Compactness of Riesz transform commutator associated with Bessel operators

TL;DR

This paper characterizes the compactness of Riesz transform commutators related to Bessel operators on weighted spaces, linking BMO and CMO spaces with operator compactness in a novel Bessel setting.

Contribution

It introduces an equivalent characterization of CMO spaces associated with Bessel operators and proves the equivalence between CMO membership and the compactness of Riesz transform commutators.

Findings

Characterization of CMO space via new Fréchet-Kolmogorov theorem in Bessel setting

Equivalence between CMO membership and compactness of commutators

Extension of classical results to weighted Bessel operator context

Abstract

Let $\lambda>0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We first introduce and obtain an equivalent characterization of ${\rm CMO}(\mathbb R_+,\, x^{2\lambda}dx)$. By this equivalent characterization and establishing a new version of the Fr\'{e}chet-Kolmogorov theorem in the Bessel setting, we further prove that a function $b\in {\rm BMO}(\mathbb R_+,\, x^{2\lambda}dx)$ is in ${\rm CMO}(\mathbb R_+,\, x^{2\lambda}dx)$ if and only if the Riesz transform commutator $[b, R_{\Delta_\lambda}]$ is compact on $L^p(\mathbb R_+, x^{2\lambda}dx)$ for any $p\in(1, \infty)$.