 
Summary: THREE RESULTS IN DUNKL ANALYSIS
B´ECHIR AMRI, JEANPHILIPPE ANKER & MOHAMED SIFI
In memory of Andrzej Hulanicki (19332008),
a distinguished polish mathematician, a guide and a friend,
who has left many orphans in Wroclaw and around the world.
We miss you.
Abstract. In this article, we establish first a geometric PaleyWiener theorem for the
Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the
Lp
Lp
norm of Dunkl translations in dimension 1. Finally we describe more precisely
the support of the distribution associated to Dunkl translations in higher dimension.
1. Introduction
Dunkl theory generalizes classical Fourier analysis on RN
. It started twenty years ago
with Dunkl's seminal work [5] and was further developed by several mathematicians. See
for instance the surveys [16, 7] and the references cited therein.
In this setting, the PaleyWiener theorem is known to hold for balls centered at the
origin. In [9], a PaleyWiener theorem was conjectured for convex neighborhoods of
the origin, which are invariant under the underlying reflection group, and was partially
