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Real-Variable Theory of Local Variable Hardy Spaces

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Abstract

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterization in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show the boundedness of inhomogeneous Calderón-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.

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Acknowledgements

The author wishes to express his heartfelt thanks to the anonymous reviewers for their carefully reading and so valuable comments which significantly improve the quality of the paper.

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Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Jian Tan

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Jian Tan

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  1. Jian Tan
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Correspondence to Jian Tan.

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Supported by National Natural Science Foundation of China (Grant No. 11901309), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180734), Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 18KJB110022) and Natural Science Foundation of Nanjing University of Posts and Telecommunications (Grant Nos. NY222168, NY219114)

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Tan, J. Real-Variable Theory of Local Variable Hardy Spaces. Acta. Math. Sin.-English Ser. 39, 1229–1262 (2023). https://doi.org/10.1007/s10114-023-1524-0

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  • DOI: https://doi.org/10.1007/s10114-023-1524-0

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