$L^p$ 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, $T_{m_{\alpha }}={\mathcal{F}}_k^{-1}(m_{\alpha }{\mathcal{F}}_k(f))$ with $m(\xi )=\mathrm{\mid }\xi {\mathrm{\mid }}^{-\alpha }{e}^{\pm i\mathrm{\mid }\xi \mathrm{\mid }}\varphi (\xi )$. 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 \cite{Stein}