Lp-estimates for the variation for singular integrals on uniformly rectifiable sets

TL;DR

This paper establishes $L^p$ and weak-$L^1$ estimates for the variation of Calderón-Zygmund operators on uniformly rectifiable sets, advancing understanding of singular integrals in geometric measure theory.

Contribution

It proves $L^p$ and weak-$L^1$ bounds for variation operators on uniformly rectifiable measures, utilizing $L^2$ boundedness and corona decomposition techniques.

Findings

Proves $L^p$ estimates for variation of singular integrals.

Establishes weak-$L^1$ bounds for the variation.

Utilizes corona decomposition in the proof.

Abstract

The $L^p$ ($1<p<\infty$) and weak-$L^1$ estimates for the variation for Calder\'on-Zygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The $L^2$ boundedness and the corona decomposition method are two key ingredients of the proof.