Dunkl convolution and elliptic regularity for Dunkl operators

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2024-09-27 DOI:10.1002/mana.202300370

Dominik Brennecken

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引用次数: 0

Abstract

We discuss in which cases the Dunkl convolution $u *_{k} v$ of distributions $u, v$ , possibly both with non-compact support, can be defined and study its analytic properties. We prove results on the (singular-)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system $A_{n - 1}$ we consider the Riesz distributions ${(R_{α})}_{α \in C}$ and prove that their Dunkl convolution exists and that $R_{α} *_{k} R_{β} = R_{α + β}$ holds.

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Dunkl算子的Dunkl卷积和椭圆正则性

我们讨论在哪些情况下Dunkl卷积u * k v $u*_kv$ 分布u v $u,v$ ，可能同时具有非紧支持，可以定义并研究其解析性质。我们证明了Dunkl卷积的奇异支持性。在此基础上，我们证明了一类Dunkl算子的椭圆正则性定理，称为椭圆Dunkl算子。最后，对于根系A n−1 $A_{n-1}$ 我们考虑Riesz分布(R α) α∈C $(R_\alpha)_{\alpha \in \mathbb {C}}$ 并证明了它们的Dunkl卷积的存在以及R α * k R β = R α+ β $R_\alpha *_kR_\beta = R_{\alpha +\beta }$ hold住。

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来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index