Some estimates for commutators of Riesz transforms associated with Schr枚dinger operators

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• 作者：Yu Liu ; ^{liuyu75@pku.org.cn" class="auth_mail} ; Jielai Sheng ^{shengjielai@126.com" class="auth_mail}
• 关键词：Commutator ; Hardy space ; Campanato space ; Reverse Hö ; lder inequality ; Riesz transform ; Schrö ; dinger operator
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：1 November 2014
• 年：2014
• 卷：419
• 期：1
• 页码：298-328
• 全文大小：463 K

文摘

We consider the Schrödinger operator L=−螖+V$L=-\mathrm{螖}+V$ on Rⁿ${\mathbb{R}}^n$, where the nonnegative potential V   belongs to the reverse Hölder class B_q1$B_{q_1}$ for some $q_1\ge \frac{n}{2}$. Let q₂=1$q_2=1$ when q₁≥n$q_1\ge n$ and $\frac{1}{q_2}=1-\frac{1}{q_1}+\frac{1}{n}$ when $\frac{n}{2}<q_1<n$. Set ${未}^{\prime }=\min \{1,2-\frac{n}{q_1}\}$. Let $H_L^p({\mathbb{R}}^n)$ be the Hardy space related to the Schrödinger operator L   for $\frac{n}{n+{未}^{\prime }}<p\le 1$. The commutator [b,R]$[b,\mathcal{R}]$ is generated by a function $b\in {螞}_{谓}^{胃}$, where ${螞}_{谓}^{胃}$ is a function space which is larger than the classical Companato space, and the Riesz transform . We show that the commutator [b,R]$[b,\mathcal{R}]$ is bounded from L^p(Rⁿ)$L^p({\mathbb{R}}^n)$ into L^q(Rⁿ)$L^q({\mathbb{R}}^n)$ for $1<p<q_2^{\prime }$, where $\frac{1}{q}=\frac{1}{p}-\frac{谓}{n}$ and bounded from $H_L^p({\mathbb{R}}^n)$ into L^q(Rⁿ)$L^q({\mathbb{R}}^n)$ for $\frac{n}{n+谓}<p\le 1$, where $\frac{1}{q}=\frac{1}{p}-\frac{谓}{n}$. Moreover, we prove that the commutator [b,R]$[b,\mathcal{R}]$ maps $H_L^{\frac{n}{n+谓}}({\mathbb{R}}^n)$ continuously into weak L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. At last, we give a characterization for the boundedness of the commutator [b,R]$[b,\mathcal{R}]$ in an extreme case.