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

Abstract

Let $\alpha>-1/2$, $\alpha\ne0$, and let $$ \Delta_\alpha=-\frac{d^2}{dx^2}-\frac{2\alpha}{x}\frac{d}{dx} $$ 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, $$ \|[b,R_\alpha]\|_{L^p(\mu\,dx)\to L^p(\lambda\,dx)} \simeq \|b\|_{\rm{BMO}_{\nu,\alpha}}, $$ where the Bloom oscillation is computed with respect to $d\rho_\alpha$. Moreover $$ [b,R_\alpha]:L^p(\mu\,dx)\to L^p(\lambda\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{\nu,\alpha}. $$ 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 Calder\'on--Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $d\rho_\alpha$ is distinct from the usual Bessel measure $dm_\alpha=x^{2\alpha}\,dx$. It is the Bloom base measure selected by the Andersen--Kerman conjugation and is used throughout the two-weight theory below.