On groups of Hölder 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 Hölder $C_0^{n,\beta }$-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 $C_0^{n,\beta }$ are ODE closed, in the sense that pointwise time-dependent $C_0^{n,\beta }$-vector fields $u$ have unique flows $\Phi$ in ${\mathcal{D}}^{n,\beta }({\mathbb{R}}^d)$. This includes, in particular, all Bochner integrable functions $u\in L^1([0,1],C_0^{n,\beta }({\mathbb{R}}^d,{\mathbb{R}}^d))$. For the latter and $n\ge 2$, we show that the flow map $L^1([0,1],C_0^{n,\beta }({\mathbb{R}}^d,{\mathbb{R}}^d))\to C([0,1],{\mathcal{D}}^{n,\alpha }({\mathbb{R}}^d))$, $u\mapsto \Phi$, 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 ${\mathcal{D}}^{n,\beta }({\mathbb{R}}^d)$.

33 pages; typos corrected, accepted for publication in Trans. Amer. Math. Soc

References in corpus (2)

• An exotic zoo of diffeomorphism groups on ${\mathbb{R}}^n$
• Measurable regularity properties of infinite-dimensional Lie groups

Cited by in corpus (3)

• Manifolds of mappings for continuum mechanics
• The Trouvé group for spaces of test functions
• On the group of A-P diffeomorphisms and its exponential map