Let X be a ball quasi-Banach function space satisfying some mild assumptions and HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document} the Hardy space associated with X. In this article, the authors introduce both the Hardy space HX(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^{n+1}_+)$$\end{document} of harmonic functions and the Hardy space HX(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}_X(\mathbb {R}^{n+1}_+)$$\end{document} of harmonic vectors, associated with X, and then establish the isomorphisms among HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document}, HX,2(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$H_{X,2}(\mathbb {R}^{n+1}_+)$$\end{document}, and HX,2(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}_{X,2}(\mathbb {R}^{n+1}_+)$$\end{document}, where HX,2(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$H_{X,2}(\mathbb {R}^{n+1}_+)$$\end{document} and HX,2(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}_{X,2}(\mathbb {R}^{n+1}_+)$$\end{document} are, respectively, certain subspaces of HX(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^{n+1}_+)$$\end{document} and HX(R+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {H}_X(\mathbb {R}^{n+1}_+)$$\end{document}. Using these isomorphisms, the authors establish the first order Riesz transform characterization of HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document}. The higher order Riesz transform characterization of HX(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$H_X(\mathbb {R}^n)$$\end{document} is also obtained. The results obtained in this article have a wide range of generality and can be applied to classical Hardy spaces, weighted Hardy spaces, variable Hardy spaces, Herz–Hardy spaces, Lorentz–Hardy spaces, mixed-norm Hardy spaces, local generalized Herz–Hardy spaces, and mixed-norm Herz–Hardy spaces and all the obtained results on the aforementioned last five Hardy-type spaces are completely new.
