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Frontiers of Mathematics in China

, Volume 12, Issue 4, pp 993–1022 | Cite as

Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

  • Hui Zhang
  • Chunyan Qi
  • Baode LiEmail author
Research Article

Abstract

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A k : k ∈ ℤ}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ| > 1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let φ: ℝ n ×[0,∞) → [0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type H A φ,∞ (ℝ n ) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to H A φ,∞ (ℝ n ) and the boundedness of the anisotropic Calderón-Zygmund operator from H A φ,∞ (ℝ n ) to L A φ,∞ (ℝ n ). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of H A φ,∞ (ℝ n ) and the interpolation theorem.

Keywords

Expansive dilation Muckenhoupt weight weak Hardy space Musielak-Orlicz function atomic decomposition Calderón-Zygmund operator 

MSC

42B35 46E30 42B25 42B30 42B20 

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Notes

Acknowledgements

The authors would like to express their deep gratitude to the referees for their valuable comments and suggestions. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11461065, 11661075) and a Cultivate Project for Young Doctor from Xinjiang Uyghur Autonomous Region (No. qn2015bs003).

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Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.College of Mathematics and System ScienceXinjiang UniversityUrumqiChina

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