Intrinsic Square Function Characterizations of Several Hardy-Type Spaces — A Survey

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doi:10.4208/ata.2021.lu80.09

ATA

2021

DOI: 10.4208/ata.2021.lu80.09

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Intrinsic Square Function Characterizations of Several Hardy-Type Spaces — A Survey

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Abstract: In this article, the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardytype spaces, including (weak) Musielak-Orlicz Hardy spaces, variable (weak) Hardy spaces, and Hardy spaces associated with ball quasi-Banach function spaces. The authors also present some open problems.

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Cited by 9 publications

References 49 publications

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Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Yan

Yang

et al. 2023

Mathematische Nachrichten

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Let false(scriptX,ρ,μfalse)$({\mathcal {X}},\rho ,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Yfalse(scriptXfalse)$Y({\mathcal {X}})$ be a ball quasi‐Banach function space on X${\mathcal {X}}$, which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HY∗(X)$H_{Y}^*({\mathcal {X}})$, associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$, via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of HY∗(X)$H_{Y}^*({\mathcal {X}})$, which proves to be a ball Campanato‐type function space associated with Yfalse(scriptXfalse)$Y({\mathcal {X}})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of X${\mathcal {X}}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of HY∗(X)$H_{Y}^*({\mathcal {X}})$, the authors further prove that Yfalse(scriptXfalse)$Y({\mathcal {X}})$ can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

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Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Yan

Yang

et al. 2023

Mathematische Nachrichten

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Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Yan

Yang

et al. 2022

Acta. Math. Sin.-English Ser.

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Some applications of the dual spaces of Hardy-amalgam spaces

Ablé

Feuto

2022

Acta Math. Hungar.

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Intrinsic Square Function Characterizations of Several Hardy-Type Spaces — A Survey

Cited by 9 publications

References 49 publications

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Some applications of the dual spaces of Hardy-amalgam spaces

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