Compactness Characterizations of Commutators on Ball Banach Function Spaces

By Jin Tao, Dachun Yang, Wen Yuan, Yangyang Zhang. Potential Analysis, 2021

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Let X be a ball Banach function space on ℝ n ${\mathbb R}^{n}$ . Let Ω be a Lipschitz function on the unit sphere of ℝ n ${\mathbb R}^{n}$ , 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 M ${\mathscr{M}}$ 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 ∈ CMO ( ℝ n ) $b\in \text {CMO }({\mathbb R}^{n})$ . 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 ℳ ${\mathcal M}$ on X and its associated space as well as the geometry of ℝ n $\mathbb R^{n}$ ; 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 ( ⋅ ) ( ℝ n ) $X:=L^{p(\cdot )}({\mathbb R}^{n})$ (the variable Lebesgue space), X : = L p → ( ℝ n ) $X:=L^{\vec {p}}({\mathbb R}^{n})$ (the mixed-norm Lebesgue space), X : = L Φ ( ℝ n ) $X:=L^{\Phi }({\mathbb R}^{n})$ (the Orlicz space), and X : = ( E Φ q ) t ( ℝ n ) $X:=(E_{\Phi }^{q})_{t}({\mathbb R}^{n})$ (the Orlicz-slice space or the generalized amalgam space), all these results are new.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3310945ed942b6cd7f350c8e6673c3c226b25cb0

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