Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting

Fix $\lambda>0$. Consider the Hardy space $H^1(\mathbb{R}_+,dm_\lambda)$ in the sense of Coifman and Weiss, where $\mathbb{R_+}:=(0,\infty)$ and $dm_\lambda:=x^{2\lambda}dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators $\Delta_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$, and $S_\lambda:=-\frac{d^2}{dx^2}+\frac{\lambda^2-\lambda}{x^2}$ on $\mathbb{R_+}$. The Hardy spaces $H^1_{\Delta_\lambda}$ and $H^1_{S_\lambda}$ associated with $\Delta_\lambda$ and $S_\lambda$ are defined via the Riesz transforms $R_{\Delta_\lambda}:=\partial_x (\Delta_\lambda)^{-1/2}$ and $R_{S_\lambda}:= x^\lambda\partial_x x^{-\lambda} (S_\lambda)^{-1/2}$, respectively. It is known that $H^1_{\Delta_\lambda}$ and $H^1(\mathbb{R}_+,dm_\lambda)$ coincide but they are different from $H^1_{S_\lambda}$. In this article, we prove the following: (a) a weak factorization of $H^1(\mathbb{R}_+,dm_\lambda)$ by using a bilinear form of the Riesz transform $R_{\Delta_\lambda}$, which implies the characterization of the BMO space associated to $\Delta_\lambda$ via the commutators related to $R_{\Delta_\lambda}$; (b) the BMO space associated to $S_\lambda$ can not be characterized by commutators related to $R_{S_\lambda}$, which implies that $H^1_{S_\lambda}$ does not have a weak factorization via a bilinear form of the Riesz transform $R_{S_\lambda}$.