Abstract
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).
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The authors would like to express their sincere thanks to the referee for his comments and suggestions.
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Amri, B., Gaidi, M. \(L^p\) Estimates for an Oscillating Dunkl Multiplier. Mediterr. J. Math. 15, 85 (2018). https://doi.org/10.1007/s00009-018-1135-7
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DOI: https://doi.org/10.1007/s00009-018-1135-7