Second Microlocalization Methods for Degenerate Cauchy—Riemann Equations

  • Nicolas Lerner
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)


We study a class of degenerate Cauchy—Riemann equations and we show that the second microlocalization with respect to a hypersurface is a useful tool to formulate and prove propagation and solvability results.


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  1. [1]
    R. Beals and C. Fefferman, On local solvability of linear partial differential equations Ann. of Math. 97 (1973), 482–498.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    J.M. Bony, Second microlocalization and propagation of singularities for senil-linear hyperbolic equations. In Hyperbolic Equations and Re lated Topics (Mizohata, ed.), Kinokuniya, 1986, pp. 11–49.Google Scholar
  3. [3]
    J.M. Bony and J.Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander Bull. S.M.F. 122 (1994), 77–118.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J.M. Bony and N. Lerner, Quantification asymtotique et microlocalisations d’ordre supérieur Ann. Sc. ENS 22 (1989), 377–483.MathSciNetzbMATHGoogle Scholar
  5. [5]
    J.M. Delort FBI Transformation Second Microlocalization and Semi-Linear Caustics Lect. Notes in Math. 1522 Springer, 1992.Google Scholar
  6. [6]
    L. Hörmander, The Analysis of Linear Partial Differential Operators Springer-Verlag, 1985.Google Scholar
  7. [7]
    L. Hörmander, On the solvability of pseudodifferential equations. In Structure of Solutions of Differential Equations (M. Morimoto and T. Kawai (eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1996, pp. 183–213.Google Scholar
  8. [8]
    Lascar and R. Lascar, Propagation of singularities for a class of non-real pseudo-differential operators,C. R. Acad. Sci. Paris Sér. I Math. 321(9) (1995) 1183–1187.Google Scholar
  9. [9]
    B. Lascar, R. Lascar, and N. Lerner, Propagation of singularities for non-real pseudo-differential operators, J. Anal. Math. 64 (1994), 263–289.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    [ ]B. Lascar and N. Lerner, Résolution de l’équation de Cauchy-Riemann dans des espaces de Gevrey, Soumis à publication. Google Scholar
  11. [11]
    G. Lebeau, Deuxième microlocalisation sur les sous-variétés isotropes, Ann.Inst. Fourier 35(2) (1985), 145–216.128 N. LernerMathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    N. Lerner, When is a pseudo-differential equation solvable? Ann.Inst. Fourier (2000)50 2(Cinquantenaire), 443–460.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    L. Nirenberg and F. Treves, On local solvability of linear partial differential equations,Comm. Purr Appl. Math.,23 1–38, (1970), 459–509;24 (1971), 279–288.MathSciNetzbMATHGoogle Scholar
  14. [14]
    J. Sjöstrand Singularités analytiques microlocales,Astérisque,95 (1982).Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Nicolas Lerner
    • 1
  1. 1.University of RennesRennes cedexFrance

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