Some estimates of weighted Hardy-Littlewood operators and their commutators on mixed Morrey-type spaces in the Dunkl setting

Abstract

In this paper, we consider the weighted Hardy--Littlewood operator ${\mathcal{H}}_{\phi }$ and its commutators in the Dunkl setting. We introduce mixed-norm radial-angular Dunkl central Morrey spaces, together with their $\lambda$-central counterparts. For the operator ${\mathcal{H}}_{\phi }$, we derive the necessary and sufficient conditions on the non-negative weight $\phi$ that ensure its boundedness on these spaces, and explicitly determine the exact operator norms. For the commutator ${\mathcal{H}}_{\phi ,b}$, we establish the necessary and sufficient conditions on the weight $\phi$ such that the commutator is bounded for all symbols $b$ in the mixed-norm radial-angular Dunkl central bounded mean oscillation space ${\mathrm{CMO}}_{\vec{p},\vec{q},k}({\mathbb{R}}^n)$. Furthermore, for all symbols $b$ in the mixed-norm radial-angular Dunkl $\lambda$-central bounded mean oscillation space ${\mathrm{CMO}}_{\vec{p},\vec{q},\lambda ,k}({\mathbb{R}}^n)$ with positive index $\lambda >0$, we also obtain a valid sufficient condition for boundedness.

33 pages