Modelled distributions of Triebel-Lizorkin type

TL;DR

This paper extends the theory of regularity structures by adapting the analytic framework to Triebel-Lizorkin spaces, enabling more refined local descriptions of functions and distributions, especially in the context of singular SPDEs.

Contribution

It generalizes the analytic component of regularity structures to Triebel-Lizorkin spaces, broadening the scope of local function modeling beyond Hölder and Besov spaces.

Findings

Extended the reconstruction theorem to Triebel-Lizorkin spaces.

Provided a framework for local modeling of distributions in Triebel-Lizorkin spaces.

Enhanced the analytical tools for studying singular SPDEs.

Abstract

In order to provide a local description of a regular function in a small neighbourhood of a point $x$, it is sufficient by Taylor's theorem to know the value of the function as well as all of its derivatives up to the required order at the point $x$ itself. In other words, one could say that a regular function is locally modelled by the set of polynomials. The theory of regularity structures due to Hairer generalizes this observation and provides an abstract setup, which in the application of singular SPDE extends the set of polynomials by functionals constructed from, e.g., white noise. In this context, the notion of Taylor polynomials is lifted to the notion of so-called modelled distributions. The celebrated reconstruction theorem, which in turn was inspired by Gubinelli's sewing lemma, is of paramount importance for the theory. It enables to reconstruct a modelled distribution as a true distribution on $R^d$ which is locally approximated by this extended set of models or "monomials". In the original work of Hairer, the error is measured based on H\"{o}lder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labb\'{e} in subsequent papers. It is the aim of this work to adapt the analytic part of the theory of regularity structures to the scale of Triebel-Lizorkin spaces.