On the boundedness of Dunkl multipliers

TL;DR

This work extends classical multiplier theory to the Dunkl setting by proving two versions of Dunkl multiplier theorems under a modified Hörmander condition. The authors develop a Dunkl-compatible Littlewood–Paley–Stein framework, including new Leibniz-type formulas for Dunkl derivatives, to control multipliers via g-functions and their vector-valued analogues. They establish that radial multipliers satisfying the modified condition are bounded on $L^p(\mathbb{R}^d,h_\kappa^2)$ for all $1<p<\infty$, and that general multipliers act boundedly on radial $L^p$-domains for $p\ge2$, with an additional result for nonradial multipliers on radial functions. The results significantly extend Fourier multiplier theory to the Dunkl setting and provide a direct, self-contained approach via semigroup-based harmonic analysis. The work also clarifies the role of the Dunkl translation and radial symmetry in achieving sharp $L^p$-boundedness results.

Abstract

In this article we use Littlewood-Paley-Stein theory to prove two versions of Dunkl multiplier theorem when the multiplier $ m $ satisfies a modified Hörmander condition. When $ m $ is radial we give a simple proof of a known result. For general $ m $ we prove that the Dunkl multiplier operator takes radial functions in $ L^p $ boundedly into $ L^p $ for all $ p \geq 2.$

Theorems & Definitions (36)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Remark 1.8
• Theorem 1.9
• Proposition 2.1