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



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.


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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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