$L^p$ estimates for the maximal singular integral in terms of the singular integral

This paper continues the study, initiated in the works {MOV} and {MOPV}, of the problem of controlling the maximal singular integral $T^{\ast }f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calder'on-Zygmund singular integral operator of convolution type. We consider two forms of control, namely, in the weighted $L^p(\omega )$ norm and via pointwise estimates of $T^{\ast }f$ by $M(Tf)$ or $M^2(Tf)$,, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M\circ M$ its iteration. The novelty with respect to the aforementioned works, lies in the fact that here $p$ is different from 2 and the $L^p$ space is weighted.