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 operator lα defined on a homogeneous group G from (Formula presented.) to (Formula presented.), where (Formula presented.) 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 lα1lα2 defined on a product of two homogeneous groups G1×G2 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 G1×G2, respectively.