Weak factorization of Hardy spaces in the Bessel setting

We provide the weak factorization of the Hardy spaces $H^{p}($\mathbb{R}$+, dm{\lambda})$intheBesselsetting,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 $[b,R_{{\mathrm{\Delta }}_{\lambda }}]$ 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.