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

2026/07/19 by Liu Ronghui, Zhang Qi
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Abstract

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

Citations

• Differential-difference operators associated to reflection groups
• On the solutions of quasi-linear elliptic partial differential equations

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