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

Abstract

Let $$\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n$$ , $$\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n$$ and $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ be the anisotropic mixed-norm Hardy space associated with $$\vec {a}$$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderon–Zygmund decomposition and a discrete Calderon reproducing formula, the authors then characterize $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , respectively, by means of atoms, the Lusin area function, the Littlewood–Paley g-function or $$g_{\lambda }^*$$ -function. The obtained Littlewood–Paley g-function characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ coincidentally confirms a conjecture proposed by Hart et al. (Trans Am Math Soc, https://doi.org/10.1090/tran/7312 , 2017). Applying the aforementioned Calderon–Zygmund decomposition as well as the atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , the authors establish a finite atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , which further induces a criterion on the boundedness of sublinear operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ to itself [or to the mixed-norm Lebesgue space $$L^{\vec {p}}(\mathbb {R}^n)$$ ]. The obtained atomic characterizations of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ and boundedness of anisotropic Calderon–Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. (J Geom Anal 27:2758–2787, 2017). All these results are new even for the isotropic mixed-norm Hardy spaces on $$\mathbb {R}^n$$ .