Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 343-410

REAL-VARIABLE CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE

Xing Fu, Tao Ma and Dachun Yang

Hubei University, Faculty of Mathematics and Statistics
Hubei Key Laboratory of Applied Mathematics
Wuhan 430062, P.R. China; xingfu 'at' hubu.edu.cn

Wuhan University, School of Mathematics and Statistics
Wuhan 430072, P.R. China; tma.math 'at' whu.edu.cn

Beijing Normal University, School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems (Ministry of Education of China)
Beijing 100875, P.R. China; dcyang 'at' bnu.edu.cn

Abstract. Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak–Orlicz Hardy spaces on (X,d,μ). To be precise, the authors first introduce the atomic Musielak–Orlicz Hardy space H^φ_at(X) and then establish its various maximal function characterizations. The authors also investigate the Littlewood–Paley characterizations of H^φ_at(X) via Lusin area functions, Littlewood–Paley g-functions and Littlewood–Paley g_λ^*-functions. The authors further obtain the finite atomic characterization of H^φ_at(X) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H^φ_at(X) to a quasi-Banach space, which are also applied to the boundedness of Calderón–Zygmund operators. Moreover, the authors find the dual space of H^φ_at(X), namely, the Musielak–Orlicz BMO space BMOφ(X), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X). The main novelty of this article is that, throughout the article, except the last section, μ is not assumed to satisfy the reverse doubling condition.

2010 Mathematics Subject Classification: Primary 42B30; Secondary 42B25, 42B20, 42B35, 30L99.

Key words: Space of homogeneous type, Musielak–Orlicz Hardy space, atom, sublinear operator, pointwise multiplier.

Reference to this article: X. Fu, T. Ma and D. Yang: Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math. 45 (2020), 343-410.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4519