© 2010

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


  • A thorough exposition of pseudodifferential calculus defined by metrics on the phase space

  • Contains a proof of the Nirenberg-Treves conjecture

  • Construction of counterexamples to “optimal” solvability under condition (psi)


Part of the Pseudo-Differential Operators book series (PDO, volume 3)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Nicolas Lerner
    Pages 1-55
  3. Nicolas Lerner
    Pages 57-160
  4. Nicolas Lerner
    Pages 161-286
  5. Nicolas Lerner
    Pages 287-382
  6. Back Matter
    Pages 383-397

About this book


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.


Derivative calculus differential equation fourier integral operator operator theory ordinary differential equation phase space pseudo-differential operator

Authors and affiliations

  1. 1.Projet Analyse fonctionnelle Institut de Mathématique de JussieuUniversité Pierre et Marie Curie (Paris VI)Paris cedex 05France

Bibliographic information

  • Book Title Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators
  • Authors Nicolas Lerner
  • Series Title Pseudo-Differential Operators
  • DOI
  • Copyright Information Birkhäuser Basel 2010
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-7643-8509-5
  • eBook ISBN 978-3-7643-8510-1
  • Edition Number 1
  • Number of Pages XII, 397
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
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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)