Abstract:
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 $L^2(m_k)$.
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