Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

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Let a:=(a₁,,aₙ)∈[1,∞)ⁿ, p:=(p₁,,pₙ)∈(0,∞)ⁿ and Hap(Rⁿ) be the anisotropic mixed-norm Hardy space associated with a defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón-Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize Hap(Rⁿ), respectively, by means of atoms, the Lusin area function, the Littlewood-Paley g-function or gλ-function. The obtained Littlewood-Paley g-function characterization of Hap(Rⁿ) coincidentally confirms a conjecture proposed by Hart et al. [Trans. Amer. Math. Soc. (2017), DOI: 10.1090/tran/7312]. Applying the aforementioned Calderón-Zygmund decomposition as well as the atomic characterization of Hap(Rⁿ), the authors establish a finite atomic characterization of Hap(Rⁿ), which further induces a criterion on the boundedness of sublinear operators from Hap(Rⁿ) into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from Hap(Rⁿ) to itself [or to Lp(Rⁿ)]. The obtained atomic characterizations of Hap(Rⁿ) and boundedness of anisotropic Calderón-Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. in [J. Geom. Anal. 27 (2017), 2758-2787]. All these results are new even for the isotropic mixed-norm Hardy spaces on Rⁿ.