Tempered Manifolds and Schwartz Functions on them

Ary Shaviv

Published 2018-02-04Version 1

We define and study the category of real tempered smooth manifolds of finite type. This category, which generalizes the category of Nash manifolds, is in a sense the "largest" category that includes all open subsets of $\mathbb{R}^n$ as objects, and on which on all of its objects the Fr\'echet space of Schwartz functions can be defined. In particular we show that all the classical properties that the spaces of Schwartz functions, tempered functions, and tempered distributions have in the Nash category, as first studied in Fokko du Cloux's work, and later in the work of Aizenbud and Gourevitch, also hold in this generalized setting. We also present some possible applications, mainly in representation theory.