A Characterization of the Two-weight Inequality for Riesz Potentials on Cones of Radially Decreasing Functions

We establish necessary and sufficient conditions on a weight pair $(v,w)$ governing the boundedness of the Riesz potential operator $I_{\alpha }$ defined on a homogeneous group $G$ from $L_{dec,r}^p(w,G)$ to $L^q(v,G)$, where $L_{dec,r}^p(w,G)$ 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 $I_{{\alpha }_1,{\alpha }_2}$ defined on a product of two homogeneous groups $G_1\times G_2$. In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces.