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Invariance and stability of almost-orthogonal systems

Author: Michael Wilson
Journal: Trans. Amer. Math. Soc. 368 (2016), 2515-2546
MSC (2010): Primary 42B25; Secondary 42B20
Published electronically: August 18, 2015
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Abstract: For $ \alpha >0$, let $ \mathcal {C}_\alpha $ be the uniformly (norm $ \leq 1$) $ \alpha $-Hölder continuous functions with supports contained in the unit ball of $ \mathbf {R}^d$. Let $ \{\phi ^{(Q)}\}\subset \mathcal {C}_\alpha $ be any family indexed over the dyadic cubes $ Q$. If $ x_Q$ is the center of $ Q$ and $ \ell (Q)$ is its sidelength, we define $ z_Q\equiv (x_Q,\ell (Q)/2)$, and we set:

$\displaystyle \phi ^{(Q)}_{z_Q}(x)\equiv \phi ^{(Q)}(2(x-x_Q)/\ell (Q)).$

We show that if $ \mu $ is a Muckenhoupt $ A_\infty $ measure, then the family $ \{\phi ^{(Q)}_{z_Q}\!/\vert Q\vert ^{1/2}\}$ is almost-orthogonal in $ L^2$ (Lebesgue measure) if and only if the family
$ \{\phi ^{(Q)}_{z_Q}/\mu (Q)^{1/2}\}$ is almost-orthogonal in $ L^2(\mu )$; where we say a family $ \{\psi _k\}$ is almost-orthogonal in $ L^2(\nu )$ if there is an $ R<\infty $ such that, for all finite subsets $ \mathcal {F} \subset \{\psi _k\}$ and all linear sums $ \sum _{k:\,\psi _k\in \mathcal {F}} \lambda _k\psi _k$,

$\displaystyle \int \left \vert \sum _{k:\,\psi _k\in \mathcal {F}}\lambda _k\ps... ...eq R\sum _{k:\,\psi _k\in \mathcal {F}}\left \vert {\lambda _k}\right \vert ^2.$

We show that if $ \mu $ is a doubling measure, then $ \mu \in A_\infty $ is necessary for this equivalence. We show that almost-orthogonal expansions of $ \mu $-based Calderón-Zygmund singular integral operators are stable with respect to small dilation/translation errors in their generating kernels if the measure $ \mu $ is $ A_\infty $.

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Michael Wilson
Affiliation: Department of Mathematics, University of Vermont, Burlington, Vermont 05405

Keywords: Littlewood-Paley theory, almost-orthogonality, weighted norm inequality, bounded mean oscillation, singular integral operators
Received by editor(s): February 4, 2013
Received by editor(s) in revised form: January 21, 2014
Published electronically: August 18, 2015
Article copyright: © Copyright 2015 American Mathematical Society