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Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents

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Abstract

Let \(p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty )\) be a measurable function satisfying some decay condition and some locally log-Hölder continuity. In this article, through first establishing characterizations of the variable exponent Hardy space \(H^{p(\cdot )}(\mathbb R^n)\) in terms of the Littlewood–Paley g-function, the Lusin area function, and the \(g_\lambda ^*\)-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood–Paley g-function, the intrinsic Lusin area function, and the intrinsic \(g_\lambda ^*\)-function. The \(p(\cdot )\)-Carleson measure characterization for the dual space of \(H^{p(\cdot )}(\mathbb R^n)\), the variable exponent Campanato space \(\mathcal {L}_{1,p(\cdot ),s}(\mathbb R^n)\), in terms of the intrinsic function is also presented.

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Acknowledgments

The authors would like to thank the referees for their several valuable remarks which helped to improve the presentation of this article. This project is supported by the National Natural Science Foundation of China (Grant Nos. 11571039 & 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003), and the Fundamental Research Funds for Central Universities of China (Grant No. 2014KJJCA10). The second author, Dachun Yang, is the corresponding author of this article.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Ciqiang Zhuo, Dachun Yang & Yiyu Liang

  2. Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Ciqiang Zhuo, Dachun Yang & Yiyu Liang

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  1. Ciqiang Zhuo
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  2. Dachun Yang
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  3. Yiyu Liang
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Correspondence to Dachun Yang.

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Communicated by Rosihan M. Ali, Dato’.

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Zhuo, C., Yang, D. & Liang, Y. Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents. Bull. Malays. Math. Sci. Soc. 39, 1541–1577 (2016). https://doi.org/10.1007/s40840-015-0266-2

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  • DOI: https://doi.org/10.1007/s40840-015-0266-2

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