Compactness Characterizations of Commutators on Ball Banach Function Spaces

Let X be a ball Banach function space on R-n. Let Omega be a Lipschitz function on the unit sphere of R-n, which is homogeneous of degree zero and has mean value zero, and let T-Omega be the convolutional singular integral operator with kernel Omega(center dot)/| center dot |(n). In this article, under the assumption that the Hardy-Littlewood maximal operatorMis bounded on both X and its associated space, the authors prove that the commutator [ b, T-Omega] is compact on X if and only if b is an element of CMO(R-n). To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm of X about the commutators and the characteristic functions of some measurable subsets, which are implied by the assumed boundedness of M on X and its associated space as well as the geometry of R-n; the complete John-Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fr ' echet-Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when X := L-p(center dot)(R-n) (the variable Lebesgue space), X := L-p(R-n) (the mixed-norm Lebesgue space), X := L-Phi (R-n) (the Orlicz space), and X := (E-Phi(q))(t) (R-n) (the Orlicz-slice space or the generalized amalgam space), all these results are new.