On groups of Hölder diffeomorphisms and their regularity
aa r X i v : . [ m a t h . C A ] M a y ON GROUPS OF H ¨OLDER DIFFEOMORPHISMSAND THEIR REGULARITY
DAVID NICOLAS NENNING AND ARMIN RAINER
Abstract.
We study the set D n,β ( R d ) of orientation preserving diffeomor-phisms of R d which differ from the identity by a H¨older C n,β -mapping,where n ∈ N ≥ and β ∈ (0 , D n,β ( R d ) forms a group,but left translations in D n,β ( R d ) are in general discontinuous. The groups D n,β − ( R d ) := T α<β D n,α ( R d ) (with its natural Fr´echet topology) and D n,β + ( R d ) := S α>β D n,α ( R d ) (with its natural inductive locally convextopology) however are C ,ω Lie groups for any slowly vanishing modulus ofcontinuity ω . In particular, D n,β − ( R d ) is a topological group and a so-calledhalf-Lie group (with smooth right translations). We prove that the H¨olderspaces C n,β are ODE closed, in the sense that pointwise time-dependent C n,β -vector fields u have unique flows Φ in D n,β ( R d ). This includes, in partic-ular, all Bochner integrable functions u ∈ L ([0 , , C n,β ( R d , R d )). For thelatter and n ≥
2, we show that the flow map L ([0 , , C n,β ( R d , R d )) → C ([0 , , D n,α ( R d )), u Φ, is continuous (even C ,β − α ), for every α < β .As an application we prove that the corresponding Trouv´e group G n,β ( R d )from image analysis coincides with the connected component of the identityof D n,β ( R d ). Introduction
Let E be a Banach space of functions f : R d → R d which is continuously embed-ded in C ( R d , R d ), i.e., C -mappings which vanish together with its first derivativeat infinity. Let u : [0 , × R d → R d be a pointwise time-dependent E -vector field , i.e., u ( t, · ) ∈ E for all t , u ( · , x ) is measurable for all x , and t
7→ k u ( t, · ) k E is integrable.It is well-known that the corresponding pointwise flow(1.1) Φ( t, x ) = x + Z t u ( s, Φ( s, x )) ds, x ∈ R d , t ∈ [0 , , is a C -diffeomorphism of R d at any t . The set of all diffeomorphisms Φ(1 , · ) attime 1 which arise in this way form a group G E , which we call the Trouv´e group of E , since this constructions is due to Trouv´e [27]; details can be found in the book[28].In general, not much is known about the Trouv´e group. We are especially in-terested in precise regularity properties of its elements. This is intimately relatedto the question as to whether E is ODE closed , i.e., G E ⊆ Id + E , and if not, whatthe ODE hull of E is. We define the ODE hull to be the intersection of all ODE Date : July 9, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
H¨older spaces, composition and inversion operators, time-dependentH¨older vector fields and their flow, half-Lie groups, ODE closedness, ODE hull.Supported by FWF-Project P 26735-N25. closed spaces continuously embedded in C ( R d , R d ) and continuously containing E ,see Section 4.2; here it is reasonable to allow for locally convex spaces (mutatismutandis) instead of just Banach spaces.ODE closedness is closely related to stability and continuity or smoothness prop-erties of composition of mappings. Indeed, it has been widely studied in the contextof regular infinite dimensional Lie groups; cf. [10]. Our results are not covered bythe general theory from [10]; H¨older (diffeomorphism) groups fail to be Lie groups.In this paper we explore these questions in the case that E is a global H¨olderspace C n,β ( R d , R d ), n ∈ N ≥ , β ∈ (0 , G n,β ( R d ), it is natural to look at theset of orientation preserving diffeomorphisms of R d which differ from the identityby a C n,β -mapping, i.e., D n,β ( R d ) := (cid:8) Φ ∈ Id + C n,β ( R d , R d ) : det d Φ( x ) > ∀ x ∈ R d (cid:9) . We will show that D n,β ( R d ) is a group with respect to composition, but it is nota topological group: left translations and inversion are in general not continuous.Left translations become continuous if the outer function is slightly more regular: φ ψ ◦ (Id + φ ) is continuous from C n,α ( R d , R d ) → C n,α ( R d , R d ) if ψ ∈ C n,β ( R d , R d )and α < β (the same holds if φ ∈ C n, ( R d , R d ) and ψ ∈ C n +1 ,β ( R d , R d )). Similarly,Φ Φ − is continuous from D n,β ( R d ) to D n,α ( R d ) if α < β . This motivates thedefinitions D n,β − ( R d ) := (cid:8) Φ ∈ Id + C n,β − ( R d , R d ) : det d Φ( x ) > ∀ x ∈ R d (cid:9) , D n,β + ( R d ) := (cid:8) Φ ∈ Id + C n,β +0 ( R d , R d ) : det d Φ( x ) > ∀ x ∈ R d (cid:9) , where C n,β − := T { C n,α : 0 < α < β } and C n,β +0 := S { C n,α : β < α < } ,equipped with the natural projective, resp. inductive locally convex topology. Weprove that D n,β ± ( R d ) are C ,ω Lie groups (see Section 3.3) for every slowly vanish-ing modulus of continuity ω , i.e.,lim inf t ↓ ω ( t ) t γ > γ > . This regularity cannot be improved; see Proposition 3.13. In particular, D n,β − ( R d )are topological groups (which remains open for D n,β + ( R d ) since the underlying lo-cally convex topology and the c ∞ -topology fall apart in this case). The right trans-lations are bounded affine linear (in the chart representation) and hence smooth.Consequently, D n,β − ( R d ) are also half-Lie groups as defined in [20].In the second part of the paper we study flows of time-dependent C n,β -vectorfields. Here we distinguish between:(1) Pointwise time-dependent C n,β -vector fields , i.e., mappings u : [0 , × R d → R d such that u ( t, · ) ∈ C n,β ( R d , R d ) for all t ∈ [0 , u ( · , x ) is measurablefor all x ∈ R d , and t
7→ k u ( t, · ) k n,β is integrable. (This corresponds to thenotion defined at the beginning of the introduction.)(2) Strong time-dependent C n,β -vector fields , i.e., Bochner integrable functions u ∈ L ([0 , , C n,β ( R d , R d )).The latter notion involves strong measurability which entails that the image u ([0 , C n,β arenon-separable. If u satisfies (2) then u ∧ satisfies (1), the converse is false. (For N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 3 f ∈ Z X × Y we consider f ∨ ∈ ( Z Y ) X defined by f ∨ ( x )( y ) = f ( x, y ), and with g ∈ ( Z Y ) X we associate g ∧ ∈ Z X × Y with g ∧ ( x, y ) = g ( x )( y ).)This deficiency has the effect that Carath´eodory’s solution theory for ODEs onBanach spaces which are Bochner integrable in time is not well suited for the H¨olderspace setting. Instead we work with pointwise estimates which has the additionalbenefit that our proofs only require the weaker assumptions in (1).We show that, for all n ∈ N ≥ , β ∈ (0 , C n,β -vectorfields u have unique pointwise flows Φ ∈ C ([0 , , D n,β ( R d )); in particular, C n,β isODE closed (although composition in D n,β is not continuous!). As a consequence,for u ∈ L ([0 , , C n,β ( R d , R d )), the identity (1.1) lifts to an identity in D n,α ( R d ),for each α < β (see Theorem 5.3):Φ ∨ ( t ) = Id + Z t u ( s ) ◦ Φ ∨ ( s ) ds, t ∈ [0 , . Furthermore, we identify the corresponding Trouv´e group:(1.2) G n,β ( R d ) = D n,β ( R d ) , where D n,β ( R n ) denotes the connected component of the identity in D n,β ( R n ).Thus there seems to be no natural topology on G n,β ( R d ) which makes it a topologicalgroup. On the other hand, we also get(1.3) G n,β − ( R d ) = D n,β − ( R d ) , G n,β + ( R d ) = D n,β + ( R d ) which endows G n,β ± ( R d ) with a C ,ω Lie group structure, for every slowly vanishingmodulus of continuity ω ; on G n,β − ( R d ) we also get a topological group structureand a half-Lie group structure. We wish to point out that our proof of (1.2)subsequently shows that the equality (1.2) also holds if in the definition of theTrouv´e group G n,β ( R d ) one restricts to pointwise time-dependent C n,β -vector fieldswhich are piecewise C n in time; see Remark 4.8.In the third part we investigate the continuity of the flow map u Φ. We findthat as a mapping(1.4) L ([0 , , C n,β ( R d , R d )) → C ([0 , , D n,α ( R d ))the flow map is • bounded, if n ∈ N ≥ and 0 < α ≤ β ≤ • continuous, even C ,β − α , if n ∈ N ≥ and 0 < α < β ≤ L ([0 , , C n,β − ( R d , R d )) → C ([0 , , D n,β − ( R d ))the flow map is bounded for all n ≥ C ,ω if n ≥ β ∈ (0 , ω .In [2] similar results were obtained in the Sobolev case E = H s ( R d , R d ), for s > d/ H s is ODE closed and that G s ( R d ) = D s ( R d ) , where G s ( R d ) denotes the corresponding Trouv´e group and D s ( R d ) the group oforientation preserving diffeomorphisms of R d which differ from the identity by amapping in H s ( R d , R d ). The methods are quite different: thanks to the fact that D s ( R d ) is a topological group (cf. [11]) Carath´eodory’s solution theory for ODEs onBanach spaces which are Bochner integrable in time is well suited for this setting. D.N. NENNING AND A. RAINER
The paper is structured as follows. We fix notation and present the main tech-nical tools in Section 2. We also review some results on the composition of H¨olderfunctions essentially due to [3]; since we need slightly altered versions we give proofsbut relegate them to Appendix A. In Section 3 we investigate the groups D n,β ( R d ), D n,β − ( R d ) and D n,β + ( R d ). We prove ODE closedness of C n,β and the identities(1.2) and (1.3) in Section 4. In Section 5 we study the continuity of the flow maps(1.4) and (1.5). Acknowledgement.
We are indebted to Peter Michor who brought the Trouv´egroup to our attention and proposed the notions of ODE closedness and ODE hull.2.
Definitions and preliminary results
H¨older spaces.
Let k ∈ N , α ∈ (0 , E, F be Banach spaces and let U ⊆ E be open. We consider the global H¨older space C k,αb ( U, F ) := (cid:8) f ∈ C k ( U, F ) : k f k k,α < ∞ (cid:9) , where k f k k,α := max {k f k k , [ f ] k,α } , k f k k := sup {k f ( l ) ( x ) k L l : x ∈ U, ≤ l ≤ k } , [ f ] k,α := sup x,y ∈ U, x = y k f ( k ) ( x ) − f ( k ) ( y ) k L k k x − y k α . Here f ( l ) = d l f : E → L l ( E ; F ) is the Fr´echet derivative of order l and L l ( E ; F )denotes the vector space of continuous l -linear mappings endowed with the operatornorm k · k L l .We denote by C k,α ( E, F ) the subspace of those mappings f ∈ C k,α ( E, F ) thattend to 0 at infinity together with all their derivatives up to order k , i.e., for every ǫ > r > k f ( l ) ( x ) k L l ≤ ǫ if k x k > r and 0 ≤ l ≤ k .All these spaces are Banach spaces.Local H¨older spaces are denoted by C k,α , i.e., f ∈ C k,α ( U, F ) if each x ∈ U hasa neighborhood V in U such that f | V ∈ C k,αb ( V, F ).Let us recall interpolation and inclusion relations for H¨older spaces. In thefollowing C n, b := C nb and k · k n, := k · k n . Lemma 2.1 ([3, 3.1]) . Let n ∈ N and ≤ α < β < γ ≤ and set µ := γ − βγ − α . Then k f k n,β ≤ M α k f k µn,α k f k − µn,γ , f ∈ C n,γb ( E, F ) , where M := 2 and M α := 1 for α > . Lemma 2.2 ([3, 3.7]) . Let m, n ∈ N and α, β ∈ [0 , with m + α ≤ n + β . Then C n,βb ( E, F ) ⊆ C m,αb ( E, F ) and k f k m,α ≤ k f k n,β , f ∈ C n,βb ( E, F ) . N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 5
The Bochner integral.
Cf. [4]. Let I = [ a, b ] ⊆ R be a closed interval (withthe Lebesgue measure). Let E be a Banach space. A measurable function f : I → E is Bochner integrable if there is a sequence of integrable simple functions s n : I → E such that s n → f a.e. (i.e., f is strongly measurable ) and R ba k f − s n k dt →
0. Inthis case the
Bochner integral is defined by Z ba f dt = lim n →∞ Z ba s n dt. By the Pettis measurability theorem, f : I → E is strongly measurable if and onlyif it is weakly measurable (i.e., λ ◦ f is measurable for all λ ∈ E ∗ ) and essentiallyseparable valued (i.e., f ( I \ N ) is separable in E for some null set N ). A stronglymeasurable function f : I → E is Bochner integrable if and only if R ba k f k dt < ∞ .Then the triangle inequality holds: (cid:13)(cid:13)(cid:13) Z ba f dt (cid:13)(cid:13)(cid:13) ≤ Z ba k f k dt. If T : E → F is a bounded linear operator into another Banach space F then T f : I → F is Bochner integrable and T Z ba f dt = Z ba T f dt.
We will use the following version of the fundamental theorem of calculus.
Lemma 2.3. If f : [ a, b ] → E is continuous, then ddt Z ta f ( s ) ds = f ( t ) , for all t ∈ I . If f : [ a, b ] → E is C , then f ( b ) − f ( a ) = Z ba f ′ ( s ) ds. It is then straightforward to deduce a mean value inequality for C -mappingsbetween Banach spaces.2.3. Carath´eodory type ODEs.
Next we collect some results on Carath´eodorytype differential equations. Those are certain ODEs on Banach spaces whose righthand side is not continuous in time. We refer to [1] and to the appendix in [2].Let E be a Banach space, U ⊆ E some open subset and I = [ t , t ] some realinterval. We say that f : I × U → E satisfies the Carath´eodory property if:(i) For every t ∈ I the mapping f ( t, · ) : U → E is continuous.(ii) For every x ∈ U the mapping f ( · , x ) : I → E is strongly measurable.Also the notion of solution of such an ODE is weakened: we say a continuouscurve Φ : I → U is a solution of the initial value problem ∂ t x = f ( t, x ) , x ( t ) = x (2.1)if and only if s f ( s, Φ( s )) is Bochner integrable andΦ( t ) = x + Z tt f ( s, Φ( s )) ds, t ∈ I. (2.2) D.N. NENNING AND A. RAINER
This already implies that Φ : I → U is continuous. It is actually absolutely con-tinuous in the sense that there exists a Bochner integrable γ : I → E such thatΦ( t ) = Φ( t ) + R tt γ ( s ) ds ; in particular, Φ is differentiable a.e. and Φ ′ = γ a.e. (see[10, Lemma 1.28]).The next theorem is the central existence and uniqueness result for Carath´eodorytype differential equations; it is taken from [2, Thm. A.2]. Theorem 2.4.
Let I = [ t , t ] and let f : I × U → E have the Carath´eodoryproperty. Let B ( x , ε ) := { x ∈ E : k x − x k < ǫ } ⊆ U . In addition let m, l bepositive locally integrable functions defined on I such that the estimates k f ( t, x ) − f ( t, x ) k ≤ l ( t ) k x − x kk f ( t, x ) k ≤ m ( t ) are valid for almost all t and all x, x , x ∈ B ( x , ε ) . Let δ be such that Z t + δt m ( s ) ds < ε, then (2.1) has a unique solution φ : [ t , t + δ ] → B ( x , ε ) in the sense of (2.2) . If the ODE is linear, we have global existence in time:
Theorem 2.5.
Let I = [ t , t ] . Let A : I → L ( E ) and b : I → E be Bochnerintegrable. Then for all x ∈ E there exists a unique solution on I of ∂ t x ( t ) = A ( t ) · x ( t ) + b ( t ) , x ( t ) = x in the sense of (2.2) . Composition in H¨older spaces.
Let us review some regularity results forthe composition in H¨older spaces due to [3]. But in contrast to [3], we need theresults for mappings F : R d → R d of the form F = Id + f where f is in some H¨olderclass; note that Id is unbounded and hence not a member of any C n,βb ( R d , R d ). Forthis reason it is convenient to introduce the seminorm[ F ] n := k F ( n ) k = sup x ∈ R d k F ( n ) ( x ) k L n . If F = Id + f and n ≥
1, then(2.3) [ F ] n ≤ f ] n . It is easy to adapt the proofs in [3] to our needs; they are outlined in Appendix Afor completeness’ sake.
Proposition 2.6 ([3, 4.2]) . Let
E, F, G, H be Banach spaces and U ⊆ E open. Let m ∈ N , α ∈ (0 , , and b : F × G → H be a bilinear continuous mapping. Then b ∗ : C m,αb ( U, F ) × C m,αb ( U, G ) → C m,αb ( U, H ) , defined by b ∗ ( f, g )( x ) := b ( f ( x ) , g ( x )) , isbilinear, continuous, and k b ∗ k ≤ m +1 k b k . The following theorem shows stability under composition and continuity of theright translation. We will denote by f ⋆ the pull-back by Id + f , i.e., f ⋆ := (Id + f ) ∗ . Theorem 2.7 ([3, 6.2]) . Let m ∈ N ≥ and α ∈ (0 , . Let f ∈ C m,αb ( R d , R d ) and g ∈ C m,αb ( R d , G ) for some Banach space G . Then g ◦ (Id + f ) ∈ C m,αb ( R d , G ) andthere exists a constant M = M ( m ) ≥ such that (2.4) k g ◦ (Id + f ) k m,α ≤ M k g k m,α (1 + k f k m,α ) m + α . N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 7
In particular, for every fixed f ∈ C m,αb ( R d , R d ) , the linear mapping f ⋆ : C m,αb ( R d , G ) → C m,αb ( R d , G ) , g f ⋆ ( g ) := g ◦ (Id + f ) is continuous. Continuity of the left translation is the content of the following theorem. Wedenote by B m,α ( f, δ ) := { g ∈ C m,αb : k f − g k m,α < δ } the open ball with radius δ centered at f . By g ⋆ we mean the push-forward by g precomposed with translationby Id, i.e., g ⋆ := g ∗ ◦ (Id + · ). Theorem 2.8 ([3, 6.2]) . Let m ∈ N ≥ and α, β ∈ (0 , , α < β . Let g ∈ C m,βb ( R d , G ) where G is some Banach space. Then, for every f ∈ C m,αb ( R d , R d ) , R > , and f , f ∈ B m,α ( f , R ) , (2.5) k g ⋆ ( f ) − g ⋆ ( f ) k m,α ≤ M k g k m,β k f − f k β − αm,α , where M = M ( m, k f k m,α , R ) . In particular, g ⋆ : C m,αb ( R d , R d ) → C m,αb ( R d , G ) , f g ⋆ ( f ) := g ◦ (Id + f ) is continuous. It follows that composition is even jointly continuous.
Corollary 2.9.
Let m ∈ N ≥ , < α < β ≤ and G some Banach space. Then,for all f ∈ C m,αb ( R d , R d ) , g ∈ C m,βb ( R d , G ) , R > , f , f ∈ B m,α ( f , R ) , and g , g ∈ B m,β ( g , R ) , (2.6) k g ◦ (Id + f ) − g ◦ (Id + f ) k m,α ≤ M (cid:0) k g − g k m,α + k f − f k m,α (cid:1) β − α , where M = M ( m, k f k m,α , k g k m,β , R ) . In particular, comp : C m,βb ( R d , G ) × C m,αb ( R d , R d ) → C m,αb ( R d , G ) , ( g, f ) g ◦ (Id + f ) is continuous. We will also need the following result on C left translations. Theorem 2.10 ([3, 6.7]) . Let m ∈ N ≥ and α, β ∈ (0 , , α < β . Let g ∈ C m +1 ,βb ( R d , G ) where G is some Banach space. Then g ⋆ : C m,αb ( R d , R d ) → C m,αb ( R d , G ) is continuously differentiable. Together with Lemma 2.3, Theorem 2.10 implies Lipschitz continuity of the lefttranslation in the following cases; but see also Theorem 2.14 below.
Corollary 2.11.
Let m ∈ N ≥ and < α < β ≤ . Let g ∈ C m +1 ,βb ( R d , G ) . Then g ⋆ : C m,αb ( R d , R d ) → C m,αb ( R d , G ) satisfies for all f , f ∈ C m,αb ( R d , R d ) , k g ⋆ ( f ) − g ⋆ ( f ) k m,α ≤ M k g k m +1 ,β (1 + max i =1 , k f i k m,α ) m +1 k f − f k m,α . Remark 2.12.
Let us stress the fact that left translation ceases to be continuous,resp. differentiable, if in Theorem 2.8, resp. Theorem 2.10, g is merely of class C m,αb ,resp. C m +1 ,αb ; see [3] and also Lemma 3.5.We shall make frequent use of the Fa`a di Bruno formula for Banach spaces: Let E, F, G be Banach spaces, let f : E ⊇ U → F and g : F ⊇ V → G be k times D.N. NENNING AND A. RAINER
Fr´echet differentiable, and assume f ( U ) ⊆ V . Then g ◦ f : U → G is k times Fr´echetdifferentiable, and for all x ∈ U , d k ( g ◦ f )( x ) = sym k X l =1 X γ ∈ Γ( l,k ) c γ g ( l ) ( f ( x )) (cid:16) f ( γ ) ( x ) , . . . , f ( γ l ) ( x ) (cid:17) , (2.7)where Γ( l, k ) := { γ ∈ N l> : | γ | = k } , c γ := k ! l ! γ ! , and sym denotes symmetrizationof multilinear mappings.Fa`a di Bruno’s formula applied to a function h : U → H of the form h ( x ) = b ( f ( x ) , g ( x )), where f, g are k times Fr´echet differentiable functions defined on acommon domain U ⊆ E and b : F × G → H is a continuous bilinear map, gives d k h ( x ) = sym k X l =0 (cid:18) kl (cid:19) b ( f ( l ) ( x ) , g ( k − l ) ( x )) . (2.8)This formula is of particular use when h ( x ) = dg ( x )( f ( x )), where f, g : R d → R d ,i.e. the bilinear map takes the form b : L ( R d , R d ) × R d → R d , ( l, x ) l ( x ). Remark 2.13.
Fa`a di Bruno’s formula (2.7) implies that for f : R d → R d and g : R d → G both in C k , we have g ◦ (Id + f ) ∈ C k ( R d , G ). So the stated regularityresults for the composition hold as well for C m,αb , etc., replaced by C m,α , etc.2.5. Convenient calculus.
Occasionally, we shall use some tools from convenientcalculus which extends differential calculus beyond Banach spaces; the main refer-ence is [15], see also [9] and the three appendices in [16]. Let us briefly describe theconcepts and results we will need.Let E be a locally convex vector space. A curve c : R → E is called C ∞ if allderivatives exist and are continuous. It can be shown that the set C ∞ ( R , E ) of C ∞ -curves in E does not depend on the locally convex topology of E , only on itsassociated bornology.The c ∞ -topology on E is the final topology with respect to C ∞ ( R , E ); equiv-alently it is the final topology with respect to all Lipschitz curves or all Mackey-convergent sequences in E . In general the c ∞ -topology is finer than the givenlocally convex topology, and it is not a vector space topology; for Fr´echet spacesthe topologies coincide.A locally convex vector space E is said to be a convenient vector space if it isMackey-complete; equivalently, a curve c : R → E is C ∞ if and only if λ ◦ c is C ∞ for all continuous (equivalently bounded) linear functionals λ on E .Let E , F , and G be convenient vector spaces, and let U ⊆ E be c ∞ -open. A map-ping f : U → F is called C ∞ , if f ◦ c ∈ C ∞ ( R , F ) for all c ∈ C ∞ ( R , U ). For mappingson Fr´echet spaces this notion of smoothness coincides with all other reasonable def-initions. Multilinear mappings are C ∞ if and only if they are bounded. The space C ∞ ( U, F ) with the initial structure with respect to all mappings f λ ◦ f ◦ c , c ∈ C ∞ ( R , E ) and λ ∈ E ∗ , is again convenient. The exponential law holds: For c ∞ -open V ⊆ F , C ∞ ( U, C ∞ ( V, G )) ∼ = C ∞ ( U × V, G )is a linear diffeomorphism of convenient vector spaces. A linear mapping f : E → C ∞ ( V, G ) is C ∞ (bounded) if and only if ev v ◦ f : E → G is C ∞ for all v ∈ V .There are, however, C ∞ -mappings which are not continuous with respect to theunderlying locally convex topology; clearly they are continuous for the c ∞ -topology. N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 9
Beside the class C ∞ due to [7], [12], [13], convenient calculus was developed forthe holomorphic class [21], the real analytic class [14], and all reasonable ultradif-ferentiable classes [16], [17], [19], [26].For the classes C k,α ( k ∈ N , α ∈ (0 , α = 1) and by [6], [5]. Let E , F beconvenient vector spaces, and let U ⊆ E be c ∞ -open. A curve c : R → F is locally α -H¨older continuous , we write c ∈ C ,α ( R , F ), if for each bounded interval I ⊆ R , n c ( t ) − c ( s ) | t − s | α : t, s ∈ I, t = s o is bounded in F . A curve c : R → F is C k,α , i.e., c ∈ C k,α ( R , F ), if all derivatives upto order k exist and are locally α -H¨older continuous. A mapping f : U → F betweenconvenient vector spaces is called C k,α , if f ◦ c ∈ C k,α ( R , F ) for all c ∈ C ∞ ( R , U ).If E and F are Banach spaces, then f is C ,α in this senses if and only if it is inthe sense of Section 2.1, i.e., k f ( x ) − f ( x ) k / k x − y k α is locally bounded; see [5],[15, 12.7], or [18, Lemma], and note that this is a special case of Lemma 3.9 below.2.6. An application of convenient calculus.
We finish this section with a resultwhich is not contained in [3]: if α = β in Theorem 2.10, resp. Corollary 2.11, theleft translation g ⋆ is still locally Lipschitz. Of course, Remark 2.13 applies to thistheorem as well. In contrast to Corollary 2.11, we do not get an explicit bound forthe Lipschitz constant. Theorem 2.14.
Let m ∈ N ≥ and < α ≤ . Let g ∈ C m +1 ,αb ( R d , R d ) . Then g ⋆ : C m,αb ( R d , R d ) → C m,αb ( R d , R d ) is locally Lipschitz.Proof. It suffices to check that g ⋆ maps C ∞ -curves to C , -curves; cf. Section 2.5.That t f ( t, · ) is C ∞ in C m,αb ( R d , R d ) means, by [9, 4.1.19], that, for all k ∈ N , k ∂ kt f ( t, · ) k m,α is locally bounded in t .Let h ( t, x ) := g ( x + f ( t, x )). Then, if F := Id + f , h ( t, x ) − h ( s, x ) = Z ts ∂ τ h ( τ, x ) dτ = Z ts dg ( F ( τ, x )) ∂ τ f ( τ, x ) dτ and, by (2.8), d kx h ( t, x ) − d kx h ( s, x ) = Z ts d kx (cid:0) dg ( F ( τ, x )) ∂ τ f ( τ, x ) (cid:1) dτ = sym k X j =0 (cid:18) kj (cid:19) Z ts d jx (cid:0) dg ( F ( τ, x )) (cid:1) ∂ τ d k − jx f ( τ, x ) dτ. With Fa`a di Bruno’s formula (2.7), d jx (cid:0) dg ( F ( τ, x )) (cid:1) = sym j X l =1 X γ ∈ Γ( l,j ) c γ g ( l +1) ( F ( τ, x )) (cid:0) d γ x F ( τ, x ) , . . . , d γ l x F ( τ, x ) , (cid:1) it is easy to see that t h ( t, · ) is locally Lipschitz into C mb ( R d , R d ).It remains to prove that t h ( t, · ) is locally Lipschitz into C m,αb ( R d , R d ). Tothis end we have to show that [ h ( t, · ) − h ( s, · )] m,α t − s is locally bounded, i.e., for each bounded interval I , the set n d mx h ( t, x ) − d mx h ( t, y ) − d mx h ( s, x ) + d mx h ( s, y ) k x − y k α | t − s | : x = y ∈ R n , s = t ∈ I o must be bounded. Without loss of generality we can assume that k x − y k ≤ k x − y k ≤ k x − y k α ; if k x − y k ≥ t h ( t, · ) is locally Lipschitz into C mb ( R d , R d ). Let us define A γ,i = A γ,i ( x, y ) := (cid:0) d γ x F ( τ, x ) , . . . , d γ i x F ( τ, x ) , d γ i +1 x F ( τ, y ) , . . . , d γ l x F ( τ, y ) (cid:1) and B γ,h := ((cid:0) A γ,l , ∂ t d m − jx f ( τ, x ) (cid:1) if h = l + 1 , (cid:0) A γ,h , ∂ t d m − jx f ( τ, y ) (cid:1) if h ≤ l. Then g ( l +1) ( F ( τ, x )) (cid:0) B γ,l +1 (cid:1) − g ( l +1) ( F ( τ, y )) (cid:0) B γ, (cid:1) = g ( l +1) ( F ( τ, x ))( B γ,l +1 ) − g ( l +1) ( F ( τ, y ))( B γ,l +1 )+ l +1 X h =1 g ( l +1) ( F ( τ, y ))( B γ,h ) − g ( l +1) ( F ( τ, y ))( B γ,h − ) . For the first summand (cid:13)(cid:13) g ( l +1) ( F ( τ, x ))( B γ,l +1 ) − g ( l +1) ( F ( τ, y ))( B γ,l +1 ) (cid:13)(cid:13) L m ≤ (cid:13)(cid:13) g ( l +1) ( F ( τ, x )) − g ( l +1) ( F ( τ, y )) (cid:13)(cid:13) L l +1 (1 + k f ( τ, · ) k m ) m k ∂ t f ( τ, · ) k m − j ≤ ( k g k m +1 [ F ( τ, · )] k x − y k (1 + k f ( τ, · ) k m ) m k ∂ t f ( τ, · ) k m if l < m, k g k m +1 ,α [ F ( τ, · )] α k x − y k α (1 + k f ( τ, · ) k m ) m k ∂ t f ( τ, · ) k m if l = m. For the other summands we observe that, by multilinearity, g ( l +1) ( F ( τ, y ))( B γ,h ) − g ( l +1) ( F ( τ, y ))( B γ,h − ) = g ( l +1) ( F ( τ, y )) (cid:0) ♯ (cid:1) , where ♯ = (cid:0) . . . , d γ h − x F ( τ, x ) , d γ h x F ( τ, x ) − d γ h x F ( τ, y ) , d γ h +1 x F ( τ, y ) , . . . (cid:1) . Hence, if h ≤ l , (cid:13)(cid:13) g ( l +1) ( F ( τ, y ))( B γ,h ) − g ( l +1) ( F ( τ, y ))( B γ,h − ) (cid:13)(cid:13) L m +1 ≤ k g k m +1 (1 + k f ( τ, · ) k m ) m − k f ( τ, · ) k m,α k x − y k α k ∂ t f ( τ, · ) k m , and, if h = l + 1, (cid:13)(cid:13) g ( l +1) ( F ( τ, y ))( B γ,h ) − g ( l +1) ( F ( τ, y ))( B γ,h − ) (cid:13)(cid:13) L m +1 ≤ k g k m +1 (1 + k f ( τ, · ) k m ) m k ∂ t f ( τ, · ) k m,α k x − y k α . The theorem follows. (cid:3)
N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 11 Groups of H¨older diffeomorphisms
The (non-topological) group D n,β ( R d ) . Let n ∈ N ≥ and β ∈ (0 , R d which differ fromthe identity by a C n,β -mapping:(3.1) D n,β ( R d ) := (cid:8) Φ ∈ Id + C n,β ( R d , R d ) : det d Φ( x ) > ∀ x ∈ R d (cid:9) . We will show that D n,β ( R d ) is a group (with respect to composition).We endow D n,β ( R d ) with the topology given by the metric d (Φ , Φ ) := k Φ − Φ k n,β and denote by B n,β (Φ , r ) the open ball of radius r and center Φ in D n,β ( R d ). Weuse the same notation for balls in C n,β ( R d , R d ) which causes no problems sinceId C n,β ( R d , R d ).Since the determinant is multiplicative, it is an easy consequence of Theorem 2.7that D n,β ( R d ) is a monoid with respect to composition. Lemma 3.1. D n,β ( R d ) consists of C n -diffeomorphisms of R d . The first n deriva-tives of the inverse of an element of D n,β ( R d ) are again globally bounded.Proof. Let Φ = Id + φ ∈ D n,β ( R d ). First we have to make sure that Φ is bijective.This is an immediate consequence of [24, Cor. 4.3], which states that a C mappingconverging to infinity at infinity with non-vanishing jacobian determinant is alreadya C diffeomorphism. The inverse mapping theorem shows that Φ − is actually C n .Boundedness of the first n derivatives of Φ − − Id follows as in [22, p. 7,8]. (cid:3)
Lemma 3.2 ([22, p. 7]) . The operator norm of an invertible linear operator A : R d → R d satisfies k A − k ≤ | det A | − k A k d − . Lemma 3.3.
Let Φ = Id + φ ∈ D n,β ( R d ) . Then: (1) ε := inf x ∈ R d det d Φ ( x ) > . (2) There is δ > such that inf x ∈ R d det d Φ( x ) ≥ ε/ for all Φ ∈ Id + B n,β ( φ , δ ) . (3) There are δ, C > such that sup x ∈ R d k d Φ − ( x ) k ≤ C for all Φ ∈ B n,β (Φ , δ ) .Proof. (1) Observe that d Φ ( x ) → as k x k → ∞ . Thus det d Φ ( x ) → k x k → ∞ , which implies ε := inf x ∈ R d det d Φ ( x ) > d × d matrices.(3) Let δ > ∈ B n,β (Φ , δ ), k d Φ − (Φ( x )) k = k ( d Φ( x )) − k ≤ k d Φ( x ) k d − | det d Φ( x ) | ≤ ǫ ( k Φ k n,β + δ ) d − , by Lemma 3.2. Since Φ is bijective, the proof is complete. (cid:3) Lemma 3.3 shows that D n,β ( R d ) − Id is an open subset of C n,β ( R d , R d ). Thus,for Φ = Id + φ ∈ D n,β ( R d ) and for sufficiently small r > B n,β (Φ , r ) = Id + B n,β ( φ , r ) . We interpret D n,β ( R d ) as a Banach manifold modelled on C n,β ( R d , R d ) with globalchart Φ Φ − Id.
Theorem 3.4.
Let n ∈ N ≥ and β ∈ (0 , . Then D n,β ( R d ) is a group. In general,left translations are discontinuous. The theorem will follow from Lemma 3.5 and Proposition 3.6.
Lemma 3.5.
In general, left translations in D n,β ( R d ) are discontinuous.Proof. The construction is taken from [3, 6.4]. We prove the claim in the case d = 1. Let χ ∈ C ∞ c ( R ) be 1 on [ − , ψ ( x ) := x n | x | β χ ( x ). Then ψ ∈ C n,β ( R , R ). In addition, let Φ k ( x ) := x + χ ( x ) /k . Since D n,β ( R ) − Id is open,we have Φ k ∈ D n,β ( R ), for sufficiently large k , and Φ k → Id in D n,β ( R ) as k → ∞ .It is easy to see that, for | x | < ψ ( n ) ( x ) = ( n + β ) · · · (1 + β ) | x | β =: C n,β | x | β . Thus, for large k , ( ψ ◦ Φ k ) ( n ) (cid:16) − k (cid:17) = C n,β (cid:12)(cid:12)(cid:12) − k + 1 k (cid:12)(cid:12)(cid:12) β = 0 , and ( ψ ◦ Φ k ) ( n ) (0) = C n,β k β . Hence (cid:0) ( ψ ◦ Φ k ) ( n ) − ( ψ ◦ Id) ( n ) (cid:1)(cid:16) − k (cid:17) − (cid:0) ( ψ ◦ Φ k ) ( n ) − ( ψ ◦ Id) ( n ) (cid:1) (0) = − C n,β k β , which immediately gives k ψ ◦ Φ k − ψ ◦ Id k n,β ≥ C n,β . Since D n,β ( R ) − Id is open,there is some small r > rψ ∈ D n,β ( R ). (cid:3) The next proposition completes the proof of Theorem 3.4.
Proposition 3.6. D n,β ( R d ) is closed under inversion. The chart representation inv c : ( D n,β ( R d ) − Id) → ( D n,β ( R d ) − Id) , φ (Id + φ ) − − Id is locally bounded.Proof. For Φ = Id + φ ∈ D n,β ( R d ) and Φ − =: Id + τ we have (Id + τ ) ◦ (Id + φ ) = Id,i.e., τ ( x + φ ( x )) = − φ ( x ) , x ∈ R d . (3.3)It follows that det d Φ − ( x ) > x and that τ ◦ (Id + φ ) ∈ C n,β ( R d , R d ). ByLemma 3.1, Φ − is n -times differentiable with globally bounded derivatives.Let Φ = Id + φ ∈ D n,β ( R d ) and Φ − =: Id + τ . Choose δ > B n,β ( φ , δ ) ⊆ ( D n,β ( R d ) − Id) (recall that D n,β ( R d ) − Id is open) and such that theconclusion of Lemma 3.3 holds.
Claim 1. inv c ( B n,β ( φ , δ )) is bounded in C n ( R d , R d ) . By Lemma 3.1, we know that inv c maps into C nb ( R d , R d ). An inspection ofFa`a di Bruno’s formula (2.7) shows that it actually maps into C n ( R d , R d ). Let φ ∈ B n,β ( φ , δ ) and τ = inv c ( φ ) ∈ C n ( R d , R d ) so that (3.3) implies k τ ( x + φ ( x )) k ≤ k φ ( x ) k ≤ k φ − φ k n,β + k φ k n,β ≤ δ + k φ k n,β for all x . Since Id + φ is bijective, this gives k inv c ( φ ) k = k τ k ≤ δ + k φ k n,β , φ ∈ B n,β ( φ , δ ) . N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 13
We prove by induction on k that for all k ≤ n there are constants D k = D k ( φ , δ )such that(3.4) k τ k k = k inv c ( φ ) k k ≤ D k , φ ∈ B n,β ( φ , δ ) . By Fa`a di Bruno’s formula (2.7), d k ( τ ◦ Φ)( x ) = τ ( k ) (Φ( x ))( d Φ( x ) , . . . , d Φ( x ))+ sym k − X l =1 X γ ∈ Γ( l,k ) c γ τ ( l ) (Φ( x )) (cid:0) Φ ( γ ) ( x ) , . . . , Φ ( γ l ) ( x ) (cid:1) . (3.5)By the induction hypothesis, for φ ∈ B n,β ( φ , δ ) and l ≤ k − (cid:13)(cid:13) τ ( l ) (Φ( x )) (cid:0) Φ ( γ ) ( x ) , . . . , Φ ( γ l ) ( x ) (cid:1)(cid:13)(cid:13) L k ≤ k τ ( l ) (Φ( x )) k L l k Φ ( γ ) ( x ) k L γ · · · k Φ ( γ l ) ( x ) k L γl ≤ D k − (1 + k φ k n,β ) k ≤ D k − (1 + δ + k φ k n,β ) k . In addition, k d k ( τ ◦ Φ)( x ) k L k ≤ k φ k n,β ≤ δ + k φ k n,β , by (3.3). It follows thatthere is some constant D k = D k ( φ , δ ) such that(3.6) k τ ( k ) (Φ( x ))( d Φ( x ) , . . . , d Φ( x )) k L k ≤ D k , φ ∈ B n,β ( φ , δ ) , x ∈ R d . Since k τ ( k ) (Φ( x )) k L k ≤ k τ ( k ) (Φ( x ))( d Φ( x ) , . . . , d Φ( x )) k L k k ( d Φ( x )) − k kL , (3.6) and Lemma 3.3 imply (3.4) and hence Claim 1. Claim 2. inv c ( B n,β ( φ , δ )) is bounded in C n,β ( R d , R d ) . Observe that, since Φ is a bijection of R d ,[ τ ] n,β = sup x = y k d n τ (Φ( x )) − d n τ (Φ( y )) k L n k x − y k β k x − y k β k Φ( x ) − Φ( y ) k β ≤ sup x = y k d n τ (Φ( x )) − d n τ (Φ( y )) k L n k x − y k β (cid:16) sup x = y k x − y kk Φ( x ) − Φ( y ) k (cid:17) β = sup x = y k d n τ (Φ( x )) − d n τ (Φ( y )) k L n k x − y k β Lip(Φ − ) β ≤ sup x = y k d n τ (Φ( x )) − d n τ (Φ( y )) k L n k x − y k β (1 + D ) β , for all φ ∈ B n,β ( φ , δ ), by (3.4). For k ≤ n , let A k = A k ( x, y ) := ( d Φ( x ) , . . . , d Φ( x ) | {z } k -times , d Φ( y ) , . . . , d Φ( y ) | {z } ( n − k )-times ) . Then k d n τ (Φ( y ))( A k ) − d n τ (Φ( y ))( A k − ) k L n ≤ k d n τ (Φ( y )) k L n k d Φ( x ) k k − L k d Φ( y ) k n − kL k dφ ( x ) − dφ ( y ) k L ≤ k τ k n (1 + k φ k ) n − k φ k n,β k x − y k β ≤ D n (1 + δ + k φ k n,β ) n k x − y k β , (3.7) where we use k dφ ( x ) − dφ ( y ) k L ≤ k φ k ,β k x − y k β for the case n = 1 (which holdsby definition). For the case n ≥ k x − y k ≤ k x − y k ≤ k x − y k β , otherwise k dφ ( x ) − dφ ( y ) k L ≤ k φ k n,β ≤ k φ k n,β k x − y k .)By Lemma 3.3, k d n τ (Φ( x )) − d n τ (Φ( y )) k L n ≤ k d n τ (Φ( x ))( A n ) − d n τ (Φ( y ))( A n ) k L n k d Φ( x ) − k nL ≤ C k d n τ (Φ( x ))( A n ) − d n τ (Φ( y ))( A ) k L n + C n X k =1 k d n τ (Φ( y ))( A k ) − d n τ (Φ( y ))( A k − ) k L n . We may use (3.7) to estimate the second term on the right-hand side. Thus, to endthe proof of Claim 2, and hence of the proposition, it remains to show the following.
Claim 3.
There exists a constant C such that k d n τ (Φ( x ))( A n ) − d n τ (Φ( y ))( A ) k L n ≤ C k x − y k β , for all φ ∈ B n,β ( φ , δ ) and all x, y ∈ R d . For any γ ∈ N l> and 0 ≤ j ≤ l let A γ,j = A γ,j ( x, y ) := (cid:0) Φ ( γ ) ( x ) , . . . , Φ ( γ j ) ( x ) , Φ ( γ j +1 ) ( y ) , . . . , Φ ( γ l ) ( y ) (cid:1) . Then, by Fa`a di Bruno’s formula (2.7), d n ( τ ◦ Φ)( x ) − d n ( τ ◦ Φ)( y ) = τ ( n ) (Φ( x ))( A n ) − τ ( n ) (Φ( y ))( A )+ sym n − X l =1 X γ ∈ Γ( l,n ) c γ (cid:0) τ ( l ) (Φ( x ))( A γ,l ) − τ ( l ) (Φ( y ))( A γ, ) (cid:1) . (3.8)By (3.3), there is a constant C such that k d n ( τ ◦ Φ)( x ) − d n ( τ ◦ Φ)( y ) k L n ≤ C k x − y k β , φ ∈ B n,β ( φ , δ ) . (3.9)Moreover, (cid:13)(cid:13) τ ( l ) (Φ( x ))( A γ,l ) − τ ( l ) (Φ( y ))( A γ, ) (cid:13)(cid:13) L n ≤ (cid:13)(cid:13) τ ( l ) (Φ( x ))( A γ,l ) − τ ( l ) (Φ( y ))( A γ,l ) (cid:13)(cid:13) L n + l X k =1 (cid:13)(cid:13) τ ( l ) (Φ( y ))( A γ,k ) − τ ( l ) (Φ( y ))( A γ,k − ) (cid:13)(cid:13) L n . For the first summand, since l < n , (cid:13)(cid:13) τ ( l ) (Φ( x ))( A γ,l ) − τ ( l ) (Φ( y ))( A γ,l ) (cid:13)(cid:13) L n ≤ (cid:13)(cid:13) τ ( l ) (Φ( x )) − τ ( l ) (Φ( y )) (cid:13)(cid:13) L l (1 + k φ k n,β ) n ≤ k τ k n (1 + k φ k ) k x − y k (1 + k φ k n,β ) n , and k τ k n ≤ D n for φ ∈ B n,β ( φ , δ ), by Claim 1. For the other summands observethat τ ( l ) (Φ( y ))( A γ,k ) − τ ( l ) (Φ( y ))( A γ,k − )= τ ( l ) (Φ( y )) (cid:0) . . . , Φ ( γ k − ) ( x ) , (Φ ( γ k ) ( x ) − Φ ( γ k ) ( y )) , Φ ( γ k +1 ) ( y ) , . . . (cid:1) , N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 15 whence (cid:13)(cid:13) τ ( l ) (Φ( y ))( A γ,k ) − τ ( l ) (Φ( y ))( A γ,k − ) (cid:13)(cid:13) L n ≤ ( k τ k n (1 + k φ k n ) n − k φ k n k x − y k if l > , k τ k k φ k n,β k x − y k β if l = 1 . Altogether this means that we find a constant K such that for all φ ∈ B n,β ( φ , δ )and all x, y ∈ R d (cid:13)(cid:13) τ ( l ) (Φ( x ))( A γ,l ) − τ ( l ) (Φ( y ))( A γ, ) (cid:13)(cid:13) L n ≤ K k x − y k β ;indeed, if k x − y k ≤ k x − y k ≤ k x − y k β , otherwise the estimate follows fromthe triangle inequality and Claim 1. Together with (3.8) and (3.9) this impliesClaim 3. (cid:3) For later use we prove the following.
Proposition 3.7.
Let n ∈ N ≥ and < α < β ≤ . Then, for all φ ∈ ( D n,β ( R d ) − Id) there exists δ > such that for all φ , φ ∈ B n,β ( φ , δ ) ⊆ ( D n,β ( R d ) − Id) , (3.10) k inv c ( φ ) − inv c ( φ ) k n,α ≤ M k φ − φ k β − αn,α , where M = M ( n, φ , δ ) . In particular, inv c : ( D n,β ( R d ) − Id) → ( D n,α ( R d ) − Id) , φ (Id + φ ) − − Id is continuous.Proof. Choose δ > B n,β ( φ , δ ) ⊆ ( D n,β ( R d ) − Id). Let φ , φ ∈ B n,β ( φ , δ ). We write τ i = inv c ( φ i ) and Φ i = Id + φ i , for i = 0 , ,
2. Then τ − τ = Φ − ◦ Φ ◦ Φ − − Φ − ◦ Φ ◦ Φ − = φ ◦ (Id + τ ) − φ ◦ (Id + τ )+ τ ◦ (Id + φ ) ◦ (Id + τ ) − τ ◦ (Id + φ ) ◦ (Id + τ ) . By Theorem 2.7, k φ ◦ (Id + τ ) − φ ◦ (Id + τ ) k n,α ≤ M ( n ) k φ − φ k n,α (1 + k τ k n,α ) n +1 , and by Theorem 2.7, Theorem 2.8 (and Lemma 2.2), k τ ◦ (Id + φ ) ◦ (Id + τ ) − τ ◦ (Id + φ ) ◦ (Id + τ ) k n,α ≤ M ( n ) k τ ◦ (Id + φ ) − τ ◦ (Id + φ ) k n,α (1 + k τ k n,α ) n +1 ≤ M ( n, k φ k n,α , δ ) k τ k n,β k φ − φ k β − αn,α (1 + k τ k n,α ) n +1 . Since k τ i k n,β is uniformly bounded for φ i ∈ B n,β ( φ , δ ) if δ > (cid:3) Intermediate H¨older spaces.
As we have already seen in Lemma 3.5, thegroup D n,β ( R d ) is not topological (with respect to the topology given as a Banachmanifold modelled on C n,β ( R d , R d )). Nevertheless we know that the left translationsbecome continuous if the outer mapping is only slightly more regular than the spaceit acts on. This observation motivates the following definitions.Let E , F be Banach spaces, U ⊆ E open, and n ∈ N . For β ∈ (0 ,
1] define C n,β − b ( U, F ) := \ α ∈ (0 ,β ) C n,αb ( U, F ) , and for β ∈ [0 , C n,β + b ( U, F ) := [ α ∈ ( β, C n,αb ( U, F ) . If β ∈ (0 ,
1) we have the strict inclusions C n,β + b ( U, F ) ( C n,βb ( U, F ) ( C n,β − b ( U, F ) . We endow C n,β − b ( U, F ) and C n,β + b ( U, F ) with their natural projective and inductivelocally convex limit topologies, respectively.Then C n,β − b ( U, F ) is a Fr´echet space with a generating system of seminorms P = {k · k n,α : α ∈ (0 , β ) } , or a countable subfamily thereof, like {k · k n,β − /k : k ≥ k } .The balls B n,β − α ( f , ε ) := { f ∈ C n,β − b ( U, F ) : k f − f k n,α < ε } satisfy B n,β − α ( f , ε ) ⊆ B n,β − α ( f , ε ) if α < α , by Lemma 2.2. Thus { B n,β − α ( f , ε ) : α < β, ε > } forms a neighborhood base of f ∈ C n,β − b ( U, F ).In analogy we define C n,β ± and C n,β ± . Lemma 3.8. C n,β + b ( U, F ) and C n,β +0 ( E, F ) are compactly regular (LB)-spaces.Proof. It suffices, by [23, Satz 1], to verify condition (M) of [25]: There ex-ists a sequence of increasing 0-neighborhoods B p ⊆ C n,β +1 /pb ( U, F ) such that foreach p there exists an m ≥ p for which the topologies of C n,β +1 /kb ( U, F ) and of C n,β +1 /mb ( U, F ) coincide on B p for all k ≥ m .For α ≤ α ′ we have k f k n,α ≤ k f k n,α ′ , by Lemma 2.2. It suffices to show thatfor β < α < α < α , ǫ >
0, and f ∈ B n,α (0 ,
1) there exists δ > B n,α ( f, δ ) ∩ B n,α (0 , ⊆ B n,α ( f, ǫ ).Let g ∈ B n,α ( f, δ ) ∩ B n,α (0 , k g − f k n,α < δ and k g k n,α <
1. ByLemma 2.1, k g − f k n,α ≤ k g − f k α − α α − α n,α k g − f k α − α α − α n,α < δ α − α α − α α − α α − α . So it is clear that we may find δ as required. (cid:3) Consequently, C n,β + b ( U, F ) and C n,β +0 ( U, F ) are complete (thus convenient),webbed, and ultra-bornological.3.3. C ,ω -mappings between convenient vector spaces. Let ω : [0 , ∞ ) → [0 , ∞ ) be a subadditive increasing modulus of continuity (lim t → ω ( t ) = ω (0) = 0).By a C ,ω -curve c we mean a function defined on the real line with values in aconvenient vector space F such that for each bounded interval I ⊆ R , n c ( t ) − c ( s ) ω ( | t − s | ) : t, s ∈ I, t = s o is bounded in F . We say that a mapping between convenient vector spaces is C ,ω ,if it maps C ∞ -curves to C ,ω -curves. The c ∞ -topology coincides with the finaltopology of all C ,ω -curves (which follows from the proof of [15, 2.13]), and so a C ,ω -mapping is continuous with respect to the c ∞ -topology. The following lemmashows that between Banach spaces the notion of C ,ω -mapping coincides with theusual definition. N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 17
Lemma 3.9.
Let E , F be Banach spaces, U ⊆ E open. A mapping f : U → F is C ,ω if and only if f ( x ) − f ( y ) /ω ( k x − y k ) is locally bounded.Proof. Suppose that there is z ∈ U and x n = y n ∈ U such that k x n − z k ≤ − n , k y n − z k ≤ − n , and k f ( x n ) − f ( y n ) k ≥ n n ω ( k x n − y n k ). By [15, 12.2], there isa C ∞ -curve c and a convergent sequence of real numbers t n such that c ( t + t n ) = x n + t ( y n − x n )2 n k x n − y n k for all 0 ≤ t ≤ s n := 2 n k x n − y n k . Then, by subadditivity of ω , k ( f ◦ c )( t n + s n ) − ( f ◦ c )( t n ) k ω ( s n ) = k f ( x n ) − f ( y n ) k ω (2 n k x n − y n k ) ≥ n. The converse implication follows from subadditivity and monotonicity of ω , since C ∞ -curves are locally Lipschitz. (cid:3) This lemma can be found in [5], [15, 12.7], or [18, Lemma] in the H¨older (orLipschitz) case ω ( t ) = t γ . Definition 3.10.
We say that ω is a slowly vanishing modulus of continuity if ω is increasing, subadditive, and satisfieslim inf t ↓ ω ( t ) t γ > γ > . For instance, ω defined by ω ( t ) := − (log t ) − , if 0 < t < e − , ω ( t ) := 1 /
2, if t ≥ e − , and ω (0) := 0, is a slowly vanishing modulus of continuity.3.4. The C ,ω Lie groups D n,β − ( R d ) and D n,β + ( R d ) . Let n ∈ N ≥ . We define D n,β ± ( R d ) := (cid:8) Φ ∈ Id + C n,β ± ( R d , R d ) : det d Φ( x ) > ∀ x ∈ R d (cid:9) , where β ∈ (0 ,
1] if ± = − and β ∈ [0 ,
1) if ± = +. Then D n,β ± ( R d ) − Id is anopen subset of C n,β ± ( R d , R d ). We take this interpretation as defining propertyfor the topology, i.e., V ⊆ D n,β ± ( R d ) is open if and only if ( V − Id) is open in( D n,β ± ( R d ) − Id) ⊆ C n,β ± ( R d , R d ).Clearly, D n,β ± ( R d ) forms a group, by Theorem 3.4. We will now prove that D n,β ± ( R d ) are C ,ω Lie groups for any slowly vanishing modulus of continuity ω . Theorem 3.11.
Let n ∈ N ≥ . Let ω be a slowly vanishing modulus of continuity.Then D n,β − ( R d ) , for β ∈ (0 , , and D n,β + ( R d ) , for β ∈ [0 , , are C ,ω Lie groups.In particular, D n,β − ( R d ) , for β ∈ (0 , , is a topological group (with respect to itsnatural Fr´echet topology).Proof. Let us first consider D n,β − ( R d ), for β ∈ (0 , g, f ∈ C ∞ ( R , C n,β − ( R d , R d )) and let I ⊆ R be a compact interval. Then the sets g ( I ), f ( I ) are bounded in C n,β − ( R d , R d ) and thus in every C n,α ( R d , R d ) for α < β . If α < ˜ α < β , then, by (2.6), k g ( t ) ◦ (Id + f ( t )) − g ( s ) ◦ (Id + f ( s )) k n,α ≤ M (cid:0) k g ( t ) − g ( s ) k n,α + k f ( t ) − f ( s ) k n,α (cid:1) ˜ α − α ≤ ˜ M | t − s | ˜ α − α , for t, s ∈ I . There is ǫ > C = C ( α, ˜ α ) such that | t − s | ˜ α − α ≤ Cω ( | t − s | )if | t − s | ≤ ǫ . Since ω is increasing, we may conclude that, for t h ( t ) := g ( t ) ◦ (Id + f ( t )),(3.11) n h ( t ) − h ( s ) ω ( | t − s | ) : s = t ∈ I o is bounded in C n,α ( R d , R d ). So the composition is C ,ω on D n,β − ( R d ).Let us turn to the inversion in D n,β − ( R d ). Let f ∈ C ∞ ( R , C n,β − ( R d , R d )). Fix α < ˜ α < β and t ∈ R . Let δ > B n, ˜ α ( f ( t ) , δ ) ⊆ ( D n, ˜ α ( R d ) − Id).There is a neighborhood I of t such that f ( I ) ⊆ B n, ˜ α ( f ( t ) , δ ). By Proposition 3.7(after possibly shrinking δ ), for all t, s ∈ I , k inv c ( f ( t )) − inv c ( f ( s )) k n,α ≤ M k f ( t ) − f ( s ) k ˜ α − αn,α , where M = M ( n, f ( t ) , δ ). Finishing the arguments in the same way as for thecomposition, we conclude that the inversion is C ,ω on D n,β − ( R d ).This implies that D n,β − ( R d ) is a topological group, since the underlying Fr´echettopology and the c ∞ -topology coincide. Of course, it also follows directly fromCorollary 2.9 and Proposition 3.7.Now let us consider D n,β + ( R d ), for β ∈ [0 , g, f ∈ C ∞ ( R , C n,β +0 ( R d , R d )).For any compact interval I ⊆ R , the images g ( I ), f ( I ) are bounded in C n,β +0 ( R d , R d ). Since C n,β +0 ( R d , R d ) is a compactly regular (LB)-space, there issome α > β such that g ( I ), f ( I ) are bounded in C n,α ( R d , R d ), and thus alsoin every C n,α ( R d , R d ), for α ∈ ( β, α ]. Let α, ˜ α ∈ ( β, α ] with α < ˜ α . Then thearguments above show that the set (3.11) is bounded in C n,α ( R d , R d ), and thusin C n,β +0 ( R d , R d ). So the composition is C ,ω on D n,β + ( R d ). Similarly for theinversion. (cid:3) Remark 3.12.
We do not know whether D n,β + ( R d ) is a topological group withrespect to its natural inductive locally convex topology, since the c ∞ -topology isfiner in this case.Groups with continuous left translations and smooth right translations weredubbed half-Lie groups in [20]. The chart representations of the right translationsin D n,β ± ( R d ) are affine and bounded, by Theorem 2.7, and thus smooth. Hence, D n,β − ( R d ) is a half-Lie group.The next result shows that the C ,ω -regularity of the group operations in D n,β ± ( R d ) is optimal. Proposition 3.13.
Let n ∈ N ≥ . (1) For all β ∈ (0 , , D n,β − ( R d ) is a half-Lie group. There are left translationsin D n,β − ( R d ) which are not locally H¨older continuous of any order γ > . (2) Let β ∈ [0 , . For any γ > , there are left translations in D n,β + ( R d ) whichare not locally H¨older continuous of order γ .Proof. (1) Let χ ∈ C ∞ c ( R ) be 1 on [ − ,
1] and satisfy χ ′ ( x ) > − x ∈ R , and set ψ ( x ) := x n | x | β χ ( x ) ∈ C n,β ( R , R ) ⊆ C n,β − ( R , R ). We will show that θ ( t ) := ψ ◦ (Id + tχ ), for small t ∈ R , is not locally H¨older continuous of order γ into C n,α ( R , R ) for any α > β − γ . This implies the assertion, since Id + rψ ∈ D n,β − ( R )if r > I ∋
0, the set n θ ( t ) ( n ) ( x ) − θ ( t ) ( n ) ( y ) − θ ( s ) ( n ) ( x ) + θ ( s ) ( n ) ( y ) | x − y | α | s − t | γ : x = y ∈ R , s = t ∈ I o is unbounded. If | x | <
1, then for small t (cf. (3.2)), θ ( t ) ( n ) ( x ) = ψ ( n ) ( x + t ) = C n,β | x + t | β . N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 19
For t = x = 0 and | y | ≤ −| y | β − | s | β + | y + s | β | y | α | s | γ and upon setting y = − s , we get − | s | β − α − γ which is unbounded near s = 0.(2) Let γ > α > β let ψ α ( x ) := x n | x | α χ ( x ). Then, as seen above, θ α ( t ) := ψ α ◦ (Id + tχ ), for small t ∈ R , is not locally H¨older continuous of order γ into C n,α ( R , R ) for any α ∈ ( β, α ) with γ > α − α . It follows that ( ψ α ) ⋆ is notlocally H¨older continuous of order γ , provided that α − β < γ . Indeed, if n θ α ( t ) − θ α ( s ) | s − t | γ : s = t ∈ I o were bounded in C n,β +0 ( R , R ), then it would be so in some step C n,α ( R , R ), byLemma 3.8. (cid:3) The results of this section are summarized in Table 1. group C ,ω -Lie group topological group half-Lie group Lie group D n,β ( R d ) yes no no no no D n,β − ( R d ) yes yes yes yes no D n,β + ( R d ) yes yes ? ? no Table 1.
Here n ∈ N ≥ and ω is any slowly vanishing modulus ofcontinuity. In the first two rows β ∈ (0 , β ∈ [0 , H¨older spaces are ODE closed
Flows of time-dependent H¨older vector fields.
Let n ∈ N ≥ and β ∈ (0 , strong time-dependent C n,β -vector field we mean a Bochner integrablefunction u : [0 , → C n,β ( R d , R d ). We will write I := [0 ,
1] and k u k L ( I,C n,β ) := Z k u ( t ) k n,β dt. The space L ( I, C n,β ( R d , R d )) of (equivalence classes with respect to a.e. coinci-dence of) Bochner integrable function u : I → C n,β ( R d , R d ) equipped with thisnorm is a Banach space.Let α ≤ β . We say that a continuous mapping Φ : I → D n,α ( R d ) is a strong D n,α -flow of u if for all t ∈ I we have(4.1) Φ( t ) = Id + Z t u ( s ) ◦ Φ( s ) ds in D n,α ( R d ), where the integral is the Bochner integral.Since evaluation ev x at x ∈ R d is continuous and linear on C n,α ( R d , R d ) and itthus commutes with the Bochner integral, (4.1) entails(4.2) Φ ∧ ( t, x ) = x + Z t u ∧ ( s, Φ ∧ ( s, x )) ds, x ∈ R d . We say that Φ ∧ : I × R d → R d is the pointwise flow of u ∧ if it satisfies (4.2). So,if u has a strong D n,α -flow Φ, then u ∧ has a pointwise flow which is continuous in t and differs from the identity by a C n,α -mapping in x . Conversely, the existenceof a pointwise flow with this properties will entail the existence of a strong D n,α -flow only if the Bochner integral in (4.1) exists. Since the H¨older spaces are non-separable, strong measurability of t u ( t ) ◦ Φ( t ) may fail and the integral in (4.1)may not exist, if the left translation u ( t ) ⋆ is not continuous. This is exactly whathappens if β = α .Luckily we can work with pointwise estimates which enable us to prove thattime-dependent C n,β -vector fields have unique pointwise flows Φ such that Φ ∨ ∈ C ( I, D n,β ( R d )) (no loss of regularity!). The proof actually works for a wider classof vector fields, so-called pointwise time-dependent C n,β -vector fields , which shallbe introduced in the next subsection.We shall see in Section 5.1 that the unique pointwise flow Φ ∨ ∈ C ( I, D n,β ( R d ))of a strong time-dependent C n,β -vector field u lifts to a strong D n,α -flow, for each α < β .4.2. Trouv´e group and ODE closedness.
Let I = [0 ,
1] and let E be a Banachspace of mappings R d → R d which is continuously embedded in C ( R d , R d ). Definition 4.1.
We say that a mapping u : I × R d → R d is a pointwise time-dependent E -vector field if the following conditions are satisfied. • u ( t, · ) ∈ E for every t ∈ I . • u ( · , x ) is measurable for every x ∈ R d . • I ∋ t → k u ( t, · ) k E is (Lebesgue) integrable.Let us denote the set of all pointwise time-dependent E -vector fields by X E ( I, R d ).We remark that instead of the third condition we could also require that k u ∨ k E isdominated a.e. by some non-negative function m ∈ L ( I ).Clearly, u ∈ L ( I, E ) implies u ∧ ∈ X E ( I, R d ); the converse is in general not true,in particular, if E is non-separable and strong measurability and measurability arenot the same; see Example 4.2 below. We will continue to write k u ∨ k L ( I,E ) = Z k u ∨ ( t ) k E dt, for u ∈ X E ( I, R d ) , even though k u ∨ k L ( I,E ) might be finite while u ∨ : I → E is not Bochner integrable;this will lead to no confusion. Example 4.2.
Let χ ∈ C ∞ c ( R ) be 1 on [ − , ψ ( x ) := x n | x | β χ ( x ), then ψ lies in C n,β ( R , R ) (cf. Lemma 3.5). Let u : I × R → R be defined by u ( t, x ) = ψ ( x − t ); u is clearly a pointwise time-dependent C n,β -vector field. But u ∨ L ( I, C n,β ( R )):indeed, for fixed t, s ∈ I , t = s , (cf. (3.2))[ u ∨ ( t ) − u ∨ ( s )] n,β = sup x = y | ψ ( n ) ( x − t ) − ψ ( n ) ( y − t ) − ψ ( n ) ( x − s ) + ψ ( n ) ( y − s ) || x − y | β ≥ C n,β sup x,y ∈ I, x = y (cid:12)(cid:12) | x − t | β − | y − t | β − | x − s | β + | y − s | β (cid:12)(cid:12) | x − y | β ≥ C n,β > x = t , y = s ).It follows that the image u ∨ ( I ) is not essentially separable in C n,β ( R ), and so u ∨ is not strongly measurable, by the Pettis measurability theorem (cf. [4, p. 42]). N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 21
It is well-known that pointwise time-dependent C n -vector fields u have uniquepointwise flows Φ = Φ u : I × R d → R d such that Φ ∨ : I → Id + C n ( R d , R d ) iscontinuous, and Φ ∨ ( t ) is a C n -diffeomorphism at any time t ; see e.g. [28, 8.7, 8.8,8.9] and the arguments in the proof of Theorem 4.6 below. Definition 4.3.
Let E be a Banach space of mappings R d → R d which is continu-ously embedded in C ( R d , R d ). Then G E := (cid:8) Φ ∨ u (1) : u ∈ X E ( I, R d ) (cid:9) is a group with respect to composition; cf. [28, 8.14]. We call G E the Trouv´e group of E . Definition 4.4.
We say that E is ODE closed if G E ⊆ Id + E . Remark 4.5.
It is clear that, more generally, we could take (mutatis mutandis)any locally convex space E of mappings R d → R d which is continuously embeddedin C ( R d , R d ) in the above definitions.Furthermore, this leads to the notion of ODE hull of E , i.e., the intersection of alllocally convex spaces F of mappings R d → R d which are continuously embedded in C ( R d , R d ) and continuously contain E , endowed with the natural projective topol-ogy. The ODE hull is well-defined, because C is ODE-closed, and it is evidentlyODE closed.4.3. The Trouv´e group of C n,β ( R d , R d ) . Let n ∈ N ≥ and β ∈ (0 , C n,β ( R d , R d ), G n,β ( R d ) := G C n,β ( R d , R d ) , coincides with the connected component of the identity in D n,β ( R d ). In particular, C n,β is ODE closed. Let us use the short notation X n,β ( I, R d ) := X C n,β ( I, R d ) . We want to stress that this is an example of an ODE closed space on which lefttranslations g ⋆ are not continuous. Theorem 4.6.
Let n ∈ N ≥ and β ∈ (0 , . Let u ∈ X n,β ( I, R d ) . Then u has aunique pointwise flow Φ : I × R d → R d such that Φ ∨ : I → D n,β ( R d ) is continuous.In particular, C n,β is ODE closed.Proof. In fact, cf. [28, 8.7, 8.8, 8.9], the pointwise flow exists, x Φ( t, x ) is C nb ,and for all t ∈ I ,(4.3) k Φ( t, · ) − Id k n ≤ C e C k u k L I,Cnb ) . Set W nx ( t ) := d nx Φ( t, x ) and V nx,y ( t ) := ( W nx ( t ) − W ny ( t )) / k x − y k β . Then W nx ( t )satisfies ∂ t W nx ( t ) = d nx (cid:0) u ∨ ( t )(Φ( t, x )) (cid:1) , W nx (0) = (cid:26) , for n = 10 , for n ≥ . Upon setting A γ,j ( t ) = A γ,j ( x, y )( t ) := (cid:0) W γ x ( t ) , . . . , W γ j x ( t ) , W γ j +1 y ( t ) , . . . , W γ l y ( t ) (cid:1) . and using Fa`a di Bruno’s formula (2.7), this ODE takes the form ∂ t W nx ( t ) = sym n X l =1 X γ ∈ Γ( l,n ) c γ u ∨ ( t ) ( l ) (Φ( t, x ))( A γ,l ( t )) , and analogously, ∂ t W ny ( t ) = sym n X l =1 X γ ∈ Γ( l,n ) c γ u ∨ ( t ) ( l ) (Φ( t, y ))( A γ, ( t )) . It follows that V nx,y ( t ) satisfies V nx,y (0) = 0 and ∂ t V nx,y ( t ) = A x ( t ) · V nx,y ( t ) + b nx,y ( t )+ sym n X l =2 X γ ∈ Γ( l,n ) c γ u ∨ ( t ) ( l ) (Φ( t, x )) · A γ,l ( t ) − u ∨ ( t ) ( l ) (Φ( t, y )) · A γ, ( t ) k x − y k β , (4.4)where A x ( t ) = du ∨ ( t )(Φ( t, x )) and b nx,y ( t ) := A x ( t ) − A y ( t ) k x − y k β · W ny ( t ) . It can be easilyseen, using (4.3), that(4.5) k A x ( t ) k L ≤ k u ∨ ( t ) k , k b nx,y ( t ) k L ≤ k u ∨ ( t ) k ,β [Φ ∨ ( t )] β [Φ ∨ ( t )] n ≤ C k u ∨ ( t ) k ,β . Similar arguments (see, e.g., the proof of Proposition 3.6) show that all remainingterms in the sum can be estimated by k u ( t ) k n,β times a constant uniformly in x, y . An application of Gronwall’s inequality implies that [Φ ∨ ( t )] n,β is boundedin t , showing that Φ ∨ ( t ) ∈ Id + C n,βb ( R d , R d ) for all t . Finally, we may conclude,integrating (4.4) and using similar estimates, that[Φ ∨ ( t ) − Φ ∨ ( t )] n,β = sup x = y k V nx,y ( t ) − V nx,y ( t ) k L n ≤ C Z tt k u ∨ ( s ) k n,β ds which tends to 0 as t → t . In an analogous way one sees that k Φ ∨ ( t ) − Φ ∨ ( t ) k n → t → t . This shows continuity in time.It remains to prove that Φ ∨ ( I ) ⊆ D n,β ( R d ). For fixed x ∈ R d , the map-ping I ∋ t det d Φ( t, x ) is continuous with image in R \ { } (since Φ ∨ ( t ) is a C -diffeomorphism of R d for each t ). Since Φ ∨ (0) = Id, we may conclude thatdet d Φ( t, x ) > x ∈ R d and all t ∈ I .Finally, let us check that φ ∨ ( t ) = Φ ∨ ( t ) − Id ∈ C n,β ( R d , R d ) for all t ∈ I . Supposethat k φ ( t, x ) k 6→ k x k → ∞ . Then there is ǫ > x k ∈ R d suchthat k x k k → ∞ and k φ ( t, x k ) k ≥ ǫ . Since Φ ∨ ( s ) is a diffeomorphism of R d , for all s ∈ I , sup x ∈ R d k u ( s, x + φ ( s, x )) k = sup y ∈ R d k u ( s, y ) k = k u ∨ ( s ) k and so the dominated convergence theorem implies that k φ ( t, x k ) k ≤ Z t k u ( s, x k + φ ( s, x k )) k ds → , because k x k + φ ( s, x k ) k → ∞ as k → ∞ and k u ( s, x ) k → k x k → ∞ , for each s ∈ I ; a contradiction. To see that k d kx φ ( t, x ) k → k x k → ∞ , for 1 ≤ k ≤ n ,we argue similarly: Since s φ ∨ ( s ) is continuous into C n,βb ( R d , R d ), there is aconstant C such that sup s ∈ I k φ ∨ ( s ) k n,β < C . Thus Fa`a di Bruno’s formula (2.7)implies that sup x ∈ R d k d kx ( u ( s, x + φ ( s, x ))) k is bounded above by k u ( s ) k k times a N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 23 constant (independent of s and x ) for all s ∈ I . Then the dominated convergencetheorem implies the assertion as before. (cid:3) Theorem 4.7.
Let n ∈ N ≥ and β ∈ (0 , . Then (4.6) G n,β ( R d ) = D n,β ( R d ) , where D n,β ( R d ) denotes the connected component of the identity in D n,β ( R d ) .Proof. The inclusion G n,β ( R d ) ⊆ D n,β ( R d ) follows from Theorem 4.6.Let us prove D n,β ( R d ) ⊆ G n,β ( R d ). Since D n,β ( R d ) is connected and locallypath-connected, it is path-connected, and each Φ ∈ D n,β ( R d ) can be connected bya polygon with the identity.Let Φ = Id + φ ∈ D n,β ( R d ) be such that γ ( t ) := (1 − t ) Id + t Φ ∈ D n,β ( R d ) forall t ∈ I . Then γ ( t )( x ) = x + tφ ( x ), and u ( t, x ) := ( γ ′ ( t ) ◦ γ ( t ) − )( x ) = φ ( γ ( t ) − ( x ))is a time-dependent vector field such that: • u ( t, · ) ∈ C n,β ( R d , R d ) for all t ∈ I , since D n,β ( R d ) is a group, by Theo-rem 3.4. • u ( · , x ) is a Borel function for every x ∈ R n ; indeed, if γ ( t ) − ( x ) =: x + τ ( t, x )then τ satisfies the implicit equation τ ( t, x ) + tφ ( x + τ ( t, x )) = 0 , and is C n by the implicit function theorem. • We have R k u ( t, · ) k n,β dt < ∞ , since inversion is locally bounded on D n,β ( R d ) and left translation maps bounded sets to bounded sets, see The-orem 2.7 and Proposition 3.6.That means that u ∈ X n,β ( I, R d ) and hence Φ ∈ G n,β ( R d ).Suppose we are given a polygon in D n,β ( R d ) with vertices Id , Φ , . . . , Φ n . ThenΦ ∈ G n,β ( R d ), by the previous paragraph. Consider the line segment γ connectingΦ and Φ . Then t γ ( t ) ◦ Φ − connects Id with Φ ◦ Φ − . So, by the above,Φ ◦ Φ − ∈ G n,β ( R d ) and hence Φ ∈ G n,β ( R d ), since G n,β ( R d ) is a group. Byiteration all vertices Φ j belong to G n,β ( R d ). (cid:3) Remark 4.8.
Analyzing the proof one finds that the identity (4.6) still holds if inthe definition of the Trouv´e group we restrict to u ∈ X n,β ( I, R d ) which are piecewise C n in time t . (Discontinuities in t guarantee that G n,β ( R d ) is a group.)4.4. The Trouv´e group of C n,β ± ( R d , R d ) . We define pointwise time-dependent C n,β ± -vector fields to be the elements of X n,β − ( I, R d ) := \ α ∈ (0 ,β ) X n,α ( I, R d ) , X n,β + ( I, R d ) := [ α ∈ ( β, X n,α ( I, R d ) , respectively, and the corresponding Trouv´e groups by G n,β ± ( R d ) := (cid:8) Φ ∨ u (1) : u ∈ X n,β ± ( I, R d ) (cid:9) . Theorem 4.9.
Let n ∈ N ≥ . For β ∈ (0 , , C n,β − is ODE closed and (4.7) G n,β − ( R d ) = D n,β − ( R d ) , and, for β ∈ [0 , , C n,β +0 is ODE closed and (4.8) G n,β + ( R d ) = D n,β + ( R d ) , In particular, G n,β ± ( R d ) has a C ,ω Lie group structure, for every slowly vanishingmodulus of continuity ω . Moreover, G n,β − ( R d ) has a topological group structure anda half-Lie group structure.Proof. This follows from Theorem 4.6, Theorem 4.7, and Theorem 3.11. To see,e.g., (4.8), note that G n,β + ( R d ) = [ α>β G n,α ( R d ) = [ α>β D n,α ( R d ) is path-connected in D n,β + ( R d ) = S α>β D n,α ( R d ) and thus contained in D n,β + ( R n ) . The inclusion D n,β + ( R d ) ⊆ G n,β + ( R d ) follows from the proof ofTheorem 4.7: the line segment γ factors to some step of the inductive limit defining D n,β + ( R d ), by Lemma 3.8. (cid:3) Clearly, Remark 4.8 also applies in this situation.Let us summarize the results of this section in Table 2.
ODE closed Trouv´e group C n,β ( R d , R d ) yes D n,β ( R d ) C n,β − ( R d , R d ) yes D n,β − ( R d ) C n,β +0 ( R d , R d ) yes D n,β + ( R d ) Table 2.
Here n ∈ N ≥ . In the first two rows β ∈ (0 , β ∈ [0 , Continuity of the flow map
Let n ∈ N ≥ and β ∈ (0 , u ∈ X n,β ( I, R d ) (thus every u ∨ ∈ L ( I, C n,β ( R d , R d ))) has a unique pointwise flow Φwith Φ ∨ ∈ C ( I, D n,β ( R d )). The goal of this section is to show the following:(1) If u ∨ ∈ L ( I, C n,β ( R d , R d )) and α < β , then Φ ∨ is the unique strong D n,α -flow of u , i.e.,(5.1) Φ ∨ ( t ) = Id + Z t u ∨ ( s ) ◦ Φ ∨ ( s ) ds, t ∈ I, in D n,α ( R d ).(2) The flow map L ( I, C n,β ( R d , R d )) → C ( I, D n,α ( R d )), u ∨ Φ ∨ , is boundedfor all n ≥ C ,β − α , if n ≥ α = β , becausestrong measurability of s u ∨ ( s ) ◦ Φ ∨ ( s ) may fail if u ∨ ( s ) ⋆ is not continuous; seeRemark 5.2 below. N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 25
Existence of the strong D n,α -flow. First we show that the Bochner integralin (5.1) exists if β > α . Lemma 5.1.
Let n ∈ N ≥ and < α < β ≤ . Let u ∈ L ( I, C n,β ( R d , R d )) andlet Φ : I → D n,α ( R d ) be continuous. Then: (i) The mapping I × C n,α ( R d , R d ) → C n,α ( R d , R d ) , ( t, φ ) u ( t ) ◦ (Id + φ ) ,has the Carath´eodory property. The function t u ( t ) ◦ Φ( t ) belongs to L ( I, C n,α ( R d , R d )) . (ii) If ev x Φ( t ) =: Φ ∧ ( t, x ) is the pointwise flow of u for the initial condition Φ ∧ (0 , x ) = x for all x , then Φ is the strong D n,α -flow of u .Proof. (i) That ( t, φ ) u ( t ) ◦ (Id + φ ) has the Carath´eodory property follows easilyfrom Theorem 2.7 and Theorem 2.8. Together with continuity of Φ, an applicationof [1, Lemma 2.2] yields strong measurability of t u ( t ) ◦ Φ( t ). Integrability followsfrom Theorem 2.7.(ii) Recall that ev x is continuous and linear on C n,α ( R d , R d ) and thus commuteswith the Bochner integral. Observe also that s u ( s ) ◦ Φ( s ) is Bochner integrablein C n,α ( R d , R d ), by (i). Since Φ ∧ ( t, x ) is the pointwise flow, we haveev x Φ( t ) = Φ ∧ ( t, x ) = x + Z t u ( s )(Φ ∧ ( s, x )) ds = ev x (cid:16) Id + Z t u ( s ) ◦ Φ( s ) ds (cid:17) . Since the family of evaluation maps is point separating on C n,α ( R d , R d ), we aredone. (cid:3) Remark 5.2.
In general, the function t u ( t ) ◦ Φ( t ) cannot belong to L ( I, C n,β ( R d , R d )), even if u is a constant and Φ is continuous into D N, ( R d ) forall N ≥ n . Indeed: Let χ ∈ C ∞ c ( R ) be 1 on [ − , ψ ( x ) := x n | x | β χ ( x ) bethe C n,β -function from the proof of Lemma 3.5 and Example 4.2. Taking u ( t ) := ψ and Φ( t )( x ) := x + tχ ( x ) we get ( u ( t ) ◦ Φ( t ))( x ) = ψ ( x + t ) if x ∈ [ − , t u ( t ) ◦ Φ( t ) is not strongly measurable. Theorem 5.3.
Let n ∈ N ≥ and < α < β ≤ . Let u ∈ L ( I, C n,β ( R d , R d )) .Then u has a unique strong D n,α -flow Φ .Proof. By Theorem 4.6, u has a unique pointwise flow Φ : I × R d → R d such thatΦ ∨ ∈ C ( I, D n,β ( R d )). For α < β , we have Φ ∨ ∈ C ( I, D n,α ( R d )). By Lemma 5.1,Φ ∨ is the unique strong D n,α -flow of u , i.e., it satisfies (5.1). (cid:3) Remark 5.4.
One can use Carath´eodory’s solution theory for ODEs on Banachspaces which are Bochner integrable in time (cf. Section 2.3) to give an alternativeproof which, however, does not work for n = 1! Indeed, using Lemma 5.1, Corol-lary 2.11, and Theorem 2.4 one can show that u has a unique strong D n − ,α -flowΦ. That the flow Φ = Id + φ is actually strongly D n,α -valued follows from theobservation that dφ satisfies the linear ODE dφ ( t ) = d Z t u ( s ) ◦ (Id + φ ( s )) ds = Z t du ( s ) ◦ (Id + φ ( s )) · ( + dφ )( s ) ds, and from Theorem 2.5. Continuity of the flow map.
First we prove that the flow map is bounded.
Proposition 5.5.
Let n ∈ N ≥ and β ∈ (0 , . The flow map L ( I, C n,β ( R d , R d )) → C ( I, D n,β ( R d )) , u Φ , is bounded. In the proof we use only pointwise estimates; thus the result still holds if u ∧ varies in X n,β ( I, R d ) endowed with the norm k u k L ( I,C n,βb ) . Proof.
We first claim that u φ := Φ − Id is bounded into C ( I, C nb ( R d , R d )).We proceed by induction on n . For simplicity of notation we simply write φ ( t, x )instead of φ ∧ ( t, x ), etc. Clearly k φ ( t, x ) k ≤ Z t k u ( s, Φ( s, x )) k ds ≤ k u k L ( I,C n,βb ) and hence k φ k C ( I,C b ) ≤ k u k L ( I,C n,βb ) . Assume that u φ is bounded into C ( I, C n − b ( R d , R d )). By Fa`a di Bruno’s formula(2.7), d nx ( u ( s ) ◦ Φ( s ))( x ) = d x u ( s )(Φ( s, x ))( d nx Φ( s, x ))+ sym n X l =2 X γ ∈ Γ( l,n ) c γ u ( s ) ( l ) (Φ( s, x )) (cid:0) d γ x Φ( s, x ) , . . . , d γ l x Φ( s, x ) (cid:1) . Hence k d nx ( u ( s ) ◦ Φ( s )) k ≤ k u ( s ) k [Φ( s )] n + n X l =2 X γ ∈ Γ( l,n ) c γ k u ( s ) k l [Φ( s )] γ · · · [Φ( s )] γ l and so, by induction hypothesis and since [Φ( s )] n ≤ φ ( s )] n ,[ φ ( t )] n ≤ Z t k d nx ( u ( s ) ◦ Φ( s )) k ds ≤ Z t k u ( s ) k [ φ ( s )] n ds + C k u k L ( I,C n,βb ) . Gronwall’s lemma implies that k φ k C ( I,C nb ) ≤ C k u k L ( I,C n,βb ) exp( k u k L ( I,C n,βb ) ) , and the claim is proved.It remains to show that u φ is bounded into C ( I, C n,αb ( R d , R d )). To this endconsider d nx ( u ( s ) ◦ Φ( s ))( x ) − d nx ( u ( s ) ◦ Φ( s ))( y )= sym n X l =1 X γ ∈ Γ( l,n ) c γ (cid:0) u ( s ) ( l ) (Φ( s, x ))( A γ,l ( x, y )) − u ( s ) ( l ) (Φ( s, y ))( A γ, ( x, y )) (cid:1) , where A γ,j = A γ,j ( x, y ) := (cid:0) d γ x Φ( s, x ) , . . . , d γ j x Φ( s, x ) , d γ j +1 x Φ( s, y ) , . . . , d γ l x Φ( s, y ) (cid:1) . N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 27
Then (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, x ))( A γ,l ) − u ( s ) ( l ) (Φ( s, y ))( A γ, ) (cid:13)(cid:13) L n ≤ (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, x ))( A γ,l ) − u ( s ) ( l ) (Φ( s, y ))( A γ,l ) (cid:13)(cid:13) L n + l X k =1 (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, y ))( A γ,k ) − u ( s ) ( l ) (Φ( s, y ))( A γ,k − ) (cid:13)(cid:13) L n . For the first summand (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, x ))( A γ,l ) − u ( s ) ( l ) (Φ( s, y ))( A γ,l ) (cid:13)(cid:13) L n ≤ (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, x )) − u ( s ) ( l ) (Φ( s, y )) (cid:13)(cid:13) L n (1 + k φ ( s ) k n ) n ≤ ( k u ( s ) k n [Φ( s )] k x − y k (1 + k φ ( s ) k n ) n if l < n, k u ( s ) k n,β [Φ( s )] β k x − y k β (1 + k φ ( s ) k n ) n if l = n. For the other summands observe that u ( s ) ( l ) (Φ( s, y ))( A γ,k ) − u ( s ) ( l ) (Φ( s, y ))( A γ,k − )= u ( s ) ( l ) (Φ( s, y )) (cid:0) . . . , d γ k − x Φ( s, x ) , d γ k x Φ( s, x ) − d γ k x Φ( s, y ) , d γ k +1 x Φ( s, y ) , . . . (cid:1) , whence, if l ≥ γ k ≤ n − (cid:13)(cid:13) u ( s ) ( l ) (Φ( s, y ))( A γ,k ) − u ( s ) ( l ) (Φ( s, y ))( A γ,k − ) (cid:13)(cid:13) L n ≤ k u ( s ) k n (1 + k φ ( s ) k n ) n − k φ ( s ) k n k x − y k . For l = 1, we have (cid:13)(cid:13) du ( s )(Φ( s, y ))( d nx Φ( s, x )) − du ( s )(Φ( s, y ))( d nx Φ( s, y )) (cid:13)(cid:13) L n ≤ k u ( s ) k n k φ ( s ) k n,β k x − y k β . These estimates, together with the fact that u φ is bounded into C ( I, C nb ( R d , R d )), imply k φ ( t ) k n,β ≤ Z t k u ( s ) ◦ Φ( s ) k n,β ds ≤ C Z t k u ( s ) k n,β k φ ( s ) k n,β ds + C k u k L ( I,C n,βb ) , and Gronwall’s inequality yields the assertion. (cid:3) Theorem 5.6.
Let n ∈ N ≥ and < α < β ≤ . Then the flow map L ( I, C n,β ( R d , R d )) → C ( I, D n,α ( R d )) , u Φ , is continuous, even C ,β − α . We do not know if the theorem also holds for n = 1 or for α = β . Proof.
Fix u ∈ L ( I, C n,β ( R d , R d )) and let u, v ∈ L ( I, C n,β ( R d , R d )) be in the ballwith radius δ > u in L ( I, C n,β ( R d , R d )). Consider the correspondingflows Φ = Id + φ, Ψ = Id + ψ ∈ C ( I, D n,α ( R d )). By Proposition 5.5, there is aconstant C = C ( u , δ ) > k φ k C ( I,C n,αb ) ≤ C, k ψ k C ( I,C n,αb ) ≤ C. By Corollary 2.11, Theorem 2.7, and Theorem 5.3, k φ ( t ) − ψ ( t ) k n − ,α ≤ Z t k u ( s ) ◦ Φ( s ) − v ( s ) ◦ Ψ( s ) k n − ,α ds ≤ Z t k u ( s ) ◦ Φ( s ) − u ( s ) ◦ Ψ( s ) k n − ,α + k ( u ( s ) − v ( s )) ◦ Ψ( s ) k n − ,α ds ≤ C Z t k u ( s ) k n,β k φ ( s ) − ψ ( s ) k n − ,α + k u ( s ) − v ( s ) k n − ,α ds and so, by Gronwall’s lemma (and Lemma 2.2), k φ ( t ) − ψ ( t ) k n − ,α ≤ C k u − v k L ( I,C n,βb ) exp( C k u k L ( I,C n,βb ) )=: C k u − v k L ( I,C n,βb ) . (5.2)This proves that u Φ is continuous into C ( I, D n − ,α ( R d )). Applying d = d x , k dφ ( t ) − dψ ( t ) k n − ,α ≤ Z t k ( du ( s ) ◦ Φ( s )) d Φ( s ) − ( dv ( s ) ◦ Ψ( s )) d Ψ( s ) k n − ,α ds ≤ Z t k ( du ( s ) ◦ Φ( s ))( d Φ( s ) − d Ψ( s )) k n − ,α ds + Z t k ( du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s )) d Ψ( s )) k n − ,α ds. By Proposition 2.6 and Theorem 2.7, k ( du ( s ) ◦ Φ( s ))( d Φ( s ) − d Ψ( s )) k n − ,α ≤ n k du ( s ) ◦ Φ( s ) k n − ,α k d Φ( s ) − d Ψ( s ) k n − ,α ≤ n M k du ( s ) k n − ,α (1 + k φ ( s ) k n − ,α ) n k d Φ( s ) − d Ψ( s ) k n − ,α ≤ C k u ( s ) k n,β k d Φ( s ) − d Ψ( s ) k n − ,α and k ( du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s )) d Ψ( s ) k n − ,α ≤ n k du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s ) k n − ,α k d Ψ( s ) k n − ,α ≤ C k du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s ) k n − ,α . By Theorem 2.7, Theorem 2.8, and (5.2), k du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s ) k n − ,α ≤ k ( du ( s ) − dv ( s )) ◦ Φ( s ) k n − ,α + k dv ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s ) k n − ,α ≤ M (cid:0) k du ( s ) − dv ( s ) k n − ,α (1 + k φ ( s ) k n − ,α ) n + k v ( s ) k n,β k φ ( s ) − ψ ( s ) k β − αn − ,α (cid:1) ≤ C (cid:0) k u ( s ) − v ( s ) k n,β + k v ( s ) k n,β k φ ( s ) − ψ ( s ) k β − αn − ,α (cid:1) . Together with (5.2) this gives Z t k du ( s ) ◦ Φ( s ) − dv ( s ) ◦ Ψ( s ) k n − ,α ds ≤ C (cid:0) k u − v k L ( I,C n,βb ) + k u − v k β − αL ( I,C n,βb ) (cid:1) ≤ C k u − v k β − αL ( I,C n,βb ) , N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 29 provided that k u − v k L ( I,C n,βb ) ≤
1. Consequently, k dφ ( t ) − dψ ( t ) k n − ,α ≤ C Z t k u ( s ) k n,β k dφ ( s ) − dψ ( s ) k n − ,α ds + C k u − v k β − αL ( I,C n,βb ) . Then Gronwall’s inequality implies k dφ ( t ) − dψ ( t ) k n − ,α ≤ C k u − v k β − αL ( I,C n,βb ) exp( C k u k L ( I,C n,βb ) ) ≤ C k u − v k β − αL ( I,C n,βb ) , for all t ∈ I , and the assertion follows. (cid:3) Flows of strong time-dependent C n,β − -vector fields. Let n ∈ N ≥ .By a strong time-dependent C n,β − -vector field , for β ∈ (0 , u : I → C n,β − ( R d , R d ) such that u ∈ L ( I, C n,α ( R d , R d )) for all α < β . We denotethe space of all strong time-dependent C n,β − -vector fields by L ( I, C n,β − ( R d , R d ))and equip it with the fundamental system of seminorms {k · k L ( I,C n,αb ) : α < β } .Clearly, for every strong time-dependent C n,β − -vector field u , u ∧ is a pointwisetime-dependent C n,β − -vector field (as defined in Section 4.4); the converse is nottrue in general.By Proposition 5.5, the flow map L ( I, C n,β − ( R d , R d )) → C ( I, D n,β − ( R d )) isbounded, for all n ∈ N ≥ , β ∈ (0 , Theorem 5.7.
Let n ∈ N ≥ . For β ∈ (0 , , the flow map L ( I, C n,β − ( R d , R d )) → C ( I, D n,β − ( R d )) , u Φ , is continuous and C ,ω , for any slowly vanishing modulusof continuity ω .Proof. This is immediate from Theorem 5.6 and from the estimates in its proof. (cid:3)
Remark 5.8.
One could define strong time-dependent C n,β +0 -vector fields , for β ∈ [0 , S α ∈ ( β, L ( I, C n,α ( R d , R d )). Then, byTheorem 4.6, we have a flow map [ α ∈ ( β, L ( I, C n,α ( R d , R d )) → [ α ∈ ( β, C ( I, C n,α ( R d , R d )) ⊆ C ( I, C n,β +0 ( R d , R d )) . Note that this is not clear for u ∈ L ( I, C n,β +0 ( R d , R d )), since such u may not factorto some step in the inductive limit defining C n,β +0 ( R d , R d ). Is this flow map C ,ω ,for slowly vanishing moduli of continuity ω ? This would follow from Theorem 5.6if the (LB)-space S α ∈ ( β, L ( I, C n,α ( R d , R d )) were regular. Appendix A. Proofs for Section 2.4
Proposition 2.6 is precisely [3, 4.2].
Proof of Theorem 2.7.
We prove the assertion by induction on m . First observethat d ( g ◦ (Id + f )) = dg ◦ (Id + f ) · ( + df ) = dg ◦ (Id + f ) + dg ◦ (Id + f ) · df . Wehave k dg ( x + f ( x )) − dg ( y + f ( y )) k L ≤ k dg k ,α k x − y + f ( x ) − f ( y ) k α ≤ k g k ,α (1 + k f k ) α k x − y k α , (A.1) and k dg ( x + f ( x )) · df ( x ) − dg ( y + f ( y )) · df ( y ) k L ≤ k dg ( x + f ( x )) · df ( x ) − dg ( x + f ( x )) · df ( y ) k L + k dg ( x + f ( x )) · df ( y ) − dg ( y + f ( y )) · df ( y ) k L ≤ k dg ( x + f ( x )) k L k df ( x ) − df ( y ) k L + k dg ( x + f ( x )) − dg ( y + f ( y )) k L k df ( y ) k L ≤ k g k ,α k f k ,α k x − y k α + k g k ,α (1 + k f k ) α k x − y k α k f k ,α . Thus, k dg ◦ (Id + f ) · df k ,α ≤ k g k ,α (1 + k f k ,α ) α , and since the same bound is trivially also valid for k g ◦ (Id + f ) k , the case m = 1is proved.Now assume the statement holds for m −
1. Then k d ( g ◦ (Id + f )) k m − ,α ≤ k dg ◦ (Id + f ) k m − ,α + k dg ◦ (Id + f ) · df k m − ,α . The inductive assumption implies k dg ◦ (Id + f ) k m − ,α ≤ M k dg k m − ,α (1 + k f k m − ,α ) m − α , and using Proposition 2.6, we get k dg ◦ (Id + f ) · df k m − ,α ≤ m k dg ◦ (Id + f ) k m − ,α · k df k m − ,α which now adds up to (2.4). (cid:3) Proof of Theorem 2.8.
We proceed by induction on m . First observe that we have k g ⋆ ( f ) − g ⋆ ( f ) k ≤ k g k k f − f k . Moreover, by Proposition 2.6, k d ( g ⋆ ( f )) − d ( g ⋆ ( f )) k ,α = k dg ◦ (Id + f ) · ( + df ) − dg ◦ (Id + f ) · ( + df ) k ,α = k dg ◦ (Id + f ) · ( df − df ) − ( dg ◦ (Id + f ) − dg ◦ (Id + f )) · ( + df ) k ,α ≤ k dg ◦ (Id + f ) k ,α k df − df k ,α + k dg ◦ (Id + f ) − dg ◦ (Id + f ) k ,α (1 + 2 k df k ,α ) . As an intermediate step we use Lemma 2.1 and (A.1) to estimate k dg ◦ (Id + f ) − dg ◦ (Id + f ) k ,α ≤ k dg ◦ (Id + f ) − dg ◦ (Id + f ) k β − αβ k dg ◦ (Id + f ) − dg ◦ (Id + f ) k αβ ,β ≤ ( k g k ,β k f − f k β ) β − αβ ( k dg ◦ (Id + f ) k ,β + k dg ◦ (Id + f ) k ,β ) αβ ≤ ( k g k ,β k f − f k β ) β − αβ ( k g k ,β ((1 + k f k ) β + (1 + k f k ) β )) αβ ≤ k g k ,β (2 + k f k + k f k ) k f − f k β − α . Consequently, if
R > f , f ∈ B ,α ( f , R ), and hence k f − f k ,α ≤ (1+2 R ) k f − f k β − α ,α , then k d ( g ⋆ ( f )) − d ( g ⋆ ( f )) k ,α ≤ M k g k ,β k f − f k β − α ,α , N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 31 where M = M ( k f k ,α , R ), and hence k g ⋆ ( f ) − g ⋆ ( f ) k ,α ≤ M k g k ,β k f − f k β − α ,α which proves the case m = 1.Now assume we have already proven the desired result for m −
1. Then, as inthe case m = 1, we have k d ( g ⋆ ( f )) − d ( g ⋆ ( f )) k m − ,α ≤ m k dg ◦ (Id + f ) k m − ,α k df − df k m − ,α + k dg ◦ (Id + f ) − dg ◦ (Id + f ) k m − ,α (1 + 2 m k df k m − ,α ) . By the inductive assumption, k dg ◦ (Id + f ) − dg ◦ (Id + f ) k m − ,α ≤ M k g k m,β k f − f k β − αm − ,α . Together with Theorem 2.7, which makes it possible to extract k g k m,β from theterm k dg ◦ (Id + f ) k m − ,α , and using that k f − f k m,α ≤ (1 + 2 R ) k f − f k β − αm,α for f , f ∈ B m,α ( f , R ) , we may conclude (2.5). (cid:3) Proof of Corollary 2.9.
This follows easily from Theorem 2.7, Theorem 2.8, and g ◦ (Id + f ) − g ◦ (Id + f ) = f ⋆ ( g − g ) + ( g ) ⋆ ( f ) − ( g ) ⋆ ( f ) . (cid:3) Proof of Theorem 2.10.
By Theorem 2.8, the mapping ( dg ) ⋆ : C m,αb ( R d , R d ) → C m,αb ( R d , L ( R d , R d )), φ dg ◦ (Id + φ ) is continuous. Consider the mapping l : C m,αb ( R d , L ( R d , R d )) → L ( C m,αb ( R d , R d ) , C m,αb ( R d , R d )) u l ( u )( η ) := ( x u ( x )( η ( x )))which is continuous and linear. We claim that d ( g ⋆ ) exists and satisfies d ( g ⋆ ) = l ◦ ( dg ) ⋆ . This implies the proposition.First note that for ψ ∈ C m,αb ( R d , R d ) and φ ∈ C m,αb ( R d , R d ),( l ◦ ( dg ) ⋆ )( ψ )( φ )( x ) = dg ◦ (Id + ψ )( x ) · φ ( x ) , where · denotes the action of the linear map dg ◦ (Id + ψ )( x ) ∈ L ( R d , R d ) to thevector φ ( x ) ∈ R d .Take φ ∈ C m,αb ( R d , R d ) with k φ k m,α ≤
1. By Theorem 2.8 (applied to dg ), for ψ ∈ B m,α ( ψ , k ( dg ) ⋆ ( ψ ) − ( dg ) ⋆ ( ψ ) k m,α ≤ M k g k m +1 ,β k ψ − ψ k β − αm,α , For ε < ψ + εφ ∈ B m,α ( ψ ,
1) for all φ ∈ B m,α (0 , ε k g ⋆ ( ψ + εφ ) − g ⋆ ( ψ ) − ( l ◦ ( dg ) ⋆ )( ψ )( εφ ) k m,α = 1 ε k g ◦ (Id + ψ + εφ ) − g ◦ (Id + ψ ) − ε ( dg ◦ (Id + ψ )) · φ k m,α = (cid:13)(cid:13)(cid:13) Z ( dg ◦ (Id + ψ + sεφ ) − dg ◦ (Id + ψ )) · φ ds (cid:13)(cid:13)(cid:13) m,α ≤ Z k dg ◦ (Id + ψ + sεφ ) − dg ◦ (Id + ψ ) k m,α k φ k m,α ds ≤ Z M k g k m +1 ,β k εsφ k β − αm,α ds ≤ M k g k m +1 ,β ε β − α which tends to 0 uniformly in φ ∈ B m,α (0 ,
1) as ε →
0. The claim is proved. (cid:3)
Proof of Corollary 2.11.
Let γ ( s ) := (1 − s ) f + sf for s ∈ [0 , g ⋆ ( f ) − g ⋆ ( f ) = Z dds ( g ⋆ ◦ γ )( s ) ds = Z d ( g ⋆ )( γ ( s )) · γ ′ ( s ) ds = Z dg ◦ (Id + γ ( s )) · ( f − f ) ds. Thus, by Proposition 2.6 and Theorem 2.7, k g ⋆ ( f ) − g ⋆ ( f ) k m,α ≤ Z k dg ◦ (Id + γ ( s )) · ( f − f ) k m,α ds ≤ Z M k dg k m,β (1 + k γ ( s ) k m,α ) m +1 k f − f k m,α ds ≤ M k g k m +1 ,β (1 + max i =1 , k f i k m,α ) m +1 k f − f k m,α . (cid:3) References [1] B. Aulbach and T. Wanner,
Integral manifolds for Carath´eodory type differential equationsin Banach spaces , Six lectures on dynamical systems (Augsburg, 1994), World Sci. Publ.,River Edge, NJ, 1996, pp. 45–119.[2] M. Bruveris and F.-X. Vialard,
On completeness of groups of diffeomorphisms , J. Eur. Math.Soc. (JEMS) (2017), no. 5, 1507–1544.[3] R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H¨olderfunctions , Discrete Contin. Dynam. Systems (1999), no. 1, 157–184.[4] J. Diestel and J. J. Uhl, Jr., Vector measures , American Mathematical Society, Providence,R.I., 1977, With a foreword by B. J. Pettis, Mathematical Surveys, No. 15.[5] C.-A. Faure,
Th´eorie de la diff´erentiation dans les espaces convenables , Ph.D. thesis, Uni-versit´e de Gen´eve, 1991.[6] C.-A. Faure and A. Fr¨olicher,
H¨older differentiable maps and their function spaces , Categor-ical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), WorldSci. Publ., Teaneck, NJ, 1989, pp. 135–142.[7] A. Fr¨olicher,
Applications lisses entre espaces et vari´et´es de Fr´echet , C. R. Acad. Sci. ParisS´er. I Math. (1981), no. 2, 125–127.
N GROUPS OF H ¨OLDER DIFFEOMORPHISMS 33 [8] A. Fr¨olicher, B. Gisin, and A. Kriegl,
General differentiation theory , Category theoreticmethods in geometry (Aarhus, 1983), Various Publ. Ser. (Aarhus), vol. 35, Aarhus Univ.,Aarhus, 1983, pp. 125–153.[9] A. Fr¨olicher and A. Kriegl,
Linear spaces and differentiation theory , Pure and Applied Math-ematics (New York), John Wiley & Sons Ltd., Chichester, 1988, A Wiley-Interscience Publi-cation.[10] H. Gl¨ockner,
Measurable regularity properties of infinite-dimensional Lie groups , (2016),arXiv:1601.02568.[11] H. Inci, T. Kappeler, and P. Topalov,
On the regularity of the composition of diffeomorphisms ,Mem. Amer. Math. Soc. (2013), no. 1062, vi+60.[12] A. Kriegl,
Die richtigen R¨aume f¨ur Analysis im Unendlich-Dimensionalen , Monatsh. Math. (1982), no. 2, 109–124.[13] , Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigenlokalkonvexen Vektorr¨aumen , Monatsh. Math. (1983), no. 4, 287–309.[14] A. Kriegl and P. W. Michor, The convenient setting for real analytic mappings , Acta Math. (1990), no. 1-2, 105–159.[15] ,
The convenient setting of global analysis , Mathematical Surveys andMonographs, vol. 53, American Mathematical Society, Providence, RI, 1997, .[16] A. Kriegl, P. W. Michor, and A. Rainer,
The convenient setting for non-quasianalyticDenjoy–Carleman differentiable mappings , J. Funct. Anal. (2009), 3510–3544.[17] ,
The convenient setting for quasianalytic Denjoy–Carleman differentiable mappings ,J. Funct. Anal. (2011), 1799–1834.[18] ,
Many parameter H¨older perturbation of unbounded operators , Math. Ann. (2012), 519–522.[19] ,
The convenient setting for Denjoy–Carleman differentiable mappings of Beurlingand Roumieu type , Rev. Mat. Complut. (2015), no. 3, 549–597.[20] , An exotic zoo of diffeomorphism groups on R n , Ann. Global Anal. Geom. (2015),no. 2, 179–222.[21] A. Kriegl and L. D. Nel, A convenient setting for holomorphy , Cahiers Topologie G´eom.Diff´erentielle Cat´eg. (1985), no. 3, 273–309.[22] P. W. Michor and D. Mumford, A zoo of diffeomorphism groups on R n , Ann. Global Anal.Geom. (2013), no. 4, 529–540.[23] H. Neus, ¨Uber die Regularit¨atsbegriffe induktiver lokalkonvexer Sequenzen , ManuscriptaMath. (1978), no. 2, 135–145.[24] R. S. Palais, Natural operations on differential forms , Trans. Amer. Math. Soc. (1959),125–141.[25] V.S. Retakh, Subspaces of a countable inductive limit. , Sov. Math., Dokl. (1970), 1384–1386 (English).[26] G. Schindl, The convenient setting for ultradifferentiable mappings of Beurling- andRoumieu-type defined by a weight matrix , Bull. Belg. Math. Soc. Simon Stevin (2015),no. 3, 471–510.[27] A. Trouv´e, An infinite dimensional group approach for physics based models in pattern recog-nition , http://cis.jhu.edu/publications/papers in database/alain/trouve1995.pdf ,1995.[28] L. Younes, Shapes and diffeomorphisms , Applied Mathematical Sciences, vol. 171, Springer-Verlag, Berlin, 2010.
D.N. Nenning: Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
E-mail address : [email protected] A. Rainer: Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
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