Journal d’Analyse Mathématique

, Volume 98, Issue 1, pp 65–82

Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces

Article

Abstract

We introduce new classes of Banach algebras of pseudodifferential operators with symbols in certain modulation spaces and investigate their composition and the functional calculus. Operators in these algebras possess the spectral invariance property on the associated family of modulation spaces. These results extend and contain Sjöstrand's theory, and they are obtained with new phase-space methods instead of “hard analysis”.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Beals,Characterization of pseudodifferential operators and applications, Duke Math. J.44 (1977), 45–57.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. Boggiatto and E. Schrohe,Characterization, spectral invariance and the Fredholm property of multi-quasi-elliptic operators, Rend. Sem. Mat. Univ. Politec. Torino59 (2003), 229–242.MathSciNetGoogle Scholar
  3. [3]
    J.-M. Bony and J.-Y. Chemin,Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France122 (1994), 77–118.MATHMathSciNetGoogle Scholar
  4. [4]
    A. Boulkhemair,Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett.4 (1997), 53–67.MATHMathSciNetGoogle Scholar
  5. [5]
    E. Cordero and K. Gröchenig,Time-frequency analysis of localization operators, J. Funct. Anal.205 (2003), 107–131.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    E. Cordero and K. Gröchenig,Necessary conditions for Schatten class localization operators, Proc. Amer. Math. Soc.133 (2005), 3573–3579.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    M. Dimassi and J. Sjöstrand,Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, Cambridge, 1999.MATHGoogle Scholar
  8. [8]
    H. G. Feichtinger,Modulation spaces on locally compact abelian groups, inProceedings of “International Conference on Wavelets and Applications” 2002, Chennai, India, 2003, pp. 99–140. Updated version of a technical report, University of Vienna, 1983.Google Scholar
  9. [9]
    H. G. Feichtinger and K. Gröchenig,Gabor frames and time-frequency analysis of distributions, J. Funct. Anal.146 (1997), 464–495.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig and C. Molitor-Braun,Weighted group algebras on groups of polynomial growth, Math. Z.102 (2003), 791–821.CrossRefGoogle Scholar
  11. [11]
    G. Fendler, K. Gröchenig and M. Leinert,Symmetry of weighted L 1 -algebras and the GRS-condition, Bull. London Math. Soc., to appear.Google Scholar
  12. [12]
    G. B. Folland,Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989.MATHGoogle Scholar
  13. [13]
    I. Gel'fand, D. Raikov and G. Shilov,Commutative Normed Rings, Chelsea Publ. Co., New York, 1964.Google Scholar
  14. [14]
    B. Gramsch,Relative Inversion in der Störungstheorie von Operatoren und ϕ-Algebren, Math. Ann.269 (1984), 27–71.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Gröchenig,Foundations of Time-Frequency Analysis, Birkhäuser Boston Inc., Boston, MA, 2001.MATHGoogle Scholar
  16. [16]
    K. Gröchenig,Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl.10 (2004), 105–132.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    K. Gröchenig and C. Heil,Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory34 (1999), 439–457.CrossRefMathSciNetGoogle Scholar
  18. [18]
    K. Gröchenig and C. Heil,Modulation spaces as symbol classes for pseudodifferential operators, inWavelets and Their Applications, (S. T. M. Krishna and R. Radha, eds.), Allied Publishers, Chennai, 2003, pp. 151–170.Google Scholar
  19. [19]
    K. Gröchenig and C. Heil,Counterexamples for boundedness of pseudodifferential operators, Osaka J. Math.41 (2004), 1–11.MathSciNetGoogle Scholar
  20. [20]
    K. Gröchenig and M. Leinert,Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc.17 (2004), 1–18.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    C. Heil, J. Ramanathan and P. Topiwala,Singular values of compact pseudodifferential operators, J. Funct. Anal.150 (1997), 426–452.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    F. Hérau,Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. Mat.39 (2001), 311–338.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    L. Hörmander,The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math.32 (1979), 360–444.CrossRefMathSciNetGoogle Scholar
  24. [24]
    L. Hörmander,The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, Berlin, 1985. Pseudodifferential operators.MATHGoogle Scholar
  25. [25]
    A. Hulanicki,On the spectrum of convolution operators on groups with polynomial growth, Invent. Math.17 (1972), 135–142.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    S. Jaffard,Propriétés des matrices “bien localisées” près de leur diagonalé et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire7 (1990), 461–476.MATHMathSciNetGoogle Scholar
  27. [27]
    D. Labate,Pseudodifferential operators on modulation spaces, J. Math. Anal. Appl.262 (2001), 242–255.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    H.-G. Leopold and E. Schrohe,Spectral invariance for algebras of pseudodifferential operators on Besov-Triebel-Lizorkin spaces, Manuscripta Math.78 (1993), 99–110.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    J. Sjöstrand,An algebra of pseudodifferential operators, Math. Res. Lett.1 (1994), 185–192.MATHMathSciNetGoogle Scholar
  30. [30]
    J. Sjöstrand,Wiener type algebras of pseudodifferential operators, inSéminaire sur les Équations aux Dérivées Partielles, 1994–1995, École Polytech., Palaiseau, 1995, Exp. No. IV, 21.Google Scholar
  31. [31]
    E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.MATHGoogle Scholar
  32. [32]
    T. Strohmer,On the role of the Heisenberg group in wireless communication, technical report, 2004.Google Scholar
  33. [33]
    J. Toft,Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble)51 (2001), 1347–1383.MATHMathSciNetGoogle Scholar
  34. [34]
    J. Toft,Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus, Bull. Sci. Math.126 (2002), 115–142.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    J. Ueberberg,Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L p -Theorie, Manuscripta Math.61 (1988), 459–475.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    H. Weyl,The Theory of Groups and Quantum Mechanics, Methuen, London, 1931; reprinted by Dover Puplications, New York, 1950.MATHGoogle Scholar
  37. [37]
    L. Zadeh,The determination of the impulsive response of variable networks, J. Appl. Phys.21 (1950), 642–645.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  1. 1.Institute of Biomathematics and BiometryGSF-National Research Center for Environment and HealthNeuherbergGermany
  2. 2.Faculty of MathematicsUniversity of ViennaWienAustria

Personalised recommendations