Reconstruction Theorem for Germs of Distributions on Smooth Manifolds

TL;DR

This paper extends the reconstruction theorem from Euclidean spaces to smooth manifolds, facilitating the development of regularity structures in more general geometric settings.

Contribution

It generalizes the reconstruction theorem for distributions to arbitrary smooth manifolds, broadening the applicability of regularity structures beyond Euclidean spaces.

Findings

Reconstruction theorem proven for distributions on smooth manifolds

Framework compatible with extension of regularity structures to manifolds

Lays groundwork for future analysis on manifold-valued distributions

Abstract

The reconstruction theorem is a cornerstone of the theory of regularity structures [Hai14]. In [CZ20] the authors formulate and prove this result in the language of distributions theory on the Euclidean space $\mathbb{R}^d$, without any reference to the original framework. In this paper we generalize their constructions to the case of distributions over a generic $d$-dimensional smooth manifold $M$, proving the reconstruction theorem in this setting. This is done having in mind the extension of the theory of regularity structures to smooth manifolds.