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 \(0<p^-_{\varphi }\le p^+_{\varphi }<\infty \) , \(q\in (0,\infty ]\) , and A be a general expansive matrix on \(\mathbb {R}^{n}\) . In this paper, the authors first study the anisotropic Musielak-Orlicz-Lorentz Hardy space \(H^{\varphi ,q}_A(\mathbb {R}^{n})\) , which coincides with the known anisotropic weak Musielak-Orlicz Hardy space \(H^{\varphi ,\infty }_A(\mathbb {R}^{n})\) when \(q=\infty \) , and then establish their atomic and molecular decompositions. As some applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators on \(H^{\varphi ,q}_A(\mathbb {R}^{n})\) when \(q\in (0,\infty )\) and 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. All the ranges of the exponents considered are shown to be the best possible, significantly improving upon existing results for \(H^{\varphi ,\infty }_A(\mathbb {R}^{n})\) via widening the original assumption \(0<p^-_{\varphi }\le p^+_{\varphi }\le 1\) into the full range \(0<p^-_{\varphi }\le p^+_{\varphi }<\infty \) , and all the results for \(q\in (0,\infty )\) are novel and generalize from isotropic setting to anisotropic frameworks.
