Dunkl--BMO functions representable in terms of generalized balayage of Carleson measures and an application

We characterize compactly supported functions $f$ in Dunkl–BMO spaces as those that can be written as \begin{equation*} f=g+\widetilde{S}_{μ,$\mathcal{P}$}, \end{equation*} the sum of some bounded function $g$ and the generalized balayage ${\tilde{S}}_{\mu ,\mathcal{P}}$ of a Carleson measure on ${\mathbb{R}}^N\times (0,\mathrm{\infty })$ associated with the Dunkl–Poisson semigroup. As a direct application of this representation of Dunkl–BMO functions, we establish a new characterization of the Hardy space in the Dunkl setting via a principal value integral related to ${\tilde{S}}_{\mu ,\mathcal{P}}$. Moreover, analogous conclusions hold for the Dunkl–heat semigroup as well.