On the Weak Boundedness of Multilinear Littlewood-Paley Functions

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.