Compactness Characterizations of Commutators on Ball Banach Function Spaces

• Published in: Potential analysis Vol. 58; no. 4; pp. 645 - 679
• Main Authors: Tao, Jin, Yang, Dachun, Yuan, Wen, Zhang, Yangyang
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2023; Springer Nature B.V
• Subjects: Characteristic functions; Commutators; Function space; Functional Analysis; Geometry; Mathematics; Mathematics and Statistics; Operators (mathematics); Orlicz space; Potential Theory; Probability Theory and Stochastic Processes; Commutator; CMO; BMO; Extrapolation; 42B35; 42B25; Ball Banach function space; 42B30; Secondary 42B20; 46E30; Convolutional singular integral operator; Fréchet–Kolmogorov theorem; Primary 47B47
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-021-09953-w

Let X be a ball Banach function space on ℝ n . Let Ω 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 Ω be the convolutional singular integral operator with kernel Ω(⋅)/|⋅| n . In this article, under the assumption that the Hardy–Littlewood maximal operator 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 ∈ 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 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 ( ⋅ ) ( ℝ n ) (the variable Lebesgue space), X : = L p → ( ℝ n ) (the mixed-norm Lebesgue space), X : = L Φ ( ℝ n ) (the Orlicz space), and X : = ( E Φ q ) t ( ℝ n ) (the Orlicz-slice space or the generalized amalgam space), all these results are new.