Compactness of Commutators for Singular Integrals on Morrey Spaces

Abstract

Abstract In this paper we characterize the compactness of the commutator [ b , T ] for the singular integral operator on the Morrey spaces . More precisely, we prove that if , the -closure of , then [ b , T ] is a compact operator on the Morrey spaces for ∞ < p < ∞ and 0 < ⋋ < n . Conversely, if and [ b , T ] is a compact operator on the for some p (1 < p < ∞), then . Moreover, the boundedness of a rough singular integral operator T and its commutator [ b , T ] on are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.