Metrics on the Phase Space

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


In this chapter, we describe a general version of the pseudo-differential calculus, due to L. Hörmander ([69], Chapter 18 in [73]). That version followed some earlier generalizations due to R. Beals and C. Fefferman ([8]) and to R. Beals ([6]). It was followed by some other generalizations due to A. Unterberger [145] and to a joint work of J.-M. Bony and N. Lerner ([20]), whose presentation we follow. We also give a precised version of the Fefferman-Phong inequality, following [100] where we provide an upper bound for the number of derivatives sufficient to obtain that inequality. Finally, we study the Sobolev spaces naturally attached to an admissible metric on the phase space, essentially following the paper by J.-M. Bony and J.-Y. Chemin [19].


Phase Space Sobolev Space Wigner Function Selfadjoint Operator Composition Formula 
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Copyright information

© Birkhäuser Verlag AG 2010

Authors and Affiliations

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

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