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Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense ofCoifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximalinequality, and the boundedness of the powered Hardy--Littlewood maximaloperator on its associate space. The authors first introduce the Hardy space$H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grandmaximal function, and then establish its various real-variablecharacterizations, respectively, in terms of radial or non-tangential maximalfunctions, atoms or finite atoms, and molecules. As an application, the authorsgive the dual space of $H_{Y}^*({\mathcal X})$, which proves to be a ballCampanato-type function space associated with $Y({\mathcal X})$. All theseresults have a wide range of generality and, particularly, even when they areapplied to variable Hardy spaces, the obtained results are also new. The majornovelties of this article exist in that, to escape the reverse doublingcondition of $\mu$ and the triangle inequality of $\rho$, the authors cleverlyconstruct admissible sequences of balls, and fully use the geometricalproperties of ${\mathcal X}$ expressed by dyadic reference points or dyadiccubes and, to overcome the difficulty caused by the lack of the good densesubset of $H_{Y}^*({\mathcal X})$, the authors further prove that $Y({\mathcalX})$ can be embedded into the weighted Lebesgue space with certain specialweight, and then can fully use the known results of the weighted Lebesguespace.

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces  of Homogeneous Type: Characterizations of Maximal Functions, Decompositions,  and Dual Spaces