A Fr\'echet Lie group on distributions

TL;DR

This paper introduces a new Fréchet Lie group structure on a space of distributions related to the  product, extending classical operations and providing a rigorous foundation for non-autonomous differential systems.

Contribution

It establishes that the  product defines a Fréchet Lie group on a space of distributions, generalizing convolution and related operations.

Findings

The  product is well-defined on the weak closure of smooth functions.

A subset of this space forms a Fre9chet space .

Invertible elements form a dense subset and a Lie group.

Abstract

Solving non-autonomous systems of ordinary differential equations leads to consider a new product of bivariate distributions called the $\star$~product in the literature. This product, distinct from the convolution product, has recently been used to establish structural results concerning non-autonomous differential systems, yet its formal underpinnings remain unclear. We demonstrate that it is well-defined on the weak closure of the space of smooth functions on a compact subset of $\mathbb{R}^2$. We establish that a subset of this weak closure has the structure of a Fr\'{e}chet space $\mathcal{D}$. The $\star$~product arises from the composition of endomorphisms of that space. Invertible elements of $\mathcal{D}$ form a dense subset of it and a Fr\'{e}chet Lie group for the operation $\star$. This product generalizes the convolution, Volterra compositions of first and second type and induces Schwartz's bracket.