Abstract
To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.
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The authors Jin Tao and Dachun Yang would like to express their sincerely thanks to Professor Loukas Grafakos for his very helpful discussion on the subject of this article.
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Communicated by Loukas Grafakos.
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This project is partially supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197, 11871101 and 11871100).
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Tao, J., Xue, Q., Yang, D. et al. XMO and Weighted Compact Bilinear Commutators. J Fourier Anal Appl 27, 60 (2021). https://doi.org/10.1007/s00041-021-09854-x
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DOI: https://doi.org/10.1007/s00041-021-09854-x