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Preduals of variable Morrey–Campanato spaces and boundedness of operators

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Abstract

Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (1,\infty )\) be a variable exponent, such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space \(L^{p(\cdot )}({\mathbb {R}}^n),\) and \(\phi :\ {\mathbb {R}}^n\times (0,\infty )\rightarrow (0,\infty )\) be a function satisfying some conditions. In this article, we give some properties of variable Campanato spaces \({\mathcal {L}}_{p(\cdot ),\phi ,d}({\mathbb {R}}^n),\) with a non-negative integer d,  and variable Morrey spaces \(L_{p(\cdot ),\phi }({\mathbb {R}}^n),\) and then establish their predual spaces. As an application of duality obtained in this article, we consider the boundedness of singular integral operators on variable Morrey spaces and their predual spaces.

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Acknowledgements

The author would like to thank the referees for valuable comments and suggestions, which make this paper more readable and, especially, improve the assumptions of Theorem 4.4 of the paper. This project is supported by the Natural Science Foundation of Changsha (Grant No. kq2202237) and Hunan province (Grant No. 2022JJ30371), and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 21A0067).

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  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Ciqiang Zhuo

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Correspondence to Ciqiang Zhuo.

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Communicated by Yong Jiao.

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Zhuo, C. Preduals of variable Morrey–Campanato spaces and boundedness of operators. Ann. Funct. Anal. 14, 76 (2023). https://doi.org/10.1007/s43034-023-00298-6

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