Notes on Bi-parameter Paraproducts
Shahaboddin Shaabani
TL;DR
This work studies the sharpness of bounds for dyadic bi-parameter paraproducts acting between product Hardy spaces. It systematically analyzes unmixed forms $\pi^1_g$ and $\pi^2_g$, establishing sharp norm equivalences: $\|\pi^1_g\|_{H_d^p\to H_d^p} \simeq \|g\|_{BMO_d}$ and $\|\pi^2_g\|_{H_d^p\to H_d^q} \simeq \|g\|_{H_d^r}$ (with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$), and showing the corresponding lower bounds via testing and sparse-coverage arguments. For the mixed paraproducts, the paper proves $\| π^3_g\|_{H_d^p\to H_d^q} \lesssim \|g\|_{H_d^r}$ with the mixed-norm diagonal lower bound $\|\|g(x,y)\|_{BMO_d(\mathbb{R},dx)}\|_{L^{\infty}(\mathbb{R},dy)}$, while $\pi^4_g$ is controlled by duality through $\pi^3_{f'}$ and by a matrix-norm bound $\|G(x,y)\|_{l^2\to l^2}$, which can surpass the $BMO_d$-norm in certain constructions. The results leverage square-function characterizations of $H_d^p$, atomic decompositions, and Fefferman–Cordoba sparse/Carleson-type tools to obtain sharp, exponents-range results and to highlight when diagonal- or mixed-norm quantities provide tighter control. The findings clarify sharp thresholds for bi-parameter paraproducts and reveal limitations of standard product BMO-type norms in the mixed setting, with implications for weak factorization and related harmonic-analysis applications.
Abstract
In this note, we investigate the sharpness of existing bounds for various types of bi-parameter paraproducts acting between product Hardy spaces in the dyadic setting. We show that these bounds are sharp in most cases but fail to be so in one particular instance.