Let $\varphi: \mathbb{ R}^{ n}\times[0,\infty)\rightarrow[0,\infty)$ be a Musielak-Orlicz function satisfying the uniformly anisotropic Muckenhoupt condition and be of uniformly lower type $p^-_{ \varphi}$ and of uniformly upper type $p^+_{ \varphi}$ with $0<p^-_{ \varphi}\leq p^+_{ \varphi}<\infty$, $q\in(0,\infty]$, and $A$ be a general expansive matrix on $\mathbb{ R}^{ n}$. In this article, the authors first introduce the anisotropic Musielak-Orlicz-Lorentz Hardy space $H^{ \varphi,q}_A(\mathbb{ R}^{ n})$ which, when $q=\infty$, coincides with the known anisotropic weak Musielak-Orlicz Hardy space $H^{ \varphi,\infty}_A(\mathbb{ R}^{ n})$, and then establish atomic and molecular characterizations of $H^{ \varphi,q}_A(\mathbb{ R}^{ n})$. As applications, the authors prove the boundedness of anisotropic Calder\'{ o}n-Zygmund operators on $H^{ \varphi,q}_A(\mathbb{ R}^{ n})$ when $q\in(0,\infty)$ or from the anisotropic Musielak-Orlicz Hardy space $H^{ \varphi}_A(\mathbb{ R}^{ n})$ to $H^{ \varphi,\infty}_A(\mathbb{ R}^{ n})$ in the critical case. The ranges of all the exponents under consideration are the best possible admissible ones which particularly improve all the known corresponding results for $H^{ \varphi,\infty}_A(\mathbb{ R}^{ n})$ via widening the original assumption $0<p^-_{ \varphi}\leq p^+_{ \varphi}\leq1$ into the full range $0<p^-_{ \varphi}\leq p^+_{ \varphi}<\infty$, and all the results when $q\in(0,\infty)$ are new and generalized from isotropic setting to anisotropic setting.
Categories: math.CA math.FA