I have two questions while reading this paper: Hp spaces of several variables." Acta math 129 (1972): 167-193.
Question 1. Line $7$ on page $147$ to line $13$ on page $148$ is a proof of "Theorem 3$\Rightarrow$ Theorem 2(i) and Theorem 2(ii)", I give a sketch as follows:
Let $\varphi$ be a BMO function, then by Theorem 3(iii), $$ \left|\int_{\mathbb{R}^n}f\varphi\,\mathrm{d}x \right|=\left|2 \int_{\mathbf{R}_{+}^{n+1}} t(\nabla f(x, t))(\nabla \varphi(x, t))\,\mathrm{d}x \mathrm{d}t\right|\leqslant A\|f\|_{H^1}, \text{ whenever } f\in H^1(\mathbb{R}^n), $$ which is equivalent to that $\varphi$ is continuous as a linear functional on $H^1(\mathbb{R}^n)$. This proves Theorem 2(i).
But I think it only proves "Theorem 3$\Rightarrow$ Theorem 2(i)", where does it prove "Theorem 3 $\Rightarrow$ Theorem 2 (ii)"?
Question 2. Line $15$ on page $148$ to line $17$ on page $148$ is the proof to the last sentence"The norm of $\varphi$ as a linear functional on $H^1$ is equivalent with the BMO norm" of Theorem 2. I can't understand this proof.
Could you please help me? Thank you very much for your time on my question!
(Line $15$ on page $148$ to line $17$ on page $148$) ..., the fact that the norm of $\varphi$ as a linear functional on $H^{1}$ is equivalent with its BMO norm follows either by a priori grounds (the closed graph theorem), or when one keeps track of the various constants that arise in the proof just given. QED