  This paper continues the study, initiated in the works {MOV} and {MOPV}, of
the problem of controlling the maximal singular integral $T^{*}f$ 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 $T^{*}f$ 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.
