Hairer's Reconstruction Theorem without Regularity Structures

TL;DR

This paper provides a simplified, self-contained proof of Hairer's Reconstruction Theorem, making it accessible without prior knowledge of Regularity Structures, and explores its applications in distribution theory.

Contribution

It offers a new elementary proof of the Reconstruction Theorem and characterizes negative H"older spaces using a single test function, independent of Regularity Structures.

Findings

Elementary proof of the Reconstruction Theorem

Characterization of negative H"older spaces

Applications to distribution theory

Abstract

This survey is devoted to Martin Hairer's Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures. Our aim is to give a new self-contained and elementary proof of this Theorem, together with some applications, including a characterization, based on a single arbitrary test function, of negative H\"older spaces. We present the Reconstruction Theorem as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.