  Fix $\lambda>0$. Consider the Bessel operator
$\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ on
$\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and
$dm_\lambda:=x^{2\lambda}dx$ with $dx$ the Lebesgue measure. We provide a
deeper study of the Bessel Riesz transform and fractional integral operator via
the related Besov and Triebel--Lizorkin spaces associated with
$\triangle_\lambda$. Moreover, we investigate some possible characterization of
the commutator of fractional integral operator, which was missing in the
literature of the Bessel setting.
