# $L^p$ estimates for an oscillating Dunkl multiplier

## TL;DR

This paper establishes $L^p$ boundedness results for an oscillating Dunkl multiplier operator, extending classical wave equation estimates and Stein's theorem to the Dunkl setting.

## Contribution

It introduces new $L^p$ bounds for oscillating multipliers in Dunkl analysis, extending classical harmonic analysis results to this setting.

## Key findings

- Proved $L^p$ bounds for Dunkl oscillating multipliers.
- Extended $L^p$ estimates for wave equations in Dunkl analysis.
- Generalized Stein's theorem for maximal spherical means in Dunkl context.

## Abstract

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}