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

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 $\frac{\alpha}{n} = \sum{i=1}^{m} \frac{1}{r_{i}} +\frac{1}{q} - 1$.