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 \cite{SXY} significantly. In particular, the new proof for \cite[Theorem 1.1]{SXY} 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. %$\int\limits_{0}^{1} \frac{\varphi(t)}{t}dt<\infty$. This solves an open problem (see e.g. \cite[pp. 37--38]{CY}). Our method can be applied to the multilinear case.