\(\mathcal {A}_{p, {\mathbb {E}}}\) Weights, Maximal Operators, and Hardy Spaces Associated with a Family of General Sets

  • Yong DingEmail author
  • Ming-Yi Lee
  • Chin-Cheng Lin


Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\), \(1<p\le \infty \), for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\). As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\), and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\). Our results generalize several well-known results.


\(A_p\) weights BMO Hardy spaces maximal operator 

Mathematics Subject Classification

Primary 42B25 42B30 Secondary 35B45 



The authors would like to express their deep gratitude to the referees for their very careful reading, important comments and valuable suggestions. The first author was supported by NSFC (No: 11371057), SRFDP (No: 20130003110003), the Fundamental Research Funds for the Central Universities (No: 2012CXQT09) and NSF of Zhejiang Province (No: LY12A01011). The second and third authors were supported by NSC of Taiwan under Grant #NSC 102-2115-M-008-006 and Grant #NSC 100-2115-M-008-002-MY3, respectively.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijing Normal UniversityBeijingChina
  2. 2.Department of MathematicsNational Central UniversityChung-LiTaiwan

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