Geodesic
Some estimates for commutators of Riesz transform associated with Schrödinger type operators
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 169-191
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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 $L_{\rm weak}^1(\mathbb {R}^n)$.
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 $L_{\rm weak}^1(\mathbb {R}^n)$.
DOI : 10.1007/s10587-016-0248-z
Classification : 35J10, 42B20, 42B30, 42B35
Keywords: commutator; Hardy space; reverse Hölder inequality; Riesz transform; Schrödinger operator; Schrödinger type operator 
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     author = {Liu, Yu and Zhang, Jing and Sheng, Jie-Lai and Wang, Li-Juan},
     title = {Some estimates for commutators of {Riesz} transform associated with {Schr\"odinger} type operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {169--191},
     year = {2016},
     volume = {66},
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Liu, Yu; Zhang, Jing; Sheng, Jie-Lai; Wang, Li-Juan. Some estimates for commutators of Riesz transform associated with Schrödinger type operators. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 169-191. doi: 10.1007/s10587-016-0248-z

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