The reconstruction theorem in Besov spaces

TL;DR

This paper extends the reconstruction theorem within the theory of regularity structures to Besov spaces with non-integer regularity, providing new embeddings and estimates that align with classical Besov space properties.

Contribution

It generalizes the reconstruction theorem to Besov spaces, broadening the framework for analyzing irregular distributions in stochastic PDEs.

Findings

Extended the reconstruction theorem to Besov spaces with non-integer regularity.

Established embedding theorems for these Besov spaces.

Derived Schauder-type estimates for the extended framework.

Abstract

The theory of regularity structures sets up an abstract framework of modelled distributions generalising the usual H\"older functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of "modelled distributions" into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces $\mathcal{B}^\gamma_{p,q}$ with non-integer regularity indices. We then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and Schauder-type estimates.