Let q ∈ (0, ∞] and φ be a Musielak–Orlicz function with uniformly lower type p φ − ∈ (0, ∞) and uniformly upper type p φ + ∈ (0, ∞). In this article, the authors establish various real-variable characterizations of the Musielak–Orlicz–Lorentz Hardy space Hφ,q(ℝn), respectively, in terms of various maximal functions, finite atoms, and various Littlewood–Paley functions. As applications, the authors obtain the dual space of Hφ,q(ℝn) and the summability of Fourier transforms from Hφ,q(ℝn) to the Musielak–Orlicz–Lorentz space Lφ,q(ℝn) when q ∈ (0, ∞) or from the Musielak–Orlicz Hardy space Hφ(ℝn) to Lφ,q(ℝn) in the critical case. These results are new when q ∈ (0, ∞) and also essentially improve the existing corresponding results (if any) in the case q = ∞ via removing the original assumption that φ is concave. To overcome the essential obstacles caused by both that φ may not be concave and that the boundedness of the powered Hardy–Littlewood maximal operator on associated spaces of Musielak–Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of Hφ,q(ℝn), the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak–Orlicz spaces and weighted Lebesgue spaces.
