On the Haar shift representations of Calderón-Zygmund operators (Q2839353)

In [Ann. Math. (2) 175, No. 3, 1473--1506 (2012; Zbl 1250.42036)] represented Calderón-Zygmund operators as averages of simplier objects, the so-called Haar shifts that are denoted by \(\mathbb{S}\). The scope of this article is to prove the sharpness of Hytönen's Haar shift representation. Fix \(d \in \mathbb N\) and \(0<\delta< 1\) and set \(\Omega = (\{0,1\}^d)^\mathbb Z\). Let for every \(m,n\in\mathbb{Z}_+\) and \(\omega\in \Omega\) be given a number \(\lambda^\omega_{m,n}\in\mathbb{R}\) such that \(|\lambda^\omega_{m,n}|\leq 2^{-(m+n)\delta}\) and a Haar shift operator \(\mathbb{S}^\omega_{m,n}\) such that \(\|\mathbb{S}^\omega_{m,n}\|_{2\to 2}\leq 1\). Then the linear operator \(S: L^2(\mathbb R^d)\to L^2(\mathbb R^d)\) defined by the formula NEWLINE\[NEWLINE\langle Sf,g\rangle= \mathbb{E}_\omega \sum_{m,n\in\mathbb{Z}_+} \lambda^\omega_{m,n}\langle\mathbb{S}^\omega_{m,n} f,g\rangle \quad (f,g\in L^2(\mathbb R^d))NEWLINE\]NEWLINE is a Calderón-Zygmund operator.