LP estimates for the maximal singular integral in terms of the singular integral

This paper continues the study, initiated in [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 Calderón-Zygmund singular integral operator of convolution type. We consider two forms of control, namely, in the weighted Lp(ω) norm and via pointwise estimates of T* f by M(Tf ) or M2(Tf), where M is the Hardy-Littlewood maximal operator and M2 = M po M its iteration. The novelty with respect to the aforementioned works lies in the fact that here p is different from 2 and the Lp space is weighted.