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Volume 36, Issue 1
Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Yueshan Wang

Anal. Theory Appl., 36 (2020), pp. 99-110.

Published online: 2020-05

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  • Abstract

Let $\mathcal{L}_2=(-\Delta)^2+V^2$ be the Schrödinger type operator, where $V\neq 0$ is a nonnegative potential and belongs to the reverse Hölder class $RH_{q_1}$ for $q_1> n/2, n\geq 5.$ The higher Riesz transform associated with $\mathcal{L}_2$ is denoted by $\mathcal{R}=\nabla^2 \mathcal{L}_2^{-\frac{1}{2}}$ and its dual is denoted by $\mathcal{R}^*=\mathcal{L}_2^{-\frac{1}{2}} \nabla^2.$ In this paper, we consider the $m$-order commutators $[b^m,\mathcal{R}]$ and $[b^m,\mathcal{R}^*],$ and establish the $(L^p,L^q)$-boundedness of these commutators when $b$ belongs to the new Campanato space $\Lambda_\beta^\theta(\rho)$ and $1/q=1/p-m\beta/n.$

  • AMS Subject Headings

42B25, 35J10, 42B35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wangys1962@163.com (Yueshan Wang)

  • BibTex
  • RIS
  • TXT
@Article{ATA-36-99, author = {Wang , Yueshan}, title = {Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {1}, pages = {99--110}, abstract = {

Let $\mathcal{L}_2=(-\Delta)^2+V^2$ be the Schrödinger type operator, where $V\neq 0$ is a nonnegative potential and belongs to the reverse Hölder class $RH_{q_1}$ for $q_1> n/2, n\geq 5.$ The higher Riesz transform associated with $\mathcal{L}_2$ is denoted by $\mathcal{R}=\nabla^2 \mathcal{L}_2^{-\frac{1}{2}}$ and its dual is denoted by $\mathcal{R}^*=\mathcal{L}_2^{-\frac{1}{2}} \nabla^2.$ In this paper, we consider the $m$-order commutators $[b^m,\mathcal{R}]$ and $[b^m,\mathcal{R}^*],$ and establish the $(L^p,L^q)$-boundedness of these commutators when $b$ belongs to the new Campanato space $\Lambda_\beta^\theta(\rho)$ and $1/q=1/p-m\beta/n.$

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2017-0055}, url = {http://global-sci.org/intro/article_detail/ata/16917.html} }
TY - JOUR T1 - Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators AU - Wang , Yueshan JO - Analysis in Theory and Applications VL - 1 SP - 99 EP - 110 PY - 2020 DA - 2020/05 SN - 36 DO - http://doi.org/10.4208/ata.OA-2017-0055 UR - https://global-sci.org/intro/article_detail/ata/16917.html KW - Schrödinger operator, Campanato space, Riesz transform, commutator. AB -

Let $\mathcal{L}_2=(-\Delta)^2+V^2$ be the Schrödinger type operator, where $V\neq 0$ is a nonnegative potential and belongs to the reverse Hölder class $RH_{q_1}$ for $q_1> n/2, n\geq 5.$ The higher Riesz transform associated with $\mathcal{L}_2$ is denoted by $\mathcal{R}=\nabla^2 \mathcal{L}_2^{-\frac{1}{2}}$ and its dual is denoted by $\mathcal{R}^*=\mathcal{L}_2^{-\frac{1}{2}} \nabla^2.$ In this paper, we consider the $m$-order commutators $[b^m,\mathcal{R}]$ and $[b^m,\mathcal{R}^*],$ and establish the $(L^p,L^q)$-boundedness of these commutators when $b$ belongs to the new Campanato space $\Lambda_\beta^\theta(\rho)$ and $1/q=1/p-m\beta/n.$

Yueshan Wang. (2020). Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators. Analysis in Theory and Applications. 36 (1). 99-110. doi:10.4208/ata.OA-2017-0055
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