Weak factorization of Hardy spaces in the Bessel setting

Abstract

We provide the weak factorization of the Hardy spaces $H^p({\mathbb{R}}_{+},d{m}_{\lambda })$ in the Bessel setting, for $p\in \left(\frac{2\lambda +1}{2\lambda +2},1\right]$. As a corollary we obtain a characterization of the boundedness of the commutator $[b,R_{{\Delta }_{\lambda }}]$ from $L^q({\mathbb{R}}_{+},d{m}_{\lambda })$ to $L^r({\mathbb{R}}_{+},d{m}_{\lambda })$ when $b\in {\text{Lip}}_{\alpha }({\mathbb{R}}_{+},d{m}_{\lambda })$ provided that $\alpha =\frac{1}{q}-\frac{1}{r}$. The results are a slight generalization and modification of the work of Duong, Li, Yang, and the second named author, which in turn are based on modifications and adaptations of work by Uchiyama.

17 pages