We prove that b is in BM O(R n ) if and only if the commutator [b, Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from the classical Morrey space L p,λ (R n ) to L q,µ (R n ), where 1 < p < ∞, 0 < α < n, 0 < λ < n − αp, 1/q = 1/p − α/n and λ/p = µ/q. Also we will show that b is in Λβ (R n ) if and only if the commutator [b, Iα] is bounded from the classical Morrey space L p,λ (R n ) to L q,µ (R n ) or from L p,λ (R n ) to L q,λ (R n ), where α and β satisfy some conditions.
We study multi-commutators on the Morrey spaces generated by BMO functions and singular integral operators or by BMO functions and fractional integral operators. We place ourselves in the setting of coming with a Radon measure μ which satisfies a certain growth condition. The Morrey-boundedness of commutators is established by M. Yan and D. Yang. However, the corresponding assertion of Morrey-compactness is still missing. The aim of this paper is to prove that the multi-commutators are compact if one of the BMO functions can be approximated with compactly supported smooth functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.