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Modelled distributions of Triebel–Lizorkin type

Volume 252 / 2020

Sebastian Hensel, Tommaso Rosati Studia Mathematica 252 (2020), 251-297 MSC: Primary 46E35, 42B35; Secondary 60H15. DOI: 10.4064/sm180411-11-2 Published online: 23 December 2019

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 to 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 \textit {sewing lemma}, is of paramount importance for the theory. It enables one to reconstruct a modelled distribution as a true distribution on $\mathbb R ^d$ which is locally approximated by this extended set of models or “monomials”. In the original work of Hairer, the error is measured by means of Hölder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labbé. 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.

Authors

  • Sebastian HenselInstitut für Mathematik
    Humboldt-Universität zu Berlin
    Unter den Linden 6
    10099 Berlin, Germany
    and
    Institute of Science and Technology Austria (IST)
    Am Campus 1
    AT-3400 Klosterneuburg, Austria
    e-mail
  • Tommaso RosatiInstitut für Mathematik
    Humboldt-Universität zu Berlin
    Unter den Linden 6
    10099 Berlin, Germany
    e-mail

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