$L^p$ estimates for an oscillating Dunkl multiplier

Abstract

In this paper, we study the $L^p$ boundedness of a class of oscillating multiplier operator for the Dunkl transform, $T_{m_α}=\mathcal{F}_k^{-1}(m_α\mathcal{F}_k(f))$ with $m(ξ)=|ξ|^{-α}e^{\pm i|ξ|}ϕ(ξ)$. 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}