On the Haar Shift representations of Calder\'on-Zygmund Operators

TL;DR

This paper demonstrates that Haar shift representations, as developed by Hyt"onen for Calderón-Zygmund operators, are only applicable to these operators and cannot be extended to more general classes.

Contribution

It proves the sharpness of Hyt"onen's Haar shift representation, establishing its limitations to Calderón-Zygmund operators only.

Findings

Haar shift representations are exclusive to Calderón-Zygmund operators.

The result confirms the optimality of Hyt"onen's representation.

The paper clarifies the scope of Haar shift techniques in harmonic analysis.

Abstract

In connection with proving the A_2 conjecture in 2010, T. Hyt\"onen obtained a representation of general Cald\'eron-Zygmund operators in terms of simpler operators known as Haar shifts. In this note, we prove that the result is sharp in the sense that Haar shift representations of Hyt\"onen's type are only available for Calder\'on-Zygmund operators.