The Operator Norm of Paraproducts on Bi-parameter Hardy spaces

TL;DR

This paper characterizes the operator norm of dyadic paraproducts on bi-parameter Hardy spaces, establishing their equivalence to certain Hardy space norms of the symbol function, and extends results to Fourier paraproducts.

Contribution

It provides a precise norm equivalence for bi-parameter dyadic paraproducts and characterizes their boundedness in terms of Hardy space and BMO norms, extending to Fourier paraproducts.

Findings

Operator norm of dyadic paraproducts is comparable to the Hardy space norm of the symbol.

Boundedness of paraproducts is characterized by BMO norms.

Results extend to bi-parameter Fourier paraproducts.

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 \[ \pi_g(f) := \sum_{R \in \mathcal{D}\otimes\mathcal{D}} g_R \left\langle f \right\rangle_{R} h_R, \] 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.