An extension of the characterization of $\mathrm{CMO}$ and its application to compact commutators on Morrey spaces

Abstract

In 1978 Uchiyama gave a proof of the characterization of $\mathrm{CMO}(\mathbb{R}^n)$ which is the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in $\mathrm{BMO}(\mathbb{R}^n)$. We extend the characterization to the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in the Campanato space with variable growth condition. As an application we characterize compact commutators $[b,T]$ and $[b,I_{\alpha}]$ on Morrey spaces with variable growth condition, where $T$ is the Calderón–Zygmund singular integral operator, $I_{\alpha}$ is the fractional integral operator and $b$ is a function in the Campanato space with variable growth condition.