Compactness of Commutators for Riesz Potential on Generalized Morrey Spaces

Abstract

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces 𝑀𝑤⁡(·) 𝑝. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces 𝐿𝑝,𝑝 >0. As an application, we prove the compactness of the commutator of the Riesz potential [𝑏,𝐼𝛼] in generalized Morrey spaces, where 𝑏 ∈𝑉⁢𝑀⁢𝑂 (𝑉⁢𝑀⁢𝑂⁡(ℝ𝑛) denote the 𝐵⁢𝑀⁢𝑂-closure of 𝐶∞ 0⁡(ℝ𝑛)). We prove auxiliary statements regarding the connection between the norm of average functions and the norm of the difference of functions in the generalized Morrey spaces. Such results are also of independent interest.