# Schwartz functions on real algebraic varieties

## TL;DR

This paper extends the theory of Schwartz functions, tempered functions, and distributions to possibly singular real algebraic varieties, establishing classical properties in affine cases and partial results for non-affine cases.

## Contribution

It introduces definitions and proves that classical properties of Schwartz and tempered functions hold on affine real algebraic varieties, with partial results for non-affine varieties.

## Key findings

- Classical properties of Schwartz functions are valid on affine real algebraic varieties.
- Definitions of Schwartz and tempered functions are extended to singular varieties.
- Partial results are provided for non-affine real algebraic varieties.

## Abstract

We define Schwartz functions, tempered functions and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.