Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Miyachi's Theorem for the k-Hankel Transform on ℝd

  • Mohamed Amine Boubatra (Teacher Equcation College of Setif, Laboratory of applied mathematics, University of Ferhat Abbes of Setif)
  • Received : 2021.07.19
  • Accepted : 2022.06.08
  • Published : 2023.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

The classical Hardy Theorem on R states that a function f and its Fourier transform cannot be simultaneously very small; this fact was generalized by Miyachi in terms of L1 + L∞ and log+-functions. In this paper, we consider the k-Hankel transform, which is a deformation of the Hankel transform by a parameter k > 0 arising from Dunkl's theory. We study Miyachi's theorem for the k-Hankel transform on ℝd.

Keywords

Acknowledgement

The author is very grateful to the referee for the useful comments and suggestions given for the paper.

References

  1. S. Ben Said, A Product formula and convolution structure for a k-Hankel transform on R, J. Math. Anal. Appl., 463(2)(2018), 1132-1146. https://doi.org/10.1016/j.jmaa.2018.03.073
  2. S. Ben Said and L. Deleaval, Translation operator and maximal function for the (k, 1)-generalized Fourier transform, J. Funct. Anal., 279(8)(2020), 108706.
  3. S. Ben Said, T. Kobayashi and B. Orsted, Generalized Fourier transforms 𝓕ka, C. R. Math. Acad. Sci. Paris, 347(19-20)(2009), 1119-1124. https://doi.org/10.1016/j.crma.2009.07.015
  4. S. Ben Said, T. Kobayashi and B. Orsted, Laguerre semigroup and Dunkl operators, Compos. Math., 148(4)(2012), 1265-1336. https://doi.org/10.1112/S0010437X11007445
  5. M. A. Boubatra, S. Negzaoui and M. Sifi, A new product formula involving Bessel functions, Integral Transforms Spec. Funct., 33(3)(2022), 247-263. https://doi.org/10.1080/10652469.2021.1926454
  6. F. Chouchene, R. Daher, T. Kawazoe and H. Mejjaoli, Miyachi's theorem for the Dunkl transform, Integral Transforms Spec. Funct., 22(3)(2011), 167-173. https://doi.org/10.1080/10652469.2010.505029
  7. M. G. Cowling and J. F. Price, Generalizations of Heisenberg inequality in Harmonic analysis, (Cortona, 1982), Lecture Notes in Math., 992(1983), 443-449. https://doi.org/10.1007/BFb0069174
  8. R. Daher, On the theorems of Hardy and Miyachi for the Jacobi-Dunkl transform, Integral Transforms Spec. Funct., 18(5-6)(2007), 305-311. https://doi.org/10.1080/10652460701318244
  9. R. Daher, J. Delgado and M. Ruzhansky, Titchmarsh theorems for Fourier transforms of Holder-Lipschitz functions on compact homogeneous manifolds, Monatsh. Math., 189(1)(2019), 23-49. https://doi.org/10.1007/s00605-018-1253-0
  10. R. Daher, T. Kawazoe and H. Mejjaoli, A generalization of Miyachi's theorem, J. Math. Soc. Japan, 61(2)(2009), 551-558. https://doi.org/10.2969/jmsj/06120551
  11. C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311(1)(1989), 167-183. https://doi.org/10.1090/S0002-9947-1989-0951883-8
  12. S. El ouadih and R. Daher, Generalization of Titchmarsh's Theorem for the Dunkl Transform in the Space Lp(ℝd, wl(x)dx), Int. J. Math. Model. Comput., 6(4)(2016), 261-267.
  13. G. H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc., 8(1933), 227-231. https://doi.org/10.1112/jlms/s1-8.3.227
  14. R. Howe, The oscillator semigroup, Proc. Sympos. Pure Math., Vol. 48, American Mathematical Society, Providence, RI, (1988), 61-132.
  15. T. Kobayashi and G. Mano, The inversion formula and holomorphic extension of the minimal representation of the conformal group, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2007), 151-208.
  16. E. M. Loualid, A. Achak and R. Daher, Analogues of Miyachi and Cowling-Price theorems for the generalized Dunkl transform, Int. J. Eng. Appl. Sci., 2(1)(2022), 355-362.
  17. M. Maslouhi, An analog of Titchmarsh's Theorem for the Dunkl transform, Integral Transforms Spec. Funct., 21(9-10)(2010), 771-778. https://doi.org/10.1080/10652461003675752
  18. A. Miyachi, A generalization of theorem of Hardy, Harmonic Analysis Seminar held at Izunagaoka, Shizuoka-Ken, Japon, (1997), 44-51.