A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group

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• 作者：Yu Liu ; Guobin Tang
• 关键词：Heisenberg group ; Stratified Lie group ; Reverse Hölder class ; Riesz transform ; Schrödinger operators
• 刊名：Analysis and Mathematical Physics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：7
• 期：1
• 页码：31-45
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Mathematical Methods in Physics;
• 出版者：Springer International Publishing
• ISSN：1664-235X
• 卷排序：7

文摘

Let \({\mathbb {H}^n}\) be the Heisenberg group and \(Q=2n+2\) be its homogeneous dimension. The Schrödinger operator is denoted by \( - {\Delta _{{\mathbb {H}^n}}} + V\), where \({\Delta _{{\mathbb {H}^n}}}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \({B_{{q_1}}}\) for \({q_1} \ge \frac{Q}{2}\). Let \(H^p_L(\mathbb {H}^n)\) be the Hardy space associated with the Schrödinger operator for \(\frac{Q}{Q+\delta _0}<p\le 1\), where \(\delta _0=\min \{1,2-\frac{Q}{q_1}\}\). In this note we show that the operators \({T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}\) and \({T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}\) are bounded from \(H_L^p({\mathbb {H}^n})\) into \({L^p}({\mathbb {H}^n})\). Our results are also valid on the stratified Lie group.