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

Copyright © 2020 by Academia Scientiarum Fennica