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commutator; Hardy space; reverse Hölder inequality; Riesz transform; Schrödinger operator; Schrödinger type operator
Let $\mathcal {L}_1=-\Delta +V$ be a Schrödinger operator and let $\mathcal {L}_2=(-\Delta )^2+V^2$ be a Schrödinger type operator on ${\mathbb {R}^n}$ $(n \geq 5)$, where $V \neq 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge {n}/{2}$. The Hardy type space $H^1_{\mathcal {L}_2}$ is defined in terms of the maximal function with respect to the semigroup $\{{\rm e}^{-t \mathcal {L}_2}\}$ and it is identical to the Hardy space $H^1_{\mathcal {L}_1}$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator $\mathcal {R}_b=b\mathcal {R}f-\mathcal {R}(bf)$ generated by the Riesz transform $\mathcal {R}=\nabla ^2\mathcal {L}_2^{-{1}/{2}}$, where $b\in {\rm BMO}_\theta (\rho )$, which is larger than the space ${\rm BMO}(\mathbb {R}^n)$. Moreover, we prove that $\mathcal {R}_b$ is bounded from the Hardy space $H_{\mathcal {L}_2}^1(\mathbb {R}^n)$ into weak $L_{\rm weak}^1(\mathbb {R}^n)$.
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