Some remarks on the maximally modulated Calder\'on-Zygmund operator satisfying $L^r$-H\"ormander condition

In this work, by recent work of Lerner and Ombrasi (J. Geom. Anal. 30(1): 1011-1027, 2020), we show a maximally modulated singular integral operator which its kernel satisfying $L^r$- H\"ormander condition can be dominated by sparse operators. Also, the local exponential decay estimates for these operators are obtained.