Haar basis testing

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.