$L^p$ 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 $T^{*}f$ 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 $L^p(ω)$ 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.

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