Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(4): 885-898

Online first article July 9, 2019      Printed July 31, 2019

https://doi.org/10.4134/BKMS.b180646

Copyright © The Korean Mathematical Society.

A note on the first order commutator $\mathcal{C}_2$

Wenjuan Li, Suying Liu

Northwest Polytechnical University; Northwest Polytechnical University

Abstract

This paper gives a counterexample to show that the first order commutator $\mathcal {C}_2$ is not bounded from $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$ into $L^{1/2}(\mathbb{R})$. Then we introduce the atomic definition of abstract weighted Hardy spaces $H_{ato,\omega}^{1}(\mathbb{R})$ and study its properties. At last, we prove that $\mathcal {C}_2$ maps $H_{ato,w}^{1}(\mathbb{R})\times H^{1}_{ato,w}(\mathbb{R})$ into $L^{1/2}_{\omega}(\mathbb{R})$.

Keywords: the first order commutator $\mathcal {C}_2$, counterexample, abstract weighted Hardy space, atom, boundedness

MSC numbers: 46E30, 42B30, 42B35

Supported by: The first author was partially supported by the National Natural Science Foundation of China (No. 11601427), China Postdoctoral Science Foundation (No.2017M613193), Fundamental Research Funds for the Central Universities (No.3102017zy035). The second author is supported by the National Natural Science Foundation of China (No. 11701453)