The Operator Norm of Paraproducts on Bi-parameter Hardy spaces

It is shown that for $0<p,q,r<\infty$, with $\frac{1}{q} = \frac{1}{p} + \frac{1}{r}$, the operator norm of the dyadic paraproduct of the form $$\pi_g(f) := \sum_{R \in \mathcal{D}\otimes\mathcal{D}} g_R \left\langle f \right\rangle_{R} h_R,$$ 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 $0 < p < \infty$, there holds $$\|g\|_{BMO_d(\mathbb{R}\otimes\mathbb{R})} \simeq \|\pi_g\|_{H_d^p(\mathbb{R}\otimes\mathbb{R}) \to \dot{H}_d^p(\mathbb{R}\otimes\mathbb{R})}.$$ Similar results are obtained for bi-parameter Fourier paraproducts of the same form.