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

The paper considers Morrey-type local spaces from LMwp & theta; The main work is the proof of the commutator compactness theorem for the Riesz potential [b, I & alpha;] in local Morrey-type spaces from LMp & theta;w1 to LMw2 q & theta; . We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b, I & alpha;] in local Morrey-type spaces from LMp & theta;w1 to LMw2 q & theta; . 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 & alpha;] in local Morrey-type spaces LMwp & theta;, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LMwp & theta;. 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 & alpha;]. Similar results were obtained for global Morrey-type spaces GMwp & theta; and for generalized Morrey spaces Mpw.