关于Dunkl环境下加权Hardy-Littlewood算子及其交换子在混合Morrey型空间中的估计
Some estimates of weighted Hardy-Littlewood operators and their commutators on mixed Morrey-type spaces in the Dunkl setting
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中文总结 AI 辅助
本文在Dunkl环境下研究了加权Hardy-Littlewood算子及其交换子在混合Morrey型空间中的有界性条件及范数估计。
中文摘要 AI 辅助
在本文中，我们考虑了Dunkl环境下加权Hardy-Littlewood算子$\mathcal{H}_{\varphi}$及其交换子。我们引入了混合范数径向-角Dunkl中心Morrey空间，以及它们的$\lambda$-中心对应物。对于算子$\mathcal{H}_{\varphi}$，我们推导出非负权$\varphi$的必要且充分条件，以确保其在这些空间上的有界性，并明确确定了精确的算子范数。对于交换子$\mathcal{H}_{\varphi,b}$，我们建立了权$\varphi$的必要且充分条件，使得对于所有符号$b$在混合范数径向-角Dunkl中心有界平均振荡空间$\mathrm{CMO}_{\vec p, \vec q, k}(\mathbb{R}^n)$中都是有界的。此外，对于所有符号$b$在混合范数径向-角Dunkl$\lambda$-中心有界平均振荡空间$\mathrm{CMO}_{\vec p, \vec q,\lambda, k}(\mathbb{R}^n)$中且正指数$\lambda>0$，我们还获得了有界性的有效充分条件。
英文摘要
In this paper, we consider the weighted Hardy--Littlewood operator $\mathcal{H}_φ$ 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 $\mathcal{H}_φ$, 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 $\mathcal{H}_{φ,b}$, 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 $\mathrm{CMO}_{\vec p, \vec q, k}(\mathbb{R}^n)$. Furthermore, for all symbols $b$ in the mixed-norm radial-angular Dunkl $λ$-central bounded mean oscillation space $\mathrm{CMO}_{\vec p, \vec q,λ, k}(\mathbb{R}^n)$ with positive index $λ>0$, we also obtain a valid sufficient condition for boundedness.
