\(L^p\) Estimates for an Oscillating Dunkl Multiplier

  • Bechir Amri
  • Mohamed Gaidi


In this paper, we study the \(L^p\) boundedness of a class of oscillating multiplier operator for the Dunkl transform \(\mathcal {F}_k\), \(T_{m_\alpha }(f)=\mathcal {F}_k^{-1}(m_{\alpha }\mathcal {F}_k(f))\) with \(m_\alpha (\xi )=|\xi |^{-\alpha }e^{\pm i|\xi |}\phi ( \xi )\). We obtain an \(L^p\)-bound result for the corresponding maximal functions. As a specific applications, we give an extension of the \(L^p\) estimate for the wave equation and of Stein’s theorem for the analytic family of maximal spherical means (Stein, Proc Natl Acad Sci USA 73:2174–2175, 1976).


Dunkl transform multiplier operators wave equation 

Mathematics Subject Classification

Primary 42A38 42A45 Secondary 35L05 



The authors would like to express their sincere thanks to the referee for his comments and suggestions.


  1. 1.
    Anker, J.-Ph., Dziubanski, A.: Hejna harmonic functions, conjugate harmonic functions and the Hardy space \(H^1\) in the rational Dunkl setting. arXiv:1802.06607
  2. 2.
    Betancor, J.J., Ciaurri, Ó., Varona, J.L.: The multiplier of the interval \([-1; 1]\) for the Dunkl transform on the real line. J. Funct. Anal. 242(1), 327–336 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dai, F., Wang, H.: A transference theorem for the Dunkl transform and its applications. J. Funct. Anal. 258(12), 4052–4074 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113(1), 147–162 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Deleaval, L.: On the boundedness of the Dunkl spherical maximal operator. J. Topol. Anal. 8, 475–495 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deleaval, L., Kriegler, C.: Dunkl spectral multipliers with values in UMD lattices. J. Funct. Anal. 272(5), 2132–2175 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dunkl, C.F.: Differential-difference operators associated to reflextion groups. Trans. Am. Math. 311(1), 167–183 (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    Feffermann, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta. Math. 129, 137–193 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hirschmann, I.: On multiplier transformations. Duke Math. J. 26, 221–242 (1959)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Miyachi, A.: On some Fourier Multipliers for \(H^p(\mathbb{R}^n))\). J. Fac. Sci. Unvi. Tokyo Sect. IA 27, 157–179 (2014)Google Scholar
  11. 11.
    Peral, C.: \(L^p \)-estimates for the wave equation. J. Funct. Anal. 36(1), 114–145 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rösler, M.: Dunkl operators : theory and applications. In: Orthogonal polynomials and special functions (Leuven, 2012), Lect. Notes Math., vol. 1817. Springer, Berlin, pp. 93–135 (2002)Google Scholar
  13. 13.
    Saïd, S.B., Ørsted, B.: The wave equation for Dunkl operators. Indag. Math. (N.S.) 16, 351–391 (2005)Google Scholar
  14. 14.
    Sjostrand, S.: On the Riesz means of the solutions of the Schrodinger equation. Ann. Scuola Norm. Sup. Pisa (3) 24, 331–348 (1970)Google Scholar
  15. 15.
    Sogge, C.D.: Fourier Integrals in Classical Analysis. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  16. 16.
    Stein, E.M.: Maximal functions: spherical means. Proc. Natl. Acad. Sci. USA 73, 2174–2175 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–56 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wainger, S.: Special trigonometric series in k dimensions. Mem. Am. Math. Soc. 59, 102 (1965)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesTaibah University Al MadinahSaudi Arabia
  2. 2.Laboratoire d’Analyse Mathématique et ApplicationsUniversité Tunis El Manar, LR11ES11El Manar ITunisia

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