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No abstract

Let λ > 0 and △ λ := − d 2 dx 2 − 2λ x d dx be the Bessel operator on R + := (0, ∞). We first introduce and obtain an equivalent characterization of CMO(R + , x 2λ 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 ∈ BMO(R + , x 2λ dx) is in CMO(R + , x 2λ dx) if and only if the Riesz transform commutator [b, R ∆ λ ] is compact on L p (R + , x 2λ dx) for any p ∈ (1, ∞).

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Let [Formula: see text] and [Formula: see text] be the Bessel operator on [Formula: see text]. In this paper, the authors show that [Formula: see text] (or [Formula: see text], respectively) if and only if the Riesz transform commutator [Formula: see text] is bounded (or compact, respectively) on Morrey spaces [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. A weak factorization theorem for functions belonging to the Hardy space [Formula: see text] in the sense of Coifman–Rochberg–Weiss in Bessel setting, via [Formula: see text] and its adjoint, is also obtained.