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

TL;DR

This paper extends the understanding of how the maximal singular integral operator can be controlled by the singular integral itself in weighted L^p spaces for p different from 2, using pointwise and norm estimates.

Contribution

It introduces new weighted L^p estimates for the maximal singular integral in terms of the singular integral, generalizing previous results to p ≠ 2.

Findings

Established weighted L^p estimates for T* in terms of Tf

Extended control results to p ≠ 2 in weighted spaces

Provided pointwise bounds involving maximal functions

Abstract

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.