Compactness of Riesz transform commutator associated with Bessel operators

Abstract

Let $\lambda >0$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${\mathbb{R}}_{+}:=(0,\infty )$. We first introduce and obtain an equivalent characterization of $\mathrm{CMO}({\mathbb{R}}_{+},x^{2\lambda }dx)$. By this equivalent characterization and establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function $b\in \mathrm{BMO}({\mathbb{R}}_{+},x^{2\lambda }dx)$ is in $\mathrm{CMO}({\mathbb{R}}_{+},x^{2\lambda }dx)$ 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 $p\in (1,\infty )$.

to appear in Journal d'Analyse Mathematique