An extension of the characterization of CMO and its application to compact commutators on Morrey spaces

Abstract

In 1978 Uchiyama gave a proof of the characterization of CMO(ℝ^𝑛) which is the closure of 𝐶^∞_comp(ℝ^𝑛) in BMO(ℝ^𝑛). We extend the characterization to the closure of 𝐶^∞_comp(ℝ^𝑛) in the Campanato space with variable growth condition. As an application we characterize compact commutators [𝑏, 𝑇] and [𝑏, 𝐼_𝛼] on Morrey spaces with variable growth condition, where 𝑇 is the Calderón–Zygmund singular integral operator, 𝐼_𝛼 is the fractional integral operator and 𝑏 is a function in the Campanato space with variable growth condition.