The Necessity of A _∞ for Translation and Scale Invariant Almost-Orthogonality

Abstract

If ν is a measure, we say a set {ψ _k } _k ⊂ L ²(ν) is almost-orthogonal in L ²(ν) if there is an R < ∞ such that, for all finite linear sums ∑ λ _k ψ _k ,

$$\displaystyle{\int \left \vert \sum \lambda _{k}\psi _{k}\right \vert ^{2}\,d\nu \leq R\sum \vert \lambda _{ k}\vert ^{2}.}$$

If z = (t, y) ∈ R ₊ ^d+1 ≡ R ^d × (0, ∞) and f: R ^d → C, define f _z (x) ≡ f((x − t)∕y). If Q ⊂ R ^d is a cube with sidelength ℓ(Q), define T(Q) ≡ Q × [ℓ(Q)∕2, ℓ(Q)). We say that {ϕ _k }₁ ⁿ, a finite set of bounded, complex-valued functions with supports contained in B(0; 1), satisfies the collective non-degeneracy condition (CNDC) if there is no ray emanating from the origin on which the Fourier transform of every ϕ _k vanishes identically. We prove: If μ is a doubling measure on R ^d with the property that, for some family {ϕ _k }₁ ⁿ satisfying CNDC, it is the case that, for every 1 ≤ k ≤ n and every choice of points $\zeta (Q) \in \overline{T(Q)}$, $Q \in \mathcal{D}$ (where $\mathcal{D}$ is the family of dyadic cubes), the set

$$\displaystyle{\left \{\frac{(\phi _{k})_{\zeta (Q)}} {\sqrt{\mu (Q)}}\right \}_{Q\in \mathcal{D}}}$$

is almost-orthogonal in L ²(μ), then μ is a Muckenhoupt A _∞ measure.