A note on the first order commutator $\mathcal{C}_2$
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 885-898
https://doi.org/10.4134/BKMS.b180646
Published online July 9, 2019
Printed July 31, 2019
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)
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