Advertisement

Microlocal Analysis and Applications

  • E. Cordero
  • F. Nicola
  • L. Rodino
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 155)

Abstract

We first give a short survey on the methods of Microlocal Analysis. In particular we recall some basic facts concerning the theory of pseudodifferential operators. We then present two applications. We first discuss lower bounds for operators with multiple characteristics. Then we give a new formula for the composition of Wick operators.

Keywords

Pseudo-differential operators Wick operators Weyl calculus lower bounds Wigner distributions short-time Fourier transforms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Ando and Y. Morimoto, Wick calculus and the Cauchy problem for some dispersive equationsOsaka J. Math. 39(2002), 123–147.MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Beals, A general calculus of pseudo-differential operatorsDuke Math. J. 42(1975), 1–42.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R. Beals and C. Fefferman, Spatially inhomogeneous pseudo-differential operatorsI Comm. Pure Appl. Math. 27(1974), 1–24.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    F.A. Berezin, Wick and anti-Wick symbols of operatorsMath. Sb. (N.S.)86(128) (1971), 578–610.MathSciNetGoogle Scholar
  5. 5.
    P. Boggiatto, E. Buzano and L. RodinoGlobal Hypoellipticity and Spectral TheoryAkademie Verlag, Berlin, 1996.zbMATHGoogle Scholar
  6. 6.
    P. Boggiatto, E. Cordero and K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spacesIntegral Equations Operator Theoryto appear.Google Scholar
  7. 7.
    P. Boggiatto and L. Rodino, Quantization and pseudo-differential operatorsCubo Mat Ed. 5(2003), 237–272.MathSciNetzbMATHGoogle Scholar
  8. 8.
    J.-M. Bony, Calcul symbolique et propagations des singularités pour les équations aux dérivées partielles non linéairesAnn. Sci. École Norm. Sup. 14(1981), 209–246.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J.-M. Bony, Sur l’inégalité de Fefferman-Phong, inSéminaire sur les Équations aux Dérivé Partielles1998–1999, Exp. No. III, École Polytech., Palaiseau, 1998.Google Scholar
  10. 10.
    L. Boutet de Monvel, A. Grigis and B. Helfer, Paramétrixes d’opérateurs pseudo-différentiels à caractéristiques multiplesAstérique 34–35(1976), 93–121.Google Scholar
  11. 11.
    A.P. Calderón and R. Vaillancourt, A class of bounded pseudo-differential operatorsProc. Nat. Acad. Sci. USA 69(1972), 1185–1187.zbMATHCrossRefGoogle Scholar
  12. 12.
    L. CohenTime-Frequency AnalysisPrentice Hall, Englewood Cliffs, NJ, 1995.Google Scholar
  13. 13.
    E. Cordero and K. Gröchenig, Time-frequency analysis of localization operatorsJ. Funct. Anal. 205(2003), 107–131MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Cordero and L. Rodino, Wick calculus: a time-frequency approachOsaka J. Math.to appear.Google Scholar
  15. 15.
    A. Córdoba and C. Fefferman, Wave packets and Fourier integral operatorsComm. Partial Differential Equations 3(1978), 979–1005, 1978.Google Scholar
  16. 16.
    I.Daubechies, Time-frequency localization operators: a geometric phase space approachIEEE Trans. Inform. Theory 34(1988), 605–612.MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Du and M.W. Wong, A product formula for localization operatorsBull. Korean Math. Soc. 37(2000), 77–84.MathSciNetzbMATHGoogle Scholar
  18. 18.
    C. Fefferman and D.H. Phong, On positivity of pseudo-differential operatorsProc. Natl. Acad. Sci. USA 75(1978), 4673–4674.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    H.G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, inAdvances in Gabor AnalysisEditors: H. G. Feichtinger and T. Strohmer, Birkhäuser, Boston, 2002, 99–128.Google Scholar
  20. 20.
    G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989.zbMATHGoogle Scholar
  21. 21.
    L. Gárding, Dirichlet’s problem for linear elliptic partial differential equationsMath. Scand. 1(1953), 55–72.MathSciNetzbMATHGoogle Scholar
  22. 22.
    K. GröchenigFoundations of Time-Frequency AnalysisBirkhäuser, Boston, 2001.Google Scholar
  23. 23.
    F. HérauOpérateurs Pseudo-Differentieles Semi-BornésPh.D. Dissertation, University of Rennes, 1999.Google Scholar
  24. 24.
    F. Hérau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebraArk. Mat. 39(2001), 311–338.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    F. Hérau, Melin inequality for paradifferential operators and applicationsComm. Partial Differential Equations 27(2002), 1659–1680.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    L. Hörmander, Pseudo-differential operatorsComm. Pure Appl. Math. 18(1965), 501–517.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    L. Hörmander, Pseudo-differential operators and hypoelliptic equations, in Singular IntegralsProc. Sympos. Pure Math.Amer. Math. Soc.10(1966), 138–183.Google Scholar
  28. 28.
    L. Hörmander, Fourier integral operators IActa Math. 127(1971), 79–183.zbMATHGoogle Scholar
  29. 29.
    L. Hörmander, The Cauchy problem for differential equations with double characteristicsJ. Analyse Math. 32(1977), 118–196.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    L. Hörmander, The Weyl calculus of pseudo-differential operatorsComm. Pure Appl. Math. 32(1979), 359–443.zbMATHCrossRefGoogle Scholar
  31. 31.
    L. HörmanderThe Analysis of Linear Partial Differential Operators I—IVSpringer-Verlag, Berlin, 1983–1985.Google Scholar
  32. 32.
    Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, New York, 1968.zbMATHGoogle Scholar
  33. 33.
    J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operatorsComm. Pure Appl. Math. 18(1965), 269–305.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    N. Lerner, The Wick calculus of pseudo-differential operators and energy estimates, inNew Trends in Microlocal AnalysisSpringer, 1997, 23–37.Google Scholar
  35. 35.
    O. Liess and L. Rodino, Linear partial differential equations with multiple involutive characteristics, inMicrolocal Analysis and Spectral TheoryEditor: L. Rodino, Kluwer Academic Publishers, 1997, 227–250.Google Scholar
  36. 36.
    M. Mascarello and L. RodinoLinear Partial Differential Operators with Multiple CharacteristicsWiley-Akademie Verlag, Berlin, 1997.Google Scholar
  37. 37.
    A. Melin, Lower bounds for pseudo-differential operatorsArk. Mat.9 (1971), 117–140.MathSciNetzbMATHGoogle Scholar
  38. 38.
    ] M. Mughetti and F. Nicola, A counterexample to a lower bound for a class of pseudodifferential operatorsProc. Amer. Math. Soc.to appear.Google Scholar
  39. 39.
    M. Mughetti and F. Nicola, Ageneralization of Hörmander’s inequality IIin preparation.Google Scholar
  40. 40.
    F. Nicola and L. RodinoRemarks on lower bounds for pseudo-differential operatorspreprint.Google Scholar
  41. 41.
    C. Parenti and A. Parmeggiani, Lower bounds for pseudo-differential operators, inMicrolocal Analysis and Spectral TheoryEditor: L. Rodino, Kluwer Academic Publishers, 1997, 227–250.Google Scholar
  42. 42.
    C. Parenti and A. Parmeggiani, Some remarks on almost-positivity of pseudodifferential operatorsBoll. Un. Mat./t.1-B 8 (1998), 187–215.MathSciNetGoogle Scholar
  43. 43.
    C. Parenti and A. Parmeggiani, A generalization of Hörmander’s inequality IComm. Partial Differential Equations25 (2000), 457–506.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    C. Parenti and A. Parmeggiani, Lower bounds for systems with double characteristicsJ. Analyse. Math.86 (2002), 49–91.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    J. Ramanathan and P. Topiwala, Time-frequency localization via the Weyl correspondenceSIAM J. Math. Anal.24 (1993), 1378–1393.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    L. Schwartz, Théorie des Distributions, Hermann, Paris, 1966.zbMATHGoogle Scholar
  47. 47.
    M.A. ShubinPseudo-Differential Operators and Spectral TheorySecond Edition, Springer-Verlag, Berlin, 2001.CrossRefGoogle Scholar
  48. 48.
    J. Sjöstrand, Parametrices for pseudo-differential operators with multiple characteristicsArk. Mat.12 (1974), 85–130.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    D. Tataru, On the Fefferman-Phong inequality and related problemsComm. Partial Differential Equations27 (2002), 2101–2138.MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    J. ToftModulation spaces and pseudo-differential operatorspreprint.Google Scholar
  51. 51.
    M.W. WongLocalization OperatorsSeoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1999.Google Scholar
  52. 52.
    M.W. Wong, Wavelet Transforms and Localization Operators, Birkhäuser, Basel, 2002.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • E. Cordero
    • 1
  • F. Nicola
    • 1
  • L. Rodino
    • 1
  1. 1.Department of Mathematics University of TorinoTorinoItaly

Personalised recommendations