  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.

Classical Analysis and ODEs