# Dunkl translations, Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$

## TL;DR

This paper studies Dunkl translations and defines Dunkl-type BMO space and Riesz transforms, proving their boundedness on L-infinity and extending support descriptions of Dunkl translations for radial functions.

## Contribution

It introduces Dunkl-type BMO space and Riesz transforms, establishing their boundedness and extending support analysis of Dunkl translations to all nonnegative radial functions.

## Key findings

- Boundedness of Riesz transforms from L-infinity to Dunkl-type BMO space.
- Extended support description of Dunkl translations to all nonnegative radial functions.
- Analysis of Dunkl translations on compactly supported functions.

## Abstract

In this paper, we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$, and prove the boundedness of Riesz transforms from $L^\infty$ to Dunkl-type $BMO$ space under the uniform boundedness assumption of Dunkl translations. The proof and the definition in Dunkl setting will be harder than in the classical case for the lack of some similar properties of Dunkl translations to that of classical translations. We will also extend the preciseness of the description of support of Dunkl translations on characteristic functions by Gallardo and Rejeb to that on all nonnegative radial functions in $L^2(m_k)$.