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
Let \((\mathbb X,d,\mu)\) be a space of homogeneous type in the sense of \textit{R. R. Coifman} and \textit{G. Weiss} [Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. (Non-commutative harmonic analysis on certain homogeneous spaces. Study of certain singular integrals.). Springer, Cham (1971; Zbl 0224.43006)], and \(X(\mathbb X)\) a ball quasi-Banach function space on \(\mathbb X\). In this paper under review, the authors introduce and study the weak Hardy space on spaces of homogeneous type in the sense of Coifman and Weiss. They first introduce the weak Hardy space \(\widetilde{WH}_X(\mathbb X)\) associated with \(X(\mathbb X)\) by using the Lusin area function. Then the molecule characterization, the grand maximal function characterization, and the Littlewood-Paley \(g\)-function and \(g_\lambda^\ast\)-function characterization of the weak Hardy space are obtained in this paper. As applications, they apply the above results to the Lebesgue space, the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space. We remark that the main results of this paper are also new in the weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces.
scientific article; zbMATH DE number 7612316
scientific article