Absolute continuity of the measures of the Dunkl intetwining operator and its dualand applications

TL;DR

This paper proves the absolute continuity of measures associated with the Dunkl intertwining operator and its dual, showing they are represented by positive integrable functions with specific support properties, and discusses applications.

Contribution

It establishes the absolute continuity of the measures of the Dunkl intertwining operator and its dual, providing explicit integral representations and support properties.

Findings

Measures are absolutely continuous with positive integrable densities.

Densities have support within specific norm-bounded regions.

Results enable applications in Dunkl analysis and related fields.

Abstract

In this paper we consider the representing measures $\mu_{x},x\in \QTR{Bbb}{R}^{d}$, and $\nu_{y},y\in \QTR{Bbb}{R}^{d}$, of the Dunkl intertwining operator and of its dual. When the multiplicity function is positive, we prove that for all $x\in \QTR{Bbb}{R}_{\QTO{mbox}{reg}}^{d}$ we have $d\mu_{x}(y)=\QTR{cal}{K}(x,y)dy$ and for almost all $y\in \QTR{Bbb}{R}^{d}$ we have $d\nu_{y}(x)=\QTR{cal}{K}(x,y)\omega_{k}(x)dx,$ where $\QTR{cal}{K}(x,.)$ is a positive integrable function on $\QTR{Bbb}{R}^{d}$ with support in $\{y\in \QTR{Bbb}{R}^{d}/\Vert y\Vert \leq \Vert x\Vert \}$ and the function $\QTR{cal}{K}(.,y)$ is locally integrable on $\QTR{Bbb}{R}^{d}$ with respect to the measure $\omega_{k}(x)dx$ and with support in $\{x\in \QTR{Bbb}{R}^{d}/\Vert x\Vert \geq \Vert y\Vert \}$. Next we present some applications of this result.