On groups of Hölder diffeomorphisms and their regularity

Abstract

Abstract. We study the set D n,β (R d ) of orientation preserving diffeomorphisms of R d which differ from the identity by a Hölder C n,β 0 -mapping, where n ∈ N ≥1 and β ∈ (0, 1]. We show that with its natural Fréchet topology) andwith its natural inductive locally convex topology) however are C 0,ω Lie groups for any slowly vanishing modulus of continuity ω. In particular, D n,β− (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,βFor the latter and n ≥ 2, we show that the flow mapAs an application we prove that the corresponding Trouvé group G n,β (R d ) from image analysis coincides with the connected component of the identity of D n,β (R d ).

On time-dependent Besov vector fields and the regularity of their flows

“…As it is already stated in [6], we do not know whether we have continuity in the Hölder case without loss of regularity, i.e. of the mapping Fl : and Fl is by Theorem 5.6. in [6] (β − α − 2ε)-Hölder continuous. Since ε is arbitrary we get Hölder continuity of any order γ < β − α.…”

“…(i) For α ∈ (0, 1) and p = ∞, using Remark 5.4, the result is already proved in [6], since Z n,α ∼ = C n,α . So we are left to show the Theorem for (n, α, p) This implies that φ u (t) Z n, 3α 4 ,∞ and, since 3α…”

“…For a bare minimum of properties of this integrability notion, consult e.g. section 2.2 in [6]. Now let us consider the codomain.…”

“…• C n 0 , in [9], • Hölder spaces, in [6], • Sobolev spaces, in [2], • several classes of (un)weighted smooth function spaces, in [7], . .…”

The Trouvé group for spaces of test functions

“…There has been a recent interest in a precise description of the regularity properties of the elements of G A ; see [5] for the Sobolev and [16] for the Hölder case. In [6] similar problems are studied in the context of Banach and Fréchet Lie groups.…”