A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions

Abstract

We establish necessary and sufficient conditions on a weight pair governing the boundedness of the Riesz potential operator defined on a homogeneous group G from to , where is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels defined on a product of two homogeneous groups . In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and , respectively. MSC:42B20, 42B25.