Characterizations of VMO and CMO Spaces in the Bessel Setting

Abstract

Let λ > 0 and \(\Delta_{\lambda} := -{{d^2} \over {dx^{2}}} - {{2\lambda} \over x} {{d} \over {dx}}\) be the Bessel operator on ℝ₊:= (0, ∞). In this paper, the authors introduce and characterize the space VMO(ℝ₊, dm_λ) 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 ∈ VMO(ℝ₊, dm_λ) using \({\tilde R}_{{\Delta}_\lambda}\), the adjoint of the Riesz transform \({R}_{{\Delta}_\lambda}\). Furthermore, we obtain the characterization of CMO(ℝ₊, dm_λ) in terms of the John–Nirenberg inequality which is new even for the classical CMO(ℝⁿ) and a sufficient condition of Fefferman–Stein type for functions f ∈ CMO(ℝ₊, dm_λ).