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

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 $x\in \text{QTR}Bbb{R}_{\text{QTO}mboxreg}^d$ we have $d{\mu }_x(y)=\text{QTR}calK(x,y)dy$ and for almost all $y\in \QTR{Bbb}{R}^{d}$wehave$d\nu_{y}(x)=\QTR{cal}{K}(x,y)\omega_{k}(x)dx,$where$\QTR{cal}{K}(x,.)$isapositiveintegrablefunctionon$\QTR{Bbb}{R}^{d}$withsupportin${y\in \QTR{Bbb}{R}^{d}/\Vert y\Vert \leq \Vert x\Vert }$ and the function $\text{QTR}calK(\mathrm{.},y)$ is locally integrable on $\text{QTR}Bbb{R}^d$ 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.