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 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.b180646Published online 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 Full-Text :