The Sparse T1 Theorem

TL;DR

This paper establishes a simple and elementary sparse bound for Calderón-Zygmund operators under standard assumptions, leading to straightforward derivations of their boundedness on weighted L^p spaces.

Contribution

It introduces a short, elementary proof of a sparse bound for Calderón-Zygmund operators under standard T1 assumptions, simplifying the derivation of their L^p mapping properties.

Findings

Sparse bilinear form bounds Calderón-Zygmund operators

Boundedness on weighted L^p spaces follows from sparse bounds

Proof is short and elementary

Abstract

We impose standard $ T1 $-type assumptions on a Calder\'on-Zygmund operator $ T $, and deduce that for bounded compactly supported functions $ f, g $ there is a sparse bilinear form $ \Lambda $ so that $$ \lvert \langle T f, g \rangle\rvert \lesssim \Lambda (f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calder\'on-Zygmund on a (weighted) $ L ^{p}$ space.