Anisotropic Hardy-Lorentz Spaces and Their Applications

Abstract

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H^{p,q}_A(\mathbb{R}^n)$ associated with $A$ via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the $\infty$-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on $\mathbb{R}^n$. As applications, the authors first prove that $H^{p,q}_A(\mathbb{R}^n)$ is an intermediate space between $H^{p_1,q_1}_A(\mathbb{R}^n)$ and $H^{p_2,q_2}_A(\mathbb{R}^n)$ with $0<p_1<p<p_2<\infty$ and $q_1,\,q,\,q_2\in(0,\infty]$, and also between $H^{p,q_1}_A(\mathbb{R}^n)$ and $H^{p,q_2}_A(\mathbb{R}^n)$ with $p\in(0,\infty)$ and $0<q_1<q<q_2\leq\infty$ in the real method of interpolation. The authors then establish a criterion on the boundedness of sublinear operators from $H^{p,q}_A(\mathbb{R}^n)$ into a quasi-Banach space; moreover, the authors obtain the boundedness of $\delta$-type Calder\'{o}n-Zygmund operators from $H^p_A(\mathbb{R}^n)$ to the weak Lebesgue space $L^{p,\infty}(\mathbb{R}^n)$ (or $H^{p,\infty}_A(\mathbb{R}^n)$) in the critical case, from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,q}(\mathbb{R}^n)$ (or $H_A^{p,q}(\mathbb{R}^n)$) with $\delta\in(0,\frac{\ln\lambda_-}{\ln b}]$, $p\in(\frac1{1+\delta},1]$ and $q\in(0,\infty]$, as well as the boundedness of some Calder\'{o}n-Zygmund operators from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,\infty}(\mathbb{R}^n)$, where $b:=|\det A|$, $\lambda_-:=\min\{|\lambda|:\ \lambda\in\sigma(A)\}$ and $\sigma(A)$ denotes the set of all eigenvalues of $A$.