On the boundedness of Dunkl multipliers

TL;DR

This paper establishes boundedness results for Dunkl multipliers using Littlewood-Paley-Stein theory, including new proofs and generalizations for radial and non-radial multipliers under Hörmander conditions.

Contribution

It provides new proofs and extends boundedness results for Dunkl multipliers, especially for non-radial multipliers, under modified Hörmander conditions.

Findings

Dunkl multipliers are bounded on L^p for p ≥ 2 under certain conditions.

A simple proof for the radial case of Dunkl multipliers is provided.

Generalization of boundedness results to non-radial multipliers is achieved.

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 H\"ormander 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.$