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Volume 34, Issue 4
Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Yueshan Wang & Yuexiang He

Anal. Theory Appl., 34 (2018), pp. 306-322.

Published online: 2018-11

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  • Abstract

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 $L^1(\mathbb{R}^n)$.

  • Keywords

Riesz transform, Schrödinger operator, commutator, Campanato space, Hardy space.

  • AMS Subject Headings

42B30, 42B25, 35J10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-34-306, author = {}, 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 = {

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 $L^1(\mathbb{R}^n)$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2017-0071}, url = {http://global-sci.org/intro/article_detail/ata/12844.html} }
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 $L^1(\mathbb{R}^n)$.

Yueshan Wang & Yuexiang He. (1970). Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators. Analysis in Theory and Applications. 34 (4). 306-322. doi:10.4208/ata.OA-2017-0071
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