Let X be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document}, the Hardy space associated with X, via the Littlewood–Paley g-functions and gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_\lambda ^*$$\end{document}-functions. Moreover, the authors obtain the boundedness of Calderón–Zygmund operators on HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document}. For the local Hardy-type space hX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$h_X(\mathbb {R}^n)$$\end{document} associated with X, the authors also obtain the boundedness of S1,00(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$S^0_{1,0}(\mathbb {R}^n)$$\end{document} pseudo-differential operators on hX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$h_X(\mathbb {R}^n)$$\end{document} via first establishing the atomic characterization of hX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$h_X(\mathbb {R}^n)$$\end{document}. Furthermore, the characterizations of hX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$h_X(\mathbb {R}^n)$$\end{document} by means of local molecules and local Littlewood–Paley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz–Hardy space, the Lorentz–Hardy space, the Morrey–Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_\lambda ^*$$\end{document}-function characterization obtained in this article improves the known results via widening the range of λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}.
