The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 2961–2983 | Cite as

Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

  • Jun Cao
  • Liguang Liu
  • Dachun Yang
  • Wen Yuan


For any \(p\in (0,\,1]\), let \(H^{\Phi _p}(\mathbb {R}^n)\) be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function \(\Phi _p\), defined by setting, for any \(x\in \mathbb {R}^n\) and \(t\in [0,\,\infty )\),
$$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when}\ n(1/p-1)\notin \mathbb N \cup \{0\},\\ \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when}\ n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$
which is the sharp target space of the bilinear decomposition of the product of the Hardy space \(H^p(\mathbb {R}^n)\) and its dual. Moreover, \(H^{\Phi _1}(\mathbb {R}^n)\) is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space \(H^{\Phi _p}(\mathbb {R}^n)\) by showing that, for any \(p\in (0,\,1]\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\) and, for any \(p\in (0,\,1)\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\), where \(H^1(\mathbb {R}^n)\) denotes the classical real Hardy space, \(H^{\phi _0}({{{\mathbb {R}}}^n})\) the Orlicz–Hardy space associated with the Orlicz function \(\phi _0(t):=t/\log (e+t)\) for any \(t\in [0,\infty )\), and \(H_{W_p}^p(\mathbb {R}^n)\) the weighted Hardy space associated with certain weight function \(W_p(x)\) that is comparable to \(\Phi _p(x,1)\) for any \(x\in \mathbb {R}^n\). As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.


Hardy space Musielak–Orlicz function Muckenhoupt weight Interpolation Atom Calderón–Zygmund decomposition 

Mathematics Subject Classification

Primary 42B30 Secondary 42B35 46E30 46B70 



This project was supported by the National Natural Science Foundation of China (Grant Nos. 11501506, 11771446, 11471042, 11571039, and 11671185). Jun Cao was partially supported by the Natural Science Foundation of Zhejiang University of Technology (Grant No. 2014XZ011).


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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsZhejiang University of TechnologyHangzhouPeople’s Republic of China
  2. 2.Department of Mathematics, School of InformationRenmin University of ChinaBeijingPeople’s Republic of China
  3. 3.Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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