The Operator Norm of Paraproducts on Bi-parameter Hardy spaces

Abstract

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 \[ π_g(f) := \sum_{R \in \mathcal{D}\otimes\mathcal{D}} g_R \left\langle f \right\rangle_{R} h_R, \] 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 $\Vert g{\Vert }_{{\dot{H}}_d^r(\mathbb{R}\otimes \mathbb{R})}$. We also prove that for all $0<p<\infty$, there holds \[ \|g\|_{BMO_d(\mathbb{R}\otimes\mathbb{R})} \simeq \|π_g\|_{H_d^p(\mathbb{R}\otimes\mathbb{R}) \to \dot{H}_d^p(\mathbb{R}\otimes\mathbb{R})}. \] Similar results are obtained for bi-parameter Fourier paraproducts of the same form.