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

Xianjie Yan, Ziyi He, Dachun Yang and Wen Yuan

Mathematische Nachrichten, 2023, vol. 296, issue 7, 3056-3116

Abstract: Let (X,ρ,μ)$({\mathcal {X}},\rho ,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Y(X)$Y({\mathcal {X}})$ 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 Y(X)$Y({\mathcal {X}})$, 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 Y(X)$Y({\mathcal {X}})$. 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 Y(X)$Y({\mathcal {X}})$ 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.

Date: 2023
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