Abstract
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.
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Acknowledgments
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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Communicated by Rodolfo H. Torres.
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Ding, Y., Lee, MY. & Lin, CC. \(\mathcal {A}_{p, {\mathbb {E}}}\) Weights, Maximal Operators, and Hardy Spaces Associated with a Family of General Sets. J Fourier Anal Appl 20, 608–667 (2014). https://doi.org/10.1007/s00041-014-9321-x
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DOI: https://doi.org/10.1007/s00041-014-9321-x