Weak factorization of the Hardy space $H^p$ for small values of $p$, in the multilinear setting

TL;DR

This paper proves a weak factorization of the Hardy space $H^p$ in the multilinear setting for small $p$, leading to new characterizations of commutator boundedness involving Lipschitz functions.

Contribution

It introduces a weak factorization approach for $H^p$ spaces in the multilinear context for small $p$, extending classical results and characterizing commutator boundedness.

Findings

Weak factorization of $H^p$ for $rac{n}{n+1}<p<1$ in multilinear setting.

Characterization of boundedness of commutators $[b,T]$ with Lipschitz functions.

New tools for analyzing multilinear Hardy spaces and commutator operators.

Abstract

We give a weak factorization proof of the Hardy space $H^{p}(\mathbb{R}^{n})$ in the multilinear setting, for $\frac{n}{n+1} < p <1$. As a consequence, we obtain a characterization of the boundedness of the commutator $[b,T]$ from $L^{r_{1}}(\mathbb R^n) \text{~x ... x~} L^{r_{m}} (\mathbb R^n) \text{ to } L^{q^\prime} (\mathbb R^n)$, where $b \in \text{Lip}_\alpha (\mathbb R^n)$, and $\frac{\alpha}{n} = \sum_{i=1}^{m} \frac{1}{r_{i}} +\frac{1}{q} - 1$.