Besov Reconstruction

TL;DR

This paper establishes a new reconstruction theorem in the Besov space setting, extending previous results and providing a more elementary approach that avoids wavelet analysis and paraproducts.

Contribution

It generalizes existing reconstruction theorems to Besov spaces using distribution theory, simplifying proofs and broadening applicability.

Findings

Proves a Besov reconstruction theorem extending prior results.

Provides an elementary proof avoiding wavelet analysis.

Offers an alternative proof of a Besov Young multiplication theorem.

Abstract

Reconstruction theorems tackle the problem of building a global distribution on $\mathbb{R}^d$ or on a manifold, given a sufficiently coherent family of local approximations, see [M.Hairer, Invent. Math. 198 (2014), no. 2,269--504], [M,Hairer and Labb\'e, C, J. Funct. Anal. 273 (2017), no. 8, 2578--2618], [Caravenna, F and Zambotti, L, EMS Surv. Math. Sci. 7 (2020), no. 2, 207--251], [Rinaldi, P and Sclavi, F, J. Math. Anal. Appl. 501 (2021), no. 2, 125215, 14 pp] for examples of such results. In this paper, we establish a reconstruction theorem in the Besov setting, generalising results both of [Caravenna, F and Zambotti, L, EMS Surv. Math. Sci. 7 (2020), no. 2, 207--251] and [M,Hairer and Labb\'e, C, J. Funct. Anal. 273 (2017), no. 8, 2578--2618]. While [M,Hairer and Labb\'e, C, J. Funct. Anal. 273 (2017), no. 8, 2578--2618] is written in the context of regularity structures and exploits nontrivial results from wavelet analysis, our calculations follow the more elementary and more general approach of [Caravenna, F and Zambotti, L, EMS Surv. Math. Sci. 7 (2020), no. 2, 207--251]. In particular, as in [Caravenna, F and Zambotti, L, EMS Surv. Math. Sci. 7 (2020), no. 2, 207--251], our results are both stated and proved with tools from the theory of distributions. As an application, we present an alternative proof of a (Besov) Young multiplication theorem which does not require the use of paraproducts.