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Real H1-BMO duality

Statement

Assume the Axiom of Choice, with the fixed $H^{1}$ kernel $φ$ and auxiliary order $\tilde{N}$ used in Mean-zero L2 functions on a cube embed continuously into H1. The map $Φ : B M O (R^{n}) / C \to (H^{1} (R^{n}))^{*}$ , $b \mapsto Λ_{b}$ of the preceding theorem is a linear bijection, and there are constants $0 < c_{n, \tilde{N}, φ} \leq C_{n, \tilde{N}, φ} < \infty$ with $c_{n, \tilde{N}, φ} ∥ b ∥_{B M O} \leq ∥ Λ_{b} ∥ \leq C_{n, \tilde{N}, φ} ∥ b ∥_{B M O}$ for every class $b$ . Thus $(H^{1} (R^{n}))^{*}$ is isomorphic to $B M O (R^{n}) / C$ with equivalent norms.

Facts & Assumptions

Given: The Axiom of Choice and a bounded functional $Λ \in (H^{1} (R^{n}))^{*}$ , with the map $Φ$ of BMO classes define bounded functionals on H1.

[F1]

The map $Φ$ is linear and bounded: for every $b \in B M O (R^{n})$ the functional $Λ_{b}$ satisfies $Λ_{b} (a) = \int a b$ on atoms and $∥ Λ_{b} ∥ \leq C_{n, \tilde{N}, φ}^{+} ∥ b ∥_{B M O}$ , and $Λ_{b} = 0$ for constant $b$ (BMO classes define bounded functionals on H1).

[F2]

The atom pairings determine the class: if $\int a b = 0$ for every atom $a$ , then $b$ is constant almost everywhere; equivalently $Φ$ is injective (BMO classes are determined by their pairings with H1 atoms).

[F3]

For every $Λ \in (H^{1})^{*}$ the preceding local representatives produce a locally integrable $u$ with $u - u_{Q}$ constant almost everywhere on every cube $Q$ , where $u_{Q} \in L_{0}^{2} (Q)$ represents $Λ ∣_{L_{0}^{2} (Q)}$ , and $∥ u ∥_{B M O} \leq C_{n, \tilde{N}, φ}^{-} ∥ Λ ∥$ (Bounded H1 functionals have compatible local L2 representatives, The dual representative has uniformly bounded BMO oscillation).

[F4]

The finite atomic sums are dense in $H^{1}$ (Finite atomic sums are dense in H1), and two bounded functionals agreeing on a dense subspace agree everywhere.

Proof

technique · direct

1.1F1F2

The map $Φ$ is linear by [F1]; it is bounded with $∥ Λ_{b} ∥ \leq C_{n, \tilde{N}, φ}^{+} ∥ b ∥_{B M O}$ by [F1]; and it is injective because a class in its kernel has vanishing pairings with all atoms and is therefore the class of the constants by [F2].

2.1step 1.1F1F3F4

Surjectivity. Let $Λ \in (H^{1})^{*}$ and let $u$ be the representative of [F3], so that $∥ u ∥_{B M O} \leq C_{n, \tilde{N}, φ}^{-} ∥ Λ ∥$ . For an atom $a$ supported in a cube $Q$ one has $a \in L_{0}^{2} (Q)$ , hence $Λ (a) = \int_{Q} u_{Q} a = \int_{Q} (u - c_{Q}) a = \int u a$ because $\int a = 0$ and $u - u_{Q} = c_{Q}$ almost everywhere on $Q$ ; meanwhile $Λ_{u} (a) = \int a u$ by the definition of $Φ$ [F1]. Thus $Λ$ and $Λ_{u}$ agree on every atom, hence on every finite atomic sum by linearity, and therefore on all of $H^{1}$ by density and continuity [F4]; that is, $Λ = Φ (u)$ and $Φ$ is surjective.

3.1step 1.1step 2.1F2

The reverse norm bound. Given a class $b$ , apply step 2.1 to $Λ_{b}$ : there is $u$ with $Λ_{b} = Φ (u)$ and $∥ u ∥_{B M O} \leq C_{n, \tilde{N}, φ}^{-} ∥ Λ_{b} ∥$ . By injectivity of $Φ$ from step 1.1 and [F2], $b - u$ is constant almost everywhere, so $∥ b ∥_{B M O} = ∥ u ∥_{B M O} \leq C_{n, \tilde{N}, φ}^{-} ∥ Λ_{b} ∥$ ; combined with step 1.1 this gives $c_{n, \tilde{N}, φ} ∥ b ∥_{B M O} \leq ∥ Λ_{b} ∥ \leq C_{n, \tilde{N}, φ} ∥ b ∥_{B M O}$ with $c_{n, \tilde{N}, φ} : = 1 / C_{n, \tilde{N}, φ}^{-}$ and $C_{n, \tilde{N}, φ} : = C_{n, \tilde{N}, φ}^{+}$ .

4.1step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 show that $Φ$ is a linear bijection with the two-sided norm bound, so $(H^{1} (R^{n}))^{*}$ is isomorphic to $B M O (R^{n}) / C$ with equivalent norms. The Axiom of Choice is inherited from the construction of $Λ_{b}$ and from the local representatives.

Depends on

Used by

Littlewood-Paley endpoint scope and the H1-BMO dual pair Remark

Dependency tree · two levels

32 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 21 (standard reference, not scraped)
Brooke Wilson, Math 581A Classical and Multilinear Harmonic Analysis (University of Washington, Fall 2024), lecture 22 (standard reference, not scraped)