2021/11/16 by Hormozi, Mahdi, Sawano, Yoshihiro, Yabuta, Kozo
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2111.08241
In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author's paper \citeSXY 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. %∫01 (φ(t))/(t)dt<∞. This solves an open problem (see e.g. \cite[pp. 37--38]CY). Our method can be applied to the multilinear case.