Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting (Journal of Mathematical Study, 2020)
On the real line, the Dunkl operators$$D_{ u}(f)(x):=\frac{d f(x)}{dx} + (2 u+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, u \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{ u})$ if and only if the maximal commutator $M_{b, u}$ is bounded on Orlicz spaces $L_{Φ}(\mathbb{R},dm_{ u})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{Φ}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.