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In this article, via first establishing a Calder\\'{o}n-Zygmund decomposition and a discrete Calder\\'{o}n reproducing formula, the authors then characterize $H_{\\vec{a}}^{\\vec{p}}(\\mathbb{R}^n)$, respectively, by means of atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\\lambda}^\\ast$-function. 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