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