Hardy Spaces Associated with Some Anisotropic Mixed-Norm Herz Spaces and Their Applications

Abstract

In this paper, we introduce anisotropic mixed-norm Herz spaces ${\dot{K}}_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$ and $K_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$ and investigate some basic properties of those spaces. Furthermore, establishing the Rubio de Francia extrapolation theory, which resolves the boundedness problems of Calderón-Zygmund operators and fractional integral operator and their commutators, on the space ${\dot{K}}_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$ and the space $K_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$. Especially, the Littlewood-Paley characterizations of anisotropic mixed-norm Herz spaces also are gained. As the generalization of anisotropic mixed-norm Herz spaces, we introduce anisotropic mixed-norm Herz-Hardy spaces $H{\dot{K}}_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$ and $H{K}_{\vec{q},\vec{a}}^{\alpha ,p}({\mathbb{R}}^n)$, on which atomic decomposition and molecular decomposition are obtained. Moreover, we gain the boundedness of classical Calderón-Zygmund operators.