On groups of Hölder diffeomorphisms and their regularity

Abstract

We study the set $\mathcal D^{n,β}(\mathbb R^d)$ of orientation preserving diffeomorphisms of $\mathbb R^d$ which differ from the identity by a Hölder $C^{n,β}_0$-mapping, where $n \in \mathbb N_{\ge 1}$ and $β\in (0,1]$. We show that $\mathcal D^{n,β}(\mathbb R^d)$ forms a group, but left translations in $\mathcal D^{n,β}(\mathbb R^d)$ are in general discontinuous. The groups $\mathcal D^{n,β-}(\mathbb R^d) := \bigcap_{α< β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural Fréchet topology) and $\mathcal D^{n,β+}(\mathbb R^d) := \bigcup_{α> β} \mathcal D^{n,α}(\mathbb R^d)$ (with its natural inductive locally convex topology) however are $C^{0,ω}$ Lie groups for any slowly vanishing modulus of continuity $ω$. In particular, $\mathcal D^{n,β-}(\mathbb R^d)$ is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces $C^{n,β}_0$ are ODE closed, in the sense that pointwise time-dependent $C^{n,β}_0$-vector fields $u$ have unique flows $Φ$ in $\mathcal D^{n,β}(\mathbb R^d)$. This includes, in particular, all Bochner integrable functions $u \in L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d))$. For the latter and $n\ge 2$, we show that the flow map $L^1([0,1],C^{n,β}_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,α}(\mathbb R^d))$, $u \mapsto Φ$, is continuous (even $C^{0,β-α}$), for every $α< β$. As an application we prove that the corresponding Trouvé group $\mathcal G_{n,β}(\mathbb R^d)$ from image analysis coincides with the connected component of the identity of $\mathcal D^{n,β}(\mathbb R^d)$.