Let X be a ball Banach function space on ℝn\documentclass[12pt]{minimal}
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\begin{document}${\mathbb R}^{n}$\end{document}. Let Ω be a Lipschitz function on the unit sphere of ℝn\documentclass[12pt]{minimal}
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\begin{document}${\mathbb R}^{n}$\end{document}, which is homogeneous of degree zero and has mean value zero, and let TΩ be the convolutional singular integral operator with kernel Ω(⋅)/|⋅|n. In this article, under the assumption that the Hardy–Littlewood maximal operator M\documentclass[12pt]{minimal}
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\begin{document}${\mathscr{M}}$\end{document} 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(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$b\in \text {CMO }({\mathbb R}^{n})$\end{document}. 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\documentclass[12pt]{minimal}
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\begin{document}${\mathcal M}$\end{document} on X and its associated space as well as the geometry of ℝn\documentclass[12pt]{minimal}
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\begin{document}$\mathbb R^{n}$\end{document}; 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)\documentclass[12pt]{minimal}
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\begin{document}$X:=L^{p(\cdot )}({\mathbb R}^{n})$\end{document} (the variable Lebesgue space), X:=Lp→(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$X:=L^{\vec {p}}({\mathbb R}^{n})$\end{document} (the mixed-norm Lebesgue space), X:=LΦ(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$X:=L^{\Phi }({\mathbb R}^{n})$\end{document} (the Orlicz space), and X:=(EΦq)t(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$X:=(E_{\Phi }^{q})_{t}({\mathbb R}^{n})$\end{document} (the Orlicz-slice space or the generalized amalgam space), all these results are new.
