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

The paper considers Morrey-type local spaces from LM^w_pθ. The main work is the proof of the commutator compactness theorem for the Riesz potential [b, I_α] in local Morrey-type spaces from LM^w1_pθ to LM^w2_qθ. We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b, I_α] in local Morrey-type spaces from LM^w1_pθ to LM^w2_qθ. 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 LM^w_pθ, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LM^w_pθ. 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_α]. Similar results were obtained for global Morrey-type spaces GM^w_pθ and for generalized Morrey spaces M^w_p.