Astérisque - Parutions - 342 (2012) < Parutions < 2012

Phase-space analysis and pseudodifferential calculus on the Heisenberg group
Hajer Bahouri, Clotilde Fermanian-Kammerer, Isabelle Gallagher
Astérisque 342 (2012), vi+128 pages

Résumé :
Analyse dans l'espace des phases, et calcul pseudodifférentiel sur le groupe de Heisenberg
Nous définissons une classe d'opérateurs pseudo-différentiels sur le groupe de Heisenberg. Comme il se doit, cette classe constitue une algèbre contenant les opérateurs différentiels. De plus, ces opérateurs pseudo-différentiels sont continus sur les espaces de Sobolev et l'on peut contrôler la perte de dérivée par leur ordre. Si un grand nombre de travaux ont été déjà consacrés à la construction et à l'étude d'algèbres d'opérateurs à coefficients variables, y compris des travaux très intéressants sur le groupe de Heisenberg, notre approche est différente et en particulier elle conduit à la notion de direction microlocale, et complète l'élaboration d'une analyse microlocale sur le groupe de Heisenberg commencée par Bahouri, Gérard et Xu en 2000 par le développement d'une théorie de Littlewood-Paley.

Mots-clefs : Groupe de Heisenberg, calcul pseudo-différentiel, calcul de Weyl-Hörmander

Abstract:
A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microlocal directions and completes, with the Littlewood-Paley theory initiated in 2000 by Bahouri, Gérard and Xu, a microlocal analysis of the Heisenberg group.

Keywords: Heisenberg group, pseudo-differential calculus, Weyl-Hörmander calculus

Class. math. : 35S05, 43A80, 35A27

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