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