The maximal singular integral : estimates in terms of the singular integral

TL;DR

This paper explores conditions under which the maximal singular integral can be controlled by the singular integral in L^2 norm, focusing on special cases like higher Riesz transforms to clarify the main ideas.

Contribution

It provides an expository analysis of the characterization of Calderón-Zygmund operators where the maximal singular integral is dominated by the singular integral, emphasizing special cases for clarity.

Findings

Characterization of even and odd homogeneous Calderón-Zygmund operators

Control of maximal singular integrals by singular integrals in L^2

Focus on higher Riesz transforms for illustrative purposes

Abstract

This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calder\'on-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral. We focuss attention on special cases of the general statements to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms. The exposition is based on two papers by J.Mateu, J.Orobitg, C.P\'erez and J.Verdera published in 2010 and 2011.