Characterizations of VMO and CMO spaces in the Bessel setting (Q6661073)

Let \(\lambda > 0\) and \(\triangle_{\lambda}=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx}\) be the Bessel operator on \(\mathbb{R}_{+}=(0, \infty)\). In this paper the authors introduce \(\mathrm{BMO}(\mathbb{R}_{+}, dm_{\lambda}), \ \mathrm{VMO}(\mathbb{R}_{+}, dm_{\lambda}),\ \mathrm{CMO}(\mathbb{R}_{+}, dm_{\lambda})\) in the Besel setting and characterize \(\mathrm{VMO}(\mathbb{R}_{+}, dm_{\lambda})\) in terms of the Hankel translation, the Hankel convolution and John-Nirenberg inequality, and obtain some sufficient condition for \(f \in \mathrm{VMO}(\mathbb{R}_{+}, dm_{\lambda})\) using \(\tilde{R}_{\triangle_{\lambda}}\), adjoint of the Riesz transform \(R_{\triangle_{\lambda}}\). Furthermore, the authors obtain a characterization of \(\mathrm{CMO}(\mathbb{R}_{+}, dm_{\lambda})\) in terms of the John-Nirenberg inequality, and some sufficient condition for \(f \in \mathrm{CMO}(\mathbb{R}_{+}, dm_{\lambda})\) using \(\tilde{R}_{\triangle_{\lambda}}\) and continuous functions vanishing at infinity and origin on \(\mathbb{R}_{+}\).