@Article{ATA-34-306,
author = {Yueshan Wang and Yuexiang He},
title = {Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators},
journal = {Analysis in Theory and Applications},
year = {2018},
volume = {34},
number = {4},
pages = {306--322},
abstract = {<p style="text-align: justify;">Let $\mathcal{L} = −∆+V$ be a Schrödinger operator on $\mathbb{R}^n(n ≥ 3)$, where the nonnegative
potential $V$ belongs to reverse Hölder class&nbsp; $RH_{q_1}$&nbsp;for $q_1 &gt; \frac{n}{2}$. Let $H^p_{\mathcal{L}}(\mathbb{R}^n)$&nbsp;be the Hardy space associated with $\mathcal{L}$. In this paper, we consider the commutator $[b,T_α]$, which associated with the Riesz transform $T_α = V^α(−∆+V)^{-\alpha}$&nbsp;with $0&lt;α≤ 1$,
and a locally integrable function $b$ belongs to the new Campanato space $Λ^θ_β(ρ)$. We
establish the boundedness of $[b,T_α]$ from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for $1&lt;p&lt;q_1/α$ with $1/q=1/p−β/n$. We also show that $[b,T_α]$ is bounded from $H^p_{\mathcal{L}}(R^n)$ to $L^q(\mathbb{R}^n)$ when $n/ (n+β) &lt; p ≤ 1$, $1/q=1/p−β/n$. Moreover, we prove that $[b,T_α]$ maps $H^{\frac{n}{n+\beta}}_{\mathcal{L}}(\mathbb{R}^n)$ continuously into weak $L^1(\mathbb{R}^n)$.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.OA-2017-0071},
url = {http://global-sci.org/intro/article_detail/ata/12844.html}
}