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.
Categories: math.CA math.FA