Compactness of Commutators for Riesz Potential on Local Morrey-type spaces

Abstract

The paper considers Morrey-type local spaces from LMw p The main work is the proof of the commutator compactness theorem for the Riesz potential [b; I ] in local Morrey-type spaces from LMw1 p to LMw2 q . We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b; I ] in local Morrey-type spaces from LMw1 p to LMw2 q . In the proof of the commutator compactness theorem for the Riesz potential, we essentially use the boundedness condition for the commutator for the Riesz potential [b; I ] in local Morrey-type spaces LMw p , and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LMw p . In the course of proving the commutator compactness theorem for the Riesz potential, we prove lemmas for the commutator ball for the Riesz potential [b; I ]. Similar results were obtained for global Morrey-type spaces GMw p and for generalized Morrey spaces Mwp .