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Applications of Hardy spaces associated with ball quasi-Banach function spaces. (English) Zbl 1431.42040

Summary: Let \(X\) be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of \(H_X(\mathbb{R}^n)\), the Hardy space associated with \(X\), via the Littlewood-Paley \(g\)-functions and \(g_\lambda^\ast\)-functions. Moreover, the authors obtain the boundedness of Calderón-Zygmund operators on \(H_X(\mathbb{R}^n)\). For the local Hardy-type space \(h_X(\mathbb{R}^n)\) associated with \(X\), the authors also obtain the boundedness of \(S^0_{1,0}(\mathbb{R}^n)\) pseudo-differential operators on \(h_X(\mathbb{R}^n)\) via first establishing the atomic characterization of \(h_X(\mathbb{R}^n)\). Furthermore, the characterizations of \(h_X(\mathbb{R}^n)\) 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 \(g_\lambda^\ast\)-function characterization obtained in this article improves the known results via widening the range of \(\lambda \).

MSC:

42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G30 Pseudodifferential operators
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