Abstract
For \(1\le p\le \infty \) and \(\alpha >0\), Besov spaces \(B^p_\alpha \) play a key role in the theory of \(\alpha \)-Möbius invariant function spaces. In some sense, \(B^1_\alpha \) is the minimal \(\alpha \)-Möbius invariant function space, \(B^2_\alpha \) is the unique \(\alpha \)-Möbius invariant Hilbert space, and \(B^\infty _\alpha \) is the maximal \(\alpha \)-Möbius invariant function space. In this paper, under the \(\alpha \)-Möbius invariant pairing and by the space \(B^\infty _\alpha \), we identify the predual and dual spaces of \(B^1_\alpha \). In particular, the corresponding identifications are isometric isomorphisms. The duality theorem via the \(\alpha \)-Möbius invariant pairing for \(B^p_\alpha \) with \(p>1\) is also given.
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The authors want to thank Professor Kehe Zhu for interesting discussions on the subject, and the referee for his/her valuable comments.
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Communicated by Mieczyslaw Mastylo.
G. Bao was supported by the National Natural Science Foundation of China (No. 12271328) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012117). Z. Lou and X. Zhou were supported by National Natural Science Foundation of China (No. 12071272).
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Bao, G., Lou, Z. & Zhou, X. Duality for \(\alpha \)-Möbius invariant Besov spaces. Banach J. Math. Anal. 17, 60 (2023). https://doi.org/10.1007/s43037-023-00285-y
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DOI: https://doi.org/10.1007/s43037-023-00285-y