Skip to main content
SpringerLink
Log in

Microlocal properties of Shubin pseudodifferential and localization operators

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The STFT is also called Gabor transform in case \(\psi =\psi _0\) and is closely connected to the so-called Bargmann and Fourier–Bros–Iagolnitzer transforms.

  2. This notion is also called the wave front set of \(a^w(x,D)\) in [8, Chapter 18.1] and the essential support of \(a^w(x,D)\) in [19].

References

  1. Boggiatto, P., Cordero, E., Gröchenig, K.: Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equ. Oper. Theory 48, 427–442 (2004)

    Article  MATH  Google Scholar 

  2. Boggiatto, P., Toft, J.: Embeddings and compactness for generalized Sobolev–Shubin spaces and modulation spaces. Appl. Anal. 84(3), 269–282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cordero, E., Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205(1), 107–131 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cordero, E., Nicola, F., Rodino, L.: Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials. Rev. Math. Phys. 27(1), 33 p (2015). doi:10.1142/S0129055X15500014

  5. Cordes, H.O.: The Technique of Pseudodifferential Operators. In: London Mathematical Society Lecture Note Series, vol. 202. Cambridge University Press, Cambridge (1995)

  6. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  7. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  8. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols. I, III. Springer, Berlin (1990)

  9. Hörmander, L.: Quadratic hyperbolic operators. In: Cattabriga, L., Rodino, L. (eds.) Microlocal Analysis and Applications. LNM, vol. 1495, pp. 118–160. Springer, Berlin (1991)

  10. Lerner, N.: Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators. Birkhäuser, Basel (2010)

    Book  MATH  Google Scholar 

  11. Nakamura, S.: Propagation of the homogeneous wave front set for Schrödinger equations. Duke Math. J. 126(2), 349–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces. Birkhäuser, Basel (2010)

    Book  MATH  Google Scholar 

  13. Nicola, F., Rodino, L.: Propagation of Gabor singularities for semilinear Schrödinger equations. Nonlinear Differ. Equ. Appl. 22(6), 1715–1732 (2015)

    Article  MathSciNet  Google Scholar 

  14. Pravda-Starov, K., Rodino, L., Wahlberg, P.: Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians (2015). arXiv:1411.0251 [math.AP]

  15. Rodino, L., Wahlberg, P.: The Gabor wave front set. Mon. Math. 173(4), 625–655 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schulz, R.: Microlocal Analysis of Tempered Distributions. Diss. Niedersächsische Staats-und Universitätsbibliothek Göttingen (2014)

  17. Schulz, R., Wahlberg, P.: The equality of the homogeneous and the Gabor wave front set (2013). arXiv:1304.7608

  18. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  19. Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  20. Wahlberg, P.: Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians (2015). arXiv:1411.6518 [math.AP]

Download references

Author information

Authors and Affiliations

  1. Institut für Analysis, Leibniz Universität Hannover, Welfenplatz 1, 30167, Hannover, Germany

    René Schulz

  2. Department of Mathematics, Linnæus University, 351 95, Växjö, Sweden

    Patrik Wahlberg

Authors
  1. René Schulz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrik Wahlberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrik Wahlberg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schulz, R., Wahlberg, P. Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl. 7, 91–111 (2016). https://doi.org/10.1007/s11868-015-0143-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-015-0143-7

Keywords

Mathematics Subject Classification

Search

Navigation