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On the Weinstein–Wigner transform and Weinstein–Weyl transform

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Abstract

In this paper, we define and study the Weinstein–Wigner transform and we prove its inversion formula. Next, we introduce and study the Weinstein–Weyl transform \({\mathcal {W}}_\sigma \) with symbol \(\sigma \) and we give an integral relation between it and the Weinstein–Wigner transform. At last, we give criteria in terms of \(\sigma \) for boundedness and compactness of the transform \({\mathcal {W}}_\sigma \).

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References

  1. Brelot, M.: Equation de Weinstein et potentiels de Marcel Riesz. In: Séminaire de Théorie du Potentiel Paris, no. 3, pp. 18–38. Springer, (1978)

  2. Dachraoui, A.: Weyl–Bessel transforms. J. Comput. Appl. Math. 133(1–2), 263–276 (2001)

    Article  MathSciNet  Google Scholar 

  3. Dachraoui, A.: Weyl transforms associated with Laguerre functions. J. Comput. Appl. Math. 153(1–2), 151–162 (2003)

    Article  MathSciNet  Google Scholar 

  4. Dachraoui, A.: Weyl–Dunkl transforms. Glob. J. Pure Appl. Math. 2(3), 206–225 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Dasgupta, A., Wong, M.W.: Weyl transforms for H-type groups. J. Pseudo Differ. Oper. Appl. 6(1), 11–19 (2015)

    Article  MathSciNet  Google Scholar 

  6. de Gosson, M.: A transformation property of the wigner distribution under Hamiltonian symplectomorphisms. J. Pseudo Differ. Oper. Appl. 2(1), 91–99 (2011)

    Article  MathSciNet  Google Scholar 

  7. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, Hoboken (1984)

    MATH  Google Scholar 

  8. Kim, M., Ben-Benjamin, J., Cohen, L.: Inverse Weyl transform/operator. J. Pseudo Differ. Oper. Appl. 8(4), 661–678 (2017)

    Article  MathSciNet  Google Scholar 

  9. Mehrez, K.: Paley–Wiener theorem for the Weinstein transform and applications. Integral Transforms Special Funct. 28(8), 616–628 (2017)

    Article  MathSciNet  Google Scholar 

  10. Mejjaoli, H., Salem, A.O.A.: New results on the continuous Weinstein wavelet transform. J. Inequal. Appl. 2017(1), 270 (2017)

    Article  MathSciNet  Google Scholar 

  11. Mejjaoli, H., Salhi, M.: Uncertainty principles for the Weinstein transform. Czechoslov. Math. J. 61(4), 941–974 (2011)

    Article  MathSciNet  Google Scholar 

  12. Molahajloo, S., Wong, M.W.: Hierarchical Weyl transforms and the heat semigroup of the hierarchical twisted Laplacian. J. Pseudo Differ. Oper. Appl. 1(1), 35–53 (2010)

    Article  MathSciNet  Google Scholar 

  13. Nahia, Z.B., Salem, N.B.: On a mean value property associated with the Weinstein operator. In: Proceedings of the International Conference on Potential Theory held in Kouty, Czech Republic (ICPT’94), pp. 243–253 (1996)

  14. Nahia, Z.B., Salem, N.B.: Spherical harmonics and applications associated with the Weinstein operator. In: Proceedings of the International Conference on Potential Theory—ICPT ’94, Kouty, Czech Republic

  15. Radha, R., Kumar, N.S.: Weyl transform and Weyl multipliers associated with locally compact abelian groups. J. Pseudo Differ. Oper. Appl. 9(2), 229–245 (2018)

    Article  MathSciNet  Google Scholar 

  16. Salem, N.B., Nasr, A.R.: Heisenberg-type inequalities for the Weinstein operator. Integral Transforms Special Funct. 26(9), 700–718 (2015)

    Article  MathSciNet  Google Scholar 

  17. Saoudi, A.: A variation of the \({L}^p\) uncertainty principles for the Weinstein transform. ar**v preprint ar**v:1810.04484 (2018)

  18. Saoudi, A.: Calderón’s reproducing formulas for the weinstein \({L}^2\)-multiplier operators. Acceoted Asian Eur. J. Math. (2019). https://doi.org/10.1142/S1793557121500030

    Article  Google Scholar 

  19. Saoudi, A., Kallel, I.A.: \( L^2 \)-uncertainty principle for the Weinstein-Multiplier Operators. Int. J. Anal. Appl. 17(1), 64–75 (2019)

    MATH  Google Scholar 

  20. Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83(2), 482–492 (1956)

    Article  MathSciNet  Google Scholar 

  21. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 10, p. 297. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  22. Weinstein, A.: Singular partial differential equations and their applications. Fluid Dyn. Appl. Math. 67, 29–49 (1962)

    MathSciNet  MATH  Google Scholar 

  23. Weyl, H.: The Theory of Groups and Quantum Mechanics. Courier Corporation, North Chelmsford (1950)

    MATH  Google Scholar 

  24. Wong, M.W.: The Weyl Transforms. Springer, Berlin (1998)

    MATH  Google Scholar 

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Authors and Affiliations

  1. College of Science, Northern Border University, Arar, P.O. Box 1631, Saudi Arabia

    Ahmed Saoudi

  2. Faculté des sciences de Tunis, Université de Tunis El Manar, Tunis, Tunisie

    Ahmed Saoudi

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  1. Ahmed Saoudi
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Correspondence to Ahmed Saoudi.

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Saoudi, A. On the Weinstein–Wigner transform and Weinstein–Weyl transform. J. Pseudo-Differ. Oper. Appl. 11, 1–14 (2020). https://doi.org/10.1007/s11868-019-00313-2

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  • DOI: https://doi.org/10.1007/s11868-019-00313-2

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