Qing Dong GUO, Jorge J. BETANCOR, Dong Yong YANG
Let $\lambda>0$ and $\Delta_{\lambda}:=-\frac{d^{2}}{dx^{2}}-\frac{2\lambda}{x}\frac{d}{dx}$ be the Bessel operator on ${\mathbb R_+}:=(0, \infty)$. In this paper, the authors introduce and characterize the space $\mathrm{VMO}(\mathbb{R}_{+},dm_{\lambda})$ in terms of the Hankel translation, the Hankel convolution and a John-Nirenberg inequality, and obtain a sufficient condition of Fefferman-Stein type for functions $f\in \mathrm{VMO}(\mathbb{R}_{+}, dm_{\lambda})$ using $\mathrm{\widetilde{R}}_{\Delta_{\lambda}}$, the adjoint of the Riesz transform $\mathrm{R}_{\Delta_{\lambda}}$. Furthermore, we obtain the characterization of $\mathrm{CMO}(\mathbb{R}_{+}, dm_{\lambda})$ in terms of the John-Nirenberg inequality which is new even for the classical $\mathrm{CMO}(\mathbb{R}^{n})$ and a sufficient condition of Fefferman-Stein type for functions $f\in \mathrm{CMO}(\mathbb{R}_{+}, dm_{\lambda})$.
