Skip to main content
SpringerLink
Log in

Micro-local analysis in Fourier Lebesgue and modulation spaces: part II

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

Abstract

We consider different types of (local) products f 1 f 2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form

$$P(x,D)f = G(x,J_k f),$$

with appropriate polynomials P and G, where J k denotes the k-jet of f. If the solution locally belongs to appropriate weighted Fourier Lebesgue space \({\fancyscript{F}L^q_{(\omega )} (\mathbf{R}^d)}\) and P is non-characteristic at (x 0, ξ 0), then we prove that \({(x_0,\xi_0)\not \in {\rm WF}_{\fancyscript{F}L^q_{(\widetilde {\omega })}} (f)}\), where \({\widetilde{\omega }}\) depends on ω, P and G.

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

Similar content being viewed by others

References

  1. Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin (1976)

    MATH  Google Scholar 

  2. Feichtinger, H.G.: Banach convolution algebras of Wiener’s type. In: Proceedings on Functions, Series, Operators in Budapest, Colloquia Math. Soc. J. Bolyai, North-Holland, Amsterdam, Oxford, New York, (1980)

  3. Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. Technical Report, University Vienna (1983). and also in Krishna, M., Radha, R., Thangavelu, S. (eds.), Wavelets and Their Applications. Allied Publishers, New Delhi, pp. 99–140 (2003)

  4. Feichtinger H.G., Gröchenig K.H.: Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86, 307–340 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Feichtinger H.G., Gröchenig K.H.: Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108, 129–148 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Feichtinger H.G., Gröchenig K.H.: Gabor frames and time–frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I, III. Springer, Berlin (1983, 1985)

  9. Hörmander L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)

    MATH  Google Scholar 

  10. Pilipović, S., Teofanov, N., Toft, J.: Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I. J. Fourier Anal. Appl. (to appear)

  11. Pilipović S., Teofanov N., Toft J.: Wave-front sets in Fourier Lebesgue space. Rend. Sem. Mat. Univ. Politec. Torino 66(4), 41–61 (2008)

    Google Scholar 

  12. Ruzhansky, M., Sugimoto, M., Tomita, N., Toft, J.: Changes of variables in modulation and Wiener amalgam spaces (2008) Preprint. arXiv:0803.3485v1

  13. Toft J.: Continuity properties for modulation spaces with applications to pseudo-differential calculus, I. J. Funct. Anal. 207, 399–429 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Toft J.: Convolution and embeddings for weighted modulation spaces. In: Boggiatto, P., Ashino, R., Wong, M.W. (eds) Advances in Pseudo-Differential Operators, Operator Theory: Advances and Applications, vol. 155, pp. 165–186. Birkhäuser, Basel (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

    Stevan Pilipović & Nenad Teofanov

  2. Department of Computer Science, Mathematics and Physics, Linnaeus University, Växjö, Sweden

    Joachim Toft

Authors
  1. Stevan Pilipović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nenad Teofanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joachim Toft
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stevan Pilipović.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pilipović, S., Teofanov, N. & Toft, J. Micro-local analysis in Fourier Lebesgue and modulation spaces: part II. J. Pseudo-Differ. Oper. Appl. 1, 341–376 (2010). https://doi.org/10.1007/s11868-010-0013-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-010-0013-2

Keywords

Search

Navigation