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 - <p style="text-align: justify;">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 $q_1&gt; n/2, n\geq 5.$ 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 $1/q=1/p-m\beta/n.$</p>