Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p → q -boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces L p , φ ( R n , w ) defined by an almost increasing function φ ( r ) and radial type weight w ( | x | ) . We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p → q -boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p → q -boundedness of the Riesz potential operator.