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Transactions of the American Mathematical Society

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On groups of Hölder diffeomorphisms and their regularity


Authors: David Nicolas Nenning and Armin Rainer
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
Published electronically: April 25, 2018
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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)$.


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Additional Information

David Nicolas Nenning
Affiliation: Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 A-1090 Wien, Austria
Email: david.nicolas.nenning@univie.ac.at

Armin Rainer
Affiliation: Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 A-1090 Wien, Austria
Email: armin.rainer@univie.ac.at

DOI: https://doi.org/10.1090/tran/7269
Keywords: H\"older spaces, composition and inversion operators, time-dependent H\"older vector fields and their flow, half-Lie groups, ODE closedness, ODE hull
Received by editor(s): December 18, 2016
Published electronically: April 25, 2018
Additional Notes: This work was supported by FWF-Project P 26735-N25
Article copyright: © Copyright 2018 American Mathematical Society

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