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Quantization on nilpotent lie groups / Veronique Fischer (2016)    

Public Titre : Quantization on nilpotent lie groups Type de document : Monographie Auteurs : Veronique Fischer, Auteur ; Michael Ruzhansky, Auteur Editeur : Berlin, Zurich, Stuttgart : Birkhaüser Année de publication : 2016 Collection : Progress in Mathematics, ISSN 0743-1643 num. 314 Importance : 557 p. ISBN/ISSN/EAN : 978-3-319-29558-9 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Analyse mathématique [Termes IGN] analyse fonctionnelle (mathématiques) [Termes IGN] groupe de Lie [Termes IGN] topologie Index. décimale : 23.30 Analyse mathématique Résumé : (Auteur) [Preface] The purpose of this monograph is to give an exposition of the global quantization of operators on nilpotent homogeneous Lie groups. We also present the background analysis on homogeneous and graded nilpotent Lie groups. The analysis on homogeneous nilpotent Lie groups drew a considerable attention from the 70’s onwards. Research went in several directions, most notably in harmonic analysis and in the study of hypoellipticity and solvability of partial differential equations. Over the decades the subject has been developing on different levels with advances in the analysis on the Heisenberg group, stratified Lie groups, graded Lie groups, and general homogeneous Lie groups. In the last years analysis on homogeneous Lie groups and also on other types of Lie groups has received another boost with newly found applications and further advances in many topics. Examples of this boost are subelliptic estimates, multiplier theorems, index formulae, nonlinear problems, potential theory, and symbolic calculi tracing full symbols of operators. In particular, the latter has produced further applications in the study of linear and nonlinear partial differential equations, requiring the knowledge of lower order terms of the operators. Because of the current advances, it seems to us that a systematic exposition of the recently developed quantizations on Lie groups is now desirable. This requires bringing together various parts of the theory in the right generality, and extending notions and techniques known in particular cases, for instance on compact Lie groups or on the Heisenberg group. In order to do so, we start with a review of the recent developments in the global quantization on compact Lie groups. In this, we follow mostly the development of this subject in the monograph [RT10a] by Turunen and the second author, as well as its further progress in subsequent papers. After a necessary exposition of the background analysis on graded and homogeneous Lie groups, we present the quantization on general graded Lie groups. As the final part of the monograph, we work out details of the general theory developed in this book in the particular case of the Heisenberg group. In the introduction, we will provide a link between, on one hand, the symbolic calculus of matrix valued symbols on compact Lie groups with, on the other hand, different approaches to the symbolic calculus on the Heisenberg group for instance. We will also motivate further our choices of presentation from the point of view of the development of the theory and of its applications. We would like to thank Fulvio Ricci for discussions and for useful comments on the historical overview of parts of the subject that we tried to present in the introduction. We would also like to thank Gerald Folland for comments leading to improvements of some parts of the monograph. Finally, it is our pleasure to acknowledge the financial support by EPSRC (grant EP/K039407/1), Marie Curie FP7 (Project PseudodiffOperatorS - 301599), and by the Leverhulme Trust (grant RPG-2014-02) at different stages of preparing this monograph. Note de contenu : Introduction Notation and conventions 1 Preliminaries on Lie groups 1.1 Lie groups, representations, and Fourier transform 1.2 Liealgebrasandvectorfields 1.3 Universalenvelopingalgebraanddifferentialoperators 1.4 DistributionsandSchwartzkerneltheorem 1.5 Convolutions 1.6 NilpotentLiegroupsandalgebras 1.7 Smooth vectors and infinitesimal representations . . 1.8 Planchereltheorem 2 Quantization on compact Lie groups 2.1 FourieranalysisoncompactLiegroups 2.2 Pseudo-differentialoperatorsoncompactLiegroups 3 Homogeneous Lie groups 3.1 GradedandhomogeneousLiegroups 3.2 OperatorsonhomogeneousLiegroups 4 Rockland operators and Sobolev spaces 4.1 Rocklandoperators 4.2 PositiveRocklandoperators 4.3 FractionalpowersofpositiveRocklandoperators 4.4 SobolevspacesongradedLiegroups 4.5 Hulanicki’s theorem 5 Quantization on graded Lie groups 5.1 Symbolsandquantization 5.2 Symbol classes 5.3 SpectralmultipliersinpositiveRocklandoperators 5.4 Kernelsofpseudo-differentialoperators 5.5 Symboliccalculus 5.6 Amplitudesandamplitudeoperators 5.7 Calderon-Vaillancourt theorem 5.8 Parametrices, ellipticity and hypoellipticity 6 Pseudo-differential operators on the Heisenberg group 6.1 Preliminaries 6.2 DualoftheHeisenberggroup 6.3 Differenceoperators 6.4 Shubin classes 6.5 Quantization and symbol classes on the Heisenberg group 6.6 Parametrices A Miscellaneous B Group C∗ and von Neumann algebras Numéro de notice : 22745 Affiliation des auteurs : non IGN Thématique : MATHEMATIQUE Nature : Monographie En ligne : http://dx.doi.org/10.1007/978-3-319-29558-9 Format de la ressource électronique : URL Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=86056

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