BMO classes define bounded functionals on H1

The bounded case. Let $b\in L^{\mathrm{\infty }}\cap \mathrm{B}\mathrm{M}\mathrm{O}$ and let $F$ be the linear span of the atoms, viewed as a subspace of $H^1$ by [F3]. Every $g\in F$ is a finite sum of bounded compactly supported atoms, hence an $L^1$ function with $g\in H^1$, so $\int gb$ converges absolutely and $\mathrm{\mid }\int gb\mathrm{\mid }\le C_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^1}$ by [F2]. The value $\int gb$ depends only on the element $g\in H^1$: if two finite sums $g,g^{\mathrm{\prime }}$ represent the same element, then the locally integrable function $g-g^{\mathrm{\prime }}$ has zero regular distribution, so $g=g^{\mathrm{\prime }}$ almost everywhere by [F7] and the two integrals agree. Thus $g\mapsto \int gb$ is a well-defined linear functional on $F$, bounded by $C_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$, and it extends uniquely to a bounded ${\mathrm{\Lambda }}_b\in (H^1{)}^{\ast }$ by density [F3]; the extension is the unique bounded functional whose value at every atom $a$ is $\int ab$, since two such functionals agree on $F$ and $F$ is dense.

The general case, existence of a cluster point. For general $b\in \mathrm{B}\mathrm{M}\mathrm{O}$ let $b_M$ be the componentwise truncation of [F4]; step 1.1 gives bounded functionals ${\mathrm{\Lambda }}_{b_M}$ with $\mathrm{\parallel }{\mathrm{\Lambda }}_{b_M}\mathrm{\parallel }\le \frac{9}{2}{C}_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$. The Axiom of Choice yields the ultrafilter lemma [F5], so the closed ball of radius $\frac{9}{2}{C}_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$ in $(H^1{)}^{\ast }$ is weak-star compact [F5]; by the compactness characterization [F5] the sequence, viewed as a net, $({\mathrm{\Lambda }}_{b_M}{)}_{M\in \mathbb{N},M\ge 1}$ has a weak-star cluster point $\mathrm{\Lambda }$. For every atom $a$ the evaluations converge: ${\mathrm{\Lambda }}_{b_M}(a)=\int a{b}_M\to \int ab$ by [F6]. Evaluation at $a$ is weak-star continuous [F5], so $\mathrm{\Lambda }(a)$ is a cluster point of the convergent net $({\mathrm{\Lambda }}_{b_M}(a){)}_M$ in $\mathbb{C}$ and therefore equals its limit, $\mathrm{\Lambda }(a)=\int ab$.

Uniqueness and the norm bound. If $\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\prime }}$ are bounded functionals with $\mathrm{\Lambda }(a)={\mathrm{\Lambda }}^{\mathrm{\prime }}(a)=\int ab$ for every atom $a$, then by linearity they agree on the span $F$ of the atoms and hence, by density [F3] and continuity, on all of $H^1$; so the functional of step 2.1 is the unique bounded functional with the required atom values, and $\mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }\le \frac{9}{2}{C}_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

Linearity, constants and finite sums. For $b,b^{\mathrm{\prime }}\in \mathrm{B}\mathrm{M}\mathrm{O}$ and $\lambda \in \mathbb{C}$, the functionals ${\mathrm{\Lambda }}_{b+b^{\mathrm{\prime }}}$ and ${\mathrm{\Lambda }}_b+{\mathrm{\Lambda }}_{b^{\mathrm{\prime }}}$ both assign to every atom $a$ the value $\int a(b+b^{\mathrm{\prime }})=\int ab+\int a{b}^{\mathrm{\prime }}$, so they are equal by the uniqueness of step 3.1; the same argument gives ${\mathrm{\Lambda }}_{\lambda b}=\lambda {\mathrm{\Lambda }}_b$. If $b$ is constant almost everywhere, then $\int ab=0$ for every atom because $\int a=0$ by [F1], so ${\mathrm{\Lambda }}_b=0$ by uniqueness. Hence $b\mapsto {\mathrm{\Lambda }}_b$ is linear with image of the constants in the zero functional, so it factors through $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$. Finally, for a finite atomic sum $g={\sum }_{j\le N}{\lambda }_j{a}_j$, linearity and step 1.1 give ${\mathrm{\Lambda }}_b(g)={\sum }_{j\le N}{\lambda }_j\int a_{j}b=\int gb$.

Steps 1.1 and 2.1 construct, for every $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$, a bounded functional with the required atom values, step 3.1 gives uniqueness and the bound $\mathrm{\parallel }{\mathrm{\Lambda }}_b\mathrm{\parallel }\le \frac{9}{2}{C}_{n,\tilde{N},\phi }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$, and step 4.1 gives linearity, the annihilation of constants, the factorisation through the quotient and the finite-sum identity. The Axiom of Choice is spent exactly at the ultrafilter lemma and the Banach-Alaoglu cluster point in step 2.1.