Dunkl Translations, Dunkl-Type BMO Space, and Riesz Transforms for the Dunkl Transform on $$L^\infty$$

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