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

Abstract

Let $\alpha >-1/2$, $\alpha \ne 0$, and let $${\Delta }_{\alpha }=-\frac{d^2}{d{x}^2}-\frac{2\alpha }{x}\frac{d}{dx}$$ be the Bessel operator on ${\mathbb{R}}_{+}=(0,\infty )$. We characterize boundedness and compactness of commutators of $R_{\alpha }=\frac{d}{dx}{\Delta }_{\alpha }^{-1/2}$ on the Andersen--Kerman two-weight setting. For $1<p<\infty$ and $\mu ,\lambda \in A_{p,\alpha }$, put $\nu ={\left(\frac{\mu }{\lambda }\right)}^{1/p},d{\rho }_{\alpha }(x)=x^{2\alpha +1}dx.$ For real-valued symbols, $$\Vert [b,R_{\alpha }]{\Vert }_{L^p(\mu dx)\to L^p(\lambda dx)}\simeq \Vert b{\Vert }_{{\mathrm{BMO}}_{\nu ,\alpha }},$$ where the Bloom oscillation is computed with respect to $d{\rho }_{\alpha }$. Moreover $$[b,R_{\alpha }]:L^p(\mu dx)\to L^p(\lambda dx)\text{ is compact}⟺b\in {\mathrm{VMO}}_{\nu ,\alpha }.$$ 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 $({\mathbb{R}}_{+},|x-y|,{\rho }_{\alpha })$. 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.