Invariance and stability of almost-orthogonal systems

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: \[ \phi ^{(Q)}_{z_Q}(x)\equiv \phi ^{(Q)}(2(x-x_Q)/\ell (Q)).\] 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 $\sum _{k: \psi _k\in \mathcal {F}} \lambda _k\psi _k$, \[ \int \left \vert \sum _{k: \psi _k\in \mathcal {F}}\lambda _k\psi _k\right \vert ^2 d\nu \leq R\sum _{k: \psi _k\in \mathcal {F}}\left \vert {\lambda _k}\right \vert ^2.\] 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 $A_\infty$.