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 Astérisque 2012; 127 pp; softcover Number: 342 ISBN-10: 2-85629-334-4 ISBN-13: 978-2-85629-334-8 List Price: US$52 Member Price: US$41.60 Order Code: AST/342 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 controlled 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, the authors' 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. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in pure mathematics. Table of Contents Introduction and main results Fundamental properties of pseudodifferential operators The algebra of pseudodifferential operators Littlewood-Paley theory The action of pseudodifferential operators on Sobolev spaces Appendix A. Some useful results on the Heisenberg group Appendix B. Weyl-Hörmander symbolic calculus on the Heisenberg group Bibliography