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

Abstract

The paper considers Morrey-type local spaces from LMpθw The main work is the proof of the commutator compactness theorem for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. 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,Iα] in local Morrey-type spaces LMpθw, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LMpθw. 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,Iα]. Similar results were obtained for global Morrey-type spaces GMpθw and for generalized Morrey spaces Mpw.