The dual representative has uniformly bounded BMO oscillation

Given: Countable Choice, the fixed $\phi ,\tilde{N}$, $\mathrm{\Lambda }\in (H^1({\mathbb{R}}^n){)}^{\ast }$, the locally integrable representative $u$ and the local representatives $u_Q$ of Bounded H1 functionals have compatible local L2 representatives, and a nondegenerate cube $Q$.

For every nondegenerate cube $Q$ the representative $u_Q\in L_0^2(Q)$ has mean $0$ on $Q$, satisfies $\mathrm{\Lambda }(f)={\int }_Q{u}_{Q}f$ for $f\in L_0^2(Q)$, and $u-u_Q$ is almost everywhere constant on $Q$ (Bounded H1 functionals have compatible local L2 representatives).

The restriction of $\mathrm{\Lambda }$ to $L_0^2(Q)$ obeys $\mathrm{\mid }\mathrm{\Lambda }(f)\mathrm{\mid }\le C_{n,\tilde{N},\phi }\mathrm{\mid }Q{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}$ for $f\in L_0^2(Q)$: this is the mean-zero embedding $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}\le C_{n,\tilde{N},\phi }\mathrm{\mid }Q{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}$ composed with $\mathrm{\mid }\mathrm{\Lambda }(f)\mathrm{\mid }\le \mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}$ (Mean-zero L2 functions on a cube embed continuously into H1).

Riesz representation is an isometry: the representing vector $u_Q$ of $\mathrm{\Lambda }{\mathrm{\mid }}_{L_0^2(Q)}$ has $\mathrm{\parallel }{u}_Q{\mathrm{\parallel }}_{L^2}=\mathrm{\parallel }\mathrm{\Lambda }{\mathrm{\mid }}_{L_0^2(Q)}\mathrm{\parallel }$, the operator norm on the subspace (Riesz representation for Hilbert spaces).

On the positive finite-measure cube $Q$, $\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }{u}_Q\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1\mathrm{/}2}\mathrm{\parallel }{u}_Q{\mathrm{\parallel }}_{L^2}$ (Cauchy-Schwarz inequality for $L^2$).