Besov Reconstruction

The reconstruction theorem tackles the problem of building a global distribution, on ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document} or on a manifold, for a given family of sufficiently coherent local approximations. This theorem is a critical tool within Hairer’s theory of Regularity Structures. In this paper, we establish a reconstruction theorem in the Besov setting, extending recent results of Caravenna and Zambotti. A Besov reconstruction theorem was first formulated by Hairer and Labbé in the context of regularity structures, exploiting nontrivial results from wavelet analysis. Our calculations follow the more elementary approach of coherent germs due to Caravenna and Zambotti. With this formulation our results are both stated and proved with tools from the theory of distributions without the need of the theory of Regularity Structures. As an application, we present an alternative proof of a (Besov) Young multiplication theorem which does not require the use of para-differential calculus.