Compactness of commutators of fractional integral operators on ball Banach function spaces

Heng Yang; Jiang Zhou; Heng Yang; Jiang Zhou

doi:10.3934/math.2024152

AIMS Mathematics

2024, Volume 9, Issue 2: 3126-3149. doi: 10.3934/math.2024152

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Research article

Compactness of commutators of fractional integral operators on ball Banach function spaces

Heng Yang ,
Jiang Zhou ^,

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 13 September 2023 Revised: 22 November 2023 Accepted: 27 November 2023 Published: 03 January 2024
MSC : 42B20, 42B25, 42B35, 47B47

Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.
- ball Banach function space,
- fractional integral operator,
- commutator,
- compactness
Citation: Heng Yang, Jiang Zhou. Compactness of commutators of fractional integral operators on ball Banach function spaces[J]. AIMS Mathematics, 2024, 9(2): 3126-3149. doi: 10.3934/math.2024152

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Abstract

Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.

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