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Volume 32, Issue 1
Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group

Y. Liu & G. B. Tang

Anal. Theory Appl., 32 (2016), pp. 78-89.

Published online: 2016-01

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  • Abstract

Let $\mathbb{H}^n$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. In this paper, we consider the Schrödinger operator $−∆_{\mathbb{H}^n} +V$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölder class $B_{q_1}$ for $q_1 ≥ Q/2$. We show that the operators $T_1 = V(−∆_{\mathbb{H}^n} +V)^{−1}$ and $T_2 = V^{1/2}(−∆_{\mathbb{H}^n} +V)^{−1/2}$ are both bounded from $H^1_L(\mathbb{H}^n)$ into $L^1(\mathbb{H}^n)$. Our results are also valid on the stratified Lie group.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

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COPYRIGHT: © Global Science Press

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@Article{ATA-32-78, author = {}, title = {Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group}, journal = {Analysis in Theory and Applications}, year = {2016}, volume = {32}, number = {1}, pages = {78--89}, abstract = {

Let $\mathbb{H}^n$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. In this paper, we consider the Schrödinger operator $−∆_{\mathbb{H}^n} +V$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölder class $B_{q_1}$ for $q_1 ≥ Q/2$. We show that the operators $T_1 = V(−∆_{\mathbb{H}^n} +V)^{−1}$ and $T_2 = V^{1/2}(−∆_{\mathbb{H}^n} +V)^{−1/2}$ are both bounded from $H^1_L(\mathbb{H}^n)$ into $L^1(\mathbb{H}^n)$. Our results are also valid on the stratified Lie group.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2016.v32.n1.7}, url = {http://global-sci.org/intro/article_detail/ata/4656.html} }
TY - JOUR T1 - Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group JO - Analysis in Theory and Applications VL - 1 SP - 78 EP - 89 PY - 2016 DA - 2016/01 SN - 32 DO - http://doi.org/10.4208/ata.2016.v32.n1.7 UR - https://global-sci.org/intro/article_detail/ata/4656.html KW - Heisenberg group, stratified Lie group, reverse Hölder class, Riesz transform, Schrödinger operator. AB -

Let $\mathbb{H}^n$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. In this paper, we consider the Schrödinger operator $−∆_{\mathbb{H}^n} +V$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölder class $B_{q_1}$ for $q_1 ≥ Q/2$. We show that the operators $T_1 = V(−∆_{\mathbb{H}^n} +V)^{−1}$ and $T_2 = V^{1/2}(−∆_{\mathbb{H}^n} +V)^{−1/2}$ are both bounded from $H^1_L(\mathbb{H}^n)$ into $L^1(\mathbb{H}^n)$. Our results are also valid on the stratified Lie group.

Y. Liu & G. B. Tang. (1970). Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group. Analysis in Theory and Applications. 32 (1). 78-89. doi:10.4208/ata.2016.v32.n1.7
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