Abstract
In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author’s paper (Shi et al in J Math Pures Appl 101:394–413, 2014) significantly. In particular, the new proof for Shi et al (J Math Pures Appl 101:394–413, 2014, Theorem 1.1) is quite short and, unlike the original proof, does not rely on the properties of the “Marcinkiewicz function”. This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewood–Paley operators by well-known techniques. In other words, we relax the log-Dini condition in the pointwise bound to the classical Dini condition. This solves an open problem (see e.g. Cao and Yabuta in J Fourier Anal Appl 25(3):1203–1247, 2019, pp. 37–38). Our method can be applied to the multilinear case.
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The authors are grateful to the anonymous referees for their careful reading of the paper and for making several valuable comments which have improved the quality of this paper.
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Communicated by Dachun Yang.
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M. Hormozi is supported by a grant from IPM. Y. Sawano was supported by Grant-in-Aid for Scientific Research (C) (19K03546), the Japan Society for the Promotion of Science and People’s Friendship University of Russia.
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Hormozi, M., Sawano, Y. & Yabuta, K. On the Weak Boundedness of Multilinear Littlewood–Paley Functions. J Fourier Anal Appl 29, 49 (2023). https://doi.org/10.1007/s00041-023-10030-6
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DOI: https://doi.org/10.1007/s00041-023-10030-6