Commutators of Hardy operators in vanishing Morrey spaces

摘要

In this paper we study boundedness of commutators of the multi-dimensional Hardy type operators with BMO coefficients, in weighted global and/or local generalized Morrey spaces L_∏~(p,Φ)(R~n, w) and vanishing local Morrey spaces VL_(loc)~(p,Φ)(R~n,w) defined by an almost increasing function Φ(r) and radial type weight w(|x|). This study is made in the perspective of posterior applications of the weighted results to some problems in the theory of PDE. We obtain sufficient conditions, in terms of some integral inequalities imposed on Φ and w, and also in terms of the Matuszewska-Orlicz indices of Φ and w, for such a boundedness.