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In this note, notwithstanding the generalization, we simplify and shorten
the proofs of the main results of the third authorÃ¢??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 Ã¢??Marcinkiewicz functionÃ¢?ï¿½. This allows us to get a precise
linear dependence on Dini constants with a subsequent application to LittlewoodÃ¢??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. 37Ã¢??38]). Our method can be applied to the multilinear case.
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