Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On groups of Hölder diffeomorphisms and their regularity
HTML articles powered by AMS MathViewer

by David Nicolas Nenning and Armin Rainer PDF
Trans. Amer. Math. Soc. 370 (2018), 5761-5794 Request permission

Abstract:

We study the set $\mathcal D^{n,\beta }(\mathbb R^d)$ of orientation preserving diffeomorphisms of $\mathbb R^d$ which differ from the identity by a Hölder $C^{n,\beta }_0$-mapping, where $n \in \mathbb N_{\ge 1}$ and $\beta \in (0,1]$. We show that $\mathcal D^{n,\beta }(\mathbb R^d)$ forms a group, but left translations in $\mathcal D^{n,\beta }(\mathbb R^d)$ are in general discontinuous. The groups $\mathcal D^{n,\beta -}(\mathbb R^d) := \bigcap _{\alpha < \beta } \mathcal D^{n,\alpha }(\mathbb R^d)$ (with its natural Fréchet topology) and $\mathcal D^{n,\beta +}(\mathbb R^d) := \bigcup _{\alpha > \beta } \mathcal D^{n,\alpha }(\mathbb R^d)$ (with its natural inductive locally convex topology) however are $C^{0,\omega }$ Lie groups for any slowly vanishing modulus of continuity $\omega$. In particular, $\mathcal D^{n,\beta -}(\mathbb R^d)$ is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces $C^{n,\beta }_0$ are ODE closed, in the sense that pointwise time-dependent $C^{n,\beta }_0$-vector fields $u$ have unique flows $\Phi$ in $\mathcal D^{n,\beta }(\mathbb R^d)$. This includes, in particular, all Bochner integrable functions $u \in L^1([0,1],C^{n,\beta }_0(\mathbb R^d,\mathbb R^d))$. For the latter and $n\ge 2$, we show that the flow map $L^1([0,1],C^{n,\beta }_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,\alpha }(\mathbb R^d))$, $u \mapsto \Phi$, is continuous (even $C^{0,\beta -\alpha }$), for every $\alpha < \beta$. As an application we prove that the corresponding Trouvé group $\mathcal G_{n,\beta }(\mathbb R^d)$ from image analysis coincides with the connected component of the identity of $\mathcal D^{n,\beta }(\mathbb R^d)$.
References
Similar Articles
Additional Information
  • David Nicolas Nenning
  • Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
  • MR Author ID: 1270132
  • Email: david.nicolas.nenning@univie.ac.at
  • Armin Rainer
  • Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
  • MR Author ID: 752266
  • ORCID: 0000-0003-3825-3313
  • Email: armin.rainer@univie.ac.at
  • Received by editor(s): December 18, 2016
  • Published electronically: April 25, 2018
  • Additional Notes: This work was supported by FWF-Project P 26735-N25
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 5761-5794
  • MSC (2010): Primary 37C10, 58C07, 58D05; Secondary 58B10, 58B25
  • DOI: https://doi.org/10.1090/tran/7269
  • MathSciNet review: 3803147