On the Dunkl intertwining operator

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• 作者：Mostafa Maslouhi ^{mostafa.maslouhi@univ-ibntofail.ac.ma" class="auth_mail" title="E-mail the corresponding author mostafa.maslouhi@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Dunkl operators ; Dunkl kernel ; Dunkl intertwining operator ; Root systems ; Weyl group
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：846-859
• 全文大小：340 K

文摘

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∘e^Δ/2$V_k\circ e^{\mathrm{\Delta }/2}$ for an arbitrary Weyl group and a large class of regular weights k   containing those of nonnegative real parts. Our representing measures are absolutely continuous with respect the Lebesgue measure in R^d${\mathbb{R}}^d$, which allows us to derive out new results about the intertwining operator V_k$V_k$ and the Dunkl kernel E_k$E_k$. We show in particular that the operator V_k∘e^Δ/2$V_k\circ e^{\mathrm{\Delta }/2}$ extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables. In the case of nonnegative weights, this operator is shown to be positivity-preserving.