article

Title: Invariance of almost-orthogonal systems between weighted spaces: the non-compact support case

Authors: James Wilson

Issue: Volume 64 (2015), Issue 1, 275-293

Abstract:

$If $Q\subset\mathbb{R}^d$ is a cube with center $x_Q$ and sidelength $\ell(Q)$, and $f:\mathbb{R}^d\to\mathbb{C}$, define $f_{z_Q}(x)\equiv f((x-x_Q)/\ell(Q))$ ("$f$ adapted to $Q$"). We show that if $\{\phi^{(Q)}\}_{Q\in\mathcal{D}}$ is any family of functions indexed over the dyadic cubes, satisfying certain weak decay and smoothness conditions, then the set \[ \left\{\frac{\phi^{(Q)}_{z_Q}}{v(Q)^{1/2}}\right\}_{Q\inmathcal{D}} \] is almost-orthogonal in $L^2(v)$ for one $A_{\infty}$ weight $v$ if and only if it is almost-orthogonal in $L^2(v)$ for all $A_{\infty}$ weights $v$. In the special case where every $\phi^{(Q)}=\psi$, a fixed Schwartz function, this universal almost-orthogonality holds if and only if $\displaystyle{\int}\psi\,\mathrm{d}x=0$.$