The Riesz Transform and Fractional Integral Operators in the Bessel Setting

Fix $\lambda >0$. Consider the Bessel operator ${\mathrm{△}}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ on ${\mathbb{R}}_{+}$, where ${\mathbb{R}}_{+}:=(0,\mathrm{\infty })$ and $d{m}_{\lambda }:=x^{2\lambda }dx$ 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 ${\mathrm{△}}_{\lambda }$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.