On groups of H\"older diffeomorphisms and their regularity

TL;DR

This paper investigates the structure and regularity properties of groups of H"older diffeomorphisms, establishing their group properties, regularity, and flow behavior, with applications to image analysis and geometric group theory.

Contribution

It introduces new regularity results for H"older diffeomorphism groups, showing their group structure, Lie group properties, and flow regularity, including the coincidence with Trouvé groups.

Findings

H"older diffeomorphism groups form groups but have discontinuous left translations.

Certain subgroups are $C^{0, ext{modulus}}$ Lie groups with smooth right translations.

Flow maps for H"older vector fields are continuous and even $C^{0,eta- ext{alpha}}$.

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 H\"older $C^{n,\beta}_0$-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 Fr\'echet 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 H\"older 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 Trouv\'e group $\mathcal G_{n,\beta}(\mathbb R^d)$ from image analysis coincides with the connected component of the identity of $\mathcal D^{n,\beta}(\mathbb R^d)$.