The Operator Norm of Paraproducts on Bi-parameter Hardy spaces

It is shown that for $0<p,q,r<\mathrm{\infty }$, with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$, the operator norm of the dyadic paraproduct of the form $${\pi }_g(f):=\sum _{R\in \mathcal{D}\otimes \mathcal{D}}{g}_R{⟨f⟩}_R{h}_R,$$ 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 $\mathrm{\parallel }g{\mathrm{\parallel }}_{{\dot{H}}_d^r(\mathbb{R}\otimes \mathbb{R})}$. We also prove that for all $0<p<\mathrm{\infty }$, there holds $$\mathrm{\parallel }g{\mathrm{\parallel }}_{BM{O}_d(\mathbb{R}\otimes \mathbb{R})}\simeq \mathrm{\parallel }{\pi }_g{\mathrm{\parallel }}_{H_d^p(\mathbb{R}\otimes \mathbb{R})\to {\dot{H}}_d^p(\mathbb{R}\otimes \mathbb{R})}\mathrm{.}$$ Similar results are obtained for bi-parameter Fourier paraproducts of the same form.