# Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in L^1_{loc}(\mathbb{R}^n)$ on $Q$. We say that $f\in L^1_{loc}(\mathbb{R}^n)$ is in $BMO$ if $$\sup_{Q\in\Delta} m_Q(|f-m_Qf|)<\infty$$ where $\Delta$ is the set of the non-degenerate cubes in $\mathbb{R}^n$ (with sides parallel to the axis). The inequality of John-Nirenberg allows us to defined equivalent norms on $BMO$ for all $1<p<\infty$: $$\Vert f\Vert_{\star,p}=\sup_{Q\in\Delta} m_Q(|f-m_Qf|^p)^{\frac{1}{p}}<\infty$$ Then let us defne the Hardy spaces, $H^{1,p}(\mathbb{R}^n)$, for $1<p\leq \infty$. Firstly, an p-atom on a cube $Q$ is a (Lebesgue)-measurable function $a:\mathbb{R}^n\rightarrow \mathbb{C}$ such that : $(i)$ $\text{supp}\, a\subset Q$, $$(ii)\Vert a\Vert_{L^p}\leq \frac{1}{\mathscr{L}^n(Q)^{1-\frac{1}{p}}}$$ $$(iii)\int_{\mathbb{R}^n}a(x) d\mathscr{L}^n(x)=0$$ $H^{1,p}(\mathbb{R}^n)= \{f\in L^1(\mathbb{R}^n), \exists (\lambda_j)_{j\in\mathbb{N}}\;\text{and p-atoms}\; (a_j)_{j\in\mathbb{N}}\in l^1(\mathbb{N}), f=\sum_{j\in\mathbb{N}}\lambda_j\, a_j \}$. $H^{1,p}(\mathbb{R}^n)$ is a Banach space equiped with the norm $$\Vert f\Vert_{H^{1,p}}=\inf\left\{\sum_{j\in\mathbb{N}}|\lambda_j|, \lambda\in l^1(\mathbb{N}), \exists (a_j)_{j\in\mathbb{N}}\, \text{p-atoms}, f=\sum_{j\in\mathbb{N}}\lambda_j\,a_j\right\}.$$ We can prove that all the spaces $H^{1,p}$ are the same with equivalent norms (recall that $1<p\leq\infty$), and we note the common space $H^1$.

My question concerns a precision about the theorem of Fefferman and Stein : the dual of $H^1$ is isomorphic to $BMO/\mathbb{R}$.

When we fix $1<p<\infty$ $\Vert\; \Vert_{\star,p'}$ and the norms $\Vert\;\Vert_{H^{1,p}}$ on $BMO/\mathbb{R}$ and $H^{1,p}$ respectively, can we construct an isometric isomorphism $BMO/\mathbb{R}\rightarrow (H^{1,p})^{\star}$?