A pseudo-differential calculus on the Heisenberg group

Véronique Fischer; Michael Ruzhansky

doi:10.1016/j.crma.2013.12.006

Partial differential equations/Harmonic analysis

A pseudo-differential calculus on the Heisenberg group

Presented by: Jean-Michel Bony

Véronique Fischer ¹; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 197-204.

Abstract
Résumé

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,2,3] of pseudo-differential calculi on graded groups. The relation between the Weyl quantisation and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.

Dans cette note, nous présentons un calcul pseudo-différentiel symbolique sur le groupe de Heisenberg. Nous particularisons à ce groupe notre construction générale [4,2,3] des calculs pseudo-différentiels sur les groupes de Lie gradués. Le lien entre la quantification de Weyl et les représentations du groupe de Heisenberg nous permet de considérer, dans ce cas, des symboles à valeurs scalaires. Nous trouvons que les conditions qui définissent les classes de symboles sont similaires, mais différentes de celles dans [1]. Comme applications, nous obtenons des résultats d'hypoellipticité de type Schwartz ainsi que des estimations subelliptiques sur le groupe de Heisenberg.

Received: 2013-08-19
Accepted: 2013-12-06
Published online: 2014-02-03

DOI: 10.1016/j.crma.2013.12.006

Author's affiliations:

Véronique Fischer ¹; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

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     title = {A pseudo-differential calculus on the {Heisenberg} group},
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Véronique Fischer; Michael Ruzhansky. A pseudo-differential calculus on the Heisenberg group. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 197-204. doi : 10.1016/j.crma.2013.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.006/

References
Cited by

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[4] V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie Groups, monograph in preparation.

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