## Abstract and Applied Analysis

### Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

#### Abstract

Let $L=-\mathrm{\Delta }+V$ be a Schrödinger operator, where $\mathrm{\Delta }$ is the laplacian on ${\Bbb R}^{n}$ and the nonnegative potential $V$ belongs to the reverse Hölder class ${B}_{{s}_{\mathrm{1}}}$ for some ${s}_{1}\ge (n/2)$. Assume that $\omega \in {A}_{\mathrm{1}}({\Bbb R}^{n})$. Denote by ${H}_{L}^{\mathrm{1}}(\omega )$ the weighted Hardy space related to the Schrödinger operator $L=-\mathrm{\Delta }+V$. Let ${\scr R}_{b}=[b,\scr R]$ be the commutator generated by a function $b\in {\text{BMO}}_{\theta }({\Bbb R}^{n})$ and the Riesz transform ${\scr R=\nabla (-\mathrm{\Delta }+V)}^{-(\mathrm{1}/\mathrm{2})}$. Firstly, we show that the operator $\scr R$ is bounded from ${L}^{\mathrm{1}}(\omega )$ into ${L}_{\text{weak}}^{1}(\omega )$. Secondly, we obtain the endpoint estimates for the commutator $[b,\scr R]$. Namely, it is bounded from the weighted Hardy space ${H}_{L}^{\mathrm{1}}(\omega )$ into ${L}_{\text{weak}}^{1}(\omega )$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 281562, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450330

Digital Object Identifier
doi:10.1155/2013/281562

Mathematical Reviews number (MathSciNet)
MR3139466

Zentralblatt MATH identifier
1293.35063

#### Citation

Liu, Yu; Sheng, Jielai; Wang, Lijuan. Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 281562, 10 pages. doi:10.1155/2013/281562. https://projecteuclid.org/euclid.aaa/1393450330

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