On the regularity of the composition of diffeomorphisms
aa r X i v : . [ m a t h . A P ] F e b On the regularity of the composition ofdiffeomorphisms
H. Inci ∗ , T. Kappeler ∗ , P. Topalov † October 14, 2018
Abstract
For M a closed manifold or the Euclidean space R n we present adetailed proof of regularity properties of the composition of H s -regulardiffeomorphisms of M for s > dim M + 1. In this paper we are concerned with groups of diffeomorphisms on a smoothmanifold M . Our interest in these groups stems from Arnold’s seminal pa-per [4] on hydrodynamics. He suggested that the Euler equation modeling aperfect fluid on a (oriented) Riemannian manifold M can be reformulated asthe equation for geodesics on the group of volume (and orientation) preserv-ing diffeomorphims of M . In this way properties of solutions of the Eulerequation can be expressed in geometric terms – see [4]. In the sequel, Ebinand Marsden [14], [15] used this approach to great success to study the initialvalue problem for the Euler equation on a compact manifold, possibly withboundary. Later it was observed that other nonlinear evolution equationssuch as Burgers equation [6], KdV, or the Camassa Holm equation [7], [17]can be viewed in a similar way – see [22], [32], as well as [5], [19], and [23].In particular, for the study of the solutions of the Camassa Holm equation, ∗ Supported in part by the Swiss National Science Foundation † Supported in part by NSF DMS-0901443 C ∞ -smooth. A straightforward formal computation shows thatthe differential of the left translation L ψ : ϕ ψ ◦ ϕ of a diffeomorphism ϕ by a diffeomorphism ψ in direction h : M → T M can be formally computedto be ( d ϕ L ψ )( h )( x ) = ( d ϕ ( x ) ψ )( h ( x )) , x ∈ M and hence involves a loss of derivative of ψ . As a consequence, for a spaceof diffeomorphisms of M to be a Lie group it is necessary that they are C ∞ -smooth and hence such a group cannot have the structure of a Banachmanifold, but only of a Fr´echet manifold. It is well known that the calculus inFr´echet manifolds is quite involved as the classical inverse function theoremdoes not hold, cf. e.g. [18], [24]. Various aspects of Fr´echet Lie groups ofdiffeomorphisms have been investigated – see e.g. [18], [31], [34], [35]. Inparticular, Riemann exponential maps have been studied in [10], [11], [20],[21].However, in many situations, one has to consider diffeomorphisms ofSobolev type – see e.g. [12], [13], [14]. In this paper we are concernedwith composition of maps in H s ( M ) ≡ H s ( M, M ). It seems to be unknownwhether, in general, the composition of two maps in H s ( M ) with s an integersatisfying s > n/ H s ( M ). In all known proofs one needs thatone of the maps is a diffeomorphism or, alternatively, is C ∞ -smooth.First we consider the case where M is the Euclidean space R n , n ≥ ( R n ) the space of orientation preserving C -diffeomorphismsof R n , i.e. the space of bijective C -maps ϕ : R n → R n so that det( d x ϕ ) > x ∈ R n and ϕ − : R n → R n is a C -map as well. For any integer s s > n/ D s ( R n ) := { ϕ ∈ Diff ( R n ) | ϕ − id ∈ H s ( R n ) } where H s ( R n ) = H s ( R n , R n ) and H s ( R n , R d ) is the Hilbert space H s ( R n , R d ) := { f = ( f , . . . , f d ) | f i ∈ H s ( R n , R ) , i = 1 , . . . , d } with H s -norm k · k s given by k f k s = (cid:0) d X i =1 k f i k s (cid:1) / and H s ( R n , R ) is the Hilbert space of elements g ∈ L ( R n , R ) with theproperty that the distributional derivatives ∂ α g , α ∈ Z n ≥ , up to order | α | ≤ s are in L ( R n , R ). Its norm is given by k g k s = (cid:0) X | α |≤ s Z R n | ∂ α g | dx (cid:1) / . (1)Here we used multi-index notation, i.e. α = ( α , . . . , α n ) ∈ Z n ≥ , | α | = P ni =1 α i , x = ( x , . . . , x n ), and ∂ α ≡ ∂ αx = ∂ α x · · · ∂ α n x n . As s > n/ D s ( R n ) − id = { ϕ − id | ϕ ∈ D s ( R n ) } is an open subset of H s ( R n ) – see Corollary 2.1 below. In this way D s ( R n )becomes a Hilbert manifold modeled on H s ( R n ). In Section 2 of this paperwe present a detailed proof of the following Theorem 1.1.
For any r ∈ Z ≥ and any integer s with s > n/ µ : H s + r ( R n , R d ) × D s ( R n ) → H s ( R n , R d ) , ( u, ϕ ) u ◦ ϕ (2) and inv : D s + r ( R n ) → D s ( R n ) , ϕ ϕ − (3) are C r -maps. emark 1.1. To the best of our knowledge there is no proof of Theorem1.1 available in the literature. Besides being of interest in itself we will useTheorem 1.1 and its proof to show Theorem 1.2 stated below. Note that thecase r = 0 was considered in [8]. Remark 1.2.
The proof for the C r -regularity of the inverse map is valid ina much more general context: using that D s ( R n ) is a topological group andthat the composition D s + r ( R n ) × D s ( R n ) → D s ( R n ) , ( ψ, ϕ ) ψ ◦ ϕ is C r -smooth we apply the implicit function theorem to show that the inversemap D s + r ( R n ) → D s ( R n ) , ϕ ϕ − is a C r -map as well. Remark 1.3.
By considering lifts to R n of diffeomorphisms of T n = R n / Z n ,the same arguments as in the proof of Theorem 1.1 can be used to showcorresponding results for the group D s ( T n ) of H s -regular diffeomorphismson T n . In Section 3 and Section 4 of this paper we discuss various classes of diffeo-morphisms on a closed manifold M . For any integer s with s > n/ H s ( M ) of Sobolev maps is defined by using coordinate charts of M . Moreprecisely, let M be a closed manifold of dimension n and N a C ∞ -manifoldof dimension d . We say that a continuous map f : M → N is an element in H s ( M, N ) if for any x ∈ M there exists a chart χ : U → U ⊆ R n of M with x ∈ U , and a chart η : V → V ⊆ R d of N with f ( x ) ∈ V , such that f ( U ) ⊆ V and η ◦ f ◦ χ − : U → V is an element in the Sobolev space H s ( U, R d ). Here H s ( U, R d ) – similarlydefined as H s ( R n , R d ) – is the Hilbert space of elements in L ( U, R d ) whosedistributional derivatives up to order s are L -integrable. In Section 3 weintroduce a C ∞ -differentiable structure on the space H s ( M, N ) in terms ofa specific cover by open sets which is especially well suited for proving regu-larity properties of the composition of mappings as well as other applications i.e., a compact C ∞ -manifold without boundary H s ( M, N )is that each of its open sets can be embedded into a finite cartesian productof Sobolev spaces of H s -maps between Euclidean spaces.It turns out that this cover makes H s ( M, N ) into a C ∞ -Hilbert manifold– see Section 4 for details. In addition, we show in Section 4 that the C ∞ -differentiable structure for H s ( M, N ) defined in this way coincides with theone, introduced by Ebin and Marsden in [14], [15] and defined in termsof a Riemannian metric on N . In particular it follows that the standarddifferentiable structure does not depend on the choice of the metric. Nowassume in addition that M is oriented. Then, for any linear isomorphism A : T x M → T y M between the tangent spaces of M at arbitrary points x and y of M , the determinant det( A ) has a well defined sign. For any integer s with s > n + 1 define D s ( M ) := (cid:8) ϕ ∈ Diff ( M ) ϕ ∈ H s ( M, M ) (cid:9) where Diff ( M ) denotes the set of all orientation preserving C smooth dif-feomorphisms of M . We will show that D s ( M ) is open in H s ( M, M ) andhence is a C ∞ -Hilbert manifold. Elements in D s ( M ) are referred to as ori-entation preserving H s -diffeomorphisms.In Section 3 we prove the following Theorem 1.2.
Let M be a closed oriented manifold of dimension n , N a C ∞ -manifold, and s an integer satisfying s > n/ . Then for any r ∈ Z ≥ ,(i) µ : H s + r ( M, N ) × D s ( M ) → H s ( M, N ) , ( f, ϕ ) f ◦ ϕ and(ii) inv : D s + r ( M ) → D s ( M ) , ϕ ϕ − are both C r -maps. Remark 1.4.
Various versions of Theorem 1.2 can be found in the literature,however mostly without proofs – see e.g. [13], [14], [16], [34], [35], [36], [37];cf. also [30]. A complete, quite involved proof of statement ( i ) of Theorem1.2 can be found in [35], Proposition 3.3 of Chapter 3 and Theorem 2.1 ofChapter 6. Using the approach sketched above we present an elementary proofof Theorem 1.2. In particular, our approach allows us to apply elements ofthe proof of Theorem 1.1 to show statement ( i ) . emark 1.5. Actually Theorem 1.1 and Theorem 1.2 continue to hold ifinstead of s being an integer it is an arbitrary real number s > n/ . Inorder to keep the exposition as elementary as possible we prove Theorem 1.1and Theorem 1.2 as stated in the main body of the paper and discuss theextension to the case where s > n/ is real in Appendix B. We finish this introduction by pointing out results on compositions ofmaps in function spaces different from the ones considered here and someadditional literature. In the paper [26], de la Llave and Obaya prove a versionof Theorem 1.1 for H¨older continuous maps between open sets of Banachspaces. Using the paradifferential calculus of Bony, Taylor [39] studies thecontinuity of the composition of maps of low regularity between open sets in R n – see also [3]. Acknowledgment : We would like to thank Gerard Misiolek and TudorRatiu for very valuable feedback on an earlier version of this paper. R n In this section we present a detailed and elementary proof of Theorem 1.1.First we prove that the composition map µ is a C r -map (Proposition 2.9)and then, using this result, we show that the inverse map is a C r -map aswell (Proposition 2.13). To simplify notation we write D s ≡ D s ( R n ) and H s ≡ H s ( R n ). Throughout this section, s denotes a nonnegative integer ifnot stated otherwise. H s ( R n , R ) In this subsection we discuss properties of the Sobolev spaces H s ( R n , R )needed later. First let us introduce some more notation. For any x, y ∈ R n denote by x · y the Euclidean inner product, x · y = P nk =1 x k y k , and by | x | the corresponding norm , | x | = ( x · x ) / . Recall that for s ∈ Z ≥ , H s ( R n , R )consists of all L -integrable functions f : R n → R with the property that thedistributional derivatives ∂ α f, α ∈ Z n ≥ , up to order | α | ≤ s are L -integrableas well. Then H s ( R n , R ), endowed with the norm (1), is a Hilbert space and6or any multi-index α ∈ Z n ≥ with | α | ≤ s , the differential operator ∂ α is abounded linear map, ∂ α : H s ( R n , R ) → H s −| α | ( R n , R ) . Alternatively, one can characterize the spaces H s ( R n , R ) via the Fouriertransform. For any f ∈ L ( R n , R ) ≡ H ( R n , R ), denote by ˆ f its Fouriertransform ˆ f ( ξ ) := (2 π ) − n/ Z R n f ( x ) e − ix · ξ dx. Then ˆ f ∈ L ( R n , R ) and k ˆ f k = k f k , where k f k ≡ k f k denotes the L -normof f . The formula for the inverse Fourier transform reads f ( x ) = (2 π ) − n/ Z R n ˆ f ( ξ ) e ix · ξ dξ. When expressed in terms of the Fourier transform ˆ f of f , the operator ∂ α , α ∈ Z n ≥ is the multiplication operatorˆ f ( iξ ) α ˆ f where ξ α = ξ α · · · ξ α n n and one can show that f ∈ L ( R n , R ) is an elementin H s ( R n , R ) iff (1 + | ξ | ) s ˆ f is in L ( R n , R ) and the H s -norm of f , k f k s = (cid:0) P | α |≤ s k ξ α ˆ f k (cid:1) / , satisfies C − s k f k s ≤ k f k ∼ s ≤ C s k f k s (4)for some constant C s ≥ k f k ∼ s := (cid:18)Z R n (1 + | ξ | ) s | ˆ f ( ξ ) | dξ (cid:19) / . (5)In this way the Sobolev space H s ( R n , R ) can be defined for s ∈ R ≥ arbitrary.See Appendix B for a study of these spaces.Using the Fourier transform one gets the following approximation propertyfor functions in H s ( R n , R ). Lemma 2.1.
For any s in Z ≥ , the subspace C ∞ c ( R n , R ) of C ∞ functionswith compact support is dense in H s ( R n , R ) . emark 2.1. The proof shows that Lemma 2.1 actually holds for any s realwith s ≥ .Proof. In a first step we show that C ∞ ( R n , R ) ∩ H s ′ ( R n , R ) is dense in H s ( R n , R ) for any integer s ′ ≥ s . Let χ : R → R be a decreasing C ∞ function satisfying χ ( t ) = 1 ∀ t ≤ χ ( t ) = 0 ∀ t ≥ . For any f ∈ H s ( R n , R ) and N ∈ Z ≥ define f N ( x ) = (2 π ) − n/ Z R n χ (cid:0) | ξ | N (cid:1) ˆ f ( ξ ) e ix · ξ dξ. The support of χ (cid:0) | ξ | N (cid:1) ˆ f ( ξ ) is contained in the ball {| ξ | ≤ N } . Hence f N ( x )is in C ∞ ( R n , R ) ∩ H s ′ ( R n , R ) for any s ′ ≥
0. In addition, by the Lebesgueconvergence theorem,lim N →∞ Z R n (1 + | ξ | ) s (cid:0) − χ ( | ξ | N ) (cid:1) | ˆ f ( ξ ) | dξ = 0 . In view of (5), we have f N → f in H s ( R n , R ). In a second step we showthat C ∞ c ( R n , R ) is dense in C ∞ ( R n , R ) ∩ H s ′ ( R n , R ) for any integer s ′ ≥ f ∈ C ∞ ( R n , R ) ∩ H s ′ ( R n , R ) by truncation in the x -space. For any N ∈ Z ≥ , let˜ f N ( x ) = χ (cid:0) | x | N (cid:1) · f ( x ) . The support of ˜ f N is contained in the ball {| x | ≤ N } and thus ˜ f N ∈ C ∞ c ( R n , R ). To see that f − ˜ f N = (1 − χ (cid:0) | x | N (cid:1) ) f converges to 0 in H s ′ ( R n , R ),note that f ( x ) − ˜ f N ( x ) = 0 for any x ∈ R n with | x | ≤ N . Furthermore it iseasy to see that sup x ∈ R n | α |≤ s ′ (cid:12)(cid:12) ∂ α (cid:0) − χ (cid:0) | x | N (cid:1)(cid:1)(cid:12)(cid:12) ≤ M s ′ for some constant M s ′ > N . Hence for any α ∈ Z n ≥ with | α | ≤ s ′ , by Leibniz’ rule, k ∂ α f − ∂ α ˜ f N k = k ∂ α (cid:16)(cid:0) − χ (cid:0) | x | N (cid:1)(cid:1) · f ( x ) (cid:17) k≤ X β + γ = α k ∂ β (cid:0) − χ (cid:0) | x | N (cid:1)(cid:1) · ∂ γ f k . − χ (cid:0) | x | N (cid:1) = 0 for any | x | ≤ N we conclude that k ∂ β (cid:0) − χ (cid:0) | x | N (cid:1)(cid:1) · ∂ γ f k ≤ M s ′ (cid:18)Z | x |≥ N | ∂ γ f | dx (cid:19) / and hence, as f ∈ H s ′ ( R n , R ),lim N →∞ k ∂ α f − ∂ α ˜ f N k = 0 . To state regularity properties of elements in H s ( R n , R ), introduce for any r ∈ Z ≥ the space C r ( R n , R ) of functions f : R n → R with continuouspartial derivatives up to order r . Denote by k f k C r the C r -norm of f , k f k C r = sup x ∈ R n sup | α |≤ r | ∂ α f ( x ) | . By C rb ( R n , R ) we denote the Banach space of functions f in C r ( R n , R ) with k f k C r < ∞ and by C r ( R n , R ) the subspace of functions f in C r ( R n , R )vanishing at infinity. These are functions in C r ( R n , R ) with the propertythat for any ε > M ≥ | α |≤ r sup | x |≥ M | ∂ α f ( x ) | < ε. Then C r ( R n , R ) ⊆ C rb ( R n , R ) ⊆ C r ( R n , R ) . By the triangle inequality one sees that C r ( R n , R ) is a closed subspace of C rb ( R n , R ). The following result is often referred to as Sobolev embeddingtheorem. Proposition 2.2.
For any r ∈ Z ≥ and any integer s with s > n/ ,the space H s + r ( R n , R ) can be embedded into C r ( R n , R ) . More precisely H s + r ( R n , R ) ⊆ C r ( R n , R ) and there exists K s,r ≥ so that k f k C r ≤ K s,r k f k s + r ∀ f ∈ H s + r ( R n , R ) . Remark 2.2.
The proof shows that Proposition 2.2 holds for any real s with s > n/ . roof. As for s > n/ Z R n (1 + | ξ | ) − s dξ < ∞ one gets by the Cauchy-Schwarz inequality for any f ∈ C ∞ c ( R n , R ) and α ∈ Z n ≥ with | α | ≤ r sup x ∈ R n | ∂ α f ( x ) | ≤ (2 π ) − n/ Z R n | ˆ f ( ξ ) | | ξ | α dξ ≤ (cid:16) Z R n (1 + | ξ | ) − s dξ (cid:17) / (2 π ) − n/ (cid:16) Z R n | ˆ f ( ξ ) | (1 + | ξ | ) s + r dξ (cid:17) / ≤ K r,s k f k r + s (6)for some K r,s >
0. By Lemma 2.1, an arbitrary element f ∈ H s + r ( R n , R )can be approximated by a sequence ( f N ) N ≥ in C ∞ c ( R n , R ). As C r ( R n , R ) isa Banach space, it then follows from (6) that ( f N ) N ≥ is a Cauchy sequencein C r ( R n , R ) which converges to some function ˜ f in C r ( R n , R ). In particular,for any compact subset K ⊆ R n , f N | K → ˜ f | K in L ( K, R ) . This shows that ˜ f ≡ f a.e. and hence f ∈ C r ( R n , R ).As an application of Proposition 2.2 one gets the following Corollary 2.1.
Let s be an integer with s > n/ . Then the followingstatements hold:(i) For any ϕ ∈ D s , the linear operators d x ϕ, d x ϕ − : R n → R n arebounded uniformly in x ∈ R n † . In particular, inf x ∈ R n det d x ϕ > . (ii) D s − id = { ϕ − id | ϕ ∈ D s } is an open subset of H s . Hence the map D s → H s , ϕ ϕ − id provides a global chart for D s , giving D s the structure of a C ∞ -Hilbertmanifold modeled on H s . † Here d x ϕ − ≡ d x ( ϕ − ) where ϕ ◦ ϕ − = id . iii) For any ϕ • ∈ D s such that inf x ∈ R n det d x ϕ • > M > there exist an open neighborhood U ϕ • of ϕ • in D s and C > such thatfor any ϕ in U ϕ • , inf x ∈ R n det d x ϕ ≥ M and sup x ∈ R n (cid:12)(cid:12) d x ϕ − (cid:12)(cid:12) < C. ‡ Remark 2.3.
The proof shows that Corollary 2.1 holds for any real s with s > n/ .Proof. ( i ) Introduce C ( R n ) := (cid:8) ϕ ∈ Diff ( R n ) (cid:12)(cid:12) ϕ − id ∈ C ( R n ) (cid:9) where C ( R n ) ≡ C ( R n , R n ) is the space of C -maps f : R n → R n , vanishingtogether with their partial derivatives ∂ x i f (1 ≤ i ≤ n ) at infinity. ByProposition 2.2, H s continuously embeds into C ( R n ) for any integer s with s > n/ D s ֒ → C ( R n ). We now prove that for any ϕ ∈ C ( R n ), dϕ and dϕ − are bounded on R n . Clearly, for any ϕ ∈ C ( R n ), dϕ is bounded on R n . To show that dϕ − is bounded as well introduce forany f ∈ C ( R n ) the function F ( f ) : R n → R given by F ( f )( x ) := det (cid:0) id + d x f (cid:1) −
1= det (cid:0) ( δ i + ∂ x f i ) ≤ i ≤ n , . . . , ( δ in + ∂ x n f i ) ≤ i ≤ n (cid:1) − f ( x ) = (cid:0) f ( x ) , . . . , f n ( x ) (cid:1) . Aslim | x |→∞ ∂ x k f i ( x ) = 0 for any 1 ≤ i, k ≤ n one has lim | x |→∞ F ( f )( x ) = 0 . (7)It is then straightforward to verify that F is a continuous map, F : C ( R n ) → C ( R n , R ) . ‡ For a linear operator A : R n → R n , denote by | A | its operator norm, | A | :=sup | x | =1 | Ax | ϕ in C ( R n ). Then (7) implies that M := inf x ∈ R n det( d x ϕ ) > . (8)As the differential of the inverse, d x ϕ − = (cid:0) d ϕ − ( x ) ϕ (cid:1) − , can be computedin terms of the cofactors of d ϕ − ( x ) ϕ and 1 / det( d ϕ − ( x ) ϕ ) it follows from (8)that M := sup x ∈ R n | d x ϕ − | < ∞ (9)where | A | denotes the operator norm of a linear operator A : R n → R n .( ii ) Using again that D s continuously embeds into C ( R n ) it remains toprove that C ( R n ) − id is an open subset of C ( R n ). Note that the map F introduced above is continuous. Hence there exists a neighborhood U ϕ of f ϕ := ϕ − id in C ( R n ) so that for any f ∈ U ϕ sup x ∈ R n | d x f − d x f ϕ | ≤ M (10)and sup x ∈ R n (cid:12)(cid:12) F (cid:0) f (cid:1) ( x ) − F (cid:0) f ϕ (cid:1) ( x ) (cid:12)(cid:12) ≤ M M , M given as in (8)-(9). We claim that id + f ∈ C ( R n ) for any f ∈ U ϕ . As ϕ ∈ C ( R n ) was chosen arbitrarily it then would follow that C ( R n ) − id is open in C ( R n ). First note that by (11),0 < M / ≤ det( id + d x f ) ∀ x ∈ R n , ∀ f ∈ U ϕ . Hence id + f is a local diffeomorphism on R n and it remains to show that id + f is 1-1 and onto for any f in U ϕ . Choose f ∈ U ϕ arbitrarily. To seethat id + f is 1-1 it suffices to prove that ψ := ( id + f ) ◦ ϕ − is 1-1. Notethat ψ = ( id + f ϕ + f − f ϕ ) ◦ ϕ − = id + ( f − f ϕ ) ◦ ϕ − . For any x, y ∈ R n , one therefore has ψ ( x ) − ψ ( y ) = x − y + ( f − f ϕ ) ◦ ϕ − ( x ) − ( f − f ϕ ) ◦ ϕ − ( y ) .
12y (9) and (10) | ( f − f ϕ ) ◦ ϕ − ( x ) − ( f − f ϕ ) ◦ ϕ − ( y ) | ≤ M | ϕ − ( x ) − ϕ − ( y ) |≤ | x − y | and thus | ( x − y ) − (cid:0) ψ ( x ) − ψ ( y ) (cid:1) | ≤ | x − y | ∀ x, y ∈ R n which implies that ψ is 1-1. To prove that id + f is onto we show that R f := { x + f ( x ) | x ∈ R n } is an open and closed subset of R n . Being nonempty, onethen has R f = R n . As id + f is a local diffeomorphism on R n , R f is open.To see that it is closed, consider a sequence ( x k ) k ≥ in R n so that y k := x k + f ( x k ), k ≥
1, converges. Denote the limit by y . As lim | x |→∞ f ( x ) = 0,the sequence (cid:0) f ( x k ) (cid:1) k ≥ is bounded, hence x k = y k − f ( x k ) is a boundedsequence and therefore admits a convergent subsequence ( x k i ) i ≥ whose limitis denoted by x . Then y = lim i →∞ x k i + lim i →∞ f ( x k i )= x + f ( x )i.e. y ∈ R f . This shows that R f is closed and finishes the proof of item ( ii ).The proof of ( iii ) is straightforward and we leave it to the reader.The following properties of multiplication of functions in Sobolev spaces arewell known – see e.g. [2]. Lemma 2.3.
Let s, s ′ be integers with s > n/ and ≤ s ′ ≤ s . Then thereexists K > so that for any f ∈ H s ( R n , R ) , g ∈ H s ′ ( R n , R ) , the product f · g is in H s ′ ( R n , R ) and k f · g k s ′ ≤ K k f k s k g k s ′ . (12) In particular, H s ( R n , R ) is an algebra. Remark 2.4.
The proof shows that Lemma 2.3 remains true for any real s and s ′ with s > n/ and ≤ s ′ ≤ s . roof. First we show that(1 + | ξ | ) s ′ / d f · g ( ξ ) = (1 + | ξ | ) s ′ / ( ˆ f ∗ ˆ g )( ξ ) ∈ L ( R n , R )where ∗ denotes the convolution( ˆ f ∗ ˆ g )( ξ ) = Z R n ˆ f ( ξ − η )ˆ g ( η ) dη. By assumption,˜ f ( ξ ) := ˆ f ( ξ ) (1 + | ξ | ) s/ and ˜ g ( ξ ) = ˆ g ( ξ ) (1 + | ξ | ) s ′ / are in L ( R n , R ). Note that in view of definition (5), k ˜ f k = k f k ∼ s and k ˜ g k = k g k ∼ s ′ . It is to show that ξ (1 + | ξ | ) s ′ / Z R n | ˜ f ( ξ − η ) | (1 + | ξ − η | ) s/ | ˜ g ( η ) | (1 + | η | ) s ′ / dη is square-integrable. We split the domain of integration into two subsets {| η | > | ξ | / } and {| η | ≤ | ξ | / } . Then(1 + | ξ | ) s ′ / Z | η | > | ξ | / | ˜ f ( ξ − η ) | (1 + | ξ − η | ) s/ | ˜ g ( η ) | (1 + | η | ) s ′ / dη ≤ s ′ (1 + | ξ | ) s ′ / Z | η | > | ξ | / | ˜ f ( ξ − η ) | (1 + | ξ − η | ) s/ | ˜ g ( η ) | (1 + | ξ | ) s ′ / dη ≤ s ′ Z R n | ˜ f ( ξ − η ) | (1 + | ξ − η | ) s/ | ˜ g ( η ) | dη = 2 s ′ | ˆ f | ∗ | ˜ g | ( ξ ) . By Young’s inequality (see e.g. Theorem 1.2.1 in [28]), (cid:12)(cid:12)(cid:12)(cid:12) | ˆ f | ∗ | ˜ g | (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˆ f k L k ˜ g k and k ˆ f k L ≤ (cid:18)Z R n (1 + | ξ | ) s | ˆ f ( ξ ) | dξ (cid:19) / (cid:18)Z R n (1 + | ξ | ) − s dξ (cid:19) / . (cid:12)(cid:12)(cid:12)(cid:12) | ˆ f | ∗ | ˜ g | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k s k g k ∼ s ′ . Similarly, one argues for the integral over the remaining subset. Note thaton the domain {| η | ≤ | ξ | / } one has(1 + | ξ − η | ) ≥ (1 + | η | ) and (1 + | ξ − η | ) ≥
14 (1 + | ξ | )and hence (1 + | ξ − η | ) s/ ≥ (1 + | η | ) ( s − s ′ ) / − s ′ (1 + | ξ | ) s ′ / Hence (1 + | ξ | ) s ′ / Z | η |≤| ξ | / | ˜ f ( ξ − η ) | (1 + | ξ − η | ) s/ | ˜ g ( η ) | (1 + | η | ) s ′ / dη ≤ s ′ Z | η |≤| ξ | / | ˜ f ( ξ − η ) | | ˜ g ( η ) | (1 + | η | ) s/ dη and the L -norm of the latter convolution is bounded by k ˜ f k k ˜ g ( η ) / (1 + | η | ) s/ k L ≤ C k f k s k g k ≤ C k f k s k g k s ′ with an appropriate constant C > H ( R n , R ). Lemma 2.4.
Let ϕ ∈ Diff ( R n ) with dϕ and dϕ − bounded on all of R n .Then the following statements hold:(i) The right translation by ϕ , f R ϕ ( f ) := f ◦ ϕ is a bounded linearmap on L ( R n , R ) .(ii) For any f ∈ H ( R n , R ) , the composition f ◦ ϕ is again in H ( R n , R ) and the differential d ( f ◦ ϕ ) is given by the map df ◦ ϕ · dϕ ∈ L ( R n , R n ) , d ( f ◦ ϕ ) = ( df ) ◦ ϕ · dϕ. (13)15 roof. ( i ) For any f ∈ L ( R n , R ), the composition f ◦ ϕ is measurable. As M := inf x ∈ R n det( d x ϕ ) = (cid:0) sup x ∈ R n det d x ϕ − (cid:1) − > Z R n (cid:12)(cid:12) f (cid:0) ϕ ( x ) (cid:1)(cid:12)(cid:12) dx ≤ M Z R n (cid:12)(cid:12) f (cid:0) ϕ ( x ) (cid:1)(cid:12)(cid:12) det( d x ϕ ) dx = 1 M Z R n | f ( x ) | dx and thus f ◦ ϕ ∈ L ( R n , R ) and the right translation R ϕ is a bounded linearmap on L ( R n , R ).( ii ) For any f ∈ C ∞ c ( R n , R ), f ◦ ϕ ∈ H ( R n , R ) and (13) holds by thestandard chain rule of differentiation. Furthermore for any f ∈ H ( R n , R ), df ∈ L ( R n , R n ) and hence by ( i ), ( df ) ◦ ϕ ∈ L ( R n , R n ). As dϕ is continuousand bounded by assumption it then follows that for any 1 ≤ i ≤ n n X k =1 (cid:0) ∂ x k f (cid:1) ◦ ϕ · ∂ x i ϕ k ∈ L ( R n , R )where ϕ k ( x ) is the k ’th component of ϕ ( x ), ϕ ( x ) = (cid:0) ϕ ( x ) , . . . , ϕ n ( x ) (cid:1) . ByLemma 2.1, f can be approximated by ( f N ) N ≥ in C ∞ c ( R n , R ). By the chainrule, for any 1 ≤ i ≤ n , one has ∂ x i ( f N ◦ ϕ ) = n X k =1 ( ∂ x k f N ) ◦ ϕ · ∂ x i ϕ k and in view of ( i ), in L , n X k =1 ( ∂ x k f N ) ◦ ϕ · ∂ x i ϕ k −→ N →∞ n X k =1 ( ∂ x k f ) ◦ ϕ · ∂ x i ϕ k . (14)Moreover, for any test function g ∈ C ∞ c ( R n , R ), − Z R n ∂ x i g · f N ◦ ϕdx = n X k =1 Z R n g · (cid:0) ∂ x k f N (cid:1) ◦ ϕ · ∂ x i ϕ k dx.
16y taking the limit N → ∞ and using (14), one sees that the distributionalderivative ∂ x i ( f ◦ ϕ ) equals P nk =1 ( ∂ x k f ) ◦ ϕ · ∂ x i ϕ k for any 1 ≤ i ≤ n .Therefore, f ◦ ϕ ∈ H ( R n , R ) and d ( f ◦ ϕ ) = df ◦ ϕ · dϕ as claimed.The next result concerns the product rule of differentiation in Sobolevspaces. To state the result, introduce for any integer s with s > n/ ε > U sε := (cid:8) g ∈ H s ( R n , R ) (cid:12)(cid:12) inf x ∈ R n (cid:0) g ( x ) (cid:1) > ε (cid:9) . By Proposition 2.2, U sε is an open subset of H s ( R n , R ) and so is U s := [ ε> U sε . Note that U s is closed under multiplication. More precisely, if g ∈ U sε and h ∈ U sδ , then g + h + gh ∈ U sεδ . Indeed, by Lemma 2.3, gh ∈ H s ( R n , R ), andhence so is g + h + gh . In addition, 1 + g + h + gh = (1 + g )(1 + h ) satisfiesinf x ∈ R n (1 + g )(1 + h ) > εδ and thus g + h + gh is in U sεδ . Lemma 2.5.
Let s, s ′ be integers with s > n/ and ≤ s ′ ≤ s . Then for any ε > and K > there exists a constant C ≡ C ( ε, K ; s, s ′ ) > so that for any f ∈ H s ′ ( R n , R ) and g ∈ U sε with k g k s < K , one has f / (1 + g ) ∈ H s ′ ( R n , R ) and k f / (1 + g ) k s ′ ≤ C k f k s ′ . (15) Moreover, the map H s ′ ( R n , R ) × U s → H s ′ ( R n , R ) , ( f, g ) f / (1 + g ) (16) is continuous. Remark 2.5.
The proof shows that Lemma 2.5 continues to hold for any s real with s > n/ . The case where in addition s ′ is real is treated in AppendixB.Proof. We prove the claimed statement by induction with respect to s ′ . For s ′ = 0, one has for any f in L ( R n , R ) and g ∈ U sε (cid:13)(cid:13)(cid:13)(cid:13) f g (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε k f k . f , f ∈ L ( R n , R ), g , g ∈ U sε ε k f g − f g k ≤ k ( f − f ) + f ( g − g ) + ( f − f ) g k≤ (cid:0) k g k C (cid:1) k f − f k + k f k k g − g k C . Hence by Proposition 2.2, ε k f g − f g k ≤ (cid:0) K s, k g k s (cid:1) k f − f k + K s, k f k k g − g k s and it follows that for any ε > L ( R n , R ) × U sε → L ( R n , R ) , ( f, g ) f / (1 + g )is continuous. As ε > s ′ = 0.Now, assuming that (15) and (16) hold for all 1 ≤ s ′ ≤ k −
1, we willprove that they hold also for s ′ = k . Take f ∈ H s ′ ( R n , R ) and g ∈ U sε . First,we will prove that f / (1 + g ) ∈ H s ′ ( R n , R ) and ∂ x i (cid:18) f g (cid:19) = ∂ x i f g − ∂ x i ( f g ) − g · ∂ x i f (1 + g ) (1 ≤ i ≤ n ) . Indeed, by Lemma 2.1, there exists ( f N ) N ≥ , ( g N ) N ≥ ⊆ C ∞ c ( R n , R ) sothat f N → f in H s ′ ( R n , R ) and g N → g in H s ( R n , R ). As U sε is openin H s ( R n , R ) we can assume that ( g N ) N ≥ ⊆ U sε . By the product rule ofdifferentiation, one has for any N ≥
1, 1 ≤ i ≤ n∂ x i (cid:18) f N g N (cid:19) = ∂ x i f N g N − ∂ x i ( f N g N ) − g N · ∂ x i f N (1 + g N ) . (17)As ∂ x i f N −→ N →∞ ∂ x i f in H s ′ − ( R n , R ) it follows by the induction hypothesisthat ∂ xi f g ∈ H s ′ − ( R n , R ) and ∂ x i f N g N −→ N →∞ ∂ x i f g in H s ′ − ( R n , R ) . (18)By Lemma 2.3, 2 g N + g N ( N ≥
1) and 2 g + g are in H s ( R n , R ) and2 g N + g N −→ N →∞ g + g in H s ( R n , R ) . (19)18s inf x ∈ R n (cid:0) g N ( x ) (cid:1) > ε and inf x ∈ R n (cid:0) g ( x ) (cid:1) > ε it follows that 2 g N + g N ( N ≥
1) and 2 g + g are elements in U sε . By Lemma2.3, f N · g N ( N ≥ , f · g are in H s ′ ( R n , R ) and f N · g N −→ N →∞ f · g in H s ′ ( R n , R ). Therefore ∂ x i ( f N · g N ) → ∂ x i ( f · g ) in H s ′ − ( R n , R ) . (20)Similarly, as ∂ x i f N −→ N →∞ ∂ x i f in H s ′ − ( R n , R ) it follows again by Lemma2.3 that g N · ∂ x i f N ( N ≥ , g · ∂ x i f are in H s ′ − ( R n , R ) and g N · ∂ x i f N −→ N →∞ g · ∂ x i f in H s ′ − ( R n , R ) . (21)It follows from (19)-(21), and the induction hypothesis that ∂ x i ( f N g N ) − g N · ∂ x i f n (1 + g N ) −→ N →∞ ∂ x i ( f g ) − g · ∂ x i f (1 + g ) in H s ′ − ( R n , R ) . (22)In view of (18) and (22), for any test function h ∈ C ∞ c ( R n , R ), one has forthe distributional derivative of f / (1 + g ) ∈ L ( R n , R ), D ∂ x i (cid:16) f g (cid:17) , h E = − R R n ∂ x i h · f g dx = − lim N →∞ R R n ∂ x i h · f N g N dx = lim N →∞ R R n h · h ∂ xi f N g N − ∂ xi ( f N g N ) − g N · ∂ xi f n (1+ g N ) i dx = R R n h · (cid:16) ∂ xi f g − ∂ xi ( fg ) − g · ∂ xi f (1+ g ) (cid:17) dx. This shows that for any 1 ≤ i ≤ n , ∂ x i (cid:18) f g (cid:19) = ∂ x i f g − ∂ x i ( f g ) − g · ∂ x i f (1 + g ) ∈ H s ′ − ( R n , R ) . (23)Hence, f / (1 + g ) ∈ H s ′ ( R n , R ). Let us rewrite (23) in the following form ∂ x i (cid:18) f g (cid:19) = ∂ x i f g − ∂ xi ( fg )1+ g − g · ∂ xi f g g . (24)19y the induction hypothesis there exists C = C ( ε, K ; s, s ′ ) > ∀ f ∈ H s ′ − ( R n , R ), k f / (1 + g ) k s ′ − ≤ C k f k s ′ − . This together with (24) and the triangle inequality imply (15). The continu-ity of (16) follows immediately from the induction hypothesis, Lemma 2.3,and (24). D s ( R n ) In this subsection we show
Proposition 2.6.
For any integer s with s > n/ , ( D s , ◦ ) is a topologicalgroup. First we show that the composition map is continuous. Actually we provethe following slightly stronger statement.
Lemma 2.7.
Let s, s ′ be integers with s > n/ and ≤ s ′ ≤ s . Then µ s ′ : H s ′ ( R n , R ) × D s → H s ′ ( R n , R ) , ( f, ϕ ) f ◦ ϕ is continuous. Moreover, given any ≤ s ′ ≤ s, M > and C > there existsa constant C s ′ = C s ′ ( M, C ) > so that for any ϕ ∈ D s satisfying inf x ∈ R n det( d x ϕ ) ≥ M, k ϕ − id k s ≤ C and for any f ∈ H s ′ ( R n , R ) , one has k f ◦ ϕ k s ′ ≤ C s ′ k f k s ′ . (25) Remark 2.6.
The proof shows that Lemma 2.7 continues to hold for any s real with s > n/ . The case where in addition s ′ is real is treated inAppendix B.Proof. We prove the claimed statement by induction with respect to s ′ . Firstconsider the case s ′ = 0. By item ( i ) of Corollary 2.1 and item ( i ) of Lemma20.4, the range of µ is contained in L ( R n , R ). To show the continuity of µ at ( f • , ϕ • ) ∈ L ( R n , R ) × D s write for ( f, ϕ ) ∈ L ( R n , R ) × D s | f ◦ ϕ − f • ◦ ϕ • | ≤ | f ◦ ϕ − f • ◦ ϕ | + | f • ◦ ϕ − f • ◦ ϕ • | . By Corollary 2.1 ( iii ) one can choose a neighborhood U ϕ • of ϕ • in D s so thatfor any ϕ ∈ U ϕ • inf x ∈ R n (det d x ϕ ) ≥ M for some constant M >
0. The term | f ◦ ϕ − f • ◦ ϕ | can then be estimatedby Z R n | f ◦ ϕ − f • ◦ ϕ | dx ≤ M Z R n | f − f • | dy To estimate the term | f • ◦ ϕ − f • ◦ ϕ • | apply Lemma 2.1 to approximate f • by ˜ f • ∈ C ∞ c ( R n , R ) and use the triangle inequality | f • ◦ ϕ − f • ◦ ϕ • | ≤ | f • ◦ ϕ − ˜ f • ◦ ϕ | + | ˜ f • ◦ ϕ − ˜ f • ◦ ϕ • | + | ˜ f • ◦ ϕ • − f • ◦ ϕ • | . For any ϕ ∈ U ϕ • , one has Z R n | f • ◦ ϕ − ˜ f • ◦ ϕ | dx ≤ M Z R n | ˜ f • − f • | dy and Z R n | ˜ f • ◦ ϕ • − f • ◦ ϕ • | dx ≤ M Z R n | ˜ f • − f • | dy. To estimate the term | ˜ f • ◦ ϕ − ˜ f • ◦ ϕ • | use that ˜ f • is Lipschitz on R n , i.e. | ˜ f • ( x ) − ˜ f • ( y ) | ≤ L | x − y | for some constant L > f • , to get Z R n | ˜ f • ◦ ϕ − ˜ f • ◦ ϕ • | dx ≤ L Z R n | ϕ − ϕ • | dx. Combining the estimates obtained so far, one gets for any ϕ ∈ U ϕ • k f ◦ ϕ − f • ◦ ϕ • k ≤ M − / k f − f • k + 2 M − / k ˜ f • − f • k + L k ϕ − ϕ • k µ at ( f • , ϕ • ). Now assume 1 ≤ s ′ ≤ s . For any( f, ϕ ) ∈ H s ′ ( R n , R ) × D s one has by Lemma 2.4 and Corollary 2.1 ( i ) d ( f ◦ ϕ ) = df ◦ ϕ · dϕ. By the induction hypothesis df ◦ ϕ is an element in H s ′ − ( R n , R n ). HenceLemma 2.3 implies that df ◦ ϕ · dϕ is in H s ′ − ( R n , R n ) and we thus haveshown that the image of µ s ′ is contained in H s ′ ( R n , R ). The continuity of µ s ′ follows from the induction hypothesis, the estimate k df ◦ ϕ · dϕ − df • ◦ ϕ • · dϕ • k s ′ − ≤ k df ◦ ϕ · ( dϕ − dϕ • ) k s ′ − + k ( df ◦ ϕ − df • ◦ ϕ • ) · dϕ • k s ′ − and Lemma 2.3 on multiplication of functions in Sobolev spaces. The esti-mate (25) is obtained in a similar fashion. For s ′ = 0, Z R n | f ◦ ϕ | dx ≤ M Z R n | f | dy. For 1 ≤ s ′ ≤ s , we argue by induction. Let f ∈ H s ′ ( R n , R ). Then by theconsiderations above, d ( f ◦ ϕ ) = df ◦ ϕ · dϕ and df ◦ ϕ ∈ H s ′ − ( R n , R n ).By induction, k df ◦ ϕ k s ′ − ≤ C s ′ − k df k s ′ − . Hence in view of Lemma 2.3, k d ( f ◦ ϕ ) k s ′ − ≤ KC s ′ − k df k s ′ − and for appropriate C s ′ > k f ◦ ϕ k s ′ ≤ C s ′ k f k s ′ .To prove Proposition 2.6 it remains to show the following properties of theinverse map. Lemma 2.8.
Let s be an integer with s > n/ . Then for any ϕ ∈ D s ,its inverse ϕ − is again in D s and inv : D s → D s , ϕ ϕ − is continuous.Proof. First we prove that the inverse ϕ − of an arbitrary element ϕ in D s is again in D s . It is to show that for any multi-index α ∈ Z n ≥ with | α | ≤ s ,one has ∂ α ( ϕ − − id ) ∈ L ( R n ). Clearly, for α = 0, one has Z R n | ϕ − − id | dx = Z R n | id − ϕ | det( d y ϕ ) dy < ∞
22s det( d y ϕ ) is bounded by Corollary 2.1. In addition we conclude that D s → L ( R n ) , ϕ ϕ − − id is continuous. Indeed, for any ϕ, ϕ • ∈ D s , write ϕ − ( x ) − ϕ − • ( x ) = ϕ − ◦ ϕ • (cid:0) ϕ − • ( x ) (cid:1) − ϕ − ◦ ϕ (cid:0) ϕ − • ( x ) (cid:1) . By Corollary 2.1 ( iii ), it follows that for any x ∈ R n , (cid:12)(cid:12) ϕ − ( x ) − ϕ − • ( x ) (cid:12)(cid:12) = (cid:12)(cid:12) ϕ − ( x ) − ϕ − (cid:0) ϕ ◦ ϕ − • ( x ) (cid:1)(cid:12)(cid:12) ≤ sup x ∈ R n (cid:12)(cid:12) d x ϕ − (cid:12)(cid:12) · | x − ϕ ◦ ϕ − • ( x ) |≤ L (cid:12)(cid:12) ( ϕ • − ϕ ) (cid:0) ϕ − • ( x ) (cid:1)(cid:12)(cid:12) (26)where L > ϕ close to ϕ • . Hence Z R n | ϕ − − ϕ − • | dx ≤ L Z R n | ϕ − ϕ • | det( d y ϕ • ) dy (27)and the claimed continuity follows. Now consider α ∈ Z n ≥ with 1 ≤ | α | ≤ s .We claim that ∂ α ( ϕ − − id ) is of the form ∂ α ( ϕ − − id ) = F ( α ) ◦ ϕ − (28)where F ( α ) is a continuous map from D s with values in H s −| α | . Then ∂ α ( ϕ − − id ) is in L ( R n ) as Z R n (cid:12)(cid:12) ∂ α ( ϕ − − id ) (cid:12)(cid:12) dx = Z R n | F ( α ) | det( d y ϕ ) dy < ∞ . (29)To prove (28), first note that ϕ and hence ϕ − are in Diff ( R n ). By thechain rule, d ( ϕ − − id ) = ( dϕ ) − ◦ ϕ − − id n = (cid:0) ( dϕ ) − − id n (cid:1) ◦ ϕ − where id n is the n × n identity matrix. The expression ( dϕ ) − − id n is ofthe form ( dϕ ) − − id n = 1det( dϕ ) (Φ − det( dϕ ) id n )23here Φ( x ) is the matrix whose entries are the cofactors of d x ϕ . In particular,each entry of Φ( x ) is a polynomial expression of ( ∂ x i ϕ j ) ≤ i,j ≤ n . Hence byLemma 2.3 the off-diagonal entries of Φ( x ) are in H s − ( R n , R ). Furthermore,any diagonal entry of Φ( x ) is an element in 1 + H s − ( R n , R ) and det( d x ϕ )is of the form 1 + g with g ∈ H s − ( R n , R ) and inf x ∈ R n (cid:0) g ( x ) (cid:1) >
0. Wethus conclude that Φ( x ) − det( d x ϕ ) id n is in H s − ( R n , R n × n ) and, in turn,by Lemma 2.5 ( dϕ ) − − id n ∈ H s − ( R n , R n × n ) (30)where R n × n denotes the space of all n × n matrices with real coefficients. Inparticular, for e i = (0 , . . . , , . . . , ∈ Z n ≥ with 1 ≤ i ≤ n we have shownthat ∂ x i ( ϕ − − id ) = F ( e i ) ◦ ϕ − . We point out that by Lemma 2.3 and Lemma 2.5, F ( e i ) , when viewed as mapfrom D s to H s − , is continuous. We now prove formula (28) for any α ∈ Z n ≥ with 1 ≤ | α | ≤ s by induction. The result has already been established for | α | = 1. Assume that it has already been proved for any β ∈ Z n ≥ with | β | ≤ s ′ where 0 ≤ s ′ < s . Choose any α ∈ Z n ≥ with | α | = s ′ . Then byinduction hypothesis, ∂ α ( ϕ − − id ) = F ( α ) ◦ ϕ − with F ( α ) ∈ H s −| α | . Notethat s − | α | ≥
1. Hence by Lemma 2.4, d ( F ( α ) ◦ ϕ − ) = dF ( α ) ◦ ϕ − · ( dϕ ) − ◦ ϕ − = ( dF ( α ) · ( dϕ ) − ) ◦ ϕ − . As ∂ x i F ( α ) ∈ H s −| α |− for any 1 ≤ i ≤ n and ( dϕ ) − − id n is in the space H s − ( R n , R n × n ) it follows by Lemma 2.3 that dF ( α ) · ( dϕ ) − ∈ H s −| α |− ( R n , R n × n ) . This shows that (28) is valid for any β ∈ Z n ≥ with | β | = s ′ + 1 and theinduction step is proved. Hence formula (28) is proved and by (29), we seethat ϕ − ∈ D s if ϕ ∈ D s . Note that we proved more: It follows from (29)and the continuity of F ( α ) : D s → H s −| α | , | α | ≤ s , that the map D s → H s ( R n , R n ) ϕ ϕ − − id (31)is locally bounded. It remains to prove that the inverse map D s → D s , ϕ ϕ − is continuous. We have already seen that D s → L ( R n ), ϕ → ϕ − − id
24s continuous. Now let α ∈ Z n ≥ with 1 ≤ | α | ≤ s and ϕ • ∈ D s . Then for any ϕ ∈ D s | ∂ α ( ϕ − − ϕ − • ) | = | F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − • |≤ | F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − | + | F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • | where F ( α ) • = F ( α ) (cid:12)(cid:12) ϕ • . It follows from the local boundedness of (31), Corol-lary 2.1 ( iii ), and Lemma 2.7 with s ′ = 0 that k F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − k ≤ C k F ( α ) − F ( α ) • k where C > ϕ near ϕ • . Together with thecontinuity of F ( α ) it then follows that k F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − k → ϕ → ϕ • . To analyze the term | F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • | we argue as in theproof of Lemma 2.7. Using Lemma 2.1 one sees that ϕ • can be approximatedby ˜ ϕ ∈ D s with ˜ ϕ − id ∈ C ∞ c ( R n , R n ). Then | F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • | ≤ | F ( α ) • ◦ ϕ − − ˜ F ( α ) ◦ ϕ − | ++ | ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • | + | ˜ F ( α ) ◦ ϕ − • − F ( α ) • ◦ ϕ − • | where ˜ F ( α ) = F ( α ) (cid:12)(cid:12) ˜ ϕ . For ϕ near ϕ • one has Z R n | F ( α ) • ◦ ϕ − − ˜ F ( α ) ◦ ϕ − | dx ≤ Z R n | F ( α ) • − ˜ F ( α ) | det( d y ϕ ) dy and Z R n | F ( α ) • ◦ ϕ − • − ˜ F ( α ) ◦ ϕ − • | dx ≤ C Z R n | F ( α ) • − ˜ F ( α ) | dy where C > x ∈ R n (det d x ϕ ) ≤ C for ϕ near ϕ • . To estimate theterm | ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • | note that ˜ F ( α ) ∈ C ∞ c . In particular, ˜ F ( α ) isLipschitz continuous, i.e. | ˜ F ( α ) ( x ) − ˜ F ( α ) ( y ) | ≤ L | x − y | ∀ x, y ∈ R n for some constant L > ϕ . Thus Z R n | ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • | dx ≤ L Z R n | ϕ − − ϕ − • | dx k ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • k → ϕ → ϕ • . Altogether we have shown that k F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • k → ϕ → ϕ • . Proof of Proposition 2.6.
The claimed statement follows from Lemma 2.7and Lemma 2.8.
As a first step we will prove the following
Proposition 2.9.
For any r ∈ Z ≥ and any integer s with s > n/ µ : H s + r ( R n , R d ) × D s → H s ( R n , R d ) , ( u, ϕ ) u ◦ ϕ (32) is a C r -map. The main ingredient of the proof of Proposition 2.9 is the converse to Taylor’stheorem. To state it we first need to introduce some more notation. Givenarbitrary Banach spaces
Y, X , . . . , X k , k ≥
1, we denote by L ( X , . . . , X k ; Y )the space of continuous k -linear forms on X × . . . × X k with values in Y . In case where X i = X for any 1 ≤ i ≤ k we write L k ( X ; Y ) in-stead of L ( X, . . . , X ; Y ) and set L ( X ; Y ) = Y . Note that the spaces L ( X ; L k − ( X ; Y )) and L k ( X ; Y ) can be identified in a canonical way. Thesubspace of L k ( X ; Y ) of symmetric continuous k -linear forms is denoted by L ksym ( X ; Y ). The converse to Taylor’s theorem can then be formulated asfollows – see [1], p.6. Theorem 2.2.
Let U ⊆ X be a convex set and F : U → Y , f k : U → L ksym ( X ; Y ) , k = 0 , . . . , r . For any x ∈ U and h ∈ X so that x + h ∈ U ,define R ( x, h ) ∈ Y by F ( x + h ) = F ( x ) + r X k =1 f k ( x )( h, . . . , h ) k ! + R ( x, h ) . If for any ≤ k ≤ r , f k is continuous and for any x ∈ U , k R ( x, h ) k / k h k r → as h → then F is of class C r on U and d k F = f k for any ≤ k ≤ r . To prove Proposition 2.9 we first need to establish some auxiliary results.26 emma 2.10.
Let s be an integer with s > n/ . To shorten notation,for this lemma and its proof we write H s instead of H s ( R n , R ) . Then forany k ≥ , the map ρ k given by ρ k : H s × D s → L ksym ( H s ; H s )( u, ϕ ) " ( h , . . . , h k ) ( u ◦ ϕ ) · k Y i =1 h i is continuous.Proof of Lemma 2.10. First we note that the map ρ k is well defined. Indeedfor any ( u, ϕ ) ∈ H s × D s , the function u ◦ ϕ is in H s by Lemma 2.7. Henceby Lemma 2.3, for any ( h i ) ≤ i ≤ k ⊆ H s the function u ◦ ϕ · Q ki =1 h i is in H s . It follows that ρ k ( u, ϕ ) ∈ L ksym ( H s ; H s ). To show that ρ k is continuousconsider arbitrary sequences ( ϕ l ) l ≥ ⊆ D s and ( u l ) l ≥ ⊆ H s with ϕ l → ϕ in D s and u l → u in H s . By Lemma 2.3, one has for any ( h i ) ≤ i ≤ k ⊆ H s , k ( u ◦ ϕ ) · k Y i =1 h i − ( u l ◦ ϕ l ) · k Y i =1 h i k s ≤ K k +1 k u ◦ ϕ − u l ◦ ϕ l k s · k Y i =1 k h i k s . As k u ◦ ϕ − u l ◦ ϕ l k s → l → ∞ by Lemma 2.7, the claimed continuityfollows. Lemma 2.11.
Let s be an integer with s > n/ . Given ϕ • ∈ D s choose ε > so small that inf x ∈ R n det( d x ϕ • ) > ε . Then there exists a convexneighborhood U ⊆ D s of ϕ • and a constant C > with the property that inf x ∈ R n det( d x ϕ ) > ε and k ϕ − id k s < C ∀ ϕ ∈ U. Furthermore, there is a constant C s = C s ( ε, C ) , depending on ε and C sothat for any f ∈ H s +1 ( R n , R ) and ϕ ∈ U k f ◦ ϕ − f ◦ ϕ • k s ≤ C s k f k s +1 k ϕ − ϕ • k s . (33) Proof of Lemma 2.11.
The first statement follows from Corollary 2.1 ( iii ).With regard to the second part note that by Lemma 2.7 it suffices to proveestimate (33) for f ∈ C ∞ c ( R n , R ) as C ∞ c ( R n , R ) is dense in H s +1 ( R n , R ) by27emma 2.1. Introduce δϕ ( x ) = ϕ ( x ) − ϕ • ( x ) and note that ϕ • + tδϕ is in U for any 0 ≤ t ≤ U is assumed to be convex. By Proposition 2.2, ϕ ∈ Diff ( R n ). For any x ∈ R n consider the C -curve,[0 , → R n , t f ◦ (cid:0) ϕ • + tδϕ (cid:1) ( x ) . Clearly, for any x ∈ R n , f ◦ ϕ ( x ) − f ◦ ϕ • ( x ) = Z ddt (cid:0) f ◦ (cid:0) ϕ • + tδϕ (cid:1) ( x ) (cid:1) dt = Z (cid:0) d ( ϕ • + tδϕ )( x ) f (cid:1) · δϕ ( x ) dt. (34)By Lemma 2.7, t d ϕ • + tδϕ f · δϕ = df ◦ ( ϕ • + tδϕ ) · δϕ is a continuous path in H s , hence it is Riemann integrable in H s and we havethat equality (34) is valid in H s . Hence, k f ◦ ϕ − f ◦ ϕ • k s ≤ Z k d ϕ • + tδϕ f · δϕ k s dt. Estimate (33) then follows using Lemma 2.3 and Lemma 2.7.
Lemma 2.12.
Let s be an integer satisfying s > n/ . To shorten nota-tion, for the course of this lemma and its proof, we write again H s insteadof H s ( R n , R ) . Then for any k ≥ , the map ν k given by ν k : D s → L ( H s +1 ; L k − sym ( H s ; H s )) ϕ " ( h, h , . . . , h k − ) ( h ◦ ϕ ) · k − Y i =1 h i is continuous. Remark 2.7.
Note that L (cid:0) H s +1 ; L k − sym ( H s ; H s ) (cid:1) isometrically embeds into L ksym ( H s +1 × H s ; H s ) in a canonical way. roof of Lemma 2.12. For any h ∈ H s +1 , ( h i ) ≤ i ≤ k − ⊆ H s and ϕ, ϕ • ∈ D s ,we have in view of Lemma 2.3, k ( h ◦ ϕ ) · k − Y i =1 h i − ( h ◦ ϕ • ) · k − Y i =1 h i k s ≤ K k − k h ◦ ϕ − h ◦ ϕ • k s · k − Y i =1 k h i k s . By Lemma 2.11, there exists C s > ϕ in a sufficiently smallneighborhood of ϕ • , k h ◦ ϕ − h ◦ ϕ • k s ≤ C s k ϕ − ϕ • k s k h k s +1 . This shows the claimed continuity.
Proof of Proposition 2.9.
To keep notation as simple as possible we presentthe proof in the case where d = n . The case r = 0 is treated in Lemma 2.7,hence it remains to consider the case r ≥
1. We want to apply the converse ofTaylor’s theorem with U = H s + r × D s , viewed as subset of X := H s + r × H s and Y := H s . Let u, δu ∈ H s + r and ϕ ∈ D s , δϕ ∈ H s be given. ByProposition 2.2, u, δu ∈ C r ( R n , R n ). Hence by Taylor’s theorem, for any x ∈ R n , u ( ϕ ( x ) + δϕ ( x )) is given by u (cid:0) ϕ ( x ) (cid:1) + r X k =1 X | α | = k α ! (cid:16) ∂ α u (cid:17)(cid:0) ϕ ( x ) (cid:1) · δϕ ( x ) α + R ( u, ϕ, δϕ )( x )where δϕ ( x ) α = δϕ ( x ) α · · · δϕ n ( x ) α n and R ( u, ϕ, δϕ )( x ) is defined by X | α | = r (cid:26) rα ! Z (1 − t ) r − (cid:16)(cid:0) ∂ α u (cid:1)(cid:0) ϕ ( x ) + tδϕ ( x ) (cid:1) − ∂ α u (cid:0) ϕ ( x ) (cid:1)(cid:17) · δϕ ( x ) α dt (cid:27) . Similarly, δu ( ϕ ( x ) + δϕ ( x )) is given by δu (cid:0) ϕ ( x ) (cid:1) + r − X k =1 X | α | = k α ! (cid:16) ∂ α δu (cid:17)(cid:0) ϕ ( x ) (cid:1) · δϕ ( x ) α + R ( δu, ϕ, δϕ )( x )with R ( δu, ϕ, δϕ )( x ) defined by X | α | = r (cid:26) rα ! Z (1 − t ) r − (cid:0) ∂ α δu (cid:1)(cid:0) ϕ ( x ) + tδϕ ( x ) (cid:1) · δϕ ( x ) α dt (cid:27) . x ∈ R n the integrals appearing in the definition of theremainder terms R and R are well-defined as Riemann integrals. Indeed,as u, δu ∈ C r ( R n , R n ) we see that for any x ∈ R n these integrands arecontinuous functions of t ∈ [0 , R and R can be viewed as continuous curves in H s , parametrized by t and hence are Riemann integrable in H s . Hencethe pointwise integrals are functions in H s . Furthermore, when viewed as H s -valued curves, the integrands depend continuously on the parameters( u, ϕ, δu, δϕ ) ∈ H s + r × D s × H s + r × H s by Lemma 2.3 and Lemma 2.7.In the following we denote by B s + rε ( u • ) the ball in H s + r of radius ε ,centered at u • ∈ H s + r , B s + rε ( u • ) = { u ∈ H s + r | k u − u • k s + r < ε } . For ( u • , ϕ • ) ∈ H s + r × D s , set U = B s + rε ( u • ) ⊆ H s + r and U = B sε ( ϕ • − id ) ⊆ H s , where we choose ε small enough to ensure that id + U ⊆ D s .Furthermore, define the subset V ⊆ H s + r × D s × H s + r × H s by V = { ( u, ϕ, δu, δϕ ) ∈ H s + r ×D s × H s + r × H s | ( u + δu, ϕ + δϕ ) ∈ U × ( id + U ) } . In view of the considerations above, we get for ( u, ϕ, δu, δϕ ) ∈ V the followingidentity in H s ( u + δu ) ◦ ( ϕ + δϕ ) = u ◦ ϕ + r X k =1 η k ( u, ϕ ) k ! ( δu, δϕ ) k + R ( u, ϕ, δu, δϕ )where ( δu, δϕ ) k stands for (cid:0) ( δu, δϕ ) , . . . , ( δu, δϕ ) (cid:1) and for any 1 ≤ k ≤ r , η k ( u, ϕ ) is an element in L ksym ( H s + r × H s ; H s ), given by η k ( u, ϕ )( δu, δϕ ) k = X | α | = k k ! α ! ( ∂ α u ) ◦ ϕ · δϕ α + X | α | = k − k ! α ! ( ∂ α δu ) ◦ ϕ · δϕ α . The remainder term R ( u, ϕ, δu, δϕ ) is given by R ( u, ϕ, δu, δϕ ) = R ( u, ϕ, δϕ ) + R ( δu, ϕ, δϕ ) . k = 1 , . . . , r , η k : H s + r × D s → L ksym ( H s + r × H s ; H s ) , ( u, ϕ ) η k ( u, ϕ )is continuous. Moreover, by Lemma 2.7 and Lemma 2.3, k R ( u, ϕ, δϕ ) k s ( k δu k s + r + k δϕ k s ) r ≤ X | α | = r α ! sup ≤ t ≤ k ∂ α u ◦ ( ϕ + tδϕ ) − ∂ α u ◦ ϕ k s → k R ( δu, ϕ, δϕ ) k s ( k δu k s + r + k δϕ k s ) r ≤ X | α | = r α ! sup ≤ t ≤ k ( ∂ α δu ) ◦ ( ϕ + tδϕ ) k s → . as k δϕ k s + k δu k s + r →
0. By Theorem 2.2, it then follows that µ is a C r map.Proposition 2.9 together with the implicit function theorem can be used toprove the following result on the inverse map. Proposition 2.13.
For any r ∈ Z ≥ and any integer s with s > n/ inv : D s + r → D s , ϕ ϕ − (35) is a C r -map.Proof. The case r = 0 has been established in Lemma 2.8. In particular weknow that for any ϕ ∈ D s , its inverse ϕ − is again in D s . So let r ≥
1. ByProposition 2.9, µ : D s + r × D s → D s , ( ϕ, ψ ) ϕ ◦ ψ is a C r -map. For any ϕ ∈ D s + r , consider the differential of ψ µ ( ϕ, ψ ) at ψ = ϕ − d ψ µ ( ϕ, · ) (cid:12)(cid:12) ψ = ϕ − : H s → H s , δψ dϕ ◦ ϕ − · δψ. As r ≥
1, we get that dϕ, dϕ ◦ ϕ − ∈ H s ( R n , R n × n ). In fact, d ψ µ ( ϕ, · ) (cid:12)(cid:12) ψ = ϕ − is a linear isomorphism on H s whose inverse is given by δψ ( dϕ ) − ◦ ϕ − · ψ . Note that by Lemma 2.7, ( dϕ ) − ◦ ϕ − ∈ H s ( R n , R n × n ) and by Lemma2.3, δψ ( dϕ ) − ◦ ϕ − · δψ is a bounded linear map H s → H s . Furthermorethe equation µ ( ϕ, ψ ) = id has the unique solution ψ = ϕ − ∈ D s . Hence by the implicit functiontheorem (see e.g. [25]), the map inv : D s + r → D s , ϕ ϕ − is C r . Proof of Theorem 1.1.
Theorem 1.1 follows from Proposition 2.9 and Propo-sition 2.13. H s ( U, R ) In Section 4 we need a version of Proposition 2.9 involving the Sobolev spaces H s ( U, R ) where U ⊆ R n is an open nonempty subset with Lipschitz bound-ary. It means that locally, the boundary ∂U can be represented as the graphof a Lipschitz function – see Definition 3.4.2 in [28] § . Let s ∈ Z ≥ . Bydefinition, H s ( U, R ) is the Hilbert space of elements f in L ( U, R ), havingthe property that their distributional derivatives ∂ α f up to order | α | ≤ s are L -integrable on U , endowed with the norm k f k s where k f k s = h f, f i / s and h· , ·i s denotes the inner product defined for f, g ∈ H s ( U, R ) by h f, g i s = X | α |≤ s Z U ∂ α f ( x ) ∂ α g ( x ) dx. Further we introduce H s ( U, R m ) := H s ( U, R ) m . The spaces H s ( U, R ) and H s ( R n , R ) are closely related. Recall that a function f : U → R is said to be C r -differentiable, r ≥
1, if there exists an open neighborhood V of U in R n and a C r -function g : V → R so that f = g | U . We denote by C r ( U , R ) thespace of C r -differentiable functions f : U → R and by C r ( U , R ) the subspaceof C r ( U , R ) consisting of functions f : U → R , vanishing at ∞ , i.e. havingthe property that for any ε >
0, there exists M ≡ M ε > x ∈ U, | x |≥ M sup | α |≤ r | ∂ α f ( x ) | < ε. § cf. § C rb ( U , R ) the subspace of C r -differentiable func-tions f : U → R so that f and all its derivatives up to order r are bounded,sup x ∈ U sup | α |≤ r (cid:12)(cid:12) ∂ α f ( x ) (cid:12)(cid:12) < ∞ . In a similar fashion one defines C ∞ ( U , R ), C ∞ ( U , R ), and C ∞ b ( U , R ) andcorresponding spaces of vector valued functions f : U → R m .The following result describes how H s ( U, R ) and H s ( R n , R ) are related –see e.g. [28], Theorem 3.4.5, Theorem 5.3.1, and Theorem 6.1.1, for thesewell known results. Proposition 2.14.
Assume that the open set U ⊆ R n has a Lipschitz bound-ary and s ∈ Z ≥ . Then the following statements hold.(i) (cid:8) f | U (cid:12)(cid:12) f ∈ C ∞ c ( R n , R ) (cid:9) is dense in H s ( U, R ) .(ii) The restriction operator, H s ( R n , R ) → H s ( U, R ) , f f | U , is con-tinuous with norm ≤ ¶ . Moreover, there is a bounded linear operator E : H s ( U, R ) → H s ( R n , R ) , so that f = ( Ef ) | U for any f in H s ( U, R ) . E is referred to as extension operator.(iii) For any integers s, r ∈ Z ≥ with s > n/ , H s + r ( U, R ) ֒ → C r ( U , R ) and the embedding is a bounded linear operator. The following result is needed for the proof of Lemma 2.17 below. As usual,we denote by L q ( U, R ) the Banach space of L q -integrable functions f : U → R . For a proof of the proposition see e.g. Theorem 5.4 in [2]. Proposition 2.15.
Assume that the open set U ⊆ R n has a Lipschitz bound-ary and let s ∈ Z ≥ . Then the following statements hold:(i) If ≤ s < n/ , then for any ≤ q ≤ nn − s , H s ( U, R ) ֒ → L q ( U, R ) is continuous. ¶ This statement holds for any open set U ⊆ R n with ∂U not necessarily Lipschitz. ii) If s = n/ , then for any ≤ q < ∞ , H s ( U, R ) ֒ → L q ( U, R ) is continuous. Combining Proposition 2.14 and Lemma 2.3 one obtains the following
Lemma 2.16.
Assume that the open set U ⊆ R n has a Lipschitz boundary.Let s, s ′ be integers with s > n/ and ≤ s ′ ≤ s . Then there exists K > so that for any f ∈ H s ( U, R ) and g ∈ H s ′ ( U, R ) , the product f · g is in H s ′ ( U, R ) and k f · g k s ′ ≤ K k f k s k g k s ′ . In particular, H s ( U, R ) is an algebra. We will also need the following variant of Lemma 2.16.
Lemma 2.17.
Let U ⊆ R n be a non-empty, open, bounded set with Lipschitzboundary and let s > n/ , s ∈ Z ≥ . Then for any r ≥ and any k =( k , . . . , k r ) ∈ Z r ≥ with P rj =1 k j ≤ s , the r -linear map, H s − k ( U, R ) ×· · ·× H s − k r ( U, R ) → L ( U, R ) , ( f , . . . , f r ) f · · · f r (36) is well-defined and continuous.Proof. First note that the map C b ( U, R ) × L ( U, R ) → L ( U, R ) , ( f , f ) f · f is continuous. Combining this with Proposition 2.14 ( iii ), one sees that itremains to prove that the map (36) is well-defined and continuous for any r ≥ k = ( k , . . . , k r ) ∈ Z r ≥ with P rj =1 k j ≤ s and s − k j − n ≤ , ≤ j ≤ r. (37)In what follows we assume that (37) holds. Divide the set I := { j ∈ N | ≤ j ≤ r } into two subsets, I = I < ∪ I , I < := { j ∈ I | s − k j − n < } I := { j ∈ I | s − k j − n } . By Proposition 2.15, for any j ∈ I < , H s − k j ( U, R ) ֒ → L q j ( U, R ) , q j = 2 nn − s − k j ) (38)and for any j ∈ I , H s − k j ( U, R ) ֒ → L q j ( U, R ) , ∀ q j ≥ . (39)We choose q j as follows: If I = I then choose q j ≥ , j ∈ I , so that1 q + · · · + 1 q r < . (40)If I < = ∅ one has by (38) X j ∈ I < q j = X j ∈ I < (cid:18) − s − k j n (cid:19) ≤ r − rsn + 1 n r X j =1 k j . As by assumption, P rj =1 k j ≤ s and s > n/ X j ∈ I < q j ≤
12 + ( r −
1) 12 − ( r − sn = 12 + r − n (cid:16) n − s (cid:17) < . Hence by choosing for any j ∈ I q j ≥ I < = ∅ (40) holds. Altogether we have shown thatthere exist q j ≥ , j ∈ I so that (38),(39), and1 q + · · · q r ≤
12 (41)hold. Thus q = (cid:16) q + · · · q r (cid:17) − ≥
2. It follows from the generalized H¨olderinequality that the r -linear map L q ( U, R ) × · · · × L q r ( U, R ) → L q ( U, R ) , ( f , . . . , f r ) f · · · f r
35s continuous. As U ⊆ R n is bounded and q ≥ L q ( U, R ) ֒ → L ( U, R )and the inclusion is continuous. Hence, the r -linear map L q ( U, R ) × · · · × L q r ( U, R ) → L ( U, R ) , ( f , . . . , f r ) f · · · f r is continuous as well. This together with the continuity of the embeddings(38) and (39) implies that the map (36) is well-defined and continuous forany k ∈ Z r ≥ satisfying P rj =1 k j ≤ s and (37).Let U ⊆ R n be a bounded open set with Lipschitz boundary and s > n/ s ∈ Z ≥ . Denote by D s ( U, R n ) the subset of H s ( U, R n ) (cid:0) ⊆ C ( U , R n ) (cid:1) consisting of orientation preserving local diffeomorphisms ϕ : U → R n thatextend to bijective maps ϕ : U → ϕ ( U ) ⊆ R n and such thatinf x ∈ U det( d x ϕ ) > . (42)More precisely, D s ( U, R n ) := (cid:8) ϕ ∈ H s ( U, R n ) (cid:12)(cid:12) ϕ : U → R n is 1-1 and inf x ∈ U det( d x ϕ ) > (cid:9) . Lemma 2.18. D s ( U, R n ) is an open subset in H s ( U, R n ) .Proof. In view of Proposition 2.2 and Proposition 2.14 ( ii ), D s ( U, R n ) canbe continuously embedded into C ( R n , R n ), D s ( U, R n ) ⊆ C ( R n , R n ) . Take an arbitrary element ϕ ∈ D s ( U, R n ). For ε > B ε the open ε -ball centered at zero in H s ( U, R n ). As U is compact one gets from (42)and the inverse function theorem that there exists ε > f ∈ B ε , the map ψ : U → R n , ψ := ϕ + f (43)is a local diffeomorphism . Strengthening these arguments one sees that thereexist ε > δ > f ∈ B ε and ∀ x, y ∈ U , x = y , | x − y | < δ = ⇒ ψ ( x ) = ψ ( y ) . (44)36n fact, following the arguments of the proof of the inverse function theoremone sees that for any x ∈ U there exist ε x > U x of x in R n such that for any f ∈ B ε x the map ψ (cid:12)(cid:12) U x : U x → R n is injective. Using the compactness of U we find x , ..., x n ∈ U such that ∪ nj =1 U x j ⊇ U . Take, ε := min ≤ j ≤ n ε x j . Then, assuming that (44) does not hold,we can construct two sequences ( p j ) ≤ j ≤ n and ( q j ) ≤ j ≤ n of points in U and( f j ) ≤ j ≤ n ⊆ B ε such that0 < | p j − q j | < /j and ψ j ( p j ) = ψ j ( q j ) (45)where ψ j := ϕ + f j . By the compactness of U , we can assume that thereexists p ∈ U such that lim j →∞ p j = lim j →∞ q j = p . Taking j ≥ p j , q j ∈ U p , and therefore ψ j ( p j ) = ψ j ( q j ). As this contradicts(45), we see that implication (44) holds.Further, we argue as follows. Consider the sets∆ δ := { x, y ∈ U (cid:12)(cid:12) | x − y | < δ } and K δ := U × U \ ∆ δ . As K δ is compact and ϕ : U → R n is injective, m := min ( x,y ) ∈K δ | ϕ ( x ) − ϕ ( y ) | > . This implies that ∀ x, y ∈ U , x = y , | ϕ ( x ) − ϕ ( y ) | < m = ⇒ | x − y | < δ . (46)By taking ε > f ∈ B ε , k ψ − ϕ k C < m/ . (47)Finally, assume that there exists f ∈ B ε so that the map ψ : U → R n , ψ = ϕ + f , is not injective. Then there exist x, y ∈ U , x = y , so that ψ ( x ) = ψ ( y ) . | ϕ ( x ) − ϕ ( y ) | < m . In view (44) and (46) we get that ψ ( x ) = ψ ( y ). This contradiction showsthat ψ is injective. Proposition 2.19.
Let U be an open bounded subset in R n with Lipschitzboundary. Then for any d, r, s ∈ Z ≥ with s > n/ µ : H s + r ( R n , R d ) × D s ( U, R n ) → H s ( U, R d ) , ( f, ϕ ) f ◦ ϕ is a C r -map. In view of Proposition 2.14, the proof of Proposition 2.9 can be easily adaptedto show Proposition 2.19. We leave the details to the reader.
Corollary 2.3.
Under the assumption of Proposition 2.19, the right trans-lation by an arbitrary element ϕ ∈ D s ( U, R n ) , R ϕ : H s ( R n , R d ) → H s ( U, R d ) , f f ◦ ϕ is a C ∞ -map.Proof. By Proposition 2.19, R ϕ is well-defined and continuous. As R ϕ is alinear operator it then follows that R ϕ is a C ∞ -map.As an application of Corollary 2.3 we get the following result. Corollary 2.4.
Let
U, V ⊆ R n be open and bounded sets with Lipschitzboundary and let ϕ : U → V be a C ∞ -diffeomorphism with ϕ ∈ C ∞ ( U , R n ) and ϕ − ∈ C ∞ ( V , R n ) . Then for any given s ≥ , s ∈ Z ≥ , the righttranslation by ϕ , R ϕ : H s ( V, R ) → H s ( U, R ) , f f ◦ ϕ is a continuous linear isomorphism. Finally, we include the following result concerning the left translation. Re-call that for any given open subset U ⊆ R n , we denote by C ∞ b ( U , R n ) thesubspace of C ∞ ( U , R n ) consisting of all elements f ∈ C ∞ ( U , R n ) so that f and all its derivatives are bounded on U .38 roposition 2.20. Let m, d, s ∈ Z ≥ with m, d ≥ and s > n/ and let U be an open bounded subset of R n with Lipschitz boundary. Then for any g in C ∞ b ( R m , R d ) , the left translation by g , L g : H s ( U, R m ) → H s ( U, R d ) , f g ◦ f is a C ∞ -map.Proof. We begin by showing that L g is continuous. Note that by Proposition2.14, the extension operator E : H s ( U, R m ) → H s ( R n , R m )is a bounded linear operator, k E k < ∞ . By Proposition 2.2 the embedding H s ( R n , R m ) ֒ → C ( R n , R m ) is continuous and for any f ∈ H s ( U, R m ), k E ( f ) k C ≤ K s, k E ( f ) k s ≤ K s, k E k k f k s . (48)As g is continuous and bounded, g ◦ E ( f ) is in C b ( R n , R d ) and hence g ◦ f in C b ( U, R d ). Furthermore C b ( R n , R m ) → C b ( R n , R d ) , h g ◦ h is continuous. More precisely, for h , h ∈ C b ( R n , R m ) k g ◦ h − g ◦ h k C ≤ L k h − h k C (49)where L := sup x ∈ R m | d x g | < ∞ . As for any f , f ∈ H s ( U, R m ), g ◦ f − g ◦ f = ( g ◦ E ( f ) − g ◦ E ( f )) | U it follows from the boundedness of the restriction map, (48) and (49), that k g ◦ f − g ◦ f k C ( U ) ≤ k g ◦ E ( f ) − g ◦ E ( f ) k C ≤ L k E ( f ) − E ( f ) k C ≤ LK s, k E k k f − f k s . (50)In particular, H s ( U, R m ) → C b ( U, R d ) , f g ◦ f is Lipschitz continuous.Take f ∈ H s ( U, R m ). By Proposition 2.14 ( i ), there exists a sequence( f ( k ) ) k ≥ , f ( k ) ∈ C ∞ c ( R n , R m ), such that f ( k ) (cid:12)(cid:12)(cid:12) U → f as k → ∞ (51)39n H s ( U, R m ). Using the chain and the Leibniz rule we see that for any k ≥
1, 1 ≤ i ≤ d , and any multi-index α ∈ Z n ≥ with | α | ≤ s , ∂ α ( g i ◦ f ( k ) ) isa linear combination of products of the form ∂ β g i ◦ f ( k ) · ∂ γ f ( k ) j · · · ∂ γ r f ( k ) j r (52)where β ∈ Z m ≥ with | β | ≤ | α | , r ∈ Z ≥ with r ≤ | α | and γ , . . . , γ r ∈ Z n ≥ with γ + . . . γ r = α . It follows from (50) and (51) that for any | β | ≤ | α | ,and for any 1 ≤ i ≤ d , ∂ β g i ◦ f ( k ) (cid:12)(cid:12)(cid:12) U → ∂ β g i ◦ f in C b ( U, R ) (53)as k → ∞ . Moreover, by (51), for any 1 ≤ p ≤ r , ∂ γ p f ( k ) j p (cid:12)(cid:12)(cid:12) U → ∂ γ p f j p in H s −| γ p | ( U, R ) . (54)As P rj =1 | γ j | = | α | ≤ s , we get from (53), (54), and Lemma 2.17 that (cid:16) ∂ β g i ◦ f ( k ) · ∂ γ f ( k ) j · · · ∂ γ r f ( k ) j r (cid:17)(cid:12)(cid:12)(cid:12) U → ∂ β g i ◦ f · ∂ γ f j · · · ∂ γ r f j r in L ( U, R ) as k → ∞ . In particular, for any test function ϕ ∈ C ∞ c ( U ),lim k →∞ Z R n h ∂ β g i ◦ f ( k ) · ∂ γ f ( k ) j · · · ∂ γ r f ( k ) j r i · ϕ dx = Z R n (cid:2) ∂ β g i ◦ f · ∂ γ f j · · · ∂ γ r f j r (cid:3) · ϕ dx. Furthermore, by (50), h ∂ α ( g i ◦ f ) , ϕ i = ( − | α | Z R n (cid:0) g i ◦ f (cid:1) ( x ) ∂ α ϕ ( x ) dx = lim k →∞ ( − | α | Z R n (cid:0) g i ◦ f ( k ) (cid:1) ( x ) ∂ α ϕ ( x ) dx = lim k →∞ Z R n ∂ α (cid:0) g i ◦ f ( k ) (cid:1) ( x ) ϕ ( x ) dx. (55)Combining this with (52) and (55) we see that for any α in Z n ≥ , | α | ≤ s , theweak derivative ∂ α ( g i ◦ f ) is in L ( U, R ). As ∂ α ( g i ◦ f ) is a linear combinationof terms of the form ∂ β g i ◦ f · ∂ γ f j · · · ∂ γ r f j r ∈ L ( U, R )40ith P rj =1 γ j = α it follows from (50) and Lemma 2.17 that the map H s ( U, R m ) → L ( U, R ), f ∂ β g i ◦ f · ∂ γ f j · · · ∂ γ r f j r , is continuous. This shows that H s ( U, R m ) → H s ( U, R d ) , f g ◦ f, (56)is continuous. To see that L g is C r -smooth for any r ≥ f, δf be elements in H s ( U, R m ). Expanding g at f ( x ), x ∈ U arbitrary, up to order r ≥
1, one gets g (cid:0) f ( x ) + δf ( x ) (cid:1) = g (cid:0) f ( x ) (cid:1) + r X i =1 X | α | = i α ! (cid:0) ∂ α g (cid:1)(cid:0) f ( x ) (cid:1) · δf α ( x )+ R ( f, δf )( x )where δf α ( x ) = Q mi =1 δf i ( x ) α i and the remainder term R ( f, δf ) is given by R ( f, δf )( x ) = X | α | = r rα ! Z (1 − t ) r − (cid:16)(cid:0) ∂ α g (cid:1)(cid:0) f ( x ) + tδf ( x ) (cid:1) − (cid:0) ∂ α g (cid:1)(cid:0) f ( x ) (cid:1)(cid:17) δf α ( x ) dt. By (56), for any α ∈ Z n ≥ , H s ( U, R m ) → H s ( U, R d ) , f ∂ α g ◦ f (57)is continuous. In view of Lemma 2.16 (cf. also Lemma 2.10), ∂ α g ◦ f can beviewed as an element in L | α | sym (cid:0) H s ( U, R ) , H s ( U, R d ) (cid:1) , defined by( δh j ) ≤ j ≤| α | ∂ α g ◦ f · | α | Y j =1 δh j and the map H s ( U, R m ) → L | α | sym (cid:0) H s ( U, R ) , H s ( U, R d ) (cid:1) , f ∂ α g ◦ f
41s continuous. Similarly one sees that R ( f, δf ) is in H s ( U, R d ) and by Lemma2.16, k R ( f, δf ) k s k δf k rs ≤ K r +1 X | α | = r α ! sup ≤ t ≤ k ∂ α g ◦ ( f + tδf ) − ∂ α g ◦ f k s . By the continuity of the map (57) it then follows thatlim k δf k s → k R ( f, δf ) k s k δf k rs = 0 . Hence Theorem 2.2 applies and it follows that L g is C r -smooth for any r ≥ Lemma 2.21.
Let U ⊆ R n be a bounded domain. If g ∈ C ∞ ( U , R ) thenthere exists ˜ g ∈ C ∞ c ( R n , R ) such that ˜ g | U = g . The following result easily follows from Proposition 2.14 ( ii ). Lemma 2.22.
Let U ⊆ R n be an open subset in R n with Lipschitz boundaryand let s > n/ . Then for any f ∈ H s ( U, R d ) and ϕ ∈ C ∞ c ( R n ) , ϕ · f ∈ H s ( U, R d ) . In this section we prove Theorem 1.2. The main results used for the proof– in addition to the ones of Proposition 2.19, Proposition 2.20, and Lemma2.21 – are summarized in Section 3.1 and will be proved in Section 4.
Let M be a closed manifold of dimension n and N a manifold of dimension d . Further let s be an integer, s > n/
2. Recall that a continuous map f : M → N is said to be an element in H s ( M, N ) if for any point x ∈ M ,there exist a chart χ : U → U ⊆ R n of M , x ∈ U , and a chart η : V → V ⊆ R d of N , f ( x ) ∈ V , such that f ( U ) ⊆ V and η ◦ f ◦ χ − : U → V ⊆ R d
42s an element in H s ( U, R d ). Note that if e χ : e U → e U and e η : e V → e V are twoother charts such that x ∈ e U and f ( e U ) ⊆ e V , then e η ◦ f ◦ e χ − is not necessarilyan element in H s ( e U , R d ). As an example consider M = T = R / Z , N = R and let f : ( − / , / → R be the function f ( x ) := (cid:26) x / , x ∈ [0 , / − x ) / , x ∈ [ − / , . Extending f periodically to R we get a function on T that we denote by thesame letter. It is not hard to see that f ∈ H ( T , R ). Now, introduce a newcoordinate y = x on the open set (0 , / ⊆ T . Then ˜ f ( y ) := f ( x ( y )) = y / , y ∈ (0 , / f ′ ( y ) = 1 / (3 y / ), and hence, ˜ f ′ / ∈ L ((0 , / , R ).This shows that ˜ f / ∈ H ((0 , / , R ).With this in mind we define Definition 3.1.
An open cover ( U i ) i ∈ I of M by coordinate charts χ i : U i → U i ⊆ R n , i ∈ I , is called a cover of bounded type , if for any i, j ∈ I with U i ∩ U j = ∅ , χ j ◦ χ − i ∈ C ∞ b (cid:0) χ i ( U i ∩ U j ) , R n (cid:1) . Definition 3.2.
Assume that U I = ( U i ) i ∈ I is a cover of M and V I = ( V i ) i ∈ I is a collection of charts of N . The pair ( U I , V I ) is said to be a fine cover ifthe following conditions are satisfied:(C1) I is finite and for any i ∈ I , χ i : U i → U i ⊆ R n and η i : V i → V i ⊆ R d are coordinate charts of M respectively N ; U i and V i are bounded andhave a Lipschitz boundary.(C2) U I [ V I ] is a cover of M [ ∪ i ∈ I V i ] of bounded type.(C3) For any i, j ∈ I , the boundaries of χ i ( U i ∩ U j ) and η i ( V i ∩ V j ) are piece-wise C ∞ -smooth, i.e. they are given by a finite (possibly empty) unionof transversally intersecting C ∞ -embedded hypersurfaces in R n respec-tively R d . In particular, χ i ( U i ∩ U j ) and η i ( V i ∩ V j ) have a Lipschitzboundary. Fine covers ( U I , V I ) will be used to construct a C ∞ -differentiable structure of H s ( M, N ). To make this construction independent of any choice of metricson M and N , the notion of a fine cover does not involve any metric.43 efinition 3.3. A triple ( U I , V I , f ) consisting of f ∈ H s ( M, N ) with s >n/ and a fine cover ( U I , V I ) is said to be a fine cover with respect to f if f ( U i ) ⋐ V i for any i ∈ I , i.e., f ( U i ) is compact and contained in V i . Lemma 3.1.
Let f ∈ H s ( M, N ) and s > n/ . Then there exists a fine cover ( U I , V I ) with respect to f .Proof. To construct such a fine cover choose a Riemannian metric g M on M , a Riemannian metric g N on N , and ρ >
0, so that 2 ρ is smaller thanthe injectivity radius of the compact subset f ( M ) ⊆ N with respect to theRiemannian metric g N . Note that f ( M ) is compact as M is compact and f is continuous. Furthermore, f : M → N is uniformly continuous. Hencethere exists r > r smaller than the injectivity radius of ( M, g M ) sothat dist g N ( f ( x ) , f ( x ′ )) < ρ for any x, x ′ ∈ M with dist g M ( x, x ′ ) < r . k Forany x ∈ M define U x := exp x ( B r ) and U x := B r ⊆ T x M ∼ = R n where B r denotes the open ball in T x M of radius r with respect to the innerproduct g M ( x ) and exp x : T x M → M denotes the Riemannian exponentialmap at x . The map χ x : U x → U x is then defined to be the restrictionof the inverse of exp x to U x , which is well defined as 2 r is smaller thanthe injectivity radius. Hence χ x is a chart of M . Assume that there existpoints x, x ′ ∈ M, x = x ′ and p ∈ ∂ U x ∩ ∂ U x ′ , so that the boundaries of thegeodesic balls U x and U x ′ do not intersect transversally at p . We claim thatin this case U x ∩ U x ′ = ∅ . Indeed, as dist g M ( x, p ) = r , dist g M ( x ′ , p ) = r ,and as 2 r is smaller than the injectivity radius of ( M, g M ) there exists aminimal geodesic connecting the points x and x ′ . In view of the assumptionsthat x = x ′ and ∂ U x and ∂ U x ′ do not intersect transversally in p it thenfollows that p lies on the above geodesic between x and x ′ and dist g M ( x, x ′ ) =2 r , hence U x ∩ U x ′ = ∅ . Therefore, for any x, x ′ ∈ M , x = x ′ , ∂ U x and ∂ U x ′ either do not intersect at all or intersect transversally. In a similarway we construct charts η f ( x ) : V f ( x ) → V f ( x ) ⊆ R d , x ∈ M , where now V f ( x ) ⊆ T f ( x ) N ∼ = R d is the open ball of radius ρ in T f ( x ) N centered at 0 and η f ( x ) = ( exp f ( x ) (cid:12)(cid:12) V f ( x ) ) − . Here exp f ( x ) denotes the Riemannian exponential k Here dist g M and dist g N denote the geodesic distances on ( M, g M ) and ( N, g N ) respec-tively. N, g N ) at f ( x ). As M is compact there exist finitely many points( x i ) i ∈ I ⊆ M so that U I = ( U i ) i ∈ I with U i ≡ U x i covers M . By construction V I = ( V i ) i ∈ I with V i = V f ( x i ) is then a cover of f ( M ) and one verifies that( U I , V I , f ) is a fine cover with respect to f . Lemma 3.2.
Let ( U I , V I , h ) be fine cover with respect to h ∈ H s ( M, N ) .Then for any i ∈ I , the map h i := η i ◦ h ◦ χ − i : U i → V i ⊆ R d is in H s ( U i , R d ) .Proof. By the definition of H s ( M, N ) and the compactness of M there exista finite open cover ( W j ) j ∈ J of M by coordinate charts µ j : W j → W j ⊆ R n and for any j ∈ J an open coordinate chart ν j : Z j → Z j ⊆ R d of N with h ( W j ) ⋐ Z j and W j , Z j bounded so that for any j ∈ Jν j ◦ h ◦ µ − j ∈ H s ( W j , R d ) . Without loss of generality we may assume that I ∩ J = ∅ . In a first step weshow that for any open subset U ⋐ W j ∩ U i with Lipschitz boundary ∂ U ,the function η i ◦ h ◦ χ − i (cid:12)(cid:12) U is in H s ( U, R d ). Here U is given by χ i ( U ) ⊆ R n .Indeed, note that as U = χ i ( U ) ⋐ U i and µ j ( U ) ⋐ W j it follows that µ j ◦ χ − i : U → µ j ( U ) is in C ∞ b ( U , R n )and χ i ◦ µ − j : µ j ( U ) → U is in C ∞ b (cid:0) µ j ( U ) , R n (cid:1) . Hence by Corollary 2.4,( ν j ◦ h ◦ µ − j ) ◦ ( µ j ◦ χ − i ) (cid:12)(cid:12)(cid:12) U ∈ H s ( U, R d ) . Furthermore, one can choose
V ⊆ N open so that h ( U ) ⋐ V ⋐ Z j ∩ V i . Hence η i ◦ ν − j (cid:12)(cid:12)(cid:12) ν j ( V ) : ν j ( V ) → η i ( V ) is in C ∞ b (cid:0) ν j ( V ) , R d (cid:1) . One then canapply Proposition 2.20 and Lemma 2.21 to conclude that η i ◦ h ◦ χ − i (cid:12) (cid:12) U = ( η i ◦ ν − j ) ◦ ( ν j ◦ h ◦ µ − j ) ◦ ( µ j ◦ χ − i ) (cid:12) (cid:12) U ∈ H s ( U, R d ) .
45n view of this we can assume that the cover ( W j ) j ∈ J is a refinement of( U i ) i ∈ I , i.e., for any j ∈ J there exists σ ( j ) ∈ I such that W j ⊆ U σ ( j ) , that satisfies the following additional properties: for any j ∈ J , W j ⋐ U σ ( j ) , µ j ≡ χ σ ( j ) | W j : W j → W j ⋐ U σ ( j ) ⊆ R n (58) ν j ≡ η σ ( j ) : Z j ≡ V σ ( j ) → Z j ≡ V σ ( j ) ⊆ R d (59)and ν j ◦ h ◦ µ − j ∈ H s ( W j , R d ) . (60)Now, choose an arbitrary i ∈ I and consider the closure U i of U i in M . Let J i := { j ∈ J | W j ∩ U i = ∅} . Then ( W j ) j ∈ J i is a open cover of U i . We can choose ( W j ) j ∈ J i so that for any j ∈ J i , χ i ( W j ∩ U i ) ⊆ R n has Lipschitz boundary. Let ( ϕ j ) j ∈ J be a partitionof unity on M subordinate to the open cover ( W j ) j ∈ J . By construction, (cid:16) X j ∈ J i ϕ j (cid:17)(cid:12)(cid:12)(cid:12) U i ≡ . (61)Take an arbitrary j ∈ J i . As the cover ( U l ) l ∈ I is of bounded type, χ σ ( j ) ◦ χ − i ∈ C ∞ b ( χ i ( U σ ( j ) ∩ U i ) , R n ) (62)and η i ◦ η − σ ( j ) ∈ C ∞ b ( η σ ( j ) ( V σ ( j ) ∩ V i ) , R d ) . (63)In view of (58) and (59) µ j ◦ χ − i | χ i ( W j ∩U i ) = χ σ ( j ) ◦ χ − i | χ i ( W j ∩U i ) ∈ C ∞ b ( χ i ( W j ∩ U i ) , R n ) (64)and η i ◦ ν − j = η i ◦ η − σ ( j ) | η σ ( j ) ( V σ ( j ) ∩V i ) ∈ C ∞ b ( η σ ( j ) ( V σ ( j ) ∩ V i ) , R d ) . (65)46e have( η i ◦ h ◦ χ − i ) | χ i ( W j ∩U i ) = ( η i ◦ ν − j ) ◦ ( ν j ◦ h ◦ µ − j ) ◦ ( µ j ◦ χ − i ) | χ i ( W j ∩U i ) . (66)Then, in view of (60), (64), (65), and (66), as well as Corollary 2.4, Propo-sition 2.20, Lemma 2.21, and Lemma 2.22 one concludes that( ϕ j ◦ χ − i ) · ( η i ◦ h ◦ χ − i ) ∈ H s ( U i , R d ) (67)where the mapping above is extended from χ i ( W j ∩ U i ) to the whole neigh-borhood U i by zero. Finally, in view of (61) we get η i ◦ h ◦ χ − i = X j ∈ J i ( ϕ j ◦ χ − i ) · ( η i ◦ h ◦ χ − i ) ∈ H s ( U i , R d ) . This completes the proof of Lemma 3.2.For a given fine cover ( U I , V I ), introduce the subset O s ≡ O s ( U I , V I ) of H s ( M, N ) O s := (cid:8) h ∈ H s ( M, N ) (cid:12)(cid:12) h ( U i ) ⋐ V i ∀ i ∈ I (cid:9) and the map ı ≡ ı U I , V I : O s → ⊕ i ∈ I H s ( U i , R d ) , h ( h i ) i ∈ I where for any i ∈ I h i := η i ◦ h ◦ χ − i : U i → V i ⊆ R d . By Lemma 3.2, the map ı is well-defined and we say that h I := ( h i ) i ∈ I is therestriction of h to U I := ( U i ) i ∈ I . Definition 3.4.
A subset S of a Hilbert space H is called a C ∞ -submanifoldof H if for any p ∈ S , there exist an open neighborhood V of p in H , openneighborhoods W i ⊆ H i of of the Hilbert spaces H i , i = 1 , , and a C ∞ -diffeomorphism ψ : V → W × W , with ψ ( p ) = (0 , so that, ψ ( V ∩ S ) = W × { } . The following result will be proved in Section 4.47 roposition 3.3.
Let ( U I , V I ) be a fine cover and O s ≡ O s ( U I , V I ) with s > n/ , and ı ≡ ı U I , V I be defined as above. Then the range ı ( O s ) of ı is a C ∞ -submanifold of ⊕ i ∈ I H s ( U i , R d ) . To continue, let us recall the notion of a C ∞ -Hilbert manifold. Let M be a topological space. A pair ( U , χ : U → U ) consisting of an open subset U ⊆ M and a homeomorphism χ : U → U ⊆ H of U onto an open subset U of a Hilbert space is said to be a chart of M . Occasionally, we also refer to U or to χ : U → U as a chart. For any x ∈ U we say that ( U , χ ) is a chart at x . Two charts χ i : U i → U i ⊆ H of M are said to be compatible if χ ◦ χ − : χ ( U ∩ U ) → χ ( U ∩ U ) is a C ∞ -map between the open sets χ i ( U ∩ U ) ⊆ H . An atlas of M is a cover A of M by compatible charts. A maximal atlasof M (maximality means that any chart that is compatible with the charts in A belongs to A ) is said to be a C ∞ -differentiable structure of M . Clearly anyatlas of M induces precisely one C ∞ -differentiable structure. Assume that( U I , V I ) is a fine cover. The following result says that the C ∞ -differentiablestructure on the subset O s ≡ O s ( U I , V I ) of H s ( M, N ) obtained by pullingback the one of the submanifold ı ( O s ) does not depend on the choice of( U I , V I ). More precisely, let ( U J , V J ) be a fine cover. For convenience wechoose the index sets I, J so that I ∩ J = ∅ . As above, introduce the subset O s ≡ O s ( U J , V J ) of H s ( M, N ) together with the restriction map, ı ≡ ı U J , V J : O s ( U J , V J ) → ⊕ j ∈ J H s ( U j , R d ) , f ( f j ) j ∈ J where for any j ∈ J , f j is given by f j := η j ◦ f ◦ χ − j : U j → V j ⊆ R d . By Proposition 3.3, O s ( U J , V J ) admits a C ∞ -differentiable structure ob-tained by pulling back the one of the submanifold ı (cid:0) O s ( U J , V J ) (cid:1) ⊆ ⊕ j ∈ J H s ( U j , R d ) . In Section 4 we prove the following statements:
Lemma 3.4.
Let s be an integer, s > n/ , and let ( U I , V I ) and ( U J , V J ) befine covers. Then O s ( U I , V I ) ∩ O s ( U J , V J ) is open in O s ( U I , V I ) . roposition 3.5. Let s be an integer with s > n/ and let ( U I , V I ) and ( U J , V J ) be fine covers. Then the C ∞ -differentiable structures on the in-tersection O s ( U I , V I ) ∩ O s ( U J , V J ) induced from O s ( U I , V I ) and O s ( U J , V J ) respectively, coincide. It follows from Lemma 3.1 that the sets O s ( U I , V I ) , O s ( U J , V J ) , . . . con-structed above, with I, J, . . . finite and pairwise disjoint, form a cover C of H s ( M, N ). By Lemma 3.4, the set T of subsets S ⊆ H s ( M, N ), having theproperty that S ∩ O s ( U I , V I ) is open in O s ( U I , V I ) ∀ O s ( U I , V I ) ∈ C (68)defines a topology of H s ( M, N ). In particular, C is an open cover of H s ( M, N )in the topology T . Note that Lemma 3.6.
The topology T of H s ( M, N ) is Hausdorff.Proof. Take f, g ∈ H s ( M, N ) so that f = g . Then there exists x ∈ M such that f ( x ) = g ( x ). Using that f and g are assumed continuous oneconstructs, as in Lemma 3.1, a fine cover ( U I , V I ) with respect to f and afine cover ( U J , V J ) with respect to g , I ∩ J = ∅ , such that there exist i ∈ I and j ∈ J so that x ∈ U i , U i = U j , and V i ∩ V j = ∅ . Then, O s ( U I , V I ) ∩ O s ( U I , V I ) = ∅ . As by Lemma 3.4 the sets O s ( U I , V I )and O s ( U I , V I ) are open in T we see that T is Hausdorff.Combining Proposition 3.5 with Lemma 3.4 it follows that the cover C defines a C ∞ -differentiable structure on H s ( M, N ). Corollary 3.1.
Let M be a closed manifold of dimension n , N a C ∞ -manifold of dimension d and s an integer with s > n/ . Then the cover C induces a C ∞ -differentiable structure A s on H s ( M, N ) so that H s ( M, N ) is a Hilbert manifold.Proof. By Lemma 3.1 and Lemma 3.4, C is an open cover of H s ( M, N ). Theclaimed statement then follows from Proposition 3.5.49bin and Marsden introduced a C ∞ -differentiable structure of H s ( M, N )in terms of a Riemannian metric g ≡ g N of N – see [14] or [15]. Moreprecisely, given any f : M → N in H s ( M, N ) introduce the linear space T f H s ( M, N ) := { X ∈ H s ( M, T N ) | π N ◦ X = f } where π N : T N → N is the canonical projection of the tangent bundle T N of N to the base manifold N . Elements in T f H s ( M, N ) are referred to as vectorfields along f . On the linear space T f H s ( M, N ) we define an inner productas follows. Choose a fine cover ( U I , V I ) so that f ∈ O s ( U I , V I ). In particular, U I is an open cover of M by coordinate charts of M , χ i : U i → U i ⊆ R n and V I is a set of coordinate charts of N , η i : V i → V i ⊆ R d . The restrictionof an arbitrary element X ∈ T f H s ( M, N ) to U i induces a continuous map X i : U i → R d , X i ( x ) = (cid:0) X k (cid:0) χ − i ( x ) (cid:1)(cid:1) dk =1 , x ∈ U i where X k are the coordinates of X (cid:0) χ − i ( x ) (cid:1) in the chart V i ⊆ R d . Using that( U I , V I ) is a fine cover one concludes from Lemma 3.2 and the compactnessof X ( M ) ⊆ T N that X i ∈ H s ( U i , R d ) . The family ( X i ) i ∈ I is referred to as the restriction of X to U I = ( U i ) i ∈ I . For X, Y ∈ T f H s ( M, N ), define h X, Y i s := X i ∈ I, | α |≤ s Z U i h ∂ α X i , ∂ α Y i i dx (69)where h· , ·i denotes the Euclidean inner product in R d . Then h· , ·i s is a innerproduct, making T f H s ( M, N ) into a Hilbert space. Another choice of U I , V I will lead to a possibly different inner product, but the two Hilbert norms canbe shown to be equivalent. In this way one obtains a differential structureof T f H s ( M, N ). With the help of the exponential maps exp y : T y N → N , y ∈ N , defined in terms of the Riemannian metric g of N , Ebin and Marsden([14]) show that H s ( M, N ) is a C ∞ -Hilbert manifold. ∗∗ More specifically,charts on H s ( M, N ) are defined with the help of the exponential mapexp : O s → H s ( M, N ) , X (cid:2) x exp f ( x ) (cid:0) X ( x ) (cid:1)(cid:3) , ∗∗ Note that our arguments will give an independent proof of this fact. O s ⊆ T f H s ( M, N ) is a sufficiently small neighborhood of zero in T f H s ( M, N ) – see Section 4 for more details. We denote the C ∞ -differentiablestructure of H s ( M, N ) defined in this way by A sg . In Section 4 we prove Proposition 3.7.
Let M be a closed manifold of dimension n , N a C ∞ -manifold endowed with a Riemannian metric g , and s an integer with s >n/ . Then A s = A sg . Now let M be a closed oriented n -dimensional manifold and let s bean integer with s > n/ s > n/ H s ( M, M ) can be continuously embedded into C ( M, M ). As in Lemma 2.18 one sees that D s ( M ) := { ϕ ∈ Diff ( M ) | ϕ ∈ H s ( M, M ) } is open in H s ( M, M ). Hence D s ( M ) is a C ∞ -Hilbert manifold. Lemma 3.8.
Let M be a closed oriented manifold of dimension n and s bean integer with s > n/ . Then for any ϕ ∈ D s ( M ) , the inverse ϕ − isin D s ( M ) and the map inv : D s ( M ) → D s ( M ) , ϕ ϕ − is continuous. For the convenience of the reader we include a proof of Lemma 3.8 inAppendix A.
To prove Theorem 1.2, we first need to introduce some more notation. Let M be a closed oriented manifold of dimension n and N a C ∞ -manifold ofdimension d . Consider open covers U I := ( U i ) i ∈ I and V I = ( V i ) i ∈ I of M where I ⊆ N is finite and a set of open subsets W I := ( W i ) i ∈ I of N so thatfor any i ∈ I , U i and V i are coordinate charts of M , χ i : U i → U i ⊆ R n , η i : V i → V i ⊆ R n and W i is a coordinate chart of N , ξ i : W i → W i ⊆ R d where U i and V i are bounded, open subsets of R n with Lipschitz boundaries.51et U I = ( U i ) i ∈ I , V I = ( V i ) i ∈ I , and W I = ( W i ) i ∈ I . For such data weintroduce the subsets P s ( U I , V I ) ⊆ ⊕ i ∈ I H s ( U i , R n )and P s ( V I , W I ) ⊆ ⊕ i ∈ I H s ( V i , R d )consisting of elements ( h i ) i ∈ I ∈ ⊕ i ∈ I H s ( U i , R n ) and ( f i ) i ∈ I ∈ ⊕ i ∈ I H s ( V i , R d )respectively such that for any i ∈ I , h i ( U i ) ⋐ V i and f i ( V i ) ⋐ W i . (70)Further, for any integer s with s > n/ D s ( U I , V I )consisting of elements ( ϕ i ) i ∈ I in P s ( U I , V I ) so that for any i ∈ I , ϕ i : U i → V i is 1-1 and 0 < inf x ∈ U i det( d x ϕ i ) . By Proposition 2.14, P s ( V I , W I ) is open in ⊕ i ∈ I H s ( V i , R d ). Moreover, oneconcludes from Lemma 2.18 and Proposition 2.14 that D s ( U I , V I ) is open in ⊕ i ∈ I H s ( U i , R n ). For any integers r , s with r ≥ s > n/ µ I : P s + r ( V I , W I ) × D s ( U I , V I ) → P s ( U I , W I )(( f i ) i ∈ I , ( ϕ i ) i ∈ I ) ( f i ◦ ϕ i ) i ∈ I By Proposition 2.19, ˜ µ I is well–defined and has the following property. Lemma 3.9. ˜ µ I is a C r -map. Proposition 3.10.
Let M be a closed oriented manifold of dimension n , N a C ∞ -manifold of dimension d , and r , s integers with r ≥ and s > n/ .Then µ : H s + r ( M, N ) × D s ( M ) → H s ( M, N ) , ( f, ϕ ) f ◦ ϕ is a C r -map. roof. Let ϕ ∈ D s ( M ) and f ∈ H s + r ( M, N ) be arbitrary. Arguing as in theproof of Lemma 3.1 one constructs open covers ( U i ) i ∈ I and ( V i ) i ∈ I on M aswell as an open cover ( W i ) i ∈ I of f ( M ) in N such that ( U I , V I ) is a fine coverwith respect to ϕ and ( V I , W I ) is a fine cover with respect to f . Denoteby O s ( U I , V I ) and O s + r ( V I , W I ) the open subsets of H s ( M, M ) respectively H s + r ( M, N ), introduced in Section 3.1. Then D s ( M ) ∩ O s ( U I , V I ) is an openneighborhood of ϕ in D s ( M ) and O s + r ( V I , W I ) is an open neighborhood of f in H s + r ( M, N ). Furthermore, note that ı U I , V I (cid:0) D s ( M ) ∩ O s ( U I , V I ) (cid:1) ⊆ D s ( U I , V I )and ı V I , W I (cid:0) O s + r ( V I , W I ) (cid:1) ⊆ P s + r ( V I , W I )where ı U I , V I and ı V I , W I are the embeddings introduced in Section 3.1. Onehas the following commutative diagram: O s + r ( V I , W I ) × ( D s ( M ) ∩ O s ( U I , V I )) µ I −→ O s ( U I , W I ) y ı V I , W I × ı U I , V I y ı U I , W I P s + r ( V I , W I ) × D s ( U I , V I ) ˜ µ I −→ P s ( U I , W I )where µ I is the restriction of the composition µ : H s + r ( M, N ) × D s ( M ) → H s ( M, N )to O s + r ( V I , W I ) × ( D s ( M ) ∩ O s ( U I , V I )). In view of Lemma 3.9˜ µ I : P s + r ( V I , W I ) × D s ( U I , V I ) → P s ( U I , W I )is C r -smooth. By the definition of the differential structure on O s + r ( V I , W I )and O s ( U I , V I ) (see § µ I is C r -smooth. As ϕ , f are arbitrary, it follows that µ is C r -smooth.Next we consider the inverse map, associating to any C -diffeomorphism ϕ : M → M of a given closed manifold M its inverse. Following the argu-ments of the proof of Proposition 2.13 and using Proposition 3.10 we obtain53 roposition 3.11. For any closed oriented manifold M of dimension n andany integers r , s with r ≥ and s > n/ inv : D s + r ( M ) → D s ( M ) , ϕ ϕ − is a C r -map.Proof of Theorem 1.2. The claimed results are established by Proposition3.10, Lemma 3.8 and Proposition 3.11.As an immediate consequence of Proposition 3.10 and Lemma 3.8 weobtain the following
Corollary 3.2.
For any closed oriented manifold M of dimension n and anyinteger s > n/ , D s ( M ) is a topological group. H s ( M, N ) In Section 3.1 we outlined the construction of a C ∞ -differentiable structureof H s ( M, N ) for any integer s with s > n/
2. In this section we prove theauxiliary results stated in Subsection 3.1, which were needed for this con-struction. Throughout this section we assume that M is a closed manifoldof dimension n , s ∈ Z ≥ with s > n/ N is a C ∞ -manifold of dimension d ,and g ≡ g N is a C ∞ -Riemannian metric on N . The main purpose of this subsection is to prove Proposition 3.3. Let us beginby recalling the set-up. Choose a fine cover ( U I , V I ) as defined in Subsection3.1. In particular, U I = ( U i ) i ∈ I is a finite cover of M and V I = ( V i ) i ∈ I oneof ∪ i ∈ I V i and for any i ∈ I , U i , V i are coordinate charts χ i : U i → U i ⊆ R n respectively η i : V i → V i ⊆ R d . Recall that O s ( U I , V I ), introduced insubsection 3.1, is given by O s ( U I , V I ) = (cid:8) h ∈ H s ( M, N ) (cid:12)(cid:12) h ( U i ) ⋐ V i ∀ i ∈ I (cid:9) (71)and the map ı ≡ ı U I , V I : O s ( U I , V I ) → ⊕ i ∈ I H s ( U i , R d ) , (72)54efined by ı ( h ) := ( h i ) i ∈ I and h i = η i ◦ h ◦ χ − i : U i → V i ⊆ R d isinjective. Proposition 3.3 states that ı (cid:0) O s ( U I , V I ) (cid:1) is a submanifold of ⊕ i ∈ I H s ( U i , R d ). We will prove this by showing that for any f ∈ O s ( U I , V I )there exists a neighborhood Q s of ( f i ) i ∈ I in ⊕ i ∈ I H s ( U i , R d ) so that Q s ∩ ı (cid:0) O s ( U I , V I ) (cid:1) coincides with ı ◦ exp f ( O s ) where exp f is the exponential mapexp f : T f H s ( M, N ) → H s ( M, N ) defined below (see also the discussionof the differential structure A sg of H s ( M, N ) in Subsection 3.1) and O s is a(small) neighborhood of 0 in T f H s ( M, N ). By proving that d ( ı ◦ exp f ) splits(Lemma 4.2 below) we then conclude that ı (cid:0) O s ( U I , V I ) (cid:1) is a submanifold of ⊕ i ∈ I H s ( U i , R d ). Let us now look at the Hilbert space T f H s ( M, N ) and themap exp f in more detail. For any y ∈ N , denote by T y N the tangent spaceof N at y and by exp y the exponential map of the Riemannian metric g on N . It maps a (sufficiently small) element v ∈ T y N to the point in N on thegeodesic issuing at y in direction v at time t = 1. For any y ∈ N the expo-nential map exp y is defined in a neighborhood of 0 y in T y N . Furthermore,for any X ∈ T f H s ( M, N ), with f ∈ H s ( M, N ), and x ∈ M , X ( x ) is anelement in T f ( x ) N , hence if k X ( x ) k is sufficiently small, exp f ( x ) (cid:0) X ( x ) (cid:1) ∈ N is well defined and, for X sufficiently small, we can introduce the mapexp f ( X ) := M → N, x exp f ( x ) (cid:0) X ( x ) (cid:1) . Note that for X = 0, exp f (0) = f . To analyze the map exp f further letus express it in local coordinates provided by the fine cover ( U I , V I ). Therestriction of an arbitrary element X ∈ T f H s ( M, N ) to U i is given by themap X i : U i → R d , x X i ( x ) . (73)As X ∈ T f H s ( M, N ), X i is an element in H s ( U i , R d ). Recall that T f H s ( M, N )is a Hilbert space. Without loss of generality we assume that the inner prod-uct (69) is defined in terms of U I and V I . It is then immediate that the linearmap ρ : T f H s ( M, N ) → ⊕ i ∈ I H s ( U i , R d ) , X ( X i ) i ∈ I (74)is an isomorphism onto its image. For X (sufficiently) close to 0 we wantto describe the restriction of exp f ( X ) to U I = ( U i ) i ∈ I . To this end, let usexpress exp y ( v ) for y ∈ V i , i ∈ I , and v sufficiently close to 0 in T y N in localcoordinates provided by η i : V i → V i . For any small v ∈ T y N , η i (exp y v )is given by γ i (1; y i , v i ) where t γ i ( t ; y i , v i ) ∈ R d is the geodesic issuing55t y i := η i ( y ) in direction given by the coordinate representation v i of thevector v . The geodesic γ i ( t ; y i , v i ) satisfies the ODE on V i ,¨ γ i + Γ( γ i )( ˙ γ i , ˙ γ i ) = 0 (75)with initial data γ i (0; y i , v i ) = y i and ˙ γ i (0; y i , v i ) = v i . (76)Here ˙ stays for ddt and for any z i ∈ V i and w i = ( w pi ) ≤ p ≤ d ∈ R d ,Γ( z i )( w i , w i ) = X ≤ p,q ≤ d Γ kpq ( z i ) w pi w qi ≤ k ≤ d (77)with Γ kpq denoting the Christoffel symbols of the Riemannian metric g , ex-pressed in the local coordinates of the chart η i : V i → V i ,Γ kpq = g kl (cid:0) ∂ z qi g pl − ∂ z li g pq + ∂ z pi g lq (cid:1) (78)where g pl are the coefficients of the metric tensor and g lk · g km = δ km where δ km is the Kronecker delta. Note that Γ kpq is a C ∞ -function on V i . Thevelocity vector v i ∈ R d in (76) is chosen close to zero so that the solution γ i ( t ; y i , v i ) exists and stays in V i for any | t | <
2. Now let us return to themap X exp f ( X ). Its restriction to U i is given by the time one map of theflow X i α i ( t ; X i ), where for any Y i ∈ H s ( U i , R d ), α i ( t ; Y i ) solves the ODE( ˙ α i , ˙ Z i ) = (cid:0) Z i , − Γ( α i )( Z i , Z i ) (cid:1) (79)with initial data (cid:0) α i (0; Y i ) , Z i (0; Y i ) (cid:1) = ( f i , Y i ) . (80)As above, f i is given by f i = η i ◦ f ◦ χ − i and satisfies f i ( U i ) ⋐ V i . Lemma 4.1.
For any f ∈ O s ( U I , V I ) and i ∈ I , there exists a neighborhood O si of in H s ( U i , R d ) so that for any Y i ∈ O si , the initial value problem(79)-(80) has a unique C ∞ -solution ( − , → H s ( U i , R d ) × H s ( U i , R d ) , t (cid:0) α i ( t ; Y i ) , Z i ( t ; Y i ) (cid:1) atisfying α i ( t ; Y i )( U i ) ⋐ V i . In fact, ( α i , Z i ) ∈ C ∞ (cid:0) ( − , × O si , H s ( U i , R d ) × H s ( U i , R d ) (cid:1) . Proof.
The claimed result follows from the classical theorem for ODE’s inBanach spaces on the existence, uniqueness, and C ∞ -smooth dependence oninitial data of solutions (cf. e.g. [25]). Indeed, denote by H s ( U i , V i ) thesubset of the Hilbert space H s ( U i , R d ), H s ( U i , V i ) := { h ∈ H s ( U i , R d ) | h ( U i ) ⋐ V i } . By the Sobolev embedding theorem (Proposition 2.14 ( iii )) H s ( U i , V i ) is openin H s ( U i , R d ). We claim that the vector field H s ( U i , V i ) × H s ( U i , R d ) → H s ( U i , R d ) × H s ( U i , R d )( h i , Y i ) (cid:0) Y i , − Γ( h i )( Y i , Y i ) (cid:1) is well-defined and C ∞ -smooth. Indeed, as h i ∈ H s ( U i , V i ), one has that h i ( U i ) ⋐ V i , thus the composition Γ ◦ h i is well-defined. Furthermore, byProposition 2.20, Lemma 2.21, (77) and (78) H s ( U i , V i ) → H s ( U i , R ) , h i Γ kpq ( h i )is C ∞ -smooth. By Lemma 2.16, H s ( U i , R ) is an algebra and multiplicationof elements of H s ( U i , R ) is C ∞ -smooth. Hence the map H s ( U i , V i ) × H s ( U i , R d ) → H s ( U i , R d ) , ( h i , Y i ) Γ( h i )( Y i , Y i )is C ∞ -smooth. Summarizing our considerations we have proved that thevector field H s ( U i , V i ) × H s ( U i , R d ) → H s ( U i , R d ) × H s ( U i , R d )( h i , Y i ) (cid:0) Y i , − Γ( h i )( Y i , Y i ) (cid:1) is C ∞ -smooth. Further note that for Y i ≡ (cid:0) α i ( t, , Z i ( t, (cid:1) = ( f i , O si of 0 in H s ( U i , R d )so that for any Y i ∈ O si , the initial value problem (79)-(80) has a uniquesolution (cid:0) α i ( t, Y i ) , Z i ( t, Y i ) (cid:1) in C ∞ (cid:0) ( − , , H s ( U i , V i ) × H s ( U i , R d ) (cid:1) . Asthe solution depends C ∞ -smoothly on the initial data one concludes that( α i , Z i ) ∈ C ∞ (cid:0) ( − , × O si , H s ( U i , V i ) × H s ( U i , R d ) (cid:1) . Corollary 4.1.
For any f ∈ O s ( U I , V I ) , there exists a neighborhood O s of in T f H s ( M, N ) so that for any X ∈ O s , exp f ( X ) is in O s ( U I , V I ) and thecomposition ı f := ı ◦ exp f ( X ) , O s exp f −→ O s ( U I , V I ) ı −→ ⊕ i ∈ I H s ( U i , R d ) is C ∞ -smooth.Proof. For any i ∈ I , the i -th component of the restriction map ρ i : T f H s ( M, N ) → H s ( U i , R d ) , X X i ( x )is linear and bounded by the definition of T f H s ( M, N ), hence it is C ∞ -smooth. As a consequence O s := \ i ∈ I ρ − i ( O si ) ⊆ T f H s ( M, N ) (81)is an open neighborhood of 0 in T f H s ( M, N ) with O si being the neighborhoodof 0 in H s ( U i , R d ) of Lemma 4.1. The latter implies that for any i ∈ I , thecomposition O s ρ i −→ H s ( U i , R d ) α i (1; · ) −→ H s ( U i , V i )is C ∞ -smooth. Recall that the restriction of exp f ( X ) to U i is given by α i (1; X i ). Hence exp f ( X ) ∈ O s ( U I , V I ) and ı f ( X ) = (cid:0) α i (1; ρ i ( X )) (cid:1) i ∈ I (82)showing that ı f is C ∞ -smooth as ρ i : T f H s ( M, N ) → H s ( U i , R d ) is abounded linear map.Next we want to analyze the map ı f further.58 emma 4.2. For any f ∈ O s ( U I , V I ) , the differential d ı f : T f H s ( M, N ) →⊕ i ∈ I H s ( U i , R d ) of ı f at X = 0 is 1-1 and has closed range.Proof. We claim that for any X ∈ T f H s ( M, N ), d ı f ( X ) = (cid:0) ρ i ( X ) (cid:1) i ∈ I where for any x ∈ U i , ρ i ( X )( x ) = X i ( x ) is the i -th component of the restric-tion map. Indeed, for any λ ∈ R with | λ | < | t | <
2, any solution ofthe initial value problem (79)-(80) with Y i and λY i in O si satisfies α i ( λt ; Y i ) = α i ( t ; λY i ) . (83)As ρ i ( λX ) = λρ i ( X ) by the linearity of the map ρ i it then follows from (82)and (83) that for any X ∈ O s with λX ∈ O s , ı f ( λX ) = (cid:0) α i ( λ ; X i ) (cid:1) i ∈ I and hence ddλ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 ı f ( λX ) = (cid:0) ˙ α i (0; X i ) (cid:1) i ∈ I = ( X i ) i ∈ I . As a consequence, d ı f ( X ) = (cid:0) ρ i ( X ) (cid:1) i ∈ I for any X ∈ T f H s ( M, N ) and d ı f is 1-1. It remains to show that d ı f has closed range. Note that for any given X ∈ O s and x ∈ χ j ( U i ∩ U j ) with i, j ∈ I , the restrictions X i and X j arerelated by d f j ( x ) ( η i ◦ η − j ) · X j ( x ) = X i (cid:0) χ i ◦ χ − j ( x ) (cid:1) . (84)Conversely, if ( Y i ) i ∈ I ∈ ⊕ i ∈ I H s ( U i , R d ) satisfies the relations (84) for any x ∈ χ j ( U i ∩ U j ) and i, j ∈ I , there exists X ∈ T f H s ( M, N ) so that ρ i ( X ) = Y i (85)for any i ∈ I . As s > n/
2, it then follows from Lemma 2.16, Corollary 2.4,Proposition 2.14(ii), as well as Proposition 2.20 and Lemma 2.21, that forany i, j ∈ I , the linear map R ij : ⊕ i ∈ I H s ( U i , R d ) → H s (cid:0) χ j ( U i ∩ U j ) , R d (cid:1) , ( X i ) i ∈ I d f j ( x ) (cid:0) η i ◦ η − j (cid:1) · X j ( x ) − X i (cid:0) χ i ◦ χ − j ( x ) (cid:1) is bounded. Hence, the relations (84) define a closed linear subspace of ⊕ i ∈ I H s ( U i , R d ). 59emma 4.2 will be used to show that ı (cid:0) O s ( U I , V I ) (cid:1) is a submanifold of ⊕ i ∈ I H s ( U i , R d ) by applying the following corollary of the inverse functiontheorem. Lemma 4.3.
Let E and H be Hilbert spaces and let H be a closed subspaceof H . Furthermore let V be an open neighborhood of in E and Φ : V → H a C ∞ -map so that d Φ( E ) = H and Ker d Φ = { } . Then there exist a C ∞ -diffeomorphism Ψ of some open neighborhood of Φ(0) ∈ H to an openneighborhood of ∈ H and an open neighborhood V ⊆ V of in E so that Ψ ◦ Φ | V is a C ∞ -diffeomorphism onto an open neighborhood of in H . See e.g. [25], Chapter I, Corollary 5.5 for a proof.
Proof of Proposition 3.3.
We will show that for any f ∈ O s ( U i , V I ) thereexists an open neighborhood Q s of ı ( f ) in ⊕ i ∈ I H s ( U i , R d ) such that ı f ( O s ) = Q s ∩ ı (cid:0) O s ( U I , V I ) (cid:1) where O s is an open neighborhood of zero in T f H s ( M, N ) such that ı f ( O s )is a submanifold in ⊕ i ∈ I H s ( U i , R d ). Recall that the differential of the map Y i α i (1; Y i ) of Lemma 4.1 at Y i = 0 is the identity (cf. the proof of Lemma4.2), d α i (1; · ) = id H s ( U i , R d ) . It thus follows by the inverse function theorem that for any i ∈ I , thereexists an open neighborhood Q si of f i contained in H s ( U i , V i ) such that, aftershrinking O si , if necessary( P (cid:26) α i (1; · ) : O si → Q si ia a C ∞ -diffeomorphism ∀ Y i ∈ O si , α i (1; Y i )( U i ) ⋐ V i By shrinking the neighborhood O si of zero in H s ( U i , R d ) once more one canensure that the open neighborhood O s of zero in T f H s ( M, N ) given by (81)satisfies the following two additional properties:( P ı f ( O s ) is a submanifold in ⊕ i ∈ I H s ( U i , R d )( P ∀ ξ ∈ O s , g ( ξ, ξ ) < ε . where ε > Q s of ı ( f ) = ( f i ) i ∈ I in ⊕ i ∈ I H s ( U i , R d ) is Q s := ⊕ i ∈ I Q si . h ∈ O s ( U I , V I ) with ı ( h ) = ( h i ) i ∈ I ∈ Q s . By the definition of Q s and Q si , there exists ( Y i ) i ∈ I ∈ ⊕ i ∈ I O si such that for any i ∈ I , α i (1; Y i ) = h i .We now have to show that ( Y i ) i ∈ I is the restriction of a global vector fieldalong f . In view of (84) and (85) it is to prove that for any x ∈ χ j ( U i ∩ U j ), i, j ∈ I , the identity (84) is satisfied. Assume the contrary. Then thereexists k, l ∈ I and x ∈ U k ∩ U l so that, with x k := χ k ( x ), x l := χ l ( x ) and y = f ( x ) ∈ V k ∩ V l , the vectors ξ ∈ T y N and ¯ ξ ∈ T y N corresponding to Y k ( x k ) and Y l ( x l ) respectively do not coincide, ξ = ¯ ξ. (86)On the other hand, by the definition of h k and α k h k ( x k ) = α k (1; Y k )( x k ) = η k (exp y ξ )and, similarly, h l ( x l ) = α l (1; Y l )( x l ) = η l (exp y ¯ ξ ) . As ı ( h ) = ( h i ) i ∈ I it then follows thatexp y ξ = h ( x ) = exp y ¯ ξ. However, in view of the choice of ε in ( P
3) and Lemma 4.4 below, the latteridentity contradicts (86). Hence ( Y i ) i ∈ I satisfies (84) and ı f ( X ) = ( Y i ) i ∈ I where X ∈ O s is the vector field along f defined by (85).It remains to state and prove Lemma 4.4 used in the proof of Proposition3.3. For any ε > A ⊆ N denote by B εg A the ε -ball bundleof N restricted to AB εg A = (cid:8) ξ ∈ ∪ y ∈ A T y N (cid:12)(cid:12) g ( ξ, ξ ) / < ε (cid:9) where g is the Riemannian metric on N . Denote by π : T N → N thecanonical projection. Recall that f ∈ H s ( M, N ) implies that f is continuous.As M is assumed to be closed, f ( M ) is compact. By the classical ODEtheorem and the compactness of f ( M ) there exists a neighborhood V of f ( M ) in N and ε > B εg V → N × N, ξ (cid:0) π ( ξ ) , exp π ( ξ ) ξ (cid:1) is well-defined and C ∞ -smooth. 61 emma 4.4. For any f ∈ O s ( U I , V I ) , there exists ε > and an open neigh-borhood V of f ( M ) so that Φ : B εg V → W ⊆ N × N, ξ (cid:0) π ( ξ ) , exp π ( ξ ) ξ (cid:1) is a C ∞ -diffeomorphism onto an open neighborhood W of { ( y, y ) | y ∈ V} in N × N .Proof. Note that for any ξ ∈ T N of the form 0 y ∈ T y N with y ∈ f ( M ),Φ(0 y ) = ( y, y ) and d y Φ : T y ( T N ) → T y N × T y N is a linear isomorphism.By the inverse function theorem and the compactness of f ( M ) it then followsthat there exist an open neighborhood V of f ( M ), an open neighborhood W of the diagonal { ( y, y ) | y ∈ V} in N × N , and ε > B εg V → W ⊆ N × N, ξ (cid:0) π ( ξ ) , exp π ( ξ ) ξ (cid:1) (87)is a local diffeomorphism that is onto and that for any x ∈ V Φ (cid:12)(cid:12)(cid:12) B εg V∩ T x N : B εg V ∩ T x N → N is a diffeomorphism onto its image. The last statement and the formula forΦ in (87) imply that Φ is is injective. Hence, Φ is a bijection. As it is also alocal diffeomorphism, Φ is a diffeomorphism. Remark 4.1.
Note that we did not use the Ebin-Marsden differential struc-ture on N s ( M, N ) . In consequence, our construction gives an independentproof of Ebin-Marsden’s result. As a by-product, the proof of Proposition 3.3 leads to the following
Corollary 4.2.
For any set of the form O s ( U I , V I ) , A s ∩ O s ( U I , V I ) = A sg ∩ O s ( U I , V I ) i.e. the C ∞ -differentiable structure induced from ⊕ i ∈ I H s ( U i , R d ) coincideson O s ( U I , V I ) with the one of Ebin-Marsden, introduced in [14]. .2 Differentiable structure In this subsection we prove Proposition 3.5 and Proposition 3.7 as well asLemma 3.4. Recall that the map ı ≡ ı U I , V I : O s ( U I , V I ) → ⊕ i ∈ I H s ( U i , R d ) (88)is injective and by Proposition 3.3, the image of ı is a C ∞ -submanifold in ⊕ i ∈ I H s ( U i , R d ). Hence, by pulling back the C ∞ -differentiable structure ofthe image of ı , we get a C ∞ -differentiable structure on O s ( U I , V I ). First weprove Lemma 3.4. Proof of Lemma 3.4.
Let ( U I , V I ) and ( U J , V J ) be fine covers. For conve-nience assume that the index sets I, J are chosen in such a way that I ∩ J = ∅ .It is to show that O s ( U I , V I ) ∩ O s ( U J , V J ) is open in O s ( U I , V I ). Given h ∈ O s ( U I , V I ) ∩ O s ( U J , V J ) consider its restriction ( h i ) i ∈ I = ı U I , V I ( h ) in ⊕ i ∈ I H s ( U i , R d ) and choose a Riemannian metric g on N . In view of Propo-sition 2.14 (iii), for any ε >
0, there exists an open neighborhood W of ( h i ) i ∈ I in ⊕ i ∈ I H s ( U i , R d ) such that for any ( p i ) i ∈ I ∈ W ∩ ı U I , V I (cid:0) O s ( U I , V I ) (cid:1) andany x ∈ M dist g (cid:0) p ( x ) , h ( x ) (cid:1) < ε (89)where dist g is the geodesic distance function on ( N, g ) and p ∈ H s ( M, N ) isthe unique element of O s ( U I , V I ) such that ı U I , V I ( p ) = ( p i ) i ∈ I . It follows from(89) and the definition of O s ( U I , V I ) that the neighborhood W of ( h i ) i ∈ I in ⊕ i ∈ I H s ( U i , R d ) can be chosen so that W := ı − U I , V I (cid:0) W ∩ ı U I , V I (cid:0) O s ( U I , V I ) (cid:1)(cid:1) ⊆ O s ( U J , V J ) . (90)In view of the definition of the topology on O s ( U I , V I ), W is an open neigh-borhood of h in O s ( U I , V I ). As h ∈ O s ( U I , V I ) ∩ O s ( U J , V J ) was chosenarbitrarily, formula (90) implies that O s ( U I , V I ) ∩ O s ( U J , V J ) is open in O s ( U I , V I ).Next we prove Proposition 3.5 which says that the C ∞ -differentiablestructures of O s ( U I , V I ) ∩ O s ( U J , V J ) induced by the ones of O s ( U I , V I ) and O s ( U J , V J ) coincide. 63 roof of Proposition 3.5. Let ( U I , V I ) and ( U J , V J ) be fine covers. For con-venience we choose I, J such that I ∩ J = ∅ and assume that O sIJ := O s ( U I , V I ) ∩ O s ( U J , V J ) = ∅ . Note that the boundary ∂χ i ( U i ∩ U j ) , i ∈ I, j ∈ J , might not be Lipschitz. To address this issue we refine the covers ( U I , V I )and ( U J , V J ). For any h ∈ O sIJ there exist fine covers ( U K , V K ), ( U L , V L )with I, J, K, L pairwise disjoint such that (i) h ∈ O s ( U K , V K ) ∩ O s ( U L , V L ),(ii) there exist maps σ : K → I and τ : L → J so that for any k ∈ K , ℓ ∈ L U k ⋐ U σ ( k ) , V k ⋐ V σ ( k ) and U ℓ ⋐ U τ ( ℓ ) , V ℓ ⋐ V τ ( ℓ ) , and (iii) for any k ∈ K and ℓ ∈ L , U k ∩ U ℓ ⊆ M and V k ∩ V ℓ ⊆ N havepiecewise smooth boundary and U K ∪ U L := {U k , U ℓ } k ∈ K,ℓ ∈ L is a cover of M of bounded type.Fine covers ( U K , V K ) and ( U L , V L ) with properties (i)-(iii) can be constructedby choosing for U k , V k ( k ∈ K ) and U ℓ , V ℓ ( ℓ ∈ L ) appropriate geodesicballs defined in terms of Riemannian metrics on M and N respectively andarguing as in the proof of Lemma 3.1. Moreover, we choose for any k ∈ K thecoordinate chart χ k : U k → U k ⊆ R n to be the restriction of the coordinatechart χ σ ( k ) : U σ ( k ) → U σ ( k ) ⊆ R n to U k . In a similar way we choose thecoordinate charts η k ( k ∈ K ) and χ ℓ , η ℓ ( ℓ ∈ L ). Let O sKL := O s ( U K , V K ) ∩O s ( U L , V L ) and O sIJKL := O sIJ ∩ O sKL and define F I := ⊕ i ∈ I H s ( U i , R d ) , F J := ⊕ j ∈ J H s ( U j , R d ) , F K := ⊕ k ∈ K H s ( U k , R d ) , F L := ⊕ ℓ ∈ L H s ( U ℓ , R d ) . By Lemma 3.4, the sets O sKL , O sIJ , and O sIJKL are open sets in the topology T , defined by (68). To prove Proposition 3.5 it suffices to show that the C ∞ -differentiable structures on O sIJKL induced from the ones of O sI and O sJ F I O sIJ F J ⊆ ⊆ ⊆ ı I ( O sIJKL ) ✛ ı I O sIJKL ı J ✲ ı J ( O sIJKL ) ı K ( O sIJKL ) P I ❄ R ✲ ı K ✛ ı L ( O sIJKL ) P J ❄ ı L ✲ ⊆ ⊆ F K F L (91)where ı I , ı J , ı K , and ı L denote the corresponding restrictions of ı U I , V I , ı U J , V J , ı U K , V K , and ı U L , V L , to O sIJKL and P I , P J are the maps P I : ı I ( O sIJKL ) → ı K ( O sIJKL ) , ( f i ) i ∈ I ( f σ ( k ) (cid:12)(cid:12) U k ) k ∈ K , P J : ı J ( O sIJKL ) → ı L ( O sIJKL ) , ( f j ) j ∈ J ( f τ ( ℓ ) (cid:12)(cid:12) U ℓ ) ℓ ∈ L . (92)Finally, the map R : ı K ( O sIJKL ) → ı L ( O sIJKL ) is defined in such a way thatthe central sub-diagram in (91) is commutative. Note that by the definitionof the charts χ k , η k ( k ∈ K ) and χ ℓ , η ℓ ( ℓ ∈ L ), the left and right sub-diagrams in (91) are commutative. By Lemma 4.5 below the map R is adiffeomorphism. Proposition 3.5 then follows once we show that the maps P I and P J are diffeomorphisms, as in this case, P − J ◦R◦P I is a diffeomorphism.Consider the map P I . As P I is the restriction of the bounded linear map e P I : F I → F K , ( f i ) i ∈ I ( f σ ( k ) (cid:12)(cid:12) U k ) k ∈ K to the submanifold ı I ( O sIJKL ) ⊆ F I , P I is smooth. Take an arbitrary element f I ≡ ı I ( f ) ∈ ı I ( O sIJKL ) and consider the differential of P I at f I , d f I P I : ρ I (cid:0) T f H s ( M, N ) (cid:1) → ρ K (cid:0) T f H s ( M, N ) (cid:1) where ρ I is the restriction map (74) corresponding to ( U I , V I ) and ρ K isthe restriction map corresponding to ( U K , V K ). In view of the choice of thecoordinate charts ( χ k ) k ∈ K , d f I P I is given by d f I P I : ( X i ) i ∈ I (cid:16) X σ ( k ) (cid:12)(cid:12) U k (cid:17) k ∈ K . (93)65n particular it follows from (93) that d f I P I is injective and onto. Hence, bythe open mapping theorem d f I P I is a linear isomorphism. As f I ∈ ı I ( O sIJKL )is arbitrary, P I : ı I ( O sIJKL ) → ı K ( O sIJKL ) is a local diffeomorphism. As bythe commutativity of the left sub-diagram of (91), P I is a homeomorphismwe get that it is a diffeomorphism. Similarly, one proves that P J is a diffeo-morphism.Next we prove Lemma 4.5 used in the proof of Proposition 3.5. Let R bethe map introduced there. Lemma 4.5. R is a diffeomorphism.Proof. Throughout the proof we use the notation introduced in the proof ofProposition 3.5 without further reference. Consider the following diagram O sKL F K ⊇ ı K ( O sKL ) e R ✲ ı K ✛ ı L ( O sKL ) ⊆ F L ı L ✲ (94)where e R : ı K ( O sKL ) → ı L ( O sKL ) is the map defined by e R (cid:0) ı K ( f ) (cid:1) = ı L ( f )for any f ∈ O sKL . Clearly, the diagram (94) is commutative and R is therestriction of e R to ı K ( O sIJKL ). It suffices to show that e R is a diffeomorphism.Note that O sKL = O s ( U K ∩ U L , V K ∩ V L )where U K ∩ U L = ( U k ∩ U ℓ ) k ∈ K,ℓ ∈ L and V K ∩ V L = ( V k ∩ V ℓ ) k ∈ K,ℓ ∈ L . On U K ∩ U L and V K ∩ V L one can introduce two families of coordinate charts.For any given k ∈ K and ℓ ∈ L define α kℓ := χ k | U k ∩U ℓ : U k ∩ U ℓ → χ k ( U k ∩ U ℓ ) ⊆ U k ⊆ R n ,β kℓ := η k | V k ∩V ℓ : V k ∩ V ℓ → η k ( V k ∩ V ℓ ) ⊆ V k ⊆ R d . γ kℓ := χ ℓ | U k ∩U ℓ : U k ∩ U ℓ → χ ℓ ( U k ∩ U ℓ ) ⊆ U ℓ ⊆ R n ,δ kℓ := η ℓ | V k ∩V ℓ : V k ∩ V ℓ → η ℓ ( V k ∩ V ℓ ) ⊆ V ℓ ⊆ R d . These two choices of coordinate charts lead to the two embeddings ı and ı ı : O s ( U K ∩ U L , V K ∩ V L ) → ⊕ k ∈ K,ℓ ∈ L H s ( χ k ( U k ∩ U ℓ ) , R d ) (95) f ( f kℓ ) k ∈ K,ℓ ∈ L where f kℓ := β kℓ ◦ f ◦ α − kℓ : χ k ( U k ∩ U ℓ ) → η k ( V k ∩ V ℓ ) ⊆ R d and ı : O s ( U K ∩ U L , V K ∩ V L ) → ⊕ k ∈ K,ℓ ∈ L H s ( χ ℓ ( U k ∩ U ℓ ) , R d ) (96) f ( g kℓ ) k ∈ K,ℓ ∈ L where g kℓ := δ kℓ ◦ f ◦ γ − kℓ : χ ℓ ( U k ∩ U ℓ ) → η ℓ ( V k ∩ V ℓ ) ⊆ R d . Let G K := ⊕ k ∈ K,ℓ ∈ L H s ( χ k ( U k ∩ U ℓ ) , R d ) , G L := ⊕ k ∈ K,ℓ ∈ L H s ( χ ℓ ( U k ∩ U ℓ ) , R d )and consider the following diagram F K ⊇ ı K ( O sKL ) ✛ ı K O sKL ı L ✲ ı L ( O sKL ) ⊆ F L G K ⊇ ı ( O sKL ) R K ❄ T ✲ ı ✛ ı ( O sKL ) ⊆ G L R L ❄ ı ✲ (97)where ı K is the restriction of ı U K , V K : O s ( U K , V K ) → F K O sKL ⊆ O s ( U K , V K ), ı L is defined similarly, and the maps R K , R L , and T are defined by R K : F K → G K , ( f k ) k ∈ K ( f kℓ ) k ∈ K,ℓ ∈ L , f kℓ := f k | χ k ( U k ∩U ℓ ) ,R L : F L → G L , ( f ℓ ) ℓ ∈ L ( g kℓ ) k ∈ K,ℓ ∈ L , g kℓ := f ℓ | χ ℓ ( U k ∩U ℓ ) ,T : G K → G L , ( f kℓ ) k ∈ K,ℓ ∈ L ( g kℓ ) k ∈ K,ℓ ∈ L , with g kℓ := (cid:0) η ℓ ◦ η − k (cid:1) ◦ f kℓ ◦ (cid:0) χ k ◦ χ − ℓ (cid:1) . Note that the diagram (97) commutes. The arguments used to prove that P I in (91) is a diffeomorphism show that R K and R L are diffeomorphisms.We claim that T is a diffeomorphism. First note that T is bijective and itsinverse T − is given by T − : G L → G K , ( g kℓ ) k ∈ K,ℓ ∈ L ( f kℓ ) k ∈ K,ℓ ∈ L with f kℓ = (cid:0) η k ◦ η − ℓ (cid:1) ◦ g kℓ ◦ (cid:0) χ ℓ ◦ χ − k (cid:1)(cid:12)(cid:12) χ k ( U k ∩U ℓ ) . In view of the boundedness of the extension operator of Proposition 2.14( ii )the smoothness of T and T − then follows from Corollary 2.3, Proposition2.20. and Lemma 2.21. Comparing the diagrams (94) and (97) we concludethat e R = R K ◦ T ◦ R − L . Hence e R is a diffeomorphism. Proof of Proposition 3.7.
The claim that the C ∞ -differentiable structure on H s ( M, N ), introduced by Ebin-Marsden and the one introduced in this papercoincide follows from Corollary 4.2 and Proposition 3.5.As a consequence of Proposition 3.7 we obtain the following corollary.
Corollary 4.3.
The C ∞ -differentiable structure on H s ( M, N ) introduced in[14], is independent of the choice of the Riemannian metric on N . A Appendix
In this appendix we prove Lemma 3.8. First we need to establish an auxiliaryresult. Throughout this appendix, we will use the notation introduced in68ection 3. For bounded open subsets
U, W ⊆ R n with C ∞ -boundaries and s > n/ D sU,W the following subset of D s ( U, R n ), D sU,W := (cid:8) ϕ ∈ D s ( U, R n ) (cid:12)(cid:12) W ⊆ ϕ ( U ) (cid:9) . Arguing as in Lemma 2.18 one can prove that D sU,W is an open subset of D s ( U, R n ). Moreover, following the arguments of the proof of Lemma 2.8one gets Lemma A.1.
Let
U, W , and s be as above. Then, for any ϕ ∈ D sU,W , ϕ − (cid:12)(cid:12) W ∈ D s ( W, R n ) and the map D sU,W → D s ( W, R n ) , ϕ ϕ − (cid:12)(cid:12) W is continuous.Proof of Lemma 3.8. Let ϕ be an arbitrary element in D s ( M ). To see thatits inverse ϕ − is again in D s ( M ), it suffices to verify that when expressed inlocal coordinates, the map ϕ − is of Sobolev class H s . To be more precise,let χ : U → U ⊆ R n and η : V → V ⊆ R n be coordinate charts so that U, V are open, bounded subsets of R n with C ∞ -boundaries and ϕ ( U ) ⋐ V . By the construction of the fine cover in Lemma3.1 we can assume that ( U , V ) is a part of a fine cover ( U I , V I ) with respect to ϕ ∈ D s ( M ). Then, by Lemma 3.2, ψ := η ◦ ϕ ◦ χ − is in H s ( U, R n ). Choose W ⋐ ϕ ( U ) so that W := η ( W ) is an open bounded subset of R n with C ∞ -boundary. By Lemma A.1, it follows that ψ − (cid:12)(cid:12) W : W → R n is in D s ( W, R n ).As the chart U , V as well as W were chosen arbitrarily, we conclude that ϕ − is in D s ( M ). By the construction of the fine cover in Lemma 3.1 we canchoose a fine cover ( U I , V I ) with respect to ϕ ∈ D s ( M ) and W I ⋐ ϕ ( U I ) suchthat ( W I , U I ) is a fine cover with respect to ϕ − ∈ D s ( M ). Then, LemmaA.1 implies that the map D s ( M ) → D s ( M ), ϕ ϕ − is continuous. B Appendix
In this appendix we discuss the extension of Theorem 1.1 and Theorem 1.2to the case where s is a real number with s > n/ s ∈ R ≥ , denote by H s ( R n , R ) the Hilbert space H s ( R n , R ) := (cid:8) f ∈ L ( R n , R ) (cid:12)(cid:12) (1 + | ξ | ) s/ ˆ f ( ξ ) ∈ L ( R n , R ) (cid:9) with inner product h f, g i ∼ s = Z R n ˆ f ( ξ )ˆ g ( ξ )(1 + | ξ | ) s dξ and induced norm k f k ∼ s := ( h f, f i ∼ s ) / . By (4), the norms k f k ∼ s and k f k s are equivalent for any integer s ≥
0. Inthe sequel, by a slight abuse of notation, we will write k f k s instead of k f k ∼ s and h· , ·i s instead of h· , ·i ∼ s for any s ∈ R ≥ . In a straightforward way oneproves the following lemma. Lemma B.1.
For any f ∈ L ( R n , R ) and s ∈ R ≥ , f ∈ H s ( R n , R ) iff forany ≤ i ≤ n , the distributional derivate ∂ x i f is in H s − ( R n , R ) . Moreover k f k + P ni =1 k ∂ x i f k s − is a norm on H s ( R n , R ) which is equivalent to k f k s . For s ∈ R > \ N , elements in H s ( R n , R ) can be conveniently characterizedas follows – see e.g. [2, Theorem 7.48]. Lemma B.2.
Let s ∈ R > \ N and f ∈ L ( R n , R ) . Then f ∈ H s ( R n , R ) iff f ∈ H ⌊ s ⌋ ( R n , R ) and [ ∂ α f ] λ < ∞ for any multi-index α = ( α , . . . , α n ) with | α | = ⌊ s ⌋ where λ = s − ⌊ s ⌋ and where [ ∂ α f ] λ denotes the L -norm of thefunction R n × R n → R , ( x, y )
7→ | ∂ α f ( x ) − ∂ α f ( y ) || x − y | λ + n/ . Moreover p hh f, f ii s is a norm on H s ( R n , R ) , equivalent to k · k s , where hh· , ·ii s is the inner product hh f, g ii s = h f, g i ⌊ s ⌋ + X α ∈ Z n ≥ | α | = ⌊ s ⌋ Z R n Z R n (cid:0) ∂ α f ( x ) − ∂ α f ( y ) (cid:1)(cid:0) ∂ α g ( x ) − ∂ α g ( y ) (cid:1) | x − y | n +2 λ dxdy. roof. We argue by induction with respect to s . In view of Lemma B.1, itsuffices to prove the claimed statement in the case 0 < s <
1. Then λ = s and we have Z R n Z R n | f ( x ) − f ( y ) | | x − y | n +2 s dxdy = Z R n Z R n | f ( x + z ) − f ( x ) | | z | n +2 s dxdz = Z R n | z | n +2 s (cid:18)Z R n | f ( x + z ) − f ( x ) | dx (cid:19) dz. By Plancherel’s theorem, Z R n | f ( x + z ) − f ( x ) | dx = Z R n | \ f ( · + z )( ξ ) − ˆ f ( ξ ) | dξ = Z R n | e iz · ξ − | | ˆ f ( ξ ) | dξ. Therefore Z R n Z R n | f ( x ) − f ( y ) | | x − y | n +2 s dxdy = Z R n | ˆ f ( ξ ) | (cid:18)Z R n | e iz · ξ − | | z | n +2 s dz (cid:19) dξ = Z R n | ξ | s | ˆ f ( ξ ) | (cid:18)Z R n | e iz · ξ − | | ξ | s | z | n +2 s dz (cid:19) dξ. Let U ∈ SO ( n ) such that U ( ξ ) = | ξ | e where e = (1 , , . . . , ∈ R n . For ξ = 0 introduce the new variable y defined by z = | ξ | U − ( y ). With thischange of variable, the inner integral becomes, Z R n | e iz · ξ − | | ξ | s | z | n +2 s dz = Z R n | e iy − | | y | n +2 s dy < ∞ . Note that the latter integral converges and equals a positive constant thatis independent of ξ . Hence we conclude that for any f ∈ L ( R n , R ) one has k f k s < ∞ iff Z R n Z R n | f ( x ) − f ( y ) | | x − y | n +2 s dxdy < ∞ . The statement on the norms is easily verified.The following result extends part ( ii ) of Lemma 2.4.71 emma B.3. Let ϕ ∈ Diff ( R n ) with dϕ and dϕ − bounded on all of R n .Then for any < s ′ < , the right translation by ϕ , f R ϕ ( f ) = f ◦ ϕ is abounded linear operator on H s ′ ( R n , R ) .Proof. In view of statement ( i ) of Lemma 2.4, it remains to show that[ R ϕ f ] s ′ < ∞ . By a change of variables one gets[ f ◦ ϕ ] s ′ = Z R n Z R n | f (cid:0) ϕ ( x ) (cid:1) − f (cid:0) ϕ ( y ) (cid:1) | | x − y | n +2 s ′ dxdy ≤ M Z R n Z R n | f ( x ) − f ( y ) | | ϕ − ( x ) − ϕ − ( y ) | n +2 s ′ dxdy where M := inf x ∈ R n (det d x ϕ ). As dϕ is bounded on R n , one has for any x, y ∈ R n | x − y | = | ϕ (cid:0) ϕ − ( x ) (cid:1) − ϕ (cid:0) ϕ − ( y ) (cid:1) | ≤ L | ϕ − ( x ) − ϕ − ( y ) | where L := sup x ∈ R n | d x ϕ | < ∞ . Hence[ f ◦ ϕ ] s ′ ≤ M − L n/ s ′ [ f ] s ′ ∀ f ∈ H s ′ ( R n , R ) . (98)Hence f ◦ ϕ ∈ H s ′ ( R n , R ) and it follows that R ϕ is a bounded linear operatoron H s ′ ( R n , R ).Next we extend Lemma 2.5 to the case where s and s ′ are real. Using thenotation introduced in Section 2, one has Lemma B.4.
Let s, s ′ be real with s > n/ and ≤ s ′ ≤ s . Then for any ε > and K > there exists a constant C ≡ C ( ε, K ; s, s ′ ) > so that for any f ∈ H s ′ ( R n , R ) and g ∈ U sε with k g k s < K one has f / (1 + g ) ∈ H s ′ ( R n , R ) and k f / (1 + g ) k s ′ ≤ C k f k s ′ . (99) Moreover, the map H s ′ ( R n , R ) × U s → H s ′ ( R n , R ) , ( f, g ) f / (1 + g ) (100) is continuous. roof. In view of Lemma 2.5 and Remark 2.5, the claimed statement holdsfor real s with s > n/ s ′ satisfying 0 ≤ s ′ ≤ s . Arguing byinduction we will prove the first statement of the Lemma. Let us first showthat (99) holds for any 0 < s ′ < s ′ ≤ s . Take an arbitrary g ∈ U sε , ε > ∀ f ∈ H s ′ ( R n , R ), f / (1 + g ) ∈ L ( R n , R ) and k f / (1 + g ) k ≤ ε k f k . (101)According to Lemma B.2 it remains to show that [ f / (1 + g )] s ′ < ∞ . Write f ( x )1 + g ( x ) − f ( y )1 + g ( y ) = (cid:0) f ( x ) − f ( y ) (cid:1) g ( x ) + g ( y ) (cid:0) g ( x ) (cid:1)(cid:0) g ( y ) (cid:1) − f ( x ) g ( x ) − f ( y ) g ( y ) (cid:0) g ( x ) (cid:1)(cid:0) g ( y ) (cid:1) and note that by Remark 2.4, f · g ∈ H s ′ ( R n , R ) andsup x,y ∈ R n | g ( x ) | + | g ( y ) | (cid:0) g ( x ) (cid:1)(cid:0) g ( y ) (cid:1) ≤ C for some constant C >
0. This together with Lemma 2.3 and Remark 2.4implies (cid:20) f g (cid:21) s ′ ≤ C [ f ] s ′ + 2 C [ f g ] s ′ ≤ C ( k f k s ′ + k gf k s ′ ) ≤ C k f k s ′ < ∞ (102)where C > †† Combining (101) with (102) we see that (99) holds for any0 < s ′ < s ′ ≤ s . This completes the proof of the Lemma when s < s > ≤ s ′ ≤ k , with 1 ≤ k < s , k ∈ Z ≥ . We will show that then (99) holds for k < s ′ < k + 1, s ′ ≤ s . Takean arbitrary f ∈ H s ′ ( R n , R ). As H s ′ ( R n , R ) ⊆ H k ( R n , R ) we get from theproof of Lemma 2.5, ∂ x i (cid:18) f g (cid:19) = ∂ x i f g − ∂ xi ( fg )1+ g − g · ∂ xi f g g . (103) †† The positive constants C and C depend on the s -norm of g .
73y Remark 2.4, g · ∂ x i f and ∂ x i ( f g ) are in H s ′ − ( R n , R ). This together withthe induction hypothesis and (103) implies that f / (1 + g ) ∈ H s ′ ( R n , R ).Inequality (99) follows immediately from the induction hypothesis and (103).In order to prove that (100) is continuous we argue as follows. Take anarbitrary g ∈ U sε , ε >
0. In view of Proposition 2.2, Remark 2.2, and (99),there exists κ > δg ∈ B sκ ‡‡ , k δg/ (1 + g ) k s < k δg k C < ε/ . (104)Consider the map, H s ′ ( R n , R ) × B sκ → H s ′ ( R n , R ),( δf, δg ) δf g + δg ) . (105)In view of (104) and the first statement of the Lemma, the map (105) iswell-defined. We have δf g + δg = δf g ·
11 + δg g = δf g + δf g · ∞ X j =0 ( − j (cid:16) δg g (cid:17) j = δf g + δf g · S ( δg ) (106)where S : B sκ → H s ( R n , R ) is an analytic function. Finally, the continuityof (105) follows from (106), (99), Lemma 2.3 and Remark 2.4.The following lemma extends Lemma 2.7 to the case where s and s ′ arereal numbers instead of integers. For any real number s > n/ D s ( R n ) := (cid:8) ϕ ∈ Diff ( R n ) (cid:12)(cid:12) ϕ − id ∈ H s ( R n ) (cid:9) . Lemma B.5.
Let s, s ′ be real numbers with s > n/ and ≤ s ′ ≤ s .Then the composition µ s ′ : H s ′ ( R n , R ) × D s ( R n ) → H s ′ ( R n , R ) , ( f, ϕ ) f ◦ ϕ is continuous. ‡‡ B sκ is the open ball of radius κ centered at zero in H s ( R n , R ). roof. We argue by induction on intervals of values of s ′ , k ≤ s ′ < k + 1.Let us begin with the case where 0 ≤ s ′ <
1. Note that the case where s isreal and s ′ integer is already dealt with in Lemma 2.7 – see Remark 2.6. Inparticular, L ( R n , R ) × D s ( R n ) → L ( R n , R ) , ( f, ϕ ) f ◦ ϕ is continuous. Next assume that 0 < s ′ <
1. Then for any f, f • ∈ H s ′ ( R n , R )and ϕ, ϕ • ∈ D s ( R n ), the expression [ f ◦ ϕ − f • ◦ ϕ • ] s ′ is bounded by Z R n Z R n | f (cid:0) ϕ ( x ) (cid:1) − f • (cid:0) ϕ ( x ) (cid:1) − f (cid:0) ϕ ( y ) (cid:1) + f • (cid:0) ϕ ( y ) (cid:1) | | x − y | n +2 s ′ dxdy + Z R n Z R n | f • (cid:0) ϕ ( x ) (cid:1) − f • (cid:0) ϕ • ( x ) (cid:1) − f • (cid:0) ϕ ( y ) (cid:1) + f • (cid:0) ϕ • ( y ) (cid:1) | | x − y | n +2 s ′ dxdy. (107)By (98), the first integral in (107) can be estimated by C [ f − f • ] s ′ where C > ϕ in D s ( R n ). The second integralin (107) we write as Z R n Z R n (cid:12)(cid:12)(cid:12)(cid:16) f • (cid:0) ϕ ( x ) (cid:1) − f • (cid:0) ϕ ( y ) (cid:1)(cid:17) − (cid:16) f • (cid:0) ϕ • ( x ) (cid:1) − f • (cid:0) ϕ • ( y ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) | x − y | n +2 s ′ dxdy. By Lemma B.2, F ( x, y ) := f • ( x ) − f • ( y ) | x − y | n/ s ′ is in L ( R n × R n , R ). Hence again by Remark 2.6, F (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) → F (cid:0) ϕ • ( x ) , ϕ • ( y ) (cid:1) in L ( R n × R n , R ) . In view of the estimate (cid:12)(cid:12)(cid:12) ϕ ( y ) − ϕ ( x ) | y − x | (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z ( d x +( y − x ) t ϕ ) (cid:16) y − x | y − x | (cid:17)(cid:12)(cid:12)(cid:12) ≤ k dϕ k C and the continuity of D s ( R n ) → C ( R n ) , ϕ ϕ − id (Remark 2.2) one seesthat ϕ ( x ) − ϕ ( y ) | x − y | → ϕ • ( x ) − ϕ • ( y ) | x − y | in L ∞ ( R n × R n , R ) . f • (cid:0) ϕ ( x ) (cid:1) − f • (cid:0) ϕ ( y ) (cid:1) | x − y | n/ s ′ = F (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) | ϕ ( x ) − ϕ ( y ) | n/ s ′ | x − y | n/ s ′ it then follows that as ϕ → ϕ • in D s ( R n ) f • (cid:0) ϕ ( x ) (cid:1) − f • (cid:0) ϕ ( y ) (cid:1) | x − y | n/ s ′ → f • (cid:0) ϕ • ( x ) (cid:1) − f • (cid:0) ϕ • ( y ) (cid:1) | x − y | n/ s ′ in L ( R n × R n , R ) . Now let us prove the induction step. Assume that the continuity of thecomposition µ s ′ has been established for any s ′ with 0 ≤ s ′ ≤ k where k ∈ Z ≥ satisfies k < s . Consider s ′ ∈ R with k ≤ s ′ ≤ s (if s < k + 1) resp. k ≤ s ′ < k + 1 (if s ≥ k + 1). By Lemma 2.4( ii ), d ( f ◦ ϕ ) = df ◦ ϕ · dϕ. In view of Lemma B.1, df ∈ H s ′ − ( R n , R n ), hence by the induction hypoth-esis, if f → f • in H s ′ ( R n , R ) and ϕ → ϕ • in D s ( R n ), one has df ◦ ϕ → df • ◦ ϕ • in H s ′ − ( R n , R n ) . As dϕ ∈ H s ′ − ( R n , R n × n ) and s − > n/ df ◦ ϕ · dϕ → df • ◦ ϕ • · dϕ • in H s ′ − ( R n , R n )and Lemma B.1 implies that f ◦ ϕ → f • ◦ ϕ • in H s ′ ( R n , R ). This establishesthe continuity of µ s ′ and proves the induction step.Next we extend Lemma 2.8 to the case where s is fractional. Lemma B.6.
Let s be real with s > n/ . Then for any ϕ ∈ D s ( R n ) , itsinverse ϕ − is again in D s ( R n ) and inv : D s ( R n ) → D s ( R n ) , ϕ ϕ − is continuous. roof. Let ϕ ∈ D s ( R n ). Then ϕ is in Diff ( R n ) and so is its inverse ϕ − . Weclaim that ϕ − is in D s ( R n ). It follows from the proof of Lemma 2.8, togetherwith Remark 2.4 and Lemma B.4 that for any α ∈ Z n ≥ with 0 ≤ | α | ≤ s , ∂ α ( ϕ − − id ) is of the form ∂ α ( ϕ − − id ) = F ( α ) ◦ ϕ − where F ( α ) ∈ H s −| α | ( R n ). In addition, by Remark 2.4 and Lemma B.4, themap D s ( R n ) → H s −| α | ( R n ), ϕ F ( α ) is continuous. It then follows that Z R n | ∂ α ( ϕ − − id ) | dx = Z R n | F ( α ) | det( d y ϕ ) dy < ∞ . Moreover, in case | α | = ⌊ s ⌋ and s / ∈ N one has for 0 < λ := s − ⌊ s ⌋ < h F ( α ) ◦ ϕ − i λ = Z R n Z R n | F ( α ) (cid:0) ϕ − ( x ) (cid:1) − F ( α ) (cid:0) ϕ − ( y ) (cid:1) | | x − y | n +2 λ dxdy ≤ M Z R n Z R n | F ( α ) ( x ′ ) − F ( α ) ( y ′ ) | | x ′ − y ′ | n +2 λ | x ′ − y ′ | n +2 λ | ϕ ( x ′ ) − ϕ ( y ′ ) | n +2 λ dx ′ dy ′ where M := sup x ∈ R n (det d x ϕ ). As | ϕ − ( x ) − ϕ − ( y ) | ≤ L | x − y | for any x, y ∈ R n with L := sup z ∈ R n | d z ϕ − | < ∞ it follows that | x ′ − y ′ || ϕ ( x ′ ) − ϕ ( y ′ ) | ≤ L ∀ x ′ , y ′ ∈ R n , x ′ = y ′ . Altogether one has, for any α ∈ Z n ≥ with | α | = ⌊ s ⌋ , h F ( α ) ◦ ϕ − i λ ≤ M L n +2 λ h F ( α ) i λ . (108)By Lemma B.2 it then follows that ϕ − − id ∈ H s ( R n ). In addition, theestimates obtained show that the map D s ( R n ) → H s ( R n ), ϕ ϕ − − id is locally bounded. It remains to show that this map is continuous. By theproof of Lemma 2.8, the map D s ( R n ) → L ( R n ), ϕ ϕ − − id is continuous.Using that F ( α ) : D s ( R n ) → H s −| α | ( R n ) is continuous for any α ∈ Z n ≥ with77 α | ≤ s one shows in a similar way as in Lemma 2.8 that D s ( R n ) → L ( R n ), ϕ ∂ α ( ϕ − − id ) = F ( α ) ◦ ϕ − is continuous. Now consider the case where α ∈ Z n ≥ satisfies | α | = ⌊ s ⌋ and λ := s − ⌊ s ⌋ >
0. For any ϕ • ∈ D s ( R n )consider (cid:2) ∂ α ( ϕ − − ϕ − • ) (cid:3) λ = h F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − • i λ ≤ h F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − i λ + h F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • i λ . It follows from (108) that h F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − i λ ≤ M L λ + n/ h F ( α ) − F ( α ) • i λ . As F ( α ) : D s ( R n ) → H λ ( R n ) is continuous, h F ( α ) ◦ ϕ − − F ( α ) • ◦ ϕ − i λ → ϕ → ϕ • in D s ( R n ). Finally consider the term h F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • i λ .Arguing as in the proof of Lemma 2.8, we approximate ϕ • by ˜ ϕ ∈ D s ( R n )with ˜ ϕ − id ∈ C ∞ c ( R n , R n ). Then h F ( α ) • ◦ ϕ − − F ( α ) • ◦ ϕ − • i λ ≤ h F ( α ) • ◦ ϕ − − ˜ F ( α ) ◦ ϕ − i λ ++ h ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • i λ + h ˜ F ( α ) ◦ ϕ − • − F ( α ) • ◦ ϕ − • i λ where ˜ F ( α ) = F ( α ) (cid:12)(cid:12) ˜ ϕ . For ϕ near ϕ • one has as above, h F ( α ) • ◦ ϕ − − ˜ F ( α ) ◦ ϕ − i λ ≤ M L λ + n/ h F ( α ) • − ˜ F ( α ) i λ . Similarly, the expression h ˜ F ( α ) ◦ ϕ − • − F ( α ) • ◦ ϕ − • i λ can be bounded in termsof h ˜ F ( α ) − F ( α ) • i λ . To estimate the remaining term it suffices to show that,as ϕ → ϕ • in D s ( R n ), k ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • k → . First we show that k ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • k → ϕ → ϕ • in D s ( R n ).Indeed, arguing as in the proof of Lemma 2.8, we note that ˜ F ( α ) is Lipschitzcontinuous, i.e. | ˜ F ( α ) ( x ) − ˜ F ( α ) ( y ) | ≤ L | x − y | ∀ x, y ∈ R n L >
0. Then Z R n | ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • | dx ≤ L Z R n | ϕ − − ϕ − • | dx and therefore k ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • k → ϕ → ϕ • in D s ( R n ) . It remains to show that k d ( ˜ F ( α ) ◦ ϕ − ) − d ( ˜ F ( α ) ◦ ϕ − • ) k → ϕ → ϕ • in D s ( R n ) . By the chain rule we have d ( F ( α ) ◦ ϕ − ) = dF ( α ) ◦ ϕ − · dϕ − . Hence k d ( ˜ F ( α ) ◦ ϕ − ) − d ( ˜ F ( α ) ◦ ϕ − • ) k ≤ k d ˜ F ( α ) ◦ ϕ − − d ˜ F ( α ) ϕ − • k k dϕ − k L ∞ + k d ˜ F ( α ) ◦ ϕ − • k k dϕ − − dϕ − • k L ∞ . Arguing as above one has, as ϕ → ϕ • in D s ( R n ), k d ˜ F ( α ) ◦ ϕ − − d ˜ F ( α ) ◦ ϕ − • k → k dϕ − − dϕ − • k L ∞ → . Altogether we thus have shown that k ˜ F ( α ) ◦ ϕ − − ˜ F ( α ) ◦ ϕ − • k → ϕ → ϕ • in D s ( R n ) . This finishes the proof of the claimed statement that ϕ → ϕ − is continuouson D s ( R n ). Proposition B.7.
For any real number s > n/ , ( D s , ◦ ) is a topologicalgroup.Proof. The claimed statement follows from Lemma B.5 and Lemma B.6.79ow we have established all ingredients to show the following extensionof Theorem 1.1.
Theorem B.1.
For any r ∈ Z ≥ and any real number s with s > n/ µ : H s + r ( R n , R d ) × D s ( R n ) → H s ( R n , R d ) , ( u, ϕ ) u ◦ ϕ and inv : D s + r ( R n ) → D s ( R n ) , ϕ ϕ − are C r -maps.Proof. Using the results established above in this appendix, the proof ofTheorem 1.1, given in Subsection 2.3, extends in a straightforward way tothe case where s is real.Finally we want to extend the results of Subsection 2.4, Section 3, andSection 4 to Sobolev spaces of fractional exponents. Definition B.1.
Let U be a bounded open set in R n with Lipschitz boundary.Then f ∈ H s ( U, R ) if there exists ˜ f ∈ H s ( R n , R ) such that ˜ f (cid:12)(cid:12) U = f . Note that for our purposes it is enough to consider only the case when theboundary of U is a finite (possibly empty) union of transversally intersecting C ∞ -embedded hypersurfaces in R n (cf. Definition 3.2).As in the case where s is an integer, the spaces H s ( U, R ) and H s ( R n , R )are closely related. In view of [38], item (ii) of Proposition 2.14 holds.Note that H s ( R n , R ) = F s ( R n , R ) where F s is the corresponding Triebel-Lizorkin space. This allows us to define maps of class H s between manifoldsand extend the results in Subsection 3.1 to Sobolev spaces of fractional ex-ponents.The corresponding space of maps is denoted by H s ( M, N ). Similarly,one extends the definition of D s ( M ) for s fractional. Following the line ofarguments of Section 3 and Section 4 one then concludes that Theorem 1.2can be extended as follows Theorem B.2.
Let M be a closed oriented manifold of dimension n , N a C ∞ -manifold and s any real number satisfying s > n/ . Then for any r ∈ Z ≥ , i) µ : H s + r ( M, N ) × D s ( M ) → H s ( M, N ) , ( f, ϕ ) f ◦ ϕ (ii) inv : D s + r ( M ) → D s ( M ) , ϕ ϕ − are both C r -maps. Remark B.1.
Note that our construction can be used to prove analogousresults for maps between manifolds in Besov or Triebel-Lizorkin spaces.
References [1] R. Abraham and J. Robbin:
Transversal mappings and flows , W.A.Benjamin, Inc., New York, 1967[2] R. Adams:
Sobolev spaces , Academic Press, New York, 1975[3] S. Alinhac:
Paracomposition et operateurs paradifferentiels , Comm.Partial Differential Equations, (1986), no. 1 , 87-121[4] V. Arnold: Sur la g´eometrie differentielle des groupes de Lie de dimen-sion infinie et ses applications `a l’hydrodynamique des fluids parfaits ,Ann. Inst. Fourier, (1966), no. 1, 319-361[5] V. Arnold, B. Khesin: Topological methods in hydrodynamics , Springer,1998[6] J. Burgers:
A mathematical model illustrating the theory of turbulence ,Advances in Applied Mechanics, Academic Press, 1948, 71-199[7] R. Camassa, D. Holm:
An integrable shallow water equation with peakedsolitons , Phys. Rev. Lett, (1993), 1661-1664[8] M. Cantor: Global analysis over non-compact spaces , Thesis, U.C.Berkeley, 1973[9] A. Constantin, J. Escher:
Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation , Commun.Pure Appl. Math., (1998), 475-5048110] A. Constantin, T. Kappeler, B. Kolev, P. Topalov: On geodesic ex-ponential maps of the Virasoro group , Ann. Global Anal. Geom., (2007), no. 2, 155-180[11] A. Constantin, B. Kolev: Geodesic flow on the diffeomorphism groupof the circle , Comment. Math. Helv., (2003), 787-804[12] C. De Lellis, T. Kappeler, P. Topalov: Low regularity solutions ofthe periodic Camassa-Holm equation , Comm. Partial Differential Equa-tions, (2007), 87-126[13] D. Ebin: The manifold of Riemannian metrics , in Global Analysis,Proc. Sympos. Pure Math., , Amer. Math. Soc., eds. S.S. Chern andS. Smale, 11-40, and Bull. Amer. Math. Soc., (1968), 1001-1003[14] D. Ebin, J. Marsden: Groups of diffeomorphisms and the motion of anincompressible fluid , Ann. Math., (1970), 102-163[15] D. Ebin, J. Marsden, A. Fischer: Diffeomorphism groups, hydrodynam-ics and relativity and the notion of an incompressible fluid , Proc. 13thBiennal Seminar of Canad. Math. Congress(Montreal), 1972, 135-279[16] H. Eliasson:
Geometry of manifolds of maps , J. Differential Geometry, (1967), 169-194[17] A. Fokas, B. Fuchssteiner: Symplectic structures, their B¨acklund trans-formation and hereditary symmetries , Physica D, (1981), 47-66[18] R. Hamilton: The inverse function theorem of Nash and Moser , Bull.Amer. Math. Soc., (1982), 66-222[19] D. Holm, J. Marsden, T. Ratiu, S. Shkoller: The Euler-Poincar´e equa-tions and semidirect products with applications to continuum theories ,Adv. Math., (1998), 1-81[20] T. Kappeler, E. Loubet, P. Topalov:
Analyticity of Riemannian expo-nential maps on
Diff( T ), J. Lie Theory, (2007), 481-5038221] T. Kappeler, E. Loubet, P. Topalov: Riemannian exponential mapsof the diffeomorphism group of T , Asian J. Math., (2008), no. 3,391-420[22] B. Khesin, V. Ovsienko: The (super) KdV equation as an Euler equa-tion , Funct. Anal. Appl, (1987), 81-82[23] B. Khesin, R. Wendt: The geometry of infinite-dimensional groups ,Berlin, Springer 2009[24] A. Kriegl, P. Michor:
The convenient setting of global analysis , AMSSeries: Mathematical surveys and monographs, , 1997[25] S. Lang: Fundamentals of Differential Geometry , Springer, 1999[26] R. de la Llave, R. Obaya:
Regularity of the composition operator inspaces of H¨older functions , Discrete Contin. Dynam. Systems, (1999),no. 1, 157-184[27] J. Marsden, R. Abraham: Hamiltonian mechanics on Lie groups andhydrodynamics , Proc. Symp. Pure Math., AMS, (1970), 237-244[28] J. Marti: Introduction to Sobolev spaces and finite element solutions ofelliptic boundary value problems , Academic Press, 1986[29] H. McKean:
Breakdown of a shallow water equation , Asian J. Math., (1998), 867-874[30] P. Michor: Some geometric evolution equations arising as geodesicequations on groups of diffeomorphisms including the Hamiltonian ap-proach ∼ michor/listpubl.html.[31] J. Milnor: Remarks on infinite-dimensional Lie groups , Les Houches,Session XL, 1983, Elsevier Science Publishers B.V., 1984.[32] G. Misiolek:
A shallow water equation as a geodesic flow on the Bott-Virasoro group , J. Geom. Phys., (1998), 203-208[33] G. Misiolek: Classical solutions of the periodic Camassa-Holm equa-tion , Geom. Funct. Anal., (2002), no. 5, 1080-11048334] H. Omori: On the group of diffeomorphisms on a compact manifold , inGlobal Analysis, Proc. Sympos. Pure Math., , eds, S.S. Chern andS. Smale, AMS, 167-183[35] H. Omori: Infinite dimensional Lie groups , Translations of Mathemat-ical Monographs, AMS, , 1997[36] R. Palais:
Foundations of global non-linear analysis , Benjamin, N.Y.,1968[37] R. Palais:
Seminar on the Atiyah-Singer index theorem , Princeton,1965[38] V. Rychkov:
On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains , J. London Math.Soc., (1999), 237-257[39] M. Taylor: Tools for PDE , Math. Surveys and Monographs, AMS, ,2000Institut f¨ur Mathematik Department of MathematicsUniversit¨at Z¨urich Northeastern UniversityWinterthurerstrasse 190 Boston, MA 02115CH-8057 Z¨urich USASchwitzerland email: [email protected] email: [email protected] email:email: