A characterization of CMO˙q→ via the commutator of Hardy-type operators on mixed Herz spaces

Abstract

In this paper, we introduce and study the mixed central bounded mean oscillation space ${\dot{\mathrm{CMO}}}^{\overrightarrow{q}}({\mathbb{R}}^n)$, which is a new central version of the bounded mean oscillation spaces BMO. The pre-dual of ${\dot{\mathrm{CMO}}}^{\overrightarrow{q}}({\mathbb{R}}^n)$ is proved to be the mixed Herz–Hardy space ${\mathrm{HA}}^{\overrightarrow{q}}({\mathbb{R}}^n)$. Moreover, we also give a characterization of ${\dot{\mathrm{CMO}}}^{\overrightarrow{q}}({\mathbb{R}}^n)$ via the boundedness of the commutators of n-dimensional Hardy operator and its dual operator on mixed Herz spaces ${\dot{K}}_{\overrightarrow{q}}^{\alpha ,p}({\mathbb{R}}^n)$.

Keywords:

• Hardy operator
• commutator
• Herz space
• Herz–Hardy space
• mixed spaces

2010 Mathematics Subject Classifications:

• 42B20
• 42B35

Acknowledgements

The author would like to thank the anonymous referee for many valuable comments and suggestions. This work was supported by the Natural Science Foundation of Henan (Grant Nos. 202300410338) and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Foundation of Henan Province [grant number 202300410338] and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.