On The Dunkl Intertwining Opereator

TL;DR

This paper provides an integral representation for the Dunkl intertwining operator combined with the heat semigroup, extending its boundedness and positivity properties for a broad class of weights and functions.

Contribution

It introduces a new integral representation for the Dunkl intertwining operator for arbitrary Weyl groups and regular weights, enabling analysis of its boundedness and positivity.

Findings

Integral representation with absolute continuous measures

Extension of the operator to non-differentiable functions

Positivity-preserving property for non-negative weights

Abstract

Dunkl operators are differential-difference operators parametrized by a finite reflection group and a weight function. The commutative algebra generated by these operators generalizes the algebra of standard differential operators and intertwines with this latter by the so-called intertwining operator. In this paper, we give an integral representation for the operator $V_k\circ e^{\Delta/2}$ for an arbitrary Weyl group and a large class of regular weights $k$ containing those of non negative real parts. Our representing measures are absolute continuous with respect the Lebesgue measure in $\Rd$, which allows us to derive out new results about the intertwining operator $V_k$ and the Dunkl kernel $E_k$. We show in particular that the operator $V_k\circ e^{\Delta/2}$ extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables. In the case of non negative weights, this operator is shown to be positivity-preserving.