  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
$m(\xi)=|\xi|^{-\alpha}e^{\pm i|\xi|}\phi(\xi)$. 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 \cite{Stein}
