Haar basis testing
Michel Alexis, Jose Luis Luna-Garcia, Eric T. Sawyer

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 and on and any fixed dyadic grid in , \[ \mathfrak{N}_{\mathbf{R}^{\lambda, n}}\left( \sigma,\omega\right) \approx\mathfrak{H}_{\mathbf{R}^{\lambda, n}}^{\mathcal{D},\operatorname*{glob}}\left( \sigma,\omega\right) +\mathfrak{H}_{\mathbf{R}^{\lambda, n}}^{\mathcal{D},\operatorname*{glob}}\left( \omega, \sigma\right) \ , \] where denotes the operator norm of the vector-Riesz transform of fractional order , 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…
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Advanced Harmonic Analysis Research · Numerical methods in inverse problems
