On Compactness of Commutators of Multiplications and Fourier Multipliers


We generalise results on compactness of commutators of multiplications and Fourier multiplier operators by Cordes (J Funct Anal 18:115–131, 1975) with respect to the smoothness of multiplication function. Our prime motivation has been a particular case known as the first commutation lemma—the basic tool for defining H-measures and H-distributions. We review and improve the known results in the \(\mathrm{L}^p\) setting, with \(1<p<\infty \), illustrating these results with an application, obtaining a generalised compactness by compensation result.

This is a preview of subscription content, access via your institution.


  1. 1.

    Aleksić, J., Pilipović, S., Vojnović, I.: H-distributions via Sobolev spaces. Mediterr. J. Math. 13, 3499–3512 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Antonić, N., Ivec, I.: On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces. J. Math. Anal. Appl. 433, 176–199 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Antonić, N., Lazar, M.: Parabolic H-measures. J. Funct. Anal. 265, 1190–1239 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an \({\rm L}^p-{\rm L}^q\) setting. Abs. Appl. Anal. 2011, Article ID 901084 (2011)

  5. 5.

    Cordes, H.O.: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Edwards, R.E.: Functional Analysis: Theory and Applications. Dover, New York (1995)

    Google Scholar 

  7. 7.

    Erceg, M., Ivec, I.: On a generalisation of H-measures. Filomat 31(16), 5027–5044 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Erceg, M., Mišur, M., Mitrović, D.: Velocity averaging and strong precompactness for degenerate parabolic equations with discontinuous flux (in preparation)

  9. 9.

    Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Grafakos, L.: Classical Fourier Analysis. Springer, Berlin (2008)

    Google Scholar 

  11. 11.

    Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Krasnosel’skij, M.A.: On a theorem of M. Riesz. Dokl. Akad. Nauk SSSR 131, 246–248 (1960). [(in Russian); translated as Soviet Math. Dokl. 1 (1960) 229–231]

    Google Scholar 

  13. 13.

    Lazar, M., Mitrović, D.: Existence of solutions for a scalar conservation law with a flux of low regularity. Electron. J. Differ. Equ. 2016(325), 1–18 (2016)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Maz’ja, V.G., Šapošnikova, T.O.: Theory of Sobolev Multipliers. Springer, Berlin (2009)

    Google Scholar 

  15. 15.

    Mišur, M., Mitrović, D.: On a generalisation of compensated compactness in the \(\text{ L }^p\)\(\text{ L }^q\) setting. J. Funct. Anal. 268, 1904–1927 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Mišur, M., Mitrović, D.: On compactness of commutator of multiplication and pseudodifferential operator. J. Pseudo Differ. Oper. Appl. (2018). https://doi.org/10.1007/s11868-018-0239-y. (OnlineFirst)

  17. 17.

    Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5, 489–507 (1978)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Panov, E.J.: Ultra-parabolic H-measures and compensated compactness. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 47–62 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinb. 115A, 193–230 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Tartar, L.: The General Theory of Homogenization: A Personalized Introduction. Springer, Berlin (2009)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Nenad Antonić.

Additional information

This work was supported in part by the Croatian Science Foundation under project 9780 WeConMApp, by the bilateral Croatian–Montenegrin project Multiscale methods and calculus of variations, as well as by the project number 01-417 Advection–diffusion equations in highly heterogeneous media of the Montenegrin Ministry of Science. Part of this work was performed, while D. Mitrović was visiting University of Zagreb in the framework of the Marie Curie FP7-PEOPLE-2011-COFUND project Micro-local defect functionals and applications.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Antonić, N., Mišur, M. & Mitrović, D. On Compactness of Commutators of Multiplications and Fourier Multipliers. Mediterr. J. Math. 15, 170 (2018). https://doi.org/10.1007/s00009-018-1215-8

Download citation

Mathematics Subject Classification

  • 42B15
  • 35B40
  • 28C15


  • Commutator
  • compactness
  • Fourier multiplier
  • H-distribution