On the Regularity of the Composition of Diffeomorphisms by H. Inci, P. Topalov, and T. Kappeler


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EMOIRS M of the American Mathematical Society Volume 226 • Number 1062 (third of 5 numbers) • November 2013 On the Regularity of the Composition of Diffeomorphisms H. Inci T. Kappeler P. Topalov ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society Providence, Rhode Island Library of Congress Cataloging-in-Publication Data Inci, H., 1982- author. On the regularity of the composition of diffeomorphisms / H. Inci, T. Kappeler, P. Topalov. pages cm – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; number 1062) ”November 2013, volume 226, number 1062 (third of 5 numbers).” Includes bibliographical references. ISBN 978-0-8218-8741-7 (alk. paper) 1. Diffeomorphisms. 2. Riemannian manifolds. I. Kappeler, Thomas, 1953- author. II. Topalov, P., 1968- author. III. Title. QA613.65.I53 2013 2013025511 516.36–dc23 DOI: http://dx.doi.org/10.1090/S0065-9266-2013-00676-4 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on the Abstract page of each article. This alphanumeric string of characters uniquely identifies each article and can be used for future cataloguing, searching, and electronic retrieval. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2013 subscription begins with volume 221 and consists of six mailings, each containing one or more numbers. Subscription prices are as follows: for paper delivery, US$795 list, US$636 institutional member; for electronic delivery, US$700 list, US$560 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. 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Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. c 2013 by the American Mathematical Society. All rights reserved.  Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. R , Zentralblatt MATH, Science Citation This publication is indexed in Mathematical Reviews  R , Science Citation IndexT M -Expanded, ISI Alerting ServicesSM , SciSearch  R , Research Index  R , CompuMath Citation Index  R , Current Contents  R /Physical, Chemical & Earth Alert  Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13 Contents Chapter 1. Introduction 1 Chapter 2. Groups of diffeomorphisms on Rn 5 Chapter 3. Diffeomorphisms of a closed manifold 31 Chapter 4. Differentiable structure of H s (M, N ) 39 Appendix A 49 Appendix B 51 Bibliography 59 iii Abstract For M a closed manifold or the Euclidean space Rn we present a detailed proof of regularity properties of the composition of H s -regular diffeomorphisms of M for s > 12 dim M + 1. Received by the editor September 7, 2011, and, in revised form, January 12, 2012. Article electronically published on March 28, 2013; S 0065-9266(2013)00676-4. 2010 Mathematics Subject Classification. Primary 58D17, 35Q31, 76N10. Key words and phrases. Group of diffeomorphisms, regularity of composition, Euler equation. The first author was supported in part by the Swiss National Science Foundation. The second author was supported in part by the Swiss National Science Foundation. The third author was supported in part by NSF DMS-0901443. c 2013 American Mathematical Society v CHAPTER 1 Introduction In this paper we are concerned with groups of diffeomorphisms on a smooth manifold M . Our interest in these groups stems from Arnold’s seminal paper [4] on hydrodynamics. He suggested that the Euler equation modeling a perfect fluid on a (oriented) Riemannian manifold M can be reformulated as the equation for geodesics on the group of volume (and orientation) preserving diffeomorphims of M . In this way properties of solutions of the Euler equation can be expressed in geometric terms – see [4]. In the sequel, Ebin and Marsden [14], [15] used this approach to great success to study the initial value problem for the Euler equation on a compact manifold, possibly with boundary. Later it was observed that other nonlinear evolution equations such 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, this approach has turned out to be very useful – see e.g. [12], [33]. In addition, following Arnold’s suggestions [4], numerous papers aim at relating the stability of the flows to the geometry of the groups of diffeomorphisms considered – see e.g. [5]. In various settings, the space of diffeomorphisms of a given manifold with prescribed regularity turns out to be a (infinite dimensional) topological group with the group operation given by the composition – see e.g. [15, p 155] for a quite detailed historical account. In order for such a group of diffeomorphisms to be a Lie group, the composition and the inverse map have to be C ∞ -smooth. A straightforward formal computation shows that the differential of the left translation Lψ : ϕ → ψ ◦ϕ of a diffeomorphism ϕ by a diffeomorphism ψ in direction h : M → T M can be formally computed to be (dϕ Lψ )(h)(x) = (dϕ(x) ψ)(h(x)), x ∈ M and hence involves a loss of derivative of ψ. As a consequence, for a space of 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 Banach manifold, but only of a Fr´echet manifold. It is well known that the calculus in Fr´echet manifolds is quite involved as the classical inverse function theorem does not hold, cf. e.g. [18], [24]. Various aspects of Fr´echet Lie groups of diffeomorphisms have been investigated – see e.g. [18], [31], [34], [35]. In particular, Riemann exponential maps have been studied in [10], [11], [20], [21]. However, in many situations, one has to consider diffeomorphisms of Sobolev type – see e.g. [12], [13], [14]. In this paper we are concerned with composition of maps in H s (M ) ≡ H s (M, M ). It seems to be unknown whether, in general, the composition of two maps in H s (M ) with s an integer satisfying s > n/2 is again in 1 2 1. INTRODUCTION H s (M ). In all known proofs one needs that one of the maps is a diffeomorphism or, alternatively, is C ∞ -smooth. First we consider the case where M is the Euclidean space Rn , n ≥ 1. Denote by Diff1+ (Rn ) the space of orientation preserving C 1 -diffeomorphisms of Rn , i.e. the space of bijective C 1 -maps ϕ : Rn → Rn so that det(dx ϕ) > 0 for any x ∈ Rn and ϕ−1 : Rn → Rn is a C 1 -map as well. For any integer s with s > n/2 + 1 introduce D s (Rn ) := {ϕ ∈ Diff1+ (Rn ) | ϕ − id ∈ H s (Rn )} where H s (Rn ) = H s (Rn , Rn ) and H s (Rn , Rd ) is the Hilbert space H s (Rn , Rd ) := {f = (f1 , . . . , fd ) | fi ∈ H s (Rn , R), i = 1, . . . , d} with H s -norm · s given by f s = d  fi 2s 1/2 i=1 and H s (Rn , R) is the Hilbert space of elements g ∈ L2 (Rn , R) with the property that the distributional derivatives ∂ α g, α ∈ Zn≥0 , up to order |α| ≤ s are in L2 (Rn , R). Its norm is given by    1/2 (1) g s = |∂ α g|2 dx . |α|≤s Rn  Here we used multi-index notation, i.e. α = (α1 , . . . , αn ) ∈ Zn≥0 , |α| = ni=1 αi , x = (x1 , . . . , xn ), and ∂ α ≡ ∂xα = ∂xα11 · · · ∂xαnn . As s > n/2 + 1 it follows from the Sobolev embedding theorem that Ds (Rn ) − id = {ϕ − id | ϕ ∈ Ds (Rn )} is an open subset of H s (Rn ) – see Corollary 2.1 below. In this way D s (Rn ) becomes a Hilbert manifold modeled on H s (Rn ). In Section 2 of this paper we present a detailed proof of the following Theorem 1.1. For any r ∈ Z≥0 and any integer s with s > n/2 + 1 (2) μ : H s+r (Rn , Rd ) × D s (Rn ) → H s (Rn , Rd ), (u, ϕ) → u ◦ ϕ and (3) inv : Ds+r (Rn ) → Ds (Rn ), ϕ → ϕ−1 are C r -maps. Remark 1.1. To the best of our knowledge there is no proof of Theorem 1.1 available in the literature. Besides being of interest in itself we will use Theorem 1.1 and its proof to show Theorem 1.2 stated below. Note that the case r = 0 was considered in [8]. Remark 1.2. The proof for the C r -regularity of the inverse map is valid in a much more general context: using that Ds (Rn ) is a topological group and that the composition Ds+r (Rn ) × D s (Rn ) → Ds (Rn ), (ψ, ϕ) → ψ ◦ ϕ r is C -smooth we apply the implicit function theorem to show that the inverse map D s+r (Rn ) → Ds (Rn ), ϕ → ϕ−1 H. INCI, T. KAPPELER, and P. TOPALOV 3 is a C r -map as well. Remark 1.3. By considering lifts to Rn of diffeomorphisms of Tn = Rn /Zn , the same arguments as in the proof of Theorem 1.1 can be used to show corresponding results for the group Ds (Tn ) of H s -regular diffeomorphisms on Tn . In Section 3 and Section 4 of this paper we discuss various classes of diffeomorphisms on a closed1 manifold M . For any integer s with s > n/2 the set H s (M ) of Sobolev maps is defined by using coordinate charts of M . More precisely, let M be a closed manifold of dimension n and N a C ∞ -manifold of 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 ⊆ Rn of M with x ∈ U, and a chart η : V → V ⊆ Rd of N with f (x) ∈ V, such that f (U) ⊆ V and η ◦ f ◦ χ−1 : U → V is an element in the Sobolev space H s (U, Rd ). Here H s (U, Rd ) – similarly defined as H s (Rn , Rd ) – is the Hilbert space of elements in L2 (U, Rd ) whose distributional derivatives up to order s are L2 -integrable. In Section 3 we introduce a C ∞ differentiable structure on the space H s (M, N ) in terms of a specific cover by open sets which is especially well suited for proving regularity properties of the composition of mappings as well as other applications presented in subsequent work. The main property of this cover of H s (M, N ) is that each of its open sets can be embedded into a finite cartesian product of 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 the one, introduced by Ebin and Marsden in [14], [15] and defined in terms of a Riemannian metric on N . In particular it follows that the standard differentiable structure does not depend on the choice of the metric. Now assume in addition that M is oriented. Then, for any linear isomorphism A : Tx M → Ty 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 > n2 + 1 define ⏐  D s (M ) := ϕ ∈ Diff1+ (M )⏐ϕ ∈ H s (M, M ) where Diff1+ (M ) denotes the set of all orientation preserving C 1 smooth diffeomorphisms of M . We will show that Ds (M ) is open in H s (M, M ) and hence is a C ∞ -Hilbert manifold. Elements in D s (M ) are referred to as orientation 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/2 + 1. Then for any r ∈ Z≥0 , (i) μ : H s+r (M, N ) × Ds (M ) → H s (M, N ), (f, ϕ) → f ◦ ϕ and (ii) inv : Ds+r (M ) → Ds (M ), ϕ → ϕ−1 are both C r -maps. 1 i.e., a compact C ∞ -manifold without boundary 4 1. INTRODUCTION 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 Theorem 1.2 can be found in [35], Proposition 3.3 of Chapter 3 and Theorem 2.1 of Chapter 6. Using the approach sketched above we present an elementary proof of Theorem 1.2. In particular, our approach allows us to apply elements of the proof of Theorem 1.1 to show statement (i). Remark 1.5. Actually Theorem 1.1 and Theorem 1.2 continue to hold if instead of s being an integer it is an arbitrary real number s > n/2 + 1. In order to keep the exposition as elementary as possible we prove Theorem 1.1 and Theorem 1.2 as stated in the main body of the paper and discuss the extension to the case where s > n/2 + 1 is real in Appendix B. We finish this introduction by pointing out results on compositions of maps in function spaces different from the ones considered here and some additional literature. In the paper [26], de la Llave and Obaya prove a version of Theorem 1.1 for H¨ older continuous maps between open sets of Banach spaces. Using the paradifferential calculus of Bony, Taylor [39] studies the continuity of the composition of maps of low regularity between open sets in Rn – see also [3]. Acknowledgment: We would like to thank Gerard Misiolek and Tudor Ratiu for very valuable feedback on an earlier version of this paper. CHAPTER 2 Groups of diffeomorphisms on Rn 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 as well (Proposition 2.13). To simplify notation we write D s ≡ Ds (Rn ) and H s ≡ H s (Rn ). Throughout this section, s denotes a nonnegative integer if not stated otherwise. 2.1. Sobolev spaces H s (Rn , R). In this subsection we discuss properties of the Sobolev spaces H s (Rn , R) needed later. First let us introduce some more notation. For any x, y ∈ Rn denote by x · y the Euclidean inner product, x · y =  n 1/2 . Recall that for k=1 xk yk , and by |x| the corresponding norm , |x| = (x · x) s n 2 s ∈ Z≥0 , H (R , R) consists of all L -integrable functions f : Rn → R with the property that the distributional derivatives ∂ α f, α ∈ Zn≥0 , up to order |α| ≤ s are L2 -integrable as well. Then H s (Rn , R), endowed with the norm (1), is a Hilbert space and for any multi-index α ∈ Zn≥0 with |α| ≤ s, the differential operator ∂ α is a bounded linear map, ∂ α : H s (Rn , R) → H s−|α| (Rn , R). Alternatively, one can characterize the spaces H s (Rn , R) via the Fourier transform. For any f ∈ L2 (Rn , R) ≡ H 0 (Rn , R), denote by fˆ its Fourier transform  fˆ(ξ) := (2π)−n/2 f (x)e−ix·ξ dx. Rn Then fˆ ∈ L2 (Rn , R) and fˆ = f , where f ≡ f 0 denotes the L2 -norm of f . The formula for the inverse Fourier transform reads  −n/2 fˆ(ξ)eix·ξ dξ. f (x) = (2π) Rn When expressed in terms of the Fourier transform fˆ of f , the operator ∂ α , α ∈ Zn≥0 is the multiplication operator fˆ → (iξ)α fˆ where ξ α = ξ1α1 · · · ξnαn and one can show f ∈ L2 (Rn , R) is an element in H s (Rn , R)   α ˆ 2 1/2 , iff (1 + |ξ|)s fˆ is in L2 (Rn , R) and the H s -norm of f , f s = |α|≤s ξ f satisfies (4) Cs−1 f s ≤ f ∼ s ≤ Cs f s for some constant Cs ≥ 1 where  ∼ (5) f s := Rn 1/2 (1 + |ξ| ) |fˆ(ξ)|2 dξ 2 s 5 . 2. GROUPS OF DIFFEOMORPHISMS ON Rn 6 In this way the Sobolev space H s (Rn , R) can be defined for s ∈ R≥0 arbitrary. See Appendix B for a study of these spaces. Using the Fourier transform one gets the following approximation property for functions in H s (Rn , R). Lemma 2.1. For any s in Z≥0 , the subspace Cc∞ (Rn , R) of C ∞ functions with compact support is dense in H s (Rn , R). Remark 2.1. The proof shows that Lemma 2.1 actually holds for any s real with s ≥ 0.  Proof. In a first step we show that C ∞ (Rn , R) ∩ H s (Rn , R) is dense in s H (Rn , R) for any integer s ≥ s. Let χ : R → R be a decreasing C ∞ function satisfying χ(t) = 1 ∀t ≤ 1 and χ(t) = 0 ∀t ≥ 2. For any f ∈ H s (Rn , R) and N ∈ Z≥1 define    ix·ξ ˆ fN (x) = (2π)−n/2 χ |ξ| dξ. N f (ξ)e Rn   ˆ The support of χ |ξ| N f (ξ) is contained in the ball {|ξ| ≤ 2N }. Hence fN (x) is in ∞ n s n C (R , R) ∩ H (R , R) for any s ≥ 0. In addition, by the Lebesgue convergence theorem,   2 ˆ 2 lim (1 + |ξ|)2s 1 − χ( |ξ| N ) |f (ξ)| dξ = 0. N →∞ Rn In view of (5), we have fN → f in H s (Rn , R). In a second step we show that  Cc∞ (Rn , R) is dense in C ∞ (Rn , R) ∩ H s (Rn , R) for any integer s ≥ 0. We get  the desired approximation of an arbitrary function f ∈ C ∞ (Rn , R) ∩ H s (Rn , R) by truncation in the x-space. For any N ∈ Z≥1 , let   f˜N (x) = χ |x| N · f (x). The support of f˜N is contained in the ball {|x| ≤ 2N } and thus f˜N ∈ Cc∞ (Rn , R).   s n To see that f − f˜N = (1 − χ |x| N )f converges to 0 in H (R , R), note that f (x) − f˜N (x) = 0 for any x ∈ Rn with |x| ≤ N . Furthermore it is easy to see that    ≤ Ms sup ∂ α 1 − χ |x| N x∈Rn |α|≤s for some constant Ms > 0 independent on N . Hence for any α ∈ Zn≥0 with |α| ≤ s , by Leibniz’ rule,     · f (x) ∂ α f − ∂ α f˜N = ∂ α 1 − χ |x| N     ≤ · ∂ γ f . ∂ β 1 − χ |x| N β+γ=α  |x|  = 0 for any |x| ≤ N we conclude that 1/2    |x|  γ β γ 2 ∂ 1 − χ N · ∂ f ≤ Ms |∂ f | dx Using that 1 − χ N |x|≥N H. INCI, T. KAPPELER, and P. TOPALOV 7  and hence, as f ∈ H s (Rn , R), lim ∂ α f − ∂ α f˜N = 0. N →∞  To state regularity properties of elements in H s (Rn , R), introduce for any r ∈ Z≥0 the space C r (Rn , R) of functions f : Rn → R with continuous partial derivatives up to order r. Denote by f C r the C r -norm of f , f C r = sup sup |∂ α f (x)|. x∈Rn |α|≤r By Cbr (Rn , R) we denote the Banach space of functions f in C r (Rn , R) with f C r < ∞ and by C0r (Rn , R) the subspace of functions f in C r (Rn , R) vanishing at infinity. r n These are functions in C (R , R) with the property that for any ε > 0 there exists M ≥ 1 so that sup sup |∂ α f (x)| < ε. |α|≤r |x|≥M Then C0r (Rn , R) ⊆ Cbr (Rn , R) ⊆ C r (Rn , R). By the triangle inequality one sees that C0r (Rn , R) is a closed subspace of Cbr (Rn , R). The following result is often referred to as Sobolev embedding theorem. Proposition 2.2. For any r ∈ Z≥0 and any integer s with s > n/2, the space H s+r (Rn , R) can be embedded into C0r (Rn , R). More precisely H s+r (Rn , R) ⊆ C0r (Rn , R) and there exists Ks,r ≥ 1 so that f C r ≤ Ks,r f s+r ∀f ∈ H s+r (Rn , R). Remark 2.2. The proof shows that Proposition 2.2 holds for any real s with s > n/2. Proof. As for s > n/2  Rn (1 + |ξ|2 )−s dξ < ∞ one gets by the Cauchy-Schwarz inequality for any f ∈ Cc∞ (Rn , R) and α ∈ Zn≥0 with |α| ≤ r  sup |∂ α f (x)| ≤ (2π)−n/2 |fˆ(ξ)| |ξ|α dξ n x∈Rn R  1/2 1/2  2 −s −n/2 2 (1 + |ξ| ) dξ (2π) |fˆ(ξ)| (1 + |ξ|2 )s+r dξ ≤ Rn Rn (6) ≤ Kr,s f r+s for some Kr,s > 0. By Lemma 2.1, an arbitrary element f ∈ H s+r (Rn , R) can be approximated by a sequence (fN )N ≥1 in Cc∞ (Rn , R). As C0r (Rn , R) is a Banach space, it then follows from (6) that (fN )N ≥1 is a Cauchy sequence in C0r (Rn , R) which converges to some function f˜ in C0r (Rn , R). In particular, for any compact subset K ⊆ Rn , fN |K → f˜ |K in L2 (K, R). This shows that f˜ ≡ f a.e. and hence f ∈ C r (Rn , R).  0 As an application of Proposition 2.2 one gets the following 2. GROUPS OF DIFFEOMORPHISMS ON Rn 8 Corollary 2.1. Let s be an integer with s > n/2 + 1. Then the following statements hold: (i) For any ϕ ∈ D s , the linear operators dx ϕ, dx ϕ−1 : Rn → Rn are bounded uniformly in x ∈ Rn∗ . In particular, inf det dx ϕ > 0. x∈Rn (ii) D s − id = {ϕ − id | ϕ ∈ Ds } is an open subset of H s . Hence the map Ds → H s , ϕ → ϕ − id provides a global chart for D s , giving Ds the structure of a C ∞ -Hilbert manifold modeled on H s . (iii) For any ϕ• ∈ D s such that inf det dx ϕ• > M > 0 x∈Rn there exist an open neighborhood Uϕ• of ϕ• in Ds and C > 0 such that for any ϕ in Uϕ• , inf det dx ϕ ≥ M and sup dx ϕ−1 < C.† x∈Rn x∈Rn Remark 2.3. The proof shows that Corollary 2.1 holds for any real s with s > n/2 + 1. Proof. (i) Introduce  C 1 (Rn ) := ϕ ∈ Diff1+ (Rn ) ϕ − id ∈ C01 (Rn ) where C01 (Rn ) ≡ C01 (Rn , Rn ) is the space of C 1 -maps f : Rn → Rn , vanishing together with their partial derivatives ∂xi f (1 ≤ i ≤ n) at infinity. By Proposition 2.2, H s continuously embeds into C01 (Rn ) for any integer s with s > n/2 + 1. In particular, D s → C 1 (Rn ). We now prove that for any ϕ ∈ C 1 (Rn ), dϕ and dϕ−1 are bounded on Rn . Clearly, for any ϕ ∈ C 1 (Rn ), dϕ is bounded on Rn . To show that dϕ−1 is bounded as well introduce for any f ∈ C01 (Rn ) the function F (f ) : Rn → R given by   F (f )(x) := det id + dx f − 1   = det (δi1 + ∂x1 fi )1≤i≤n , . . . , (δin + ∂xn fi )1≤i≤n − 1   where f (x) = f1 (x), . . . , fn (x) . As lim ∂xk fi (x) = 0 for any |x|→∞ 1 ≤ i, k ≤ n one has lim F (f )(x) = 0. (7) |x|→∞ It is then straightforward to verify that F is a continuous map, F : C01 (Rn ) → C00 (Rn , R). Choose an arbitrary element ϕ in C 1 (Rn ). Then (7) implies that M1 := infn det(dx ϕ) > 0. (8) x∈R ∗ Here † For dx ϕ−1 ≡ dx (ϕ−1 ) where ϕ ◦ ϕ−1 = id. a linear operator A : Rn → Rn , denote by |A| its operator norm, |A| := sup|x|=1 |Ax| H. INCI, T. KAPPELER, and P. TOPALOV 9  −1 As the differential of the inverse, dx ϕ−1 = dϕ−1 (x) ϕ , can be computed in terms of the cofactors of dϕ−1 (x) ϕ and 1/ det(dϕ−1 (x) ϕ) it follows from (8) that M2 := sup |dx ϕ−1 | < ∞ (9) x∈Rn where |A| denotes the operator norm of a linear operator A : Rn → Rn . (ii) Using again that D s continuously embeds into C 1 (Rn ) it remains to prove that C 1 (Rn ) − id is an open subset of C01 (Rn ). Note that the map F introduced above is continuous. Hence there exists a neighborhood Uϕ of fϕ := ϕ − id in C01 (Rn ) so that for any f ∈ Uϕ 1 (10) sup |dx f − dx fϕ | ≤ 2M2 x∈Rn and     M1 sup F f (x) − F fϕ (x) ≤ 2 x∈Rn (11) with M1 , M2 given as in (8)-(9). We claim that id + f ∈ C 1 (Rn ) for any f ∈ Uϕ . As ϕ ∈ C 1 (Rn ) was chosen arbitrarily it then would follow that C 1 (Rn ) − id is open in C01 (Rn ). First note that by (11), 0 < M1 /2 ≤ det(id + dx f ) ∀x ∈ Rn , ∀f ∈ Uϕ . Hence id + f is a local diffeomorphism on Rn and it remains to show that id + f is 1-1 and onto for any f in Uϕ . Choose f ∈ Uϕ arbitrarily. To see that id + f is 1-1 it suffices to prove that ψ := (id + f ) ◦ ϕ−1 is 1-1. Note that ψ = (id + fϕ + f − fϕ ) ◦ ϕ−1 = id + (f − fϕ ) ◦ ϕ−1 . For any x, y ∈ Rn , one therefore has ψ(x) − ψ(y) = x − y + (f − fϕ ) ◦ ϕ−1 (x) − (f − fϕ ) ◦ ϕ−1 (y). By (9) and (10) |(f − fϕ ) ◦ ϕ−1 (x) − (f − fϕ ) ◦ ϕ−1 (y)| ≤ ≤ 1 |ϕ−1 (x) − ϕ−1 (y)| 2M2 1 |x − y| 2 and thus   1 |(x − y) − ψ(x) − ψ(y) | ≤ |x − y| ∀x, y ∈ Rn 2 which implies that ψ is 1-1. To prove that id + f is onto we show that Rf := {x + f (x) | x ∈ Rn } is an open and closed subset of Rn . Being nonempty, one then has Rf = Rn . As id + f is a local diffeomorphism on Rn , Rf is open. To see that it is closed, consider a sequence (xk )k≥1 in Rn so that yk := xk + f (xk ), k ≥ 1, converges. Denote the limit by y. As lim|x|→∞ f (x) = 0, the sequence f (xk ) k≥1 is bounded, hence xk = yk − f (xk ) is a bounded sequence and therefore admits a convergent subsequence (xki )i≥1 whose limit is denoted by x. Then y = lim xki + lim f (xki ) i→∞ i→∞ = x + f (x) i.e. y ∈ Rf . This shows that Rf is closed and finishes the proof of item (ii). The proof of (iii) is straightforward and we leave it to the reader.  2. GROUPS OF DIFFEOMORPHISMS ON Rn 10 The following properties of multiplication of functions in Sobolev spaces are well known – see e.g. [2]. Lemma 2.3. Let s, s be integers with s > n/2 and 0 ≤ s ≤ s. Then there  exists K > 0 so that for any f ∈ H s (Rn , R), g ∈ H s (Rn , R), the product f · g is in  H s (Rn , R) and f · g s ≤ K f s g s . (12) In particular, H s (Rn , 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/2 and 0 ≤ s ≤ s. Proof. First we show that   (1 + |ξ|2 )s /2 f · g(ξ) = (1 + |ξ|2 )s /2 (fˆ ∗ gˆ)(ξ) ∈ L2 (Rn , R) where ∗ denotes the convolution (fˆ ∗ gˆ)(ξ) = By assumption, f˜(ξ) := fˆ(ξ) (1 + |ξ|2 )s/2  Rn fˆ(ξ − η)ˆ g (η)dη.  g˜(ξ) = gˆ(ξ) (1 + |ξ|2 )s /2 are in L2 (Rn , R). Note that in view of definition (5), f˜ = f ∼ g = g ∼ s and ˜ s . It is to show that  |f˜(ξ − η)| |˜ g (η)| 2 s /2 ξ → (1 + |ξ| ) dη 2 )s/2 (1 + |η|2 )s /2 (1 + |ξ − η| n R and is square-integrable. We split the domain of integration into two subsets {|η| > |ξ|/2} and {|η| ≤ |ξ|/2}. Then   |f˜(ξ − η)| |˜ g (η)| dη (1 + |ξ|2 )s /2 2 s/2 (1 + |η|2 )s /2 |η|>|ξ|/2 (1 + |ξ − η| )    |f˜(ξ − η)| |˜ g (η)| dη ≤ 2s (1 + |ξ|2 )s /2 2 s/2 (1 + |ξ|2 )s /2 |η|>|ξ|/2 (1 + |ξ − η| )  |f˜(ξ − η)| s |˜ g (η)|dη ≤ 2 2 s/2 Rn (1 + |ξ − η| )  g |(ξ). = 2s |fˆ| ∗ |˜ By Young’s inequality (see e.g. Theorem 1.2.1 in [28]), |fˆ| ∗ |˜ g g | ≤ fˆ L1 ˜ and  fˆ L1 ≤ This implies that 1/2  (1 + |ξ| ) |fˆ(ξ)|2 dξ 2 s Rn Rn 2 −s (1 + |ξ| ) 1/2 dξ . |fˆ| ∗ |˜ g | ≤ C f s g ∼ s . Similarly, one argues for the integral over the remaining subset. Note that on the domain {|η| ≤ |ξ|/2} one has 1 (1 + |ξ − η|2 ) ≥ (1 + |η|2 ) and (1 + |ξ − η|2 ) ≥ (1 + |ξ|2 ) 4 H. INCI, T. KAPPELER, and P. TOPALOV and hence   11  (1 + |ξ − η|2 )s/2 ≥ (1 + |η|2 )(s−s )/2 2−s (1 + |ξ|2 )s /2 Hence   (1 + |ξ|2 )s /2 ≤ 2s   |η|≤|ξ|/2 |η|≤|ξ|/2 |f˜(ξ − η)| |f˜(ξ − η)| |˜ g (η)| dη 2 s/2 (1 + |ξ − η| ) (1 + |η|2 )s /2 |˜ g (η)| dη (1 + |η|2 )s/2 and the L2 -norm of the latter convolution is bounded by f˜ ˜ g (η)/(1 + |η|2 )s/2 L1 ≤ C f s g ≤ C f s g s  with an appropriate constant C > 0. The following results concern the chain rule of differentiation for functions in H 1 (Rn , R). Lemma 2.4. Let ϕ ∈ Diff+1 (Rn ) with dϕ and dϕ−1 bounded on all of Rn . Then the following statements hold: (i) The right translation by ϕ, f → Rϕ (f ) := f ◦ ϕ is a bounded linear map on L2 (Rn , R). (ii) For any f ∈ H 1 (Rn , R), the composition f ◦ ϕ is again in H 1 (Rn , R) and the differential d(f ◦ ϕ) is given by the map df ◦ ϕ · dϕ ∈ L2 (Rn , Rn ), d(f ◦ ϕ) = (df ) ◦ ϕ · dϕ. (13) Proof. (i) For any f ∈ L2 (Rn , R), the composition f ◦ ϕ is measurable. As −1  M1 := infn det(dx ϕ) = sup det dx ϕ−1 >0 x∈R x∈Rn one obtains by the transformation formula       1 f ϕ(x) 2 dx ≤ f ϕ(x) 2 det(dx ϕ)dx M1 Rn Rn  1 |f (x)|2 dx = M1 Rn and thus f ◦ ϕ ∈ L2 (Rn , R) and the right translation Rϕ is a bounded linear map on L2 (Rn , R). (ii) For any f ∈ Cc∞ (Rn , R), f ◦ ϕ ∈ H 1 (Rn , R) and (13) holds by the standard chain rule of differentiation. Furthermore for any f ∈ H 1 (Rn , R), df ∈ L2 (Rn , Rn ) and hence by (i), (df ) ◦ ϕ ∈ L2 (Rn , Rn ). As dϕ is continuous and bounded by assumption it then follows that for any 1 ≤ i ≤ n n    ∂xk f ◦ ϕ · ∂xi ϕk ∈ L2 (Rn , R) k=1   where ϕk (x) is the k’th component of ϕ(x), ϕ(x) = ϕ1 (x), . . . , ϕn (x) . By Lemma 2.1, f can be approximated by (fN )N ≥1 in Cc∞ (Rn , R). By the chain rule, for any 1 ≤ i ≤ n, one has ∂xi (fN ◦ ϕ) = n  (∂xk fN ) ◦ ϕ · ∂xi ϕk k=1 2. GROUPS OF DIFFEOMORPHISMS ON Rn 12 and in view of (i), in L2 , n  (14) (∂xk fN ) ◦ ϕ · ∂xi ϕk −→ N →∞ k=1 n  (∂xk f ) ◦ ϕ · ∂xi ϕk . k=1 Moreover, for any test function g ∈ Cc∞ (Rn , R),  n     − ∂xi g · fN ◦ ϕdx = g · ∂xk fN ◦ ϕ · ∂xi ϕk dx. Rn k=1 Rn By taking the limit N →  ∞ and using (14), one sees that the distributional den rivative ∂xi (f ◦ ϕ) equals k=1 (∂xk f ) ◦ ϕ · ∂xi ϕk for any 1 ≤ i ≤ n. Therefore, 1 n f ◦ ϕ ∈ H (R , R) and d(f ◦ ϕ) = df ◦ ϕ · dϕ as claimed.  The next result concerns the product rule of differentiation in Sobolev spaces. To state the result, introduce for any integer s with s > n/2 and ε > 0 the set    Uεs := g ∈ H s (Rn , R) infn 1 + g(x) > ε . x∈R By Proposition 2.2, Uεs is an open subset of H s (Rn , R) and so is  U s := Uεs . ε>0 Note that U s is closed under multiplication. More precisely, if g ∈ Uεs and s h ∈ Uδs , then g + h + gh ∈ Uεδ . Indeed, by Lemma 2.3, gh ∈ H s (Rn , R), and hence so is g + h + gh. In addition, 1 + g + h + gh = (1 + g)(1 + h) satisfies s . inf x∈Rn (1 + g)(1 + h) > εδ and thus g + h + gh is in Uεδ Lemma 2.5. Let s, s be integers with s > n/2 and 0 ≤ s ≤ s. Then for any ε > 0 and K > 0 there exists a constant C ≡ C(ε, K; s, s ) > 0 so that for any   f ∈ H s (Rn , R) and g ∈ Uεs with g s < K, one has f /(1 + g) ∈ H s (Rn , R) and f /(1 + g) s ≤ C f s . (15) Moreover, the map   H s (Rn , R) × U s → H s (Rn , R), (16) (f, g) → f /(1 + g) is continuous. Remark 2.5. The proof shows that Lemma 2.5 continues to hold for any s real with s > n/2. The case where in addition s is real is treated in Appendix B. Proof. We prove the claimed statement by induction with respect to s . For s = 0, one has for any f in L2 (Rn , R) and g ∈ Uεs    f  1    1 + g  ≤ ε f .  Moreover, for any f1 , f2 ∈ L2 (Rn , R), g1 , g2 ∈ Uεs f1 ε2 1+g − 1 f2 1+g2 Hence by Proposition 2.2, f1 − ε2 1+g 1 f2 1+g2 ≤ (f1 − f2 ) + f1 (g2 − g1 ) + (f1 − f2 )g1   ≤ 1 + g1 C 0 f1 − f2 + f1 g2 − g1 C 0 .   ≤ 1 + Ks,0 g1 s f1 − f2 + Ks,0 f1 g2 − g1 s H. INCI, T. KAPPELER, and P. TOPALOV 13 and it follows that for any ε > 0 L2 (Rn , R) × Uεs → L2 (Rn , R), (f, g) → f /(1 + g) is continuous. As ε > 0 was taken arbitrarily, we see that the map (16) is continuous as well. Thus the claimed statements are proved in the case s = 0. Now, assuming that (15) and (16) hold for all 1 ≤ s ≤ k − 1, we will prove  that they hold also for s = k. Take f ∈ H s (Rn , R) and g ∈ Uεs . First, we will  prove that f /(1 + g) ∈ H s (Rn , R) and f ∂x (f g) − g · ∂xi f ∂xi f − i ∂xi (1 ≤ i ≤ n). = 1+g 1+g (1 + g)2 Indeed, by Lemma 2.1, there exists (fN )N ≥1 , (gN )N ≥1 ⊆ Cc∞ (Rn , R) so that fN →  f in H s (Rn , R) and gN → g in H s (Rn , R). As Uεs is open in H s (Rn , R) we can assume that (gN )N ≥1 ⊆ Uεs . By the product rule of differentiation, one has for any N ≥ 1, 1 ≤ i ≤ n fN ∂x (fN gN ) − gN · ∂xi fN ∂xi fN (17) ∂xi − i . = 1 + gN 1 + gN (1 + gN )2  As ∂xi fN −→ ∂xi f in H s −1 (Rn , R) it follows by the induction hypothesis that ∂x i f 1+g ∈H N →∞ s −1 n (R , R) and ∂xi f ∂xi fN −→ 1 + gN N →∞ 1 + g (18) in  H s −1 (Rn , R). 2 (N ≥ 1) and 2g + g 2 are in H s (Rn , R) and By Lemma 2.3, 2gN + gN 2 2gN + gN −→ 2g + g 2 (19) N →∞ As inf x∈Rn  2 1 + gN (x) > ε2 and in inf H s (Rn , R). x∈Rn  2 1 + g(x) > ε2 2 it follows that 2gN + gN (N ≥ 1) and 2g + g 2 are elements in Uεs2 . By Lemma   2.3, fN · gN (N ≥ 1), f · g are in H s (Rn , R) and fN · gN −→ f · g in H s (Rn , R). N →∞ Therefore (20) ∂xi (fN · gN ) → ∂xi (f · g)  in H s −1 (Rn , R).  Similarly, as ∂xi fN −→ ∂xi f in H s −1 (Rn , R) it follows again by Lemma 2.3 that N →∞  gN · ∂xi fN (N ≥ 1), g · ∂xi f are in H s −1 (Rn , R) and (21) gN · ∂xi fN −→ g · ∂xi f N →∞  in H s −1 (Rn , R). It follows from (19)-(21), and the induction hypothesis that (22) ∂xi (f g) − g · ∂xi f ∂xi (fN gN ) − gN · ∂xi fn −→ 2 N →∞ (1 + gN ) (1 + g)2 in  H s −1 (Rn , R).

Author H. Inci, P. Topalov, and T. Kappeler Isbn 9780821887417 File size 608.98KB Year 2013 Pages 60 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare For $M$ a closed manifold or the Euclidean space $mathbb{R}^n$, the authors present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > frac{1}{2}dim M 1$.     Download (608.98KB) Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions Kuznetsov’s Trace Formula and the Hecke Eigenvalues of Maass Forms Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three Lectures On Mathematical Logic Volume Iii The Logic Of Arithmetic Spaces of PL Manifolds and Categories of Simple Maps (AM-186 Load more posts

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