Weak factorization of Hardy spaces in the Bessel setting

We provide the weak factorization of the Hardy spaces $H^{p}(\mathbb{R}_+, dm_{\lambda})$ 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 $L^{q}(\mathbb{R}_+, dm_{\lambda})$ to $L^{r}(\mathbb{R}_+, dm_{\lambda})$ when $b\in \textrm{Lip}_{\alpha}(\mathbb{R}_+, dm_{\lambda})$ 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.