Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators: 3 - Brossura

Lerner, Nicolas

Brossura

ISBN 10: 376438509X ISBN 13: 9783764385095

Casa editrice: Birkhäuser, 2010

Riassunto

This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.

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Recensione

From the reviews:

“The present book is devoted to the theory of pseudodifferential operators and some applications. The presentation starts from basic results and continues up to the most advanced and recent developments of the local solvability theory ... . the present monograph will become a reference book for researchers in microlocal analysis. ... Ph. D. students will greatly appreciate this up-to-date overview of such a deep subject ... .” (Fabio Nicola, Mathematical Reviews, Issue 2011 b)

“This very interesting book of a well-known specialist in partial differential equations is devoted to the study of pseudo-differential operators and describes the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators, most of them belonging to the author. ... To sum up, the present book is really excellent. The first two parts are accessible to graduate students in analysis. The third chapter is highly recommended to researchers, providing an up-to-date overview of the subject.” (Viorel Iftimie, Zentralblatt MATH, Vol. 1186, 2010)

Contenuti

Preface.- 1 Basic Notions of Phase Space Analysis.- 1.1 Introduction to pseudodifferential operators.- 1.2 Pseudodifferential operators on an open subset of Rn.- 1.3 Pseudodifferential operators in harmonic .- 2 Metrics on the Phase Space.- 2.1 The structure of the phase space.- 2.2 Admissible metrics.- 2.3 General principles of pseudodifferential calculus.- 2.4 The Wick calculus of pseudodifferential operators.- 2.5 Basic estimates for pseudodifferential operators.- 2.6 Sobolev spaces attached to a pseudodifferential calculus.- 3 Estimates for Non-selfadjoint Operators.- 3.1 Introduction.- 3.2 First bracket analysis.- 3.3 The geometry of condition (Y).- 3.4 The necessity of condition (Y).- 3.5 Estimates with loss of k/k + 1 derivative.- 3.6 Estimates with loss of one derivative.- 3.7 (Y) does not imply solvability with loss of one derivative.- 3.8 (Y) implies solvability with loss of 3/2 derivatives.- 3.9 Open problems.- 4 Appendix.- 4.1 Some elements of Fourier analysis.- 4.2 Some remarks of algebra.- 4.3 Lemmas of classical analysis.- 4.4 On the symplectic and metaplectic groups.- 4.5 Composing a large number of symbols.- 4.6 A few elements of operator theory.- 4.7 On Sjöstrand algebra.- 4.8 On preparation theorems.- 4.9 On the pseudospectrum.- 4.10 More on symbolic calculus.

Dettagli dell'edizione

• Editore: Birkhäuser
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 376438509X
• ISBN 13: 9783764385095
• Rilegatura: Copertina flessibile
• Numero di pagine: 412
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile