Boundedness and Compactness of Localization Operators Associated with the Spherical Mean Wigner Transform

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Abstract

We introduce the notion of localization operators associated with the spherical mean Wigner transform, and we give a trace formula for the localization operators associated with the spherical mean Wigner transform as a bounded linear operator in the trace class from \(L^{2}(d\nu )\) into \(L^{2}(d\nu )\) in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of localization operators associated with the spherical mean Wigner transform on \(L^{p}(d\nu )\), \(1 \le p \le \infty \).

Keywords

Spherical mean operator Spherical mean Wigner transform Localisation operators 

Mathematics Subject Classification

33E30 42B10 43A32 44A20 

Notes

Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. The first author thanks the professor M.W. Wong for his help.

References

  1. 1.
    Baccar, C., Omri, S., Rachdi, L.T.: Fock spaces connected with spherical mean operator and associated operators. Mediterr. J. Math. 6(1), 1–25 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Cambridge (1988)MATHGoogle Scholar
  3. 3.
    Boggiatto, P., Wong, M.W.: Two-wavelet localization operators on \(L^{p}({\mathbb{R}}^{d})\) for the Weyl-Heisenberg group. Integr. Equ. Oper. Theory 49, 1–10 (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Calderon, J.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34(4), 605–612 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Daubechies, I., Paul, T.: Time-frequency localization operators-a geometric phase space approach: II. The use of dilations. Inverse Probl. 4(3), 661–680 (1988)CrossRefMATHGoogle Scholar
  7. 7.
    Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fawcett, J.A.: Inversion of N-dimensional spherical means. SIAM. J. Appl. Math. 45, 336–341 (1983)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)MATHGoogle Scholar
  10. 10.
    Helesten, H., Andersson, L.E.: An inverse method for the processing of synthetic aperture radar data. Inverse Probl. 3, 111–124 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    He, Z., Wong, M.W.: Localization operators associated to square integrable group representations. Panam. Math. J. 6(1), 93–104 (1996)MathSciNetMATHGoogle Scholar
  12. 12.
    Hleili, K., Omri, S.: The Littlewood–Paley g-function associated with the spherical mean operator. Mediterr. J. Math. 10(2), 887–907 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience, New York (1955)MATHGoogle Scholar
  14. 14.
    Lieb, E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31(3), 594–599 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, L.: A trace class operator inequality. J. Math. Anal. Appl. 328, 1484–1486 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ma, B., Wong, M.W.: \(L^{p}-\)boundedness of wavelet multipliers. Hokkaido Math. J. 33, 637–645 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nessibi, M.M., Rachdi, L.T., Trimèche, K.: Ranges and inversion formulas for spherical mean operator and its dual. J. Math. Anal. App. 196, 861–884 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Omri, S.: Uncertainty principle in terms of entropy for the spherical mean operator. J. Math. Inequal 5(4), 473–490 (2010)MathSciNetMATHGoogle Scholar
  19. 19.
    Rachdi, L.T., Trimèche, K.: Weyl transforms associated with the spherical mean operator. Anal. Appl. 1(2), 141–164 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Trimèche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach Science Publishers, Philadelphia (1997)MATHGoogle Scholar
  22. 22.
    Zhao, J., Peng, L.: Wavelet and Weyl transforms associated with the spherical mean operator. Integr. Equ. Oper. Theory 50, 279–290 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wong, M.W.: Localization operators on the Weyl-Heisenberg group. In: Pathak, R.S. (ed.) Geometry, Analysis and Applications, pp. 303–314. World-Scientific, Singapore (2001)Google Scholar
  24. 24.
    Wong, M.W.: \(L^{p}\) boundedness of localization operators associated to left regular representations. Proc. Am. Math. Soc. 130, 2911–2919 (2002)CrossRefMATHGoogle Scholar
  25. 25.
    Wong, M.W.: Wavelet Transforms and Localization Operators, vol. 136. Springer, Berlin (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesTaibah UniversityAl-Madinah AL-MunawarahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia

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