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Paper   IPM / M / 17320
School of Mathematics
  Title:   On the weak boundedness of multilinear Littlewood-Paley functions
  Author(s):  Mahdi Hormozi (Joint with Y. Sawani and K. Yabuta)
  Status:   Published
  Journal: Journal of Fourier Analysis and Applications
  Vol.:  29
  Year:  2023
  Pages:   1-42
  Supported by:  IPM
In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third authorâ??s paper [23] significantly. In particular, the new proof for [23, Theorem 1.1] is quite short and, unlike the original proof, does not rely on the properties of the â??Marcinkiewicz functionâ?�. This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewoodâ??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. [5, pp. 37â??38]). Our method can be applied to the multilinear case.

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