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 Calder\'on-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 $HK_{\vec{q}, \vec{a}}^{\alpha, p}(\mathbb R^n)$, on which atomic decomposition and molecular decomposition are obtained. Moreover, we gain the boundedness of classical Calder\'on-Zygmund operators.