Musielak–Orlicz–Lorentz Hardy Spaces: Maximal Function, Finite Atomic, and Littlewood–Paley Characterizations with Applications to Dual Spaces and Summability of Fourier Transforms

Abstract

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 H^φ,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 H^φ,q(ℝⁿ) and the summability of Fourier transforms from H^φ,q(ℝⁿ) to the Musielak–Orlicz–Lorentz space L^φ,q(ℝⁿ) when q ∈ (0, ∞) or from the Musielak–Orlicz Hardy space H^φ(ℝⁿ) to L^φ,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 H^φ,q(ℝⁿ), the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak–Orlicz spaces and weighted Lebesgue spaces.