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Let(,d,μ){(\mathbb{X},d,\mu)}be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and letX⁢(){X(\mathbb{X})}be a ball quasi-Banach function space on{\mathbb{X}}. In this article, the authors introduce the weak Hardy spaceW⁢H~X⁢(){\widetilde{WH}_{X}(\mathbb{X})}associated withX⁢(){X(\mathbb{X})}via the Lusin area function. Then the authors characterizeW⁢H~X⁢(){\widetilde{WH}_{X}(\mathbb{X})}by the molecule, the grand maximal function, and the Littlewood–Paley  g -function andgλ*{g^{*}_{\lambda}}-function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of{\mathbb{X}}expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm ofX⁢(){X(\mathbb{X})}and the reverse doubling condition of μ, and that the authors use the tent space on×ℤ{\mathbb{X}\times\mathbb{Z}}to characterizeW⁢H~X⁢(){\widetilde{WH}_{X}(\mathbb{X})}by the Littlewood–Paleygλ*{g^{*}_{\lambda}}-function, where the range of λ might be best possible in some cases.

forum mathematicum

Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations