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A new vanishing Lipschitz-type subspace of BMO and compactness of bilinear commutators

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Abstract

A new vanishing Lipschitz-type space called \({\mathrm {XMO}}_\alpha \,(\mathbb {R}^n)\) is introduced when \(\alpha \in [0,1]\). This space coincides with the space \({\mathrm {XMO}}\,(\mathbb {R}^n)\) of Torres and Xue (Rev Mat Iberoam 36, 939–956, 2020) when \(\alpha =0\). A characterization of \({\mathrm {XMO}}_\alpha \,(\mathbb {R}^n)\) via limit properties of mean oscillations is established in this work. The proof relies on a carefully-constructed approximation function of multilinear type, on fine geometric properties of dyadic cubes of \(\mathbb {R}^n\), and on subtle estimates of higher-order derivatives. A second result of this article concerns the compactness of commutators of bilinear Calderón–Zygmund operators with functions in \({\mathrm {XMO}}_\alpha \,(\mathbb {R}^n)\).

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Acknowledgements

The authors would like to thank the referee for her/his carefully reading and several motivating remarks which indeed improve the presentation of this article.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Jin Tao & Dachun Yang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Dongyong Yang

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  1. Jin Tao
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Correspondence to Dachun Yang.

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Communicated by Loukas Grafakos.

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Jin Tao and Dachun Yang are supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058 and 12071197). Dongyong Yang is supported by the National Natural Science Foundation of China (Grant No. 11971402) and Fundamental Research Funds for Central Universities of China (Grant No. 20720210031)

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Tao, J., Yang, D. & Yang, D. A new vanishing Lipschitz-type subspace of BMO and compactness of bilinear commutators. Math. Ann. 386, 495–531 (2023). https://doi.org/10.1007/s00208-022-02402-y

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  • DOI: https://doi.org/10.1007/s00208-022-02402-y

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