Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights

Let $\alpha >-1\mathrm{/}2$, $\alpha \mathrm{\ne }0$, and let $${\mathrm{\Delta }}_{\alpha }=-\frac{d^2}{d{x}^2}-\frac{2\alpha }{x}\frac{d}{dx}$$ be the Bessel operator on ${\mathbb{R}}_{+}=(0,\mathrm{\infty })$. We characterize boundedness and compactness of commutators of $R_{\alpha }=\frac{d}{dx}{\mathrm{\Delta }}_{\alpha }^{-1\mathrm{/}2}$ on the Andersen–Kerman two-weight setting. For $1<p<\mathrm{\infty }$ and $\mu ,\lambda \in A_{p,\alpha }$, put $\nu ={\left(\frac{\mu }{\lambda }\right)}^{1\mathrm{/}p},d{\rho }_{\alpha }(x)=x^{2\alpha +1}dx\mathrm{.}$ For real-valued symbols, $$\mathrm{\parallel }[b,R_{\alpha }]{\mathrm{\parallel }}_{L^p(\mu dx)\to L^p(\lambda dx)}\simeq \mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{B}\mathrm{M}\mathrm{O}}_{\nu ,\alpha }},$$ where the Bloom oscillation is computed with respect to $d{\rho }_{\alpha }$. Moreover $$[b,R_{\alpha }]:L^p(\mu dx)\to L^p(\lambda dx)\text{ is compact}⟺b\in {\mathrm{V}\mathrm{M}\mathrm{O}}_{\nu ,\alpha }\mathrm{.}$$ The proof uses the exact conjugation $U_w(x)=x^{p-2\alpha -1}w(x),[U_w{]}_{A_p({\rho }_{\alpha })}=[w{]}_{A_{p,\alpha }},$ which transfers the problem to the quotient Bessel kernel on $({\mathbb{R}}_{+},\mathrm{\mid }x-y\mathrm{\mid },{\rho }_{\alpha })$. We isolate the required Calderón–Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $d{\rho }_{\alpha }$ is distinct from the usual Bessel measure $d{m}_{\alpha }=x^{2\alpha }dx$. It is the Bloom base measure selected by the Andersen–Kerman conjugation and is used throughout the two-weight theory below.