Let X be a ball quasi-Banach function space on \({{\mathbb {R}}}^n\). In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space \(H_X({{\mathbb {R}}}^n)\) associated with X, under some weak assumptions on the Littlewood–Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X. All these results have wide applications. Particularly, when \(X:=M_r^p({{\mathbb {R}}}^n)\) (the Morrey space), \(X:=L^{\vec {p}}({{\mathbb {R}}}^n)\) (the mixed-norm Lebesgue space), \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space), \(X:=L_\omega ^p({{\mathbb {R}}}^n)\) (the weighted Lebesgue space) and \(X:=(E_\Phi ^r)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), the Littlewood–Paley function characterizations of \(H_X({{\mathbb {R}}}^n)\) obtained in this article improve the existing results via weakening the assumptions on the Littlewood–Paley functions and widening the range of \(\lambda \) in the Littlewood–Paley \(g_\lambda ^*\)-function characterization of \(H_X({\mathbb {R}}^n)\).

中文翻译：

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与球拟Banach函数空间相关的Hardy型空间的Littlewood-Paley刻画

令X为\（{{\ mathbb {R}}} ^ n \）上的球拟Banach函数空间。在本文中，假设有能力的Hardy-Littlewood最大算子满足X上的某些Fefferman-Stein向量值最大不等式且在相关空间上是有界的，则作者建立了Hardy空间 \（H_X（在Littlewood–Paley函数的一些弱假设下，与X相关联的{{\ mathbb {R}} ^ n）\）。为此，作者还对与X关联的帐篷空间中的角度变化建立了有用的估计。所有这些结果具有广泛的应用。特别是当\（X：= M_r ^ p（{{\ mathbb {R}}} ^ n）\）（Morrey空间），\（X：= L ^ {\ vec {p}}（{{\ mathbb {R}}} ^ n）\）（混合范数Lebesgue空间），\（X：= L ^ {p（\ cdot）}（{{\ mathbb {R}}} ^ n）\）（可变Lebesgue空间），\（X：= L_ \ omega ^ p（{{\ mathbb {R}}} ^ n）\）（加权Lebesgue空间）和\（X：=（E_ \ Phi ^ r）_t（{{\ mathbb {R}}} ^ n）\）（Orlicz切片空间），Littlewood本文获得的– （ H_X （{{\ mathbb {R}} ^ n）\）的–Paley函数特征通过削弱关于Littlewood–Paley函数的假设并扩大\（\ lambda的范围，从而改善了现有结果。 \）在Littlewood–Paley \（g_ \ lambda ^ * \） - \（H_X（{\ mathbb {R}} ^ n）\）的函数表征中。