Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications

Abstract

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

Keywords

• anisotropic
• mixed-norm
• herz space
• herz-hardy spaces
• hardy-littlewood maximal operator
• 42b35
• 42b25
• 42b20