The operator norm of paraproducts on bi-parameter Hardy spaces

It is shown that for $0 \lt p,q,r \lt \infty $, with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, the operator norm of the dyadic paraproduct of the form $$\pi_g(f) := \sum _{R \in \mathcal D\otimes \mathcal D} g_R \langle f\rangle_{R} h_R, $$ from the bi-parameter dyadic Hardy space $H_d^p(\mathbb {R}\otimes \mathbb {R})$ to $\dot {H}_d^q(\mathbb {R}\otimes \mathbb {R})$ is comparable to $\|g\|_{\dot H_d^r(\mathbb {R}\otimes \mathbb {R})}$. We also prove that for all $0 \lt p \lt \infty $, $$ \|g\|_{\mathrm{BMO}_d(\mathbb R\otimes \mathbb R)} \simeq \|\pi _g\|_{H_d^p(\mathbb {R}\otimes \mathbb {R})\to \dot H_d^p(\mathbb {R}\otimes \mathbb {R})}. $$ Similar results are obtained for bi-parameter Fourier paraproducts of the same form.