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Maximal Commutator of Conditional Expectations

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Abstract

In the paper, we prove that the maximal commutator of conditional expectations \(\sup _{n\ge 0}|[{\mathbb {E}}_n,g]|\) is bounded from \(H_1^S\) to \(L_{1,\infty }\) provided \(g\in \textrm{BMO}_2\). If \(g\in \textrm{bmo}_2\), we also show that a modified maximal operator \(\sup _{n\ge 0}{\mathbb {E}}_n(|[{\mathbb {E}}_n,g]|)\) is bounded from \(\textrm{h}_1\) to \(L_{1,\infty }\).

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Acknowledgements

The author would like to thank the reviewers for their valuable comments and suggestions.

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  1. School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

    Yanyun Li

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  1. Yanyun Li
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Correspondence to Yanyun Li.

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Communicated by See Keong Lee.

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Li, Y. Maximal Commutator of Conditional Expectations. Bull. Malays. Math. Sci. Soc. 46, 29 (2023). https://doi.org/10.1007/s40840-022-01434-6

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  • DOI: https://doi.org/10.1007/s40840-022-01434-6

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