TY - JOUR T1 - Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators JO - Analysis in Theory and Applications VL - 4 SP - 306 EP - 322 PY - 2018 DA - 2018/11 SN - 34 DO - http://doi.org/10.4208/ata.OA-2017-0071 UR - https://global-sci.org/intro/article_detail/ata/12844.html KW - Riesz transform, Schrödinger operator, commutator, Campanato space, Hardy space. AB -

Let $\mathcal{L} = −∆+V$ be a Schrödinger operator on $\mathbb{R}^n(n ≥ 3)$, where the nonnegative potential $V$ belongs to reverse Hölder class  $RH_{q_1}$ for $q_1 > \frac{n}{2}$. Let $H^p_{\mathcal{L}}(\mathbb{R}^n)$ be the Hardy space associated with $\mathcal{L}$. In this paper, we consider the commutator $[b,T_α]$, which associated with the Riesz transform $T_α = V^α(−∆+V)^{-\alpha}$ with $0<α≤ 1$, 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 $1<p<q_1/α$ 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 $n/ (n+β) < p ≤ 1$, $1/q=1/p−β/n$. Moreover, we prove that $[b,T_α]$ maps $H^{\frac{n}{n+\beta}}_{\mathcal{L}}(\mathbb{R}^n)$ continuously into weak $L^1(\mathbb{R}^n)$.