On groups of Hölder diffeomorphisms and their regularity
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- by David Nicolas Nenning and Armin Rainer PDF
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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
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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