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

In this paper, we consider the weighted Hardy--Littlewood operator $\mathcal{H}_{\varphi}$ 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 $\mathcal{H}_{\varphi}$, we derive the necessary and sufficient conditions on the non-negative weight $\varphi$ that ensure its boundedness on these spaces, and explicitly determine the exact operator norms.

For the commutator $\mathcal{H}_{\varphi,b}$, we establish the necessary and sufficient conditions on the weight $\varphi$ 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.