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On compactness of commutator of multiplication and pseudodifferential operator

Article

Abstract

In this short note we show results on the compactness of the commutator of pseudodifferential operator and operator of multiplication in both \(\mathrm{L}^2\) and \(\mathrm{L}^p\) setting. Our results use the boundedness results of pseudodifferential operators by Hwang and Hwang-Lee, and the Krasnoselskij type interpolation lemma. We use the obtained results to construct a variant of microlocal defect functionals via pseudodifferential operators and derive its localisation principle.

Keywords

Commutator Compactness Pseudodifferential operator 

Mathematics Subject Classification

35S05 47G30 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for his/her useful comments.

This work was supported in part by the Croatian Science Foundation under project 9780 WeConMApp, and by the project number 01–417 Advection–diffusion equations in highly heterogeneous media of the Montenegrin Ministry of Science. This work was initialised while the second author was visiting University of Zagreb in the framework of the Marie Curie FP7-PEOPLE-2011-COFUND project Micro-local defect functionals and applications.

References

  1. 1.
    Antonić, N., Erceg, M., Mišur, M.: Distributions of anisotropic order and applications. (In Preparation)Google Scholar
  2. 2.
    Antonić, N., Mišur, M., Mitrović, D.: On the First commutation lemma. (In Preparation)Google Scholar
  3. 3.
    Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an \({\rm L}^p-{\rm L}^q\) setting. Abs. Appl. Anal. 2011, p 12 (2011). Article ID 901084Google Scholar
  4. 4.
    Conway, J.B.: A Course in Functional Analysis. Springer, Berlin (1997)Google Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Erceg, M., Mišur, M., Mitrović, D.: Velocity averaging and strong precompactness for degenerate parabolic equations with discontinuous flux. (In Preparation)Google Scholar
  7. 7.
    Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Eq. 16, 1761–1794 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Grafakos, L.: Classical Fourier Analysis. Springer, Berlin (2008)MATHGoogle Scholar
  9. 9.
    Hwang, I.L.: The \({\rm L}^2\)-boundedness of pseudo-differential operators. Trans. Am. Math. Soc. 302, 55–76 (1987)Google Scholar
  10. 10.
    Hwang, I.L., Lee, R.B.: The \({\rm L}^p\)-boundedness of pseudo-differential operators of class \(S_{0,0}\). Trans. Am. Math. Soc. 346, 489–510 (1994)Google Scholar
  11. 11.
    Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Krasnosel’skij, M. A.: On a theorem of M. Riesz, Dokl. Akad. Nauk SSSR 131: 246–248; translated as Soviet Math. Dokl. 1, 229–231 (1960). (In Russian)Google Scholar
  13. 13.
    Mišur, M.: H-distributions and compactness by compensation, PhD thesis, University of Zagreb, (2017)Google Scholar
  14. 14.
    Mišur, M., Mitrović, D.: On a generalisation of compensated compactness in the \({\rm L}^p-{\rm L}^q\) setting. J. Funct. Anal. 268, 1904–1927 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Panov, E.J.: Ultra-parabolic H-measures and compensated compactness. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 47–62 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Persson, A.: Compact linear mappings between interpolation spaces. Ark. Mat. 5, 215–219 (1964)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rindler, F.: Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms. Arch. Ration. Mech. Anal. 215, 1–63 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinburgh 115A, 193–230 (1990)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tartar, L.: The general theory of homogenization: A personalized introduction. Springer, Berlin (2009)MATHGoogle Scholar
  20. 20.
    Tréves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, London (1967)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of ScienceUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of MathematicsUniversity of MontenegroPodgoricaMontenegro

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