New estimates for the maximal singular integral

TL;DR

This paper investigates how the maximal singular integral can be controlled by the singular integral itself, revealing fundamental differences between even and odd kernels, and providing new estimates in various function spaces.

Contribution

It extends the understanding of maximal singular integrals by analyzing the odd kernel case and establishing new control estimates, including weak (1,1) bounds for even kernels.

Findings

Weak (1,1) estimates for even kernels.

Sharp inequalities involving $L \, LogL$ for odd kernels.

Fundamental differences between even and odd kernels in control estimates.

Abstract

In this paper we pursue the study 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 of convolution type. We consider two forms of control, namely, in the $L^2(\Rn)$ 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. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.