Advertisement

Dixmier Traceability for General Pseudo-differential Operators

  • Fabio Nicola
  • Luigi Rodino
Part of the Trends in Mathematics book series (TM)

Abstract

For Hörmander’s classes OPS(m, g) of pseudo-differential operators associated with a weight m and a metric g we prove (under an additional technical condition) that, if m is in the space L 1-weak, all operators in that class have finite Dixmier trace.

Keywords

Pseudo-differential operators Dixmier trace Lorentz-Marcinkiewicz space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Boggiatto, E. Buzano, L. Rodino, Global Hypoellipticity and Spectral Theory. Akademie Verlag, Berlin, 1996.zbMATHGoogle Scholar
  2. [2]
    P. Boggiatto, F. Nicola, Non-commutative residues for anisotropic pseudo-differential operators inn. J. Funct. Anal. 203 (2003), 305–320.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. Buzano, F. Nicola, Pseudodifferential operators and Schatten-von Neumann classes. In: P. Boggiatto, R. Ashino, M.W. Wong (Eds.), Advances in Pseudodifferential Operators (Proceedings ISAAC, Toronto 2003), Operator Theory Adv. Appl., vol. 155, Birkhäuser, Basel, 2004, 117–130.Google Scholar
  4. [4]
    A. Connes, The action functional in non-commutative geometry. Comm. Math. Physics 117 (1988), 673–683.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. Connes, Noncommutative Geometry. Academic Press, New York, London, Tokyo, 1994.zbMATHGoogle Scholar
  6. [6]
    A. Connes, H. Moscovici, The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), 174–243.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    E. Cordero, K.H. Gröchenig, Time-frequency analysis of localization operators. J. Funct. Anal. 205 (2003), 107–131.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    E. Cordero, S. Pilipović, L. Rodino, N. Teofanov, Localization operators and exponential weights for modulation spaces. Mediterr. J. Math. 2 (2005), 381–394.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Dixmier, Existence de traces non normales. C.R. Acad. Sc. Paris, Série A, 262 (1966), 1107–1108.zbMATHMathSciNetGoogle Scholar
  10. [10]
    G.B. Folland, Harmonic Analysis in Phase Space. Princeton University Press, 1989.Google Scholar
  11. [11]
    V. Gayral, J.M. Gracia-Bondía, B. Iochum, T. Schücker, J.C. Vàrilly, Moyal planes are spectral triples. Comm. Math. Phys. 246 (2004), 569–623.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    K.H. Gröchenig, An uncertainty principle related to the Poisson summation formula. Studia Math. 121 (1996), 87–104.zbMATHMathSciNetGoogle Scholar
  13. [13]
    K.H. Gröchenig, Foundation of Time-Frequency Analysis. Birkhäuser, Boston, 2001.Google Scholar
  14. [14]
    K.H. Gröchenig, C. Heil, Modulation spaces and pseudo-differential operators. Integral Equations Operator Theory 34 (1999), 439–457.CrossRefMathSciNetGoogle Scholar
  15. [15]
    L. Hörmander, On the asymptotic distribution of the eigenvalues of pseudo-differential operators inn. Arkiv för Mat. 17 (1979), 297–313.zbMATHCrossRefGoogle Scholar
  16. [16]
    L. Hörmander, The analysis of linear partial differential operators III. Springer-Verlag, Berlin, 1985.Google Scholar
  17. [17]
    F. Nicola, Trace functionals for a class of pseudo-differential operators inn. Math. Phys. Anal. Geom. 6 (2003), 89–105.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    D. Robert, Autour de l’approximation semi-classique. Birkhäuser, Boston, 1987.Google Scholar
  19. [19]
    B.-W. Schulze, Boundary value problems and singular pseudo-differential operators. Pure and Applied Mathematics, John Wiley & Sons, Chichester, England, 1998.Google Scholar
  20. [20]
    B. Simon, The Weyl transfrom and L p functions on phase space. Proc. Amer.Math. Soc. 116 (1992), 1045–1047.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    E. Schrohe, Wodzicki’s noncommutative residue and traces for operator algebras on manifolds with conical singularities. In L. Rodino, editor, Microlocal Analysis and Spectral Theory, 1997 Kluwer Academic Publishers, Printed in the Netherlands, 1997, 227–250.Google Scholar
  22. [22]
    M.A. Shubin, Pseudo-differential operators and spectral theory. Springer-Verlag, Berlin, 1987.Google Scholar
  23. [23]
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971.Google Scholar
  24. [24]
    J. Toft, Schatten-von Neumann properties in the Weyl calculus, and calculus of metrics on symplectic vector spaces. Preprint 2004.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Fabio Nicola
    • 1
  • Luigi Rodino
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

Personalised recommendations