Global orthogonality implies local almost-orthogonality (Q1573712)

A family \(\{\phi_{(I)}: I\in{\mathcal D}\}\) of functions, where \({\mathcal D}\) denotes the usual family of dyadic cubes \(I\subset\mathbb{R}^d\), is said to be ``almost-orthogonal'' if for every finite-linear combination \[ f(x)= \sum_I\gamma_I \Phi_{(I)}(x), \] the inequality \[ \int|f|^2 dx\leq \sum_I |\gamma_I|^2 \] holds. Each function \(\phi_{(I)}\) is assumed to satisfy fairly weak decay, smoothness, and cancellation conditions. The author introduces a new stopping-time argument to prove a new Littlewood-Paley type result for such linear combinations \(f\), involving a weight function belonging to the class \(A_\infty\).