Real-variable Theory of Anisotropic Musielak-Orlicz-Lorentz Hardy Spaces with Applications to Calder\'{o}n-Zygmund Operators

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.