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POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

Part of: Miscellaneous topics - Partial differential equations Abstract harmonic analysis Linear function spaces and their duals Inequalities General theory of differentiable manifolds Symplectic geometry, contact geometry

Published online by Cambridge University Press:  29 June 2020

Annalisa Baldi Open the ORCID record for Annalisa Baldi [Opens in a new window] ,
Bruno Franchi Open the ORCID record for Bruno Franchi [Opens in a new window]  and
Pierre Pansu
Annalisa Baldi
Affiliation:
Università di Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40126Bologna, Italy (annalisa.baldi2@unibo.it; bruno.franchi@unibo.it)
Bruno Franchi
Affiliation:
Università di Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40126Bologna, Italy (annalisa.baldi2@unibo.it; bruno.franchi@unibo.it)
Pierre Pansu
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France (pierre.pansu@universite-paris-saclay.fr)
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Abstract

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In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

Keywords

Heisenberg groupsdifferential formsSobolev–Poincaré inequalitiescontact manifoldshomotopy formula

MSC classification

Primary: 58A10: Differential forms 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc. 53D10: Contact manifolds, general
Secondary: 26D15: Inequalities for sums, series and integrals 43A80: Analysis on other specific Lie groups 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 3 , May 2022 , pp. 869 - 920
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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