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On functions of bounded mean oscillation with bounded negative part

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Abstract

Let \(b\) be a locally integrable function and \(\mathfrak{M}\) be the bilinear maximal function

$$\mathfrak{M}(f,g)(x)=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f(y)g(2x-y)|dy.$$

In this paper, characterization of the BMO function in terms of commutator \(\mathfrak{M}^{(1)}_{b}\) is established. Also, we obtain the necessary and sufficient conditions for the boundedness of the commutator \([b, \mathfrak{M}]_{1}\). Moreover, some new characterizations of Lipschitz and non-negative Lipschitz functions are obtained.

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Acknowledgement

I would like to thank the referee for the helpful comment which improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    H. Zhao

  2. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241000, People’s Republic of China

    D. Wang

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  1. H. Zhao
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  2. D. Wang
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Correspondence to D. Wang.

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This work was supported by National Natural Science Foundation of China (No. 12101010) and Natural Science Foundation of China of Anhui Province (No. 2108085QA19).

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Zhao, H., Wang, D. On functions of bounded mean oscillation with bounded negative part. Anal Math (2024). https://doi.org/10.1007/s10476-024-00018-9

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