Abstract In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case p = 2 , as well as an extension to the coefficient p = 1 that was previously unknown. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings).

中文翻译：

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Dunkl 算子的 Sobolev 型不等式

摘要 在本文中，我们使用两种新方法研究了 Dunkl 设置中的 Sobolev 不等式，这些方法提供了经典情况 p = 2 的更简单基本证明，以及对先前未知的系数 p = 1 的扩展。我们还发现了 Dunkl 梯度的 Sobolev 不等式的尖锐常数的估计。还考虑了相关的不等式和一些改进（纳什不等式，Besov 空间嵌入）。