On groups of H\"older diffeomorphisms and their regularity

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 $\mathcal D^{n,\beta}(\mathbb R^d)$.