | Zbl

[6] Pierre Pansu Cup-products in L q,p -cohomology: discretization and quasi-isometry invariance (2017) (https://arxiv.org/abs/1702.04984)

[7] Pierre Pansu; Michel Rumin On the ℓ q,p cohomology of Carnot groups (to appear in Ann. Henri Lebesgue) | Zbl

[8] Stefano Pigola; Alberto G. Setti; Marc Troyanov The connectivity at infinity of a manifold and L q,p -Sobolev inequalities, Expo. Math., Volume 32 (2014) no. 4, pp. 365-383 | DOI | MR | Zbl

[9] Michel Rumin Differential forms on contact manifolds, J. Differ. Geom., Volume 39 (1994) no. 2, pp. 281-330 | MR | Zbl

Cité par Sources :