Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces

Abstract

In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces 𝐺⁡𝑀𝑤⁡(·) 𝑝⁢𝜃. Our main result is the compactness of the commutators of the Riesz potential [𝑏,𝐼𝛼] in global Morrey-type spaces from 𝐺⁡𝑀𝑤1⁡(·) 𝑝1⁢𝜃1 to 𝐺⁡𝑀𝑤2⁡(·) 𝑝2⁢𝜃2. We also present new sufficient conditions for the commutator [𝑏,𝐼𝛼] to be bounded from 𝐺⁡𝑀𝑤1⁡(·) 𝑝1⁢𝜃1 to 𝐺⁡𝑀𝑤2⁡(·) 𝑝2⁢𝜃2. In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential [𝑏,𝐼𝛼] in global Morrey-type spaces 𝐺⁡𝑀𝑤⁡(·) 𝑝⁢𝜃, and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces 𝐺⁡𝑀𝑤⁡(·)𝑝⁢𝜃.