On groups of Hölder diffeomorphisms and their regularity
Authors: Nenning David , Rainer Armin 
Published: May 2017

We study the set

D

n
,
β

(

R

d

)

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

of orientation preserving diffeomorphisms of

R

d

\mathbb R^d

which differ from the identity by a Hölder

C
0

n
,
β

C^{n,\beta }_0

-mapping, where

n
∈

N

≥
1

n \in \mathbb N_{\ge 1}

and

β
∈
(
0
,
1
]

\beta \in (0,1]

. We show that

D

n
,
β

(

R

d

)

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

forms a group, but left translations in

D

n
,
β

(

R

d

)

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

are in general discontinuous. The groups
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D Superscript n comma beta minus Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis colon equals intersection Underscript alpha greater-than beta Endscripts script upper D Superscript n comma alpha Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis">

D

n
,
β
−

(

R

d

)
:=

⋂

α
>
β

D

n
,
α

(

R

d

)

\mathcal D^{n,\beta -}(\mathbb R^d) := \bigcap _{\alpha > \beta } \mathcal D^{n,\alpha }(\mathbb R^d)

(with its natural Fréchet topology) and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D Superscript n comma beta plus Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis colon equals union Underscript alpha greater-than beta Endscripts script upper D Superscript n comma alpha Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis">

D

n
,
β
+

(

R

d

)
:=

⋃

α
>
β

D

n
,
α

(

R

d

)

\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
,
ω

C^{0,\omega }

Lie groups for any slowly vanishing modulus of continuity

ω
\omega

. In particular,

D

n
,
β
−

(

R

d

)

\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
,
β

C^{n,\beta }_0

are ODE closed, in the sense that pointwise time-dependent

C
0

n
,
β

C^{n,\beta }_0

-vector fields

u
u

have unique flows

Φ
\Phi

in

D

n
,
β

(

R

d

)

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

. This includes, in particular, all Bochner integrable functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u element-of upper L Superscript 1 Baseline left-parenthesis left-bracket 0 comma 1 right-bracket comma upper C 0 Superscript n comma beta Baseline left-parenthesis double-struck upper R Superscript d Baseline comma double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis">

u
∈

L
1

(
[
0
,
1
]
,

C
0

n
,
β

(

R

d

,

R

d

)
)

u \in L^1([0,1],C^{n,\beta }_0(\mathbb R^d,\mathbb R^d))

. For the latter and

n
≥
2

n\ge 2

, we show that the flow map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1 Baseline left-parenthesis left-bracket 0 comma 1 right-bracket comma upper C 0 Superscript n comma beta Baseline left-parenthesis double-struck upper R Superscript d Baseline comma double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis right-arrow upper C left-parenthesis left-bracket 0 comma 1 right-bracket comma script upper D Superscript n comma alpha Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis right-parenthesis">

L
1

(
[
0
,
1
]
,

C
0

n
,
β

(

R

d

,

R

d

)
)
→
C
(
[
0
,
1
]
,

D

n
,
α

(

R

d

)
)

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
↦
Φ

u \mapsto \Phi

, is continuous (even

C

0
,
β
−
α

C^{0,\beta -\alpha }

), for every

α
>
β

\alpha > \beta

. As an application we prove that the corresponding Trouvé group

G

n
,
β

(

R

d

)

\mathcal G_{n,\beta }(\mathbb R^d)

from image analysis coincides with the connected component of the identity of

D

n
,
β

(

R

d

)

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

.