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

In 1978 Uchiyama gave a proof of the characterization of CMO(R-n) which is the closure of C-comp(infinity) (R-n) in BMO(R-n) . We extend the characterization to the closure of C-comp(infinity) (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 Calderon-Zygmund singular integral operator, I-alpha is the fractional integral operator and b is a function in the Campanato space with variable growth condition.