Weighted Hardy spaces and BMO spaces associated with Schrödinger operators

In this paper, we characterize the weighted Hardy space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{\mathcal {L}}^{1}(\omega )$\end{document} related to the Schrödinger operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}=-\Delta +V$\end{document} , with V a non‐negative potential satisfying a reverse Hölder inequality, by atomic decomposition and Riesz transforms. We also get a characterization of its dual space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B\hspace*{-1.0pt}M\hspace*{-1.0pt}O_{\mathcal {L}}(\omega )$\end{document} through a weighted Carleson measure. Then we prove the boundedness of some classical operators associated to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}$\end{document} on the weighted BMO space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B\hspace*{-1.0pt}M\hspace*{-1.0pt}O_{\mathcal {L}}(\omega )$\end{document} .