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BMO classes are determined by their pairings with H1 atoms

Statement

Let $b \in B M O (R^{n})$ and suppose $\int a b = 0$ for every $H^{1}$ atom $a$ . Then $b$ is constant almost everywhere. Equivalently, the evaluation map on atoms is injective on $B M O / C$ .

Facts & Assumptions

Given: $b \in B M O (R^{n})$ with $\int a b = 0$ for every $(1, \infty, 0)$ -atom $a$ , and a cube $Q$ .

[F1]

A bounded mean-zero function $g$ vanishing off $Q$ is a scalar multiple of an atom: for $∥ g ∥_{\infty} > 0$ , divide by $∣ Q ∣ ∥ g ∥_{\infty}$ and choose the representative vanishing off the closed cube $Q$ ; a zero $L^{\infty}$ class has zero pairing ( $H^{p}$ atoms with a prescribed moment order).

[F2]

$b$ is locally integrable, $b_{Q} = ∣ Q ∣^{- 1} \int_{Q} b$ , and the BMO seminorm vanishes exactly on the almost-everywhere constants (BMO seminorm and the quotient by constants).

Proof

technique · direct

1.1F1F2givenconstruct

Define $h (x) = \overline{b (x) - b_{Q}} / ∣ b (x) - b_{Q} ∣$ on $Q$ where $b (x) \neq b_{Q}$ , and $h (x) = 0$ elsewhere. Then $∣ h ∣ \leq 1$ , and $g : = (h - h_{Q}) 1_{Q}$ is bounded, supported in $Q$ and mean zero. By [F1] and the hypothesis, $\int g b = 0$ , including the zero-class case. All products are integrable because $b \in L^{1} (Q)$ and $g$ is bounded.

2.1step 1.1F1F2algebra∎

Using $\int_{Q} (b - b_{Q}) = 0$ and the definition of $h$ , we obtain $0 = \int_{Q} g b = \int_{Q} g (b - b_{Q}) = \int_{Q} h (b - b_{Q}) = \int_{Q} ∣ b - b_{Q} ∣$ . Since $Q$ was arbitrary, the BMO seminorm is zero, so $b$ is constant almost everywhere by [F2]. Applying this to $b - b^{'}$ proves injectivity on classes with equal atom pairings; conversely constants pair to zero by atom cancellation. No choice principle or local $L^{2}$ estimate is used.

Depends on

Used by

Real H1-BMO duality Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)
Brooke Wilson, Math 581A Classical and Multilinear Harmonic Analysis (University of Washington, Fall 2024), lecture 20 (standard reference, not scraped)