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

Abstract

The paper considers Morrey-type local spaces from LM_pθ^w 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_pθ^w1 to LM_pθ^w2. 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_pθ^w1 to LM_pθ^w2. 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_α] in local Morrey-type spaces LM_pθ^w, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LM_pθ^w. 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_pθ^w and for generalized Morrey spaces M_p^w.