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April, 2020 An extension of the characterization of $\mathrm{CMO}$ and its application to compact commutators on Morrey spaces
Ryutaro ARAI, Eiichi NAKAI
J. Math. Soc. Japan 72(2): 507-539 (April, 2020). DOI: 10.2969/jmsj/81458145

Abstract

In 1978 Uchiyama gave a proof of the characterization of $\mathrm{CMO}(\mathbb{R}^n)$ which is the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in $\mathrm{BMO}(\mathbb{R}^n)$. We extend the characterization to the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in the Campanato space with variable growth condition. As an application we characterize compact commutators $[b,T]$ and $[b,I_{\alpha}]$ on Morrey spaces with variable growth condition, where $T$ is the Calderón–Zygmund singular integral operator, $I_{\alpha}$ is the fractional integral operator and $b$ is a function in the Campanato space with variable growth condition.

Funding Statement

The second author was supported by Grant-in-Aid for Scientific Research (B), No. 15H03621, Japan Society for the Promotion of Science.

Citation

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Ryutaro ARAI. Eiichi NAKAI. "An extension of the characterization of $\mathrm{CMO}$ and its application to compact commutators on Morrey spaces." J. Math. Soc. Japan 72 (2) 507 - 539, April, 2020. https://doi.org/10.2969/jmsj/81458145

Information

Received: 14 October 2018; Published: April, 2020
First available in Project Euclid: 28 October 2019

zbMATH: 07196911
MathSciNet: MR4090345
Digital Object Identifier: 10.2969/jmsj/81458145

Subjects:
Primary: 42B35
Secondary: 42B20, 42B25, 46E30

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 2 • April, 2020
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