Tempered distributions and Schwartz functions on definable manifolds

TL;DR

This paper extends the theory of Schwartz functions, tempered functions, and distributions to manifolds definable in polynomially bounded o-minimal structures, generalizing classical properties from the Nash setting and exploring applications in representation theory.

Contribution

It introduces a generalized framework for Schwartz and tempered functions on definable manifolds within polynomially bounded o-minimal structures, establishing their classical properties in this broader context.

Findings

Classical properties of Schwartz and tempered functions hold in polynomially bounded o-minimal structures.

The theory cannot be extended to non-polynomially bounded o-minimal structures.

Potential applications are identified in representation theory.

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.