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Frontiers of Mathematics in China

Front. Math. China    2017, Vol. 12 Issue (4) : 993-1022     DOI: 10.1007/s11464-016-0546-7
Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications
Hui ZHANG, Chunyan QI, Baode LI()
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China
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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 {Ak : 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: Rn×[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 HAφ,(?n) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to HAφ,(?n) and the boundedness of the anisotropic Calderón-Zygmund operator from HAφ,(?n) to Lφ,(?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 HAφ,(?n) and the interpolation theorem.

Keywords Expansive dilation      Muckenhoupt weight      weak Hardy space      Musielak-Orlicz function      atomic decomposition      Calderón-Zygmund operator     
Corresponding Authors: Baode LI   
Issue Date: 06 July 2017
 Cite this article:   
Hui ZHANG,Chunyan QI,Baode LI. Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications[J]. Front. Math. China, 2017, 12(4): 993-1022.
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Chunyan QI
Baode LI
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