Some properties of the Riesz potentials in Dunkl analysis

TL;DR

This paper investigates the properties of Riesz potentials within Dunkl analysis, focusing on their behavior at infinity and establishing weighted boundedness results, leading to a generalized Sobolev inequality.

Contribution

It provides new insights into the behavior and boundedness of Riesz potentials in Dunkl analysis, including conditions for weighted (Lp,Lq) boundedness and applications to Sobolev inequalities.

Findings

Behavior at infinity of Riesz potentials analyzed

Weighted (Lp,Lq) boundedness established under certain conditions

Weighted generalized Sobolev inequality proved

Abstract

In Dunkl theory on Rd which generalizes classical Fourier analysis, we study first the behavior at infinity of the Riesz potential of a non compactly supported function. Second, we give for 1<p<=q<infinite, weighted (Lp,Lq) boundedness of the Riesz potentials with sufficient conditions. As application, we prove a weighted generalized Sobolev inequality.