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

TL;DR

This paper characterizes the necessary and sufficient conditions for the boundedness of Riesz potential operators on homogeneous groups, focusing on radially decreasing functions and extending to product kernels, with new results even in Euclidean spaces.

Contribution

It provides the first comprehensive characterization of two-weight inequalities for Riesz potentials on cones of radially decreasing functions, including product kernels on product groups.

Findings

Established necessary and sufficient conditions for boundedness

Extended results to product kernels on product groups

Provided new insights even for Euclidean spaces

Abstract

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 $L^p_{dec,r}(w, G)$ to $L^q(v, G)$, where $L^p_{dec,r}(w, G)$ 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.