We consider the Schrödinger operator \({\mathcal {L}}=-\Delta +V\) on \({\mathbb {R}}^n,\) where \( n\ge 3\) and the nonnegative potential V belongs to reverse Hölder class \(RH_{s}\) for \(s>\frac{n}{2}.\) Let \(H^p_{\mathcal {L}}({\mathbb {R}}^n)\) denote the Hardy space related to \({\mathcal {L}}\) and \(BMO_{{\mathcal {L}}}({\mathbb {R}}^n)\) denote the dual space of \(H^1_{{\mathcal {L}}}({\mathbb {R}}^n).\) In this paper, we consider the operator \(T_{\alpha ,\beta }=V^\alpha {\mathcal {L}}^{-\beta }\) and its adjoint operator \(T^*_{\alpha ,\beta },\) where \(0<\alpha \le \beta < n/2.\) We show that \(T_{\alpha ,\beta }\) is bounded from \(H^{p_1}_{{\mathcal {L}}}({\mathbb {R}}^n)\) into \(L^{p_2}({\mathbb {R}}^n)\) for \(\frac{n}{n+\delta '}<p_1\le 1\) and \(\frac{1}{p_2}=\frac{1}{p_1}-\frac{2(\beta -\alpha )}{n},\) where \(\delta '=\min \{1, 2-n/q_0\},\) and \(q_0\) is the reverse Hölder index of V. Moreover, we prove that \(T^*_{\alpha ,\beta }\) is also bounded from \(L^{p_1}({\mathbb {R}}^n)\) into \(BMO_{{\mathcal {L}}}({\mathbb {R}}^n)\) for \(p_1=\frac{n}{2(\beta -\alpha )}.\)

我们考虑Schrödinger算子\（{\ mathcal {L}} =-\ Delta + V \）在\（{\ mathbb {R}} ^ n，\）上，其中\（n \ ge 3 \）和非负势V属于反向holder类\（RH_ {S} \）为\（S> \压裂{N} {2}。\）让\（H ^ p _ {\ mathcal {L}}（{\ mathbb {R} } ^ N）\）表示相关的哈迪空间\（{\ mathcal {L}} \）和\（BMO _ {{\ mathcal {L}}}（{\ mathbb {R}} ^ N）\）分别表示\（H ^ 1 _ {{\ mathcal {L}}}（{\ mathbb {R}} ^ n）。\）的对偶空间在本文中，我们考虑算子\（T _ {\ alpha，\ beta} = V ^ \ alpha {\ mathcal {L}} ^ {-\ beta} \）及其伴随运算符\（T ^ * _ {\α，\测试}，\）其中\（0 <\阿尔法\文件\测试<N / 2。\）我们表明，\（T _ {\α，\测试} \）是从\（H ^ {p_1} _ {{\ mathcal {L}}}（{\ mathbb {R}} ^ n）\）到\（L ^ {p_2}（{\ mathbb {R}} ^ n ）\）为\（\压裂{N} {N + \增量“} <P_1 \文件1 \）和\（\压裂{1} {P_2} = \压裂{1} {P_1} - \压裂{2（ \ beta-\ alpha}} {n}，\）其中\（\ delta'= \ min \ {1，2-n / q_0 \}，\）和\（q_0 \）是V的反向Hölder索引。此外，我们证明\（T ^ * _ {\ alpha，\ beta} \）也从\（L ^ {p_1}（{\ mathbb {R}} ^ n）\）到\（BMO _ {{ \ mathcal {L}}}（{\ mathbb {R}} ^ n）\）为\（p_1 = \ frac {n} {2（\ beta-\ alpha}}。\）