Haar basis testing

TL;DR

This paper establishes a characterization of the operator norm of fractional Riesz transforms using Haar testing conditions on doubling measures, extending to Alpert wavelets and Hilbert space bases.

Contribution

The paper introduces a new Haar testing characterization for fractional Riesz transforms, extending previous results to weighted Alpert wavelets and general Hilbert space bases.

Findings

Operator norm characterized by Haar testing conditions

Extension to weighted Alpert wavelets in two-weight T1 theorems

Discussion of orthonormal bases in arbitrary Hilbert spaces

Abstract

We show that for two doubling measures $\sigma$ and $\omega$ on $\mathbb{R}^{n}$ and any fixed dyadic grid $\mathcal{D}$ in $\mathbb{R}^{n}$, \[ \mathfrak{N}_{\mathbf{R}^{\lambda, n}}\left( \sigma,\omega\right) \approx\mathfrak{H}_{\mathbf{R}^{\lambda, n}}^{\mathcal{D},\operatorname*{glob}}\left( \sigma,\omega\right) +\mathfrak{H}_{\mathbf{R}^{\lambda, n}}^{\mathcal{D},\operatorname*{glob}}\left( \omega, \sigma\right) \ , \] where $\mathfrak{N}_{\mathbf{R}^{\lambda, n}} (\sigma, \omega)$ denotes the $L^2 (\sigma) \to L^2 (\omega)$ operator norm of the vector-Riesz transform $\mathbf{R}^{\lambda, n}$ of fractional order $\lambda \neq 1$, and \[ \mathfrak{H}_{\mathbf{R}^{\lambda,n}}^{\mathcal{D},\operatorname*{glob}}\left( \sigma,\omega\right) \equiv\sup_{I\in\mathcal{D}}\left\Vert \mathbf{R}^{\lambda,n} h_{I}^{\sigma}\right\Vert _{L^{2}\left( \omega\right) }\ , \] is the global Haar testing characteristic for $\mathbf{R}^{\lambda,n}$ on the grid $\mathcal{D}$, and $\left\{ h_{I}^{\sigma}\right\} _{I\in\mathcal{D}}$ is the weighted Haar orthonormal basis of $L^{2}\left( \sigma\right) $ arising in the work of Nazarov, Treil and Volberg. We also show this theorem extends more generally to weighted Alpert wavelets which replace the weighted Haar wavelets in the proofs of some recent two-weight $T1$ theorems. Finally, we briefly pose these questions in the context of orthonormal bases in arbitrary Hilbert spaces.