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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} }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.