Commutators of Hardy operators in vanishing Morrey spaces

In this paper we study boundedness of commutators of the multi-dimensional Hardy type operators with BMOcoefficients, in weighted global and/or local generalized Morrey spaces Lp,φΠ (Rn,w) and vanishing local Morrey spacesVLp,φloc(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). This study is made in theperspective of posterior applications of the weighted results to some problems in the theory of PDE. We obtain sufficientconditions, in terms of some integral inequalities imposed on φ and w, and also in terms of the Matuszewska-Orlicz indicesof φ and w, for such a boundedness.