Compactness Characterizations of Commutators on Ball Banach Function Spaces

Abstract

Let X be a ball Banach function space on ${\mathbb R}^{n}$. Let Ω be a Lipschitz function on the unit sphere of ${\mathbb R}^{n}$, which is homogeneous of degree zero and has mean value zero, and let T_Ω be the convolutional singular integral operator with kernel Ω(⋅)/|⋅|ⁿ. In this article, under the assumption that the Hardy–Littlewood maximal operator ${\mathscr{M}}$ is bounded on both X and its associated space, the authors prove that the commutator [b, T_Ω] is compact on X if and only if $b\in \text {CMO }({\mathbb R}^{n})$. 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 ${\mathcal M}$ on X and its associated space as well as the geometry of $\mathbb R^{n}$; 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 $X:=L^{p(\cdot )}({\mathbb R}^{n})$ (the variable Lebesgue space), $X:=L^{\vec {p}}({\mathbb R}^{n})$ (the mixed-norm Lebesgue space), $X:=L^{\Phi }({\mathbb R}^{n})$ (the Orlicz space), and $X:=(E_{\Phi }^{q})_{t}({\mathbb R}^{n})$ (the Orlicz-slice space or the generalized amalgam space), all these results are new.