Xian Jie YAN, Zi Yi HE, Da Chun YANG, Wen YUAN
Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, 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 $H_{Y}({\mathcal X})$ associated with $Y({\mathcal X})$, via the Lusin-area function, and then establish its various equivalent characterizations, respectively, in terms of atoms, molecules, and Littlewood--Paley $g$-functions and $g_{\lambda}^*$-functions. As an application, the authors obtain the boundedness of Calder\'on--Zygmund operators from $H_{Y}({\mathcal X})$ to $Y({\mathcal X})$, or to $H_{Y}({\mathcal X})$ via first establishing a boundedness criterion of linear operators on $H_{Y}({\mathcal X})$. 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 article exist in that, to escape the reverse doubling condition of $\mu$ and the triangle inequality of $\rho$, the authors subtly use the wavelet reproducing formula, originally establish an admissible molecular characterization of $H_{Y}({\mathcal X})$, and fully apply the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes.
