Abstract We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.

中文翻译：

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可定义流形上的调和分布和 Schwartz 函数

摘要 我们在多项式有界 o 极小结构中可定义的流形上定义 Schwartz 函数、调和函数和调和分布的空间。我们证明了这些空间在 Nash 范畴中的所有经典性质，正如在 Fokko du Cloux 的工作中首次研究的那样，也适用于这个广义设置。我们还表明，在非多项式有界的 o 极小结构中可定义的流形上，不能构建这样的理论。我们介绍了一些可能的应用，主要是在表示论中。