Sparse domination for singular integral operators and their commutators in Dunkl setting with applications
Yanping Chen, Xueting Han
TL;DR
This work develops a comprehensive sparse domination framework for Dunkl-Calderón-Zygmund operators and their commutators in the Dunkl setting, establishing pointwise sparse domination by Dunkl dyadic averages. Leveraging this, it proves weighted and two-weight bounds, defining and employing the Dunkl Muckenhoupt classes $A_p^d$ and the Dunkl-BMO spaces $\mathrm{BMO}^d_u$, to obtain boundedness results for $T$ on $L^p(\mathbb{R}^N,dud\omega)$ and for $[b,T]$ on the corresponding two-weight spaces. The authors also extend the analysis to extrapolation, showing boundedness of $T$ on extrapolation spaces $\Delta(\{X_\alpha\})$ and $\Sigma(\{X_\alpha\})$ provided the Dunkl maximal operator $M_d$ is bounded on the spaces and their duals. Key technical tools include a grand maximal truncated operator $M_T$, a Dunkl dyadic cube framework, and a Dunkl-weighted Calderón-Zygmund machinery, enabling two-weight estimates and extrapolation in this non-Euclidean setting.
Abstract
In this paper, we establish sparse dominations for the Dunkl-Calderón-Zygmund operators and their commutators in the Dunkl setting. As applications, we first define the Dunkl-Muckenhoupt $A_p$ weight and obtain the weighted bounds for the Dunkl-Calderón-Zygmund operators, as well as the two-weight bounds for their commutators. Moreover, we also obtain the boundedness of the Dunkl-Calderón-Zygmund operators on the extrapolation space of a family of Banach function spaces.