In this paper, the weak $L\log L$ estimates for the commutators of a class of Littlewood-Paley operators with real parameter are established by using a technique of the sharp function.
"Endpoint estimates for commutators of a class of Littlewood-Paley operators." Hokkaido Math. J. 36 (2) 245 - 282, May 2007. https://doi.org/10.14492/hokmj/1277472804