Compactness Characterizations of Commutators on Ball Banach Function Spaces

Abstract

Let X be a ball Banach function space on R. Let Ω be a Lipschitz function on the unit sphere of R, which is homogeneous of degree zero and has mean value zero, and let TΩ be the convolutional singular integral operator with kernel Ω(·)/| · |. In this article, under the assumption that the Hardy–Littlewood maximal operatorM is bounded on both X and its associated space, the authors prove that the commutator [b, TΩ] is compact on X if and only if b ∈ CMO(R). To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in X of the commutators and the characteristic functions of some measurable subset, which are implied by the assumed boundedness ofM on X and its associated space as well as the geometry of R; the complete John–Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fréchet–Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when X := Lp(·)(Rn) (the variable Lebesgue space), X := L(R) (the mixed-norm Lebesgue space), X := L(R) (the Orlicz space), and X := (E q Φ )t(R ) (the Orlicz-slice space or the generalized amalgam space), all these results are new.