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 (v, w) governing the boundedness of the Riesz potential operatoris 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 I α 1 ,α 2 defined on a product of two homogeneous groups G 1 × G 2 . 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 G 1 × G 2 , respectively. MSC: 42B20; 42B25