aa r X i v : . [ m a t h . S G ] A ug On the Uniqueness of Hofer’s Geometry
Lev Buhovsky, Yaron OstroverNovember 8, 2018
Abstract
We study the class of norms on the space of smooth functions on a closedsymplectic manifold, which are invariant under the action of the group of Hamil-tonian diffeomorphisms. Our main result shows that any such norm that iscontinuous with respect to the C ∞ -topology, is dominated from above by the L ∞ -norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metricon the group of Hamiltonian diffeomorphisms that is generated by an invariantnorm that satisfies the aforementioned continuity assumption, is either identi-cally zero or equivalent to Hofer’s metric. A remarkable fact, which is among the cornerstones of symplectic rigidity theory,is that the group of Hamiltonian diffeomorphisms of a symplectic manifold can beequipped with an intrinsic geometry given by a bi-invariant Finsler metric known asHofer’s metric. In contrast with finite-dimensional Lie groups, the existence of sucha metric on an infinite-dimensional group of transformations is highly unusual due tothe lack of compactness. In the past twenty years, Hofer’s metric has been intensivelystudied with many new discoveries covering a wide range of aspects in Hamiltoniandynamics and symplectic geometry.The purpose of this note is to show that under some mild assumption, Hofer’smetric is, in a sense, the only bi-invariant Finsler metric on the group of Hamiltoniandiffeomorphisms of closed symplectic manifolds. In order to state this result preciselywe proceed with some standard definitions and notations, and refer the reader to thebooks [7, 11, 15] for symplectic preliminaries, and further discussions on the group ofHamiltonian diffeomorphisms and Hofer’s geometry.Let (
M, ω ) be a closed 2 n -dimensional symplectic manifold, and denote by C ∞ ( M )the space of smooth functions that are zero-mean normalized with respect to the1anonical volume form ω n . For every smooth time-dependent Hamiltonian function H : M × [0 , → R , we associate a vector field X H t via the equation i X Ht ω = − dH t ,where H t ( x ) = H ( t, x ). The flow of X H t is denoted by φ tH and is defined for all t ∈ [0 , M, ω ) = { φ H | where φ tH is a Hamiltonian flow } When equipped with the standard C ∞ -topology, the group Ham( M, ω ) is an infinite-dimensional Fr´echet Lie group, whose Lie algebra A can be identified with the space C ∞ ( M ). Moreover, the adjoint action of Ham( M, ω ) on A is the standard actionof diffeomorphisms on functions i.e., Ad φ f = f ◦ φ − , for every f ∈ A and φ ∈ Ham(
M, ω ). Next, we define a Finsler (pseudo) distance on Ham(
M, ω ). Given anynorm k · k on the Lie algebra A , we define the length of a path α : [0 , → Ham(
M, ω )as length { α } = Z k ˙ α k dt = Z k H t k dt, where H t ( x ) = H ( t, x ) is the unique normalized Hamiltonian function generating thepath α . Here H is said to be normalized if R M H t ω n = 0 for every t ∈ [0 , d ( ψ, ϕ ) := inf length { α } , where the infimum is taken over all Hamiltonian paths α connecting ψ and ϕ . Itis not hard to check that d is non-negative, symmetric and satisfies the triangleinequality. Moreover, a norm on A which is invariant under the adjoint action yieldsa bi-invariant pseudo-distance function, i.e. d ( ψ, φ ) = d ( θ ψ, θ φ ) = d ( ψ θ, φ θ ) forevery ψ, φ, θ ∈ Ham(
M, ω ). From now on we will deal solely with such norms andwe will refer to d as the pseudo-distance generated by the norm k · k . Remark 1.1.
When one studies the geometric properties of the group of Hamilto-nian diffeomorphisms, it is convenient to consider smooth paths [0 , → Ham(
M, ω ),among which, those that start at the identity correspond to smooth Hamiltonianflows. Moreover, for a given Finsler metric on Ham(
M, ω ), a natural assumptionfrom a geometric point of view is that every smooth path [0 , → Ham(
M, ω ) is of afinite length. As it turns out, the latter assumption is equivalent to the continuity ofthe norm on A corresponding to the Finsler metric in the C ∞ -topology . We prove We remark that a fruitful study of right-invariant Finsler metrics on Ham(
M, ω ), motivatedin part by applications to hydrodynamics, was initiated in a well known paper by Arnold [1] (seealso [2], [8] and the references within). Moreover, non-Finslerian bi-invariant metrics on Ham(
M, ω )have been intensively studied in the realm of symplectic geometry, starting with the works ofViterbo [18], Schwarz [17], and Oh [12], and followed by many others. We thank A. Katok for his illuminating remark regarding the naturalness of the assumptionthat the norm is continuous in the C ∞ -topology. d (1l , φ ) > φ = 1l. In fact, for closed sym-plectic manifolds, a bi-invariant pseudo-metric d on Ham( M, ω ) is either a genuinemetric or identically zero. This is an immediate corollary of a well known theoremby Banyaga [3], which states that Ham(
M, ω ) is a simple group, combined with thefact that the null-set null( d ) = { φ ∈ Ham(
M, ω ) | d (1l , φ ) = 0 } is a normal subgroup of Ham( M, ω ). A distinguished result by Hofer [6] states thatthe L ∞ -norm on A gives rise to a genuine distance function on Ham( M, ω ) known asHofer’s metric. This was discovered and proved by Hofer for the case of R n , thengeneralized by Polterovich [14], and finally proven in full generality by Lalonde andMcDuff [10]. In a sharp contrast to the above, Eliashberg and Polterovich [5] showedthat for 1 ≤ p < ∞ , the pseudo-distances on Ham( M, ω ) corresponding to the L p -norms on A vanishes identically. A considerable generalization of the latter resultwas given by Ostrover-Wagner [13] who proved that for a closed symplectic manifold: Theorem 1.2 (Ostrover-Wagner [13]) . Let k · k be a Ham ( M, ω ) -invariant norm on A such that k · k ≤ C k · k ∞ for some constant C , but the two norms are not equivalent.Then the associated pseudo-distance d on Ham ( M, ω ) vanishes identically. In [5], the authors started a discussion regarding the uniqueness of Hofer’s metric(cf. [4], [15]). For the case of closed symplectic manifolds, one question they arose is:
Question:
Does there exist a Finsler bi-invariant metric on Ham(
M, ω ) which is notequivalent to Hofer’s metric.In this paper we provide an answer to the above question under the natural con-tinuity assumption mentioned in Remark 1.1. More precisely, our main result is:
Theorem 1.3.
Let ( M, ω ) be a closed symplectic manifold. Any Ham ( M, ω ) -invariantpseudo norm k·k on A that is continuous in the C ∞ -topology, is dominated from aboveby the L ∞ -norm i.e., k · k ≤ C k · k ∞ for some constant C . Combining together Theorem 1.3 and Theorem 1.2, we conclude that:
Corollary 1.4.
For a closed symplectic manifold ( M, ω ) , any bi-invariant Finslerpseudo metric on Ham ( M, ω ) , obtained by a pseudo norm k· k on A that is continuousin the C ∞ -topology, is either identically zero or equivalent to Hofer’s metric. In Here two metrics d , d are said to be equivalent if C d d Cd for some constant C > articular, any non-degenerate bi-invariant Finsler metric on Ham ( M, ω ) , which isgenerated by a norm that is continuous in the C ∞ -topology, gives rise to the sametopology on Ham ( M, ω ) as the one induced by Hofer’s metric. Remark 1.5.
Let us emphasize that any norm k · k on A can be turned into aHam( M, ω )-invariant pseudo-norm via the invariantization procedure k f k 7→ k f k inv ,where: k f k inv = inf nX k φ ∗ i f i k ; f = X f i , and φ i ∈ Ham ( M, ω ) o Note that k · k inv ≤ k · k . Thus, if k · k is continuous in the C ∞ -topology, then so is k · k inv . Moreover if k · k ′ is a Ham( M, ω )-invariant norm, then: k · k ′ ≤ k · k = ⇒ k · k ′ ≤ k · k inv In particular, the above invariantization procedure provides a plethora of Ham(
M, ω )-invariant genuine norms on A , e.g., by taking the homogenization of the k· k C k -norms. Structure of the paper:
In Section 2 we sketch an outline of the proof of Theo-rem 1.3. In Section 3 we prove a local version of this theorem, which would serve asthe main ingredient in the proof of the general case given in Section 4.
Notations:
Let x , . . . , x n be the Cartesian coordinates in R n . For any multi-index α = ( α , . . . , α n ), set ∂ α = ∂ α ∂ α . . . ∂ α n n , where ∂ i = ∂/∂x i . For an open set Ω ⊂ R n we denote C c (Ω) the space of compactly supported continuous functions on Ω, and let k · k ∞ stands for the L ∞ -norm. For an integer k , define C kc (Ω) the class of functions f from C c (Ω) such that ∂ α f ∈ C c (Ω) for all | α | ≤ k . The C k -norm of u ∈ C kc (Ω) isgiven by k u k C k = max | α |≤ k sup Ω | ∂ α u | As usual, C ∞ c (Ω) is the intersection of all the C kc (Ω) and is endowed with the C ∞ -topology. We denote by supp ( f ) the support of the function f i.e., the closure of theset { x | f ( x ) = 0 } , and by int ( D ) the interior of a domain D ⊂ R n . For an open do-main U ⊂ R n , we denote by Ham c ( D, ω ) the group of Hamiltonian diffeomorphismsof R n , which are generated by Hamiltonian functions H : R n × [0 , → R , whosesupport is compact and contained in U × [0 , ω is the standard symplecticform on R n given by ω = dp ∧ dq , where { q , p , . . . , q n , p n } are the canonical coor-dinates in R n . We say that a function f : R n → R is a product function , if it is ofthe form f ( q, p ) = Q ni =1 f i ( q i , p i ). Finally, the letters C, C , C , . . . are used to denotepositive constants that depend solely on the dimension of the ambient space relevantin each particular context. Acknowledgements:
Both authors are grateful to H. Hofer and L. Polterovich, fortheir interest in this work and helpful comments. This article was written duringvisits of the first author at the Institute for Advanced Study (IAS) in Princeton, and4isits of the second author at the Mathematical Sciences Research Institute (MSRI),Berkeley. We thank these institutions for their stimulating working atmospheres andfor financial support. The first author was supported by the Mathematical SciencesResearch Institute. The second author was supported by NSF Grant DMS-0635607,and by the Israel Science Foundation grant No. 1057/10. Any opinions, findings andconclusions or recommendations expressed in this material are those of the authorsand do not necessarily reflect the views of the NSF or the ISF.
Here we briefly describe the strategy of the proof of Theorem 1.3. For technicalreasons, we shall prove Theorem 1.3 for norms on the space C ∞ ( M ), instead of thespace A . The original claim would follow from this result since any Ham( M, ω )invariant pseudo-norm k · k on A can be naturally extended to an invariant pseudo-norm k · k ′ on C ∞ ( M ) by setting k f k ′ = k f − M f k , where M f = V ol ( M ) R M f ω n Note that if k · k is continuous in the C ∞ -topology, then so is k · k ′ . Moreover, thenorm k · k ′ coincides with k · k on the space A . By a standard partition of unityargument, we reduce the proof of the theorem to a “local result”, i.e., we show that itis sufficient to prove Theorem 1.3 for Ham c ( W, ω )-invariant norms on C ∞ c ( W ), where W = ( − L, L ) n is a 2 n -dimensional cube in R n . As a first step toward this end, weintroduce a special Ham c ( W, ω )-invariant norm k · k F ,max on C ∞ c ( W ), which dependson a given finite collection F ⊂ C ∞ c ( W ). More precisely: Definition I.
For a non-empty finite collection
F ⊂ C ∞ c ( W ) , let L F := nX i c i Φ ∗ i f i | c i ∈ R , Φ i ∈ Ham c ( W, ω ) , f i ∈ F , and { i | c i = 0 } < ∞ o , be equipped with the norm k f k L F = inf X | c i | , where the infimum is taken over all the representations f = P c i Φ ∗ i f i as above. Definition II.
For any compactly supported function f ∈ C ∞ c ( W ) , let k f k F , max = inf (cid:8) lim inf i →∞ k f i k L F (cid:9) , where the infimum is taken over all subsequences { f i } in L F which converge to f inthe C ∞ -topology. As usual, the infimum of the empty set is set to be + ∞ . k · k F , max is that it dominates from above any otherHam c ( W, ω )-invariant norm that is continuous in the C ∞ -topology (see Lemma 3.3).The next step, which is also the main part of the proof, is to show that for a suitablecollection of functions F ⊂ C ∞ c ( W ), the norm k · k F , max is in turn dominated fromabove by the L ∞ -norm. This is proved in Theorem 3.4, and in light of the above, itcompletes the proof of Theorem 1.3. The proof of Theorem 3.4 is divided into twomain steps which we now turn to describe: The local two-dimensional case:
Here, we shall construct a collection F of smoothcompactly supported functions on a two-dimensional cube W ⊂ R n , such that any f ∈ C ∞ c ( W ) satisfies k f k F , max C k f k ∞ for some absolute constant C . Thereare two independent components in the proof of this claim. First, we show thatone can decompose any f ∈ C ∞ c ( W ) with k f k ∞ f = P N i =1 ǫ j Ψ ∗ j g j . Here, ǫ j ∈ {− , } , Ψ j ∈ Ham c ( W , ω ), and g j are smooth radialfunctions whose L ∞ -norm is bounded by an absolute constant, and which satisfycertain other technical conditions (see Proposition 3.5 for the precise statement). Inwhat follows we call such functions by “simple functions”. We emphasize that N is a constant independent of f . Thus, we can restrict ourselves to the case where f is a “simple function”. In the second part of the proof, we construct an explicitcollection F = { f , f , f } , where f i ∈ C ∞ c ( W ) , and i = 0 , ,
2. Using an averagingprocedure (Proposition 3.6), we show that every “simple function” f ∈ C ∞ c ( W ) canbe approximated arbitrarily well in the C ∞ -topology by a sum of the form X i,k α i,k e Ψ ∗ i,k f k , where e Ψ i,k ∈ Ham c ( W , ω ) , k ∈ { , , } , and such that P | α i,k | ≤ C k f k ∞ for some absolute constant C . Combining this withthe above definiton of k · k F , max , we conclude that k f k F , max ≤ C k f k ∞ , for every f ∈ C ∞ c ( W ). This completes the proof of Theorem 3.4 in the 2-dimensional case. The local higher-dimensional case:
The proof of Theorem 3.4 for arbitrary di-mension strongly relies on the 2-dimensional case. We extend (in a natural way) theconstruction of the above mentioned collection F = { f , f , f } to the 2 n -dimensionalcase. By abuse of notation, we shall denote the new collection by F as well. Basedon the proof of Theorem 3.4 in the 2-dimensional case, and on the construction of theclass F , we show that Theorem 3.4 holds for “product functions”, i.e., for f ∈ C ∞ c ( W )of the form f = Q ni =1 f i ( q i , p i ), where f i ∈ C ∞ c ( W ). From this we derive, using aFourier series argument, that the norm k · k F ,max is dominated from above by the k · k C n +1 -norm, i.e., for any f ∈ C ∞ c ( W ) one has k f k F , max ≤ C k f k C n +1 , (2.1)for some constant C (see Proposition 3.14 for the proof of the above two claims).Next, for any ǫ >
0, we construct a partition of unity function R ǫ : R n → R , with6 upp ( R ǫ ) ⊂ ( − ǫ, ǫ ) n , and such that X v ∈ ǫ Z n R ǫ ( x − v ) = 1l( x )For any w ∈ X := { , , , } n , we consider a finite grid Γ ǫw ⊂ W given by:Γ ǫw = ǫw + 4 ǫ Z n ∩ ( − L + 3 ǫ, L − ǫ ) n , and define f w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) f ( x )Note, that for ǫ sufficiently small such that supp ( f ) ⊂ ( − L + 4 ǫ, L − ǫ ) n , one has f ( x ) = X w ∈ X f w ( x )For any w ∈ X , the function f w is a finite sum of smooth functions that lie near thepoints of the grid Γ ǫw . Moreover, these functions have mutually disjoint supports,which are spaced commodiously. Next, we fix w ∈ X , and for any v ∈ Γ ǫw weconsider the decomposition of f ∈ C ∞ c ( W ) as a Taylor polynomial of order 2 n + 1and a remainder, around the point v (this specific choice of the order ensure, basedon (2.1 ), the estimate (2.2 ) below): f ( x ) = P v n +1 ( x − v ) + R v n +1 ( x − v ) . We decompose each f w as f w ( x ) = g w ( x ) + h w ( x ), where g w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) P v n +1 ( x − v ) , and h w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) R v n +1 ( x − v ) . Based on (2.1 ), in Lemma 3.16 (cf. Corrolary 3.17) we show that the k · k F ,max -normof the reminder parts { h w } can be taken to be arbitrarily small. More precisely, k h w k F ,max C k h w k C n +1 C ǫ k f k C n +2 , (2.2)for some constants C and C . On the other hand, using a combinatorial argumentand the above mentioned fact that Theorem 3.4 holds for “product functions”, weprove the estimate k g w k F , max C (cid:0) n +1 X i =0 k f k C i ǫ i (cid:1) (2.3)for some constnat C . Combining the above estimates (2.2 ) and (2.3 ) for all w ∈ X ,and taking ǫ →
0, we conclude that for every f ∈ C ∞ c ( W ) one has k f k F , max C k f k ∞ , for some absolute constant C . This completes the proof of the theorem.7 A Local Version of the Main Result
In this section we prove a local version of our main result (Theorem 3.4 below), whichwould later serve as the main component in the proof of Theorem 1.3.Consider an open cube W = I n ⊂ R n , where I = ( − L, L ) ⊂ R is an openinterval. Endow W with linear coordinates ( q , p , . . . , q n , p n ), and with the standardsymplectic structure ω = dp ∧ dq descending from R n . For a finite non-emptycollection F of functions in C ∞ c ( W ), we define the space L F := nX i c i Φ ∗ i f i | c i ∈ R , Φ i ∈ Ham c ( W, ω ) , f i ∈ F , and { i | c i = 0 } < ∞ o We equip L F with the norm k f k L F := inf X | c i | , where the infimum is taken over all the representations f = P c i Φ ∗ i f i as above. Definition 3.1.
For any compactly supported function f ∈ C ∞ c ( W ) , let k f k F , max = inf (cid:8) lim inf i →∞ k f i k L F (cid:9) , (3.1) where the infimum is taken over all subsequences { f i } in L F which converge to f inthe C ∞ -topology. If such sequence do not exists, we set k f k F , max ≡ + ∞ . Remark 3.2.
It follows from the definition above that k · k F , max is homogeneous,Ham c ( W, ω )-invariant, and satisfies the triangle inequality . Moreover, let { f k } be asequence of smooth functions that converge in the C ∞ -topology to f , and such thatfor every k > k f k k F , max C for some constant C . Then k f k F , max C .The fact that k · k F , max is non-degenerate (i.e., k f k F , max = 0 if and only if f = 0)follows from the next lemma. Lemma 3.3.
Let
F ⊂ C ∞ c ( W ) be a non-empty finite collection of smooth compactlysupported functions in W . Then, any Ham c ( W, ω ) -invariant norm k · k on C ∞ c ( W ) which is continuous in the C ∞ -topology, satisfies k · k C k · k F , max for some absoluteconstant C . Proof of Lemma 3.3.
Let C = max {k g k ; g ∈ F } . For any f = P c i Φ ∗ i f i ∈ L F ,one has: k f k ≤ X | c i |k Φ ∗ i f i k ≤ C X | c i | ≤ C k f k F , max (3.2)The lemma now follows from combining (3.2 ), definition (3.1 ), and the fact that thenorm k · k is assumed to be continuous in the C ∞ -topology. When k · k F , max ≡ + ∞ , these statements are trivially true. F , the subspace L F ⊂ C ∞ c ( W ) is dense in the C ∞ -topology, and moreover, that the norm k · k F , max on C ∞ c ( W ) is dominated fromabove by the k · k ∞ -norm. Theorem 3.4.
There is a finite collection
F ⊂ C ∞ c ( W ) , such that k · k F , max is agenuine norm on C ∞ c ( W ) , and k · k F , max ≤ C k · k ∞ for some absolute constant C . The remainder of this section is devoted to the proof of Theorem 3.4, which wesplit into two separate cases:
We assume that n = 1, and hence W = ( − L, L ) × ( − L, L ). We set z = x + iy ,where { x, y } are local coordinates on W , and denote by D a = {| z | ≤ a } the disc withradius a centered at the origin, and by D a,A = { a ≤ | z | ≤ A } the annulus with radii a, A . The proof of Theorem 3.4 in the two-dimensional case follows from the nexttwo propositions, the proof of which we postpone to Subsections 3.1.1 and 3.1.2. Proposition 3.5.
There are positive constants a, A, C such that a < A < L ; a smoothradial function f with supp ( f ) = D A ; and an integer number N ∈ N , such that every f ∈ C ∞ c ( W ) with k f k ∞ can be decomposed as f = N X j =1 ǫ j Φ ∗ j g j , where Φ j ∈ Ham c ( W, ω ) , ǫ j ∈ {− , } , and g j are smooth radial functions that satisfy: supp ( g j ) = D A , g j ≡ f on D a , and k g j k ∞ C (3.1.3) Proposition 3.6.
Let < a < A be positive numbers. Then there exists a smoothfunction F a,A : R → R with supp ( F a,A ) ⊂ D A , such that the following holds: forevery smooth radial function f : R → R , that satisfies k f k ∞ , supp ( f ) ⊂ D a,A , and Z R f ω = 0 , (3.1.4) there exists an area-preserving diffeomorphism Φ : R → R , with supp (Φ) ⊂ D a,A ,and such that: Z D r Φ ∗ F a,A ω = Z D r f ω, for any r > roof of Theorem 3.4 (the 2-dimensional case): Let f ∈ C ∞ c ( W ) with k f k ∞ a, A, C , aninteger N , and a smooth radial function f with supp ( f ) = D A , such that f can bewritten as f = N X j =1 ǫ j Φ ∗ j g j , where Φ j ∈ Ham c ( W, ω ), ǫ j ∈ {− , } , and { g j } are smooth radial functions thatsatisfy (3.1.3 ). Next, let f be a smooth radial function with supp ( f ) = D a,A suchthat R W f ω = 1. Moreover, let f = F a,A be the function provided by Proposition 3.6above. We consider the function h j := g j − f − c j f , where c j = Z W ( g j − f ) ω Note that there exists a constant C ′ such that k h j k ∞ ≤ C ′ . Indeed: k h j k ∞ ≤ C + k f k ∞ + | c j |k f k ∞ ≤ C + k f k ∞ + k f k ∞ (cid:16) π CA + Z W | f | ω (cid:17) From Proposition 3.6 it follows that there are area-preserving diffeomorphisms e Φ j with supp ( e Φ j ) ⊂ D a,A , such that for any r > Z D r ( e Φ ∗ j f ) ω = 1 C ′ Z D r h j ω (3.1.5)To complete the proof of the theorem, we shall need the following technical lemma: Lemma 3.7.
Let f ∈ C ∞ c ( D ) be a compactly supported function in a disk D . Then N N X i =1 f ( ze πiN ) N →∞ −−−→ π Z π f ( ze iθ ) dθ, in the C ∞ topology Postponing the proof of Lemma 3.7, we first finish the proof of the theorem.Consider a compactly supported Hamiltonian isotopy T Aθ : W → W , where θ ∈ R ,and such that T Aθ ( z ) = e iθ z in D A . From Lemma 3.7 and (3.1.5 ) it follows that: C ′ N N X k =1 ( T A πkN ) ∗ e Φ ∗ j f N →∞ −−−→ h j , in the C ∞ topology (3.1.6)We set F = { f , f , f } . From (3.1.6 ) and Remark 3.2 it follows that k h j k F ,max ≤ C ′ .Moreover, by definition one has: k f k F ,max , k f k F ,max ≤
1. This implies that k g j k F ,max ≤ C ′′ , C ′′ is an absolute constant given by: C ′′ = C ′ + 1 + πCA + Z W | f | ω Thus, we conclude that k f k F ,max ≤ N C ′′ . This completes the proof of the theorem. Proof of Lemma 3.7.
We shall prove the convergence1 N N X i =1 f ( ze πiN ) N →∞ −−−→ π Z π f ( ze iθ ) dθ in C kc ( D ), for any k ∈ N . Note that the operators P N ( f ) = N P Ni =1 f ( ze πiN ), definedon the space C kc ( D ), have a bounded operator norm which is independent on N .Therefore, it is enough to check that P N f N →∞ −−−→ π Z π f ( ze iθ ) dθ, in C kc ( D ) only on some dense subspace. We choose this subspace to be consists of allthe finite sums: s m ( z ) = m X l =0 u l ( r ) cos( lθ ) + v l ( r ) sin( lθ ) , where u l and v l are smooth radial functions supported in the disk D . Note that for N > m one has P N s m ( z ) = u ( r ) = 12 π Z π s m ( ze iθ ) dθ, and hence the statement of the lemma is satisfied in a trivial way. The proof of thelemma is now complete.We now return to complete the proof of Proposition 3.5 and Proposition 3.6. For the sake of clarity, we fragment the proof of the proposition in several steps:
Step I:
We choose a = L , A = L . The area of the sector { z ∈ W | a < | z | < A ; 0 < Arg z < π } equals to π (cid:18) L − L (cid:19) = 3 π L > L Area ( W )3211sing a smooth partition of unity, one can decompose f as f = P k =1 f k , where thesupport of each f k lies in an open sub-rectangle of the square W of area Area ( W )32 ,and k f k k ∞ ≤
1. Next, we take compactly supported area-preserving diffeomorphisms e Φ k : W → W , such that f k = e Φ ∗ k f ′ k , for k = 1 , . . . ,
33, and supp ( f ′ k ) ⊂ (0 , L ) × (0 , L ).Denote L = L and L = L . From the above we conclude that it is enough to restrictourselves to the case where supp ( f ) ⊂ (0 , L ) × (0 , L ). Indeed, if the proposition holdsfor such functions, then by replacing N with 33 N , it will hold for any compactlysupported function f ∈ C ∞ c ( W ). Step II:
Following Step I, we assume that supp ( f ) ⊂ (0 , L ) × (0 , L ). Next, weapply the following lemma to the function f . Lemma 3.8.
Let R = [0 , L ] × [0 , L ] ⊂ R be a rectangle, and let f : R → R be a smooth function with supp ( f ) ⊂ int ( R ) , and k f k ∞ . Then there exists adecomposition f = P i =1 f i , and compactly supported diffeomorphisms Ψ i : R → R , i = 1 , , ..., , such that the functions g i := Ψ ∗ i f i satisfy | ∂∂x g i | L . The proof of Lemma 3.8 will be given in Subsection 3.1.3.
Remark 3.9.
Analogously to Step I, Lemma 3.8 reduces the proposition to thecase where supp ( f ) ⊂ (0 , L ) × (0 , L ), and moreover that there is a diffeomorphismΨ : W → W with supp (Ψ) ⊂ (0 , L ) × (0 , L ), such that g = Ψ ∗ f satisfies | ∂∂x g | L .Indeed, the general case would follow by replacing N with 8 · · N = 264 N . Thus,we assume in what follows the existence of f, g and Ψ as above. Step III:
Denote by R the rectangle [0 , L ] × [0 , L ]. From the fact that Area ( R ) < Area ( { z ∈ W | a < | z | < A ; 0 < Arg z < π } ) , one can easily find an area preserving diffeomorphism Φ : W → W withΦ( R ) = { z ∈ W | a < | z | < A ; 0 < Arg z < π } , for an appropriate a < A < A ; and such that on R , the diffeomorphism Φ takesthe form Φ( x + iy ) = r ( x ) e θ ( y ) , where r ( x ) is a monotone increasing function. Let C = min x ∈ [0 ,L ] r ′ ( x ) > , and define h = (Φ − ) ∗ g . Note that one can bound theradial derivative of h by:max | ∂∂r h | ≤ C max | ∂∂x g | ≤ L C Next, we set C = L C , and fix a smooth radial function f such that supp ( f ) ⊂ D A , ∂∂r f ( z ) < − C for z ∈ D a,A , ∂∂r f ( z ) < z ∈ int ( D A ) \ { } , z = 0 is a non-degenerate maximum for the function f . Wedenote H = h + f ( z ), and observe that H satisfies: supp ( H ) ⊂ D A , ∂∂r H < int ( D A ) \ { } , H ( z ) ≡ f ( z ) in D a ∪ D A ,A , and that the point z = 0 is a unique non-degenerate critical point of H , which isa maximum point. Consider the gradient flow of H . By a standard Morse theoryargument one can find a diffeomorphism Υ : W → W , with supp (Υ) ⊂ D a,A , andsuch that K := Υ ∗ H is a radial function. Finally, we have f = (Ψ − ) ∗ g = (Ψ − ) ∗ Φ ∗ h = (Ψ − ) ∗ Φ ∗ H − (Ψ − ) ∗ Φ ∗ f = (Ψ − ) ∗ Φ ∗ (Υ − ) ∗ K − (Ψ − ) ∗ Φ ∗ f . Note, that for z ∈ W \ D a,A , one hasΨΦ − Υ( z ) = ΨΦ − ( z ) = Φ − ( z )Indeed, this follows from the fact that supp (Ψ) ⊂ R ⊂ Φ − ( D a,A ), and that Υ is theidentity on the complement W \ D a,A . Thus, we conclude that(ΨΦ − Υ) ∗ ω = (ΨΦ − ) ∗ ω = ω, on the complement W \ D a,A Next, let S r = { z ∈ W | | z | = r } . We shall need the following lemma: Lemma 3.10.
Let ω ′ be a symplectic form on W which coincides with the standardsymplectic form ω on the complement W \ D a,A , and such that R W ω ′ = R W ω . Then,there exists a diffeomorphism Λ : W → W supported in D a,A , such that for every a < r < A , one has Λ( S r ) = S R , for some a < R < A , and such that Λ ∗ ω = ω ′ . Proof of Lemma 3.10.
Consider the function S : [0 , L ) → [0 , ∞ ), defined by S ( r ) = R D r ω ′ . Note that S is a smooth function, and that S ( r ) = πr for every r ∈ [0 , a ] ∪ [ A, L ). Define a diffeomorphism ∆ : W → W , supported in D a,A , by∆ ( r, θ ) = r S ( r ) π , θ ! , for r ∈ [0 , L ) , and extend it by the identity diffeomorphism to the whole W . Denote ω ′′ = (∆ − ) ∗ ω ′ ,and note that R D r ω ′′ = πr for r A , and ω ′′ = ω ′ = ω on W \ D a,A . Next, weexplicitly construct a diffeomorphism ∆ : W → W supported in D a,A , such that ω ′′ = ∆ ∗ ω , and for 0 < r < L , it takes the form ∆ ( r, θ ) = ( r, F ( r, θ )), for somesmooth map F : (0 , L ) × S → S . To this end, note that ω ′′ = Gω for some positivefunction G : W → (0 , ∞ ), such that G = 1 on W \ D a,A . Moreover, πr = Z D r ω ′′ = Z D r Gω, for all 0 < r < L Z π G ( r, θ ) dθ = 2 π, for every 0 < r < L (3.1.7)On the other hand, we require ∆ to satisfy:∆ ∗ ω = rF θ ( r, θ ) dr ∧ dθ = F θ ( r, θ ) ω, for every r ∈ (0 , L )Thus, the condition ω ′′ = ∆ ∗ ω is equivalent to F θ ( r, θ ) = G ( r, θ ), for r ∈ (0 , L ). Wedefine F ( r, θ ) = Z θ G ( r, s ) ds, for r ∈ (0 , L ) , θ ∈ [0 , π ) (3.1.8)In light of (3.1.7 ), we obtain a smooth map F : (0 , L ) × S → S . Moreover, since G = 1 on W \ D a,A , one has F ( r, θ ) = θ for r ∈ (0 , a ] ∩ [ A, L ). Therefore, defining∆ ( r, θ ) = ( r, F ( r, θ )) for 0 < r < L , where F is given in (3.1.8 ), we obtain adiffeomorphism of D L supported in D a,A . We extend ∆ to the whole W by theidentity diffeomorphism. Note that ω ′′ = ∆ ∗ ω , and hence ω ′ = ∆ ∗ ω ′′ = ∆ ∗ ∆ ∗ ω .Denoting Λ = ∆ ∆ , we conclude the statement of the lemma.We return now to the proof of the Proposition. By applying Lemma 3.10 to theforms ω ′ = (ΨΦ − Υ) ∗ ω and ω ′′ = (ΨΦ − ) ∗ ω , we obtain two diffeomorphisms Λ ′ , Λ ′′ such that Λ ′∗ ω = (ΨΦ − Υ) ∗ ω , and Λ ′′∗ ω = (ΨΦ − ) ∗ ω . Denote Φ ′ := Λ ′ Υ − ΦΨ − ,Φ ′′ := Λ ′′ ΦΨ − . Note that Φ ′ , Φ ′′ ∈ Ham c ( W, ω ), and that f = (Ψ − ) ∗ Φ ∗ (Υ − ) ∗ K − (Ψ − ) ∗ Φ ∗ f = (Ψ − ) ∗ Φ ∗ (Υ − ) ∗ (Λ ′ ) ∗ K − (Ψ − ) ∗ Φ ∗ (Λ ′′ ) ∗ f = (Φ ′ ) ∗ K − (Φ ′′ ) ∗ f The decomposition f = (Φ ′ ) ∗ K − (Φ ′′ ) ∗ f shows that the proposition holds for f asin Remark 3.9, with only two summands in the decomposition, and with C = k f k ∞ .Therefore, we obtain the conclusion of Proposition 3.5 with N = 264 · We start with a construction of a function F , such that for any smooth radial function f : R → R , satisfying the conditions (3.1.4 ) one can find a diffeomorphism (notnecessarily area-preserving) Ψ : R → R supported in D A such that for any r > Z D r Ψ ∗ ω = Z D r ω = πr , (3.1.9)and, Z D r Ψ ∗ ( F ω ) = Z D r f ω. (3.1.10)14e shall take the function F to be of the form F ( r, θ ) = φ ( r ) ψ ( θ ), where φ, ψ aresmooth functions. We assume that φ ( r ) = 0, for small enough r , and that φ ( r ) = 1 for r > a . The function ψ is assumed to satisfy R π ψ ( θ ) dθ = 0, and would be determinedin the sequel. Moreover, R ( r, θ ) = p u ( r ) µ ( θ ) + v ( r ) ν ( θ ) ,u ( r ) = v ( r ) = r for r a or r > A,u ′ ( r ) , v ′ ( r ) > r > ,µ ( θ ) , ν ( θ ) > ,µ ( θ ) + ν ( θ ) = 1 (3.1.11)Here, µ, ν, u and v , are smooth functions that would be determined explicitly inthe sequel. Note that conditions (3.1.11 ) ensure that Ψ is a diffeomorphism of R supported in D a,A . Next, we computeΨ ∗ ω = R ( r, θ ) R ′ r ( r, θ ) dr ∧ dθ = 12 (cid:0) u ′ ( r ) µ ( θ ) + v ′ ( r ) ν ( θ ) (cid:1) dr ∧ dθ, and Ψ ∗ ( F ω ) = F ( R ( r, θ ) , θ ) R ( r, θ ) R ′ r ( r, θ ) dr ∧ dθ = 12 φ ( R ( r, θ )) ψ ( θ ) (cid:0) u ′ ( r ) µ ( θ ) + v ′ ( r ) ν ( θ ) (cid:1) dr ∧ dθ. After differentiating by r and some simplification, conditions (3.1.9 ), (3.1.10 ) become u ′ ( r ) Z π µ ( θ ) dθ + v ′ ( r ) Z π ν ( θ ) dθ = 4 πr (3.1.12)and, u ′ ( r ) Z π φ ( R ( r, θ )) ψ ( θ ) µ ( θ ) dθ + v ′ ( r ) Z π φ ( R ( r, θ )) ψ ( θ ) ν ( θ ) dθ = 4 πrf ( r ) (3.1.13)Note that when r > a , one has R ( r, θ ) > a , and condition (3.1.13 ) turns to: u ′ ( r ) Z π ψ ( θ ) µ ( θ ) dθ + v ′ ( r ) Z π ψ ( θ ) ν ( θ ) dθ = 4 πrf ( r ) (3.1.14)Next, we choose the functions ψ, µ, ν to be any smooth functions satisfying: R π ψ ( θ ) µ ( θ ) dθ = 2 π, R π ψ ( θ ) ν ( θ ) dθ = − π, R π µ ( θ ) dθ = R π ν ( θ ) dθ = π,µ ( θ ) , ν ( θ ) > ,µ ( θ ) + ν ( θ ) = 1 (3.1.15)15ote that this choice of ψ, µ, ν do not depend on the function f . Moreover, with theabove choice, for r > a , equations (3.1.12 ) and (3.1.14 ) become ( u ′ ( r ) + v ′ ( r ) = 4 r,u ′ ( r ) − v ′ ( r ) = 2 rf ( r ) (3.1.16)Next, we consider equations (3.1.16 ) for every r >
0, with initial conditions u (0) = v (0) = 0. There is no difficulty in checking that the solutions of this system are ( u ( r ) = R r s (2 + f ( s )) ds,v ( r ) = R r s (2 − f ( s )) ds (3.1.17)One can easily check, that as required, the function u and v satisfy ( u ′ ( r ) , v ′ ( r ) > , for r > ,u ( r ) = v ( r ) = r , for r a and r > A (3.1.18)Moreover, by definition, they satisfy equations (3.1.12 ) and (3.1.13 ) when r > a .Let us now show that these equations hold for r < a as well. First, note thatequation (3.1.12 ) clearly holds when r < a . Second, by defintion, for r < a one has u ( r ) = v ( r ) = r , and R ( r, θ ) = r . Hence, we compute u ′ ( r ) Z π φ ( R ( r, θ )) ψ ( θ ) µ ( θ ) dθ + v ′ ( r ) Z π φ ( R ( r, θ )) ψ ( θ ) ν ( θ ) dθ = u ′ ( r ) φ ( r ) Z π ψ ( θ ) µ ( θ ) dθ + v ′ ( r ) φ ( r ) Z π ψ ( θ ) ν ( θ ) dθ = 2 rφ ( r ) (cid:16)Z π ψ ( θ ) µ ( θ ) dθ + Z π ψ ( θ ) ν ( θ ) dθ (cid:17) = 2 rφ ( r )(2 π − π ) = 0Combining this with the fact that supp ( f ) ⊂ D a,A , we obtain that the functions u and v , satisfy (3.1.12 ) and (3.1.13 ) for all r >
0. We conclude that the resultingdiffeomorphism Ψ satisfies conditions (3.1.9 ) and (3.1.10 ). Furthermore, since thediffeomorphism Ψ satisfies (3.1.9 ), and supp (Ψ) ⊂ D a,A , by using a similar argumentsas in the proof of Lemma (3.1.5) from [16], we conclude that there exists an area-preserving diffeomorphism Φ : R → R , with supp (Φ) ∈ D a,A , such that Φ( D ( r )) =Ψ( D ( r )) for any r >
0. Thus, we obtain Z D r (Φ ∗ F ) ω = Z D r Φ ∗ ( F ω ) = Z Φ( D r ) F ω = Z Ψ( D r ) F ω = Z D r Ψ ∗ ( F ω ) = Z D r f ω, and the proof of the Proposition in now complete.16 .1.3 Technical Lemmata In this subsection we prove Lemma 3.8 which was used in the proof of Proposition 3.5.We start with the following preparation:
Lemma 3.11.
There is a smooth function φ : R → R with the following properties:1. supp ( φ ) = [0 , , φ ( t ) > , for t ∈ (0 , ,3. φ ′ ( t ) > , for t ∈ (0 , / , and φ ′ ( t ) < t ∈ (3 / , , sup t ∈ (0 , (cid:16) φ ′ ( t ) φ ( t ) (cid:17) ′ = sup t ∈ (0 , φ ′′ ( t ) φ ( t ) − φ ′ ( t ) φ ( t ) < , P n ∈ Z φ ( t + n ) ≡ Proof of Lemma 3.11.
Consider first the smooth function f : R → R , defined by f ( x ) = ( e − x , for x > , , for x x >
0, one has f ′′ ( x ) = 4 x e − x (1 − x ) , and hence f ′′ ( x ) > x ∈ (0 , f ′′ (0) = f ′′ (1) = 0. Note moreover that f ′′ ( x ) f ( x ) − f ′ ( x ) = (cid:16) f ′ ( x ) f ( x ) (cid:17) ′ f ( x ) = − x e − x < , for x ∈ (0 , + ∞ )We approximate, in the C -norm, the function f ′′ | [0 , arbitrarily close by a smoothpositive function h : [0 , → [0 , ∞ ), such that h ( x ) = f ′′ ( x ) for x ∈ [0 , ], and suchthat h ( x ) = 0 near x = 1. Next, consider the smooth function F : [0 , → R , that isuniquely determined by the requirements F ′′ ( x ) = h ( x ), and F (0) = F ′ (0) = 0. Notethat the function F is arbitrary close, in the C -topology, to f | [0 , , and F ( x ) = f ( x )for x ∈ [0 , ]. Moreover, the requirement that h is C -sufficiently close to f ′′ | [0 , ensures that F ′′ ( x ) F ( x ) − F ′ ( x ) <
0, for every x ∈ (0 , F ′′ ( x ) + F ′′ (1 − x ) > x ∈ (0 , F ( x ) is a linearfunction near x = 1. Finally, we define φ : R → R as follows: φ ( x ) = F ( x )2 F (1) for x ∈ [0 , , F (1) − F ( x − − F (2 − x )2 F (1) for x ∈ (1 , , F (3 − x )2 F (1) for x ∈ (2 , , x / ∈ [0 , φ is a non-negative smooth function,with supp ( φ ) = [0 , φ ( x ) = φ (3 − x ), and that for x ∈ (1 , φ | (1 , ) ′′ ( x ) = − F ′′ ( x − − F ′′ (2 − x )2 F (1) < φ ′ (3 /
2) = 0, we obtain that φ ′ ( x ) > x ∈ (1 , / φ ′ ( x ) < x ∈ (3 / , F , it follows that φ ′ ( x ) > x ∈ (0 ,
1] and φ ′ ( x ) < x ∈ [2 , φ satisfies the first three requirements of the lemma. We next turnto show that φ satisfies the forth one. Note that φ ′′ ( x ) φ ( x ) − φ ′ ( x ) < x ∈ (0 , F for x ∈ (0 , ∪ (2 , x ∈ (1 , φ ′′ ( x ) φ ( x ) − φ ′ ( x ) = − φ ′ ( x ) < x = 1 , φ it follows that φ ( x ) ≃ e − x for x close to 0, and φ ( x ) ≃ e − − x for x close to 3, where ≃ means arbitrary close in the C -topology. Therefore, we obtain:lim x → + φ ′′ ( x ) φ ( x ) − φ ′ ( x ) φ ( x ) = lim x → − φ ′′ ( x ) φ ( x ) − φ ′ ( x ) φ ( x ) = −∞ . From the above we conclude that:sup x ∈ (0 , φ ′′ ( x ) φ ( x ) − φ ′ ( x ) φ ( x ) < , as required. Finally, there is no difficulty in checking that P n ∈ Z φ ( x + n ) = 1. Thedetails of this last step are left to the reader. Lemma 3.12.
Let R = [ α , β ] × [ α , β ] ⊂ R be a rectangle, and consider twosmooth non-negative functions u : [ α , β ] → R , and v : [ α , β ] → R , positive on ( α , β ) and ( α , β ) respectively, such that u ( x ) = e − x − α near α ; u ( x ) = e − β − x near β , v ( y ) = e − y − α near α ; and v ( y ) = e − β − y near β . Moreover, let φ ( x ) be thefunction described in Lemma 3.11 above, and let F : R → R be any smooth functionthat satisfies:1. supp ( F ) = R F ( x, y ) > for ( x, y ) ∈ int ( R ) F ( x, y ) = u ( x ) v ( y ) near the boundary of R Then there exists an ǫ > , such that for any < ǫ < ǫ , and a ∈ R , the followingholds: denote by G ( x, y ) = F ( x, y ) φ ( x − aǫ ) , and assume that G = 0 (this holds when ( α , β ) ∩ ( a, a + 3 ǫ ) = ∅ ). Moreover, set U = supp ( G ) = [ a , a ] × [ α , β ] . Then, here exists a smooth function c : [ α , β ] → ( a , a ) , which is constant near α , β ,such that for any y ∈ ( α , β ) one has: ( ∂∂x G ( x, y ) > , for a < x < c ( y ) , ∂∂x G ( x, y ) < , for c ( y ) < x < a Proof of Lemma 3.12.
From the above assumptions it follows that there exists α < γ < δ < β , such that u ( x ) = e − x − α for α < x < γ , u ( x ) = e − β − x for δ < x < β , and F ( x, y ) = u ( x ) v ( y ) when x ∈ ( α , γ ] ∪ [ δ , β ). Pick some γ ′ , δ ′ ,such that α < γ ′ < γ < δ < δ ′ < β , and denote ǫ = min { γ − γ ′ , δ ′ − δ } . Next, takeany 0 < ǫ < ǫ , and any a ∈ R , and consider the function G ( x, y ) = F ( x, y ) φ ( x − aǫ ). Case I:
Assume a ∈ [ γ ′ , δ ]. Then, one has γ ′ a < a + 3 ǫ δ ′ , and therefore supp ( G ) = [ a, a + 3 ǫ ] × [ α , β ]. Fix some y ∈ ( α , β ). Our goal is to show that forsufficiently small ǫ (which is independent of y ), there exists a value c ( y ) ∈ ( a, a +3 ǫ ),such that ∂∂x G ( x, y ) >
0, for a < x < c ( y ), and ∂∂x G ( x, y ) <
0, for c ( y ) < x < a +3 ǫ .For this end, we compute: ∂∂x G ( x, y ) G ( x, y ) = ∂∂x F ( x, y ) F ( x, y ) + 1 ǫ φ ′ ( x − aǫ ) φ ( x − aǫ ) . Note that, the function x G ( x, y ) is a positive function, supported in [ a, a + 3 ǫ ].Thus, ∂∂x G ( x,y ) G ( x,y ) = 0 at least at one point x ∈ ( a, a + 3 ǫ ) (e.g., at the maximum pointof x G ( x, y )). Let us show next that: ∂∂x ∂∂x G ( x, y ) G ( x, y ) < , for all x ∈ ( a, a + 3 ǫ ) (3.1.20)We start by claiming that ∂∂x ∂∂x F ( x,y ) F ( x,y ) is bounded on [ γ ′ , δ ′ ] × ( α , β ). Indeed, fromthe assumptions of the lemma it follows that F ( x, y ) = u ( x ) v ( y ) near the boundaryof R , and therefore there exist α < γ < δ < β , such that F ( x, y ) = u ( x ) v ( y ) for y ∈ [ α , γ ] ∪ [ δ , β ]. Thus, for a point ( x, y ) near the boundary of R , one has ∂∂x ∂∂x F ( x, y ) F ( x, y ) = ∂∂x u ′ ( x ) u ( x ) = u ′′ ( x ) u ( x ) − u ′ ( x ) u ( x ) (3.1.21)Restricting ourselves to the case where x ∈ [ γ ′ , δ ′ ] and y ∈ [ α , γ ] ∪ [ δ , β ], and bynoticing that u | ( α ,β ) is strictly positive smooth function, we obtain that the function ∂∂x ∂∂x F ( x,y ) F ( x,y ) is bounded on [ γ ′ , δ ′ ] × (( α , γ ] ∪ [ δ , β )). On the other hand, becauseof compactness, the function ∂∂x ∂∂x F ( x,y ) F ( x,y ) is bounded on [ γ ′ , δ ′ ] × [ γ , δ ]. Hence, weconclude that ∂∂x ∂∂x F ( x,y ) F ( x,y ) is bounded on [ γ ′ , δ ′ ] × ( α , β ). Next, note that ∂∂x ǫ φ ′ ( x − aǫ ) φ ( x − aǫ ) = 1 ǫ φ ′′ ( x − aǫ ) φ ( x − aǫ ) − φ ′ ( x − aǫ ) φ ( x − aǫ ) (3.1.22)19rom Lemma 3.11 it follows thatsup t ∈ (0 , φ ′′ ( t ) φ ( t ) − φ ′ ( t ) φ ( t ) < , (3.1.23)and hence (3.1.22 ) can be chosen to be arbitrarily negative. As a conclusion, weobtain that for sufficiently small ǫ , say 0 < ǫ < ǫ , one has ∂∂x ∂∂x G ( x, y ) G ( x, y ) < , for every ( x, y ) ∈ supp ( G ) = [ a, a + 3 ǫ ] × ( α , β ) (3.1.24)Moreover, for any y ∈ ( α , β ), there exists therefore a unique x := c ( y ) ∈ ( a, a + 3 ǫ ),such that ∂∂x G ( x,y ) G ( x,y ) = 0. It follows from (3.1.24 ) and the implicit function theorem,that the function y c ( y ) is smooth for y ∈ ( α , β ). Moreover, since ∂∂x G ( x,y ) G ( x,y ) isindependent of y , when y is close to α or to β , it follows that y c ( y ) is constantnear the endpoints α , β . This completes the proof of the Lemma in Case I. Case II : Assume that a < γ ′ or a > δ . Here we have [ a, a +3 ǫ ] ⊂ ( −∞ , γ ) ∪ ( δ , + ∞ ).Therefore, the function ∂∂x F ( x, y ) F ( x, y ) = u ′ ( x ) u ( x )is independent of y , as well as ∂∂x G ( x, y ) G ( x, y ) = u ′ ( x ) u ( x ) + 1 ǫ φ ′ ( x − aǫ ) φ ( x − aǫ ) , for ( x, y ) ∈ supp ( G ). Also, for ( x, y ) ∈ supp ( G ) one has ∂∂x ∂∂x F ( x, y ) F ( x, y ) = ∂∂x u ′ ( x ) u ( x ) = u ′′ ( x ) u ( x ) − u ′ ( x ) u ( x ) Thus, since u ( x ) = e − x − α for x ∈ ( α , γ ), and u ( x ) = e − β − x for x ∈ ( δ , β ), we obtain ∂∂x ∂∂x F ( x, y ) F ( x, y ) < , for ( x, y ) ∈ supp ( G )As in Case I, by combining (3.1.22 ) and (3.1.23 ), one has ∂∂x ǫ φ ′ ( x − aǫ ) φ ( x − aǫ ) < , for ( x, y ) ∈ supp ( G )Therefore, we conclude that ∂∂x ∂∂x G ( x, y ) G ( x, y ) < , for ( x, y ) ∈ supp ( G )20s in the previous case, since x → G ( x, y ) is positive in the interior of its support supp ( G ) = [ a , a ] × [ α , β ], for each fixed y ∈ ( α , β ) there exists x ∈ ( a , a )such that ∂∂x G ( x, y ) = 0. Therefore for each fixed y ∈ ( α , β ), there is a unique x = c ( y ) ∈ ( a , a ), such that ∂∂x G ( x,y ) G ( x,y ) = 0. Moreover, since the function ∂∂x G ( x,y ) G ( x,y ) is independent of y for ( x, y ) ∈ supp ( G ), we conclude that the function y c ( y ) isconstant on ( α , β ). This completes the proof of lemma 3.12. Lemma 3.13.
In the same setting as in Lemma 3.12, for any open neighborhood V of U = supp ( G ) = [ a , a ] × [ α , β ] , there exists a compactly supported diffeomorphism Φ : V → V , such that H = Φ ∗ G satisfies | ∂∂x H | k G k ∞ a − a , and supp ( H ) = supp ( G ) . Proof of Lemma 3.13.
We divide the proof of the lemma into two steps:
Step I:
Let V be an open neighborhood of U = supp ( G ) = [ a , a ] × [ α , β ]. Take e α < α < β < e β , such that [ a , a ] × [ e α , e β ] ⊂ V . Moreover, take e a , e a such that a < e a < min [ α ,β ] c ( y ) ≤ max [ α ,β ] c ( y ) < e a < a , and, e a < a + a < e a . One can easily find a smooth family of diffeomorphisms f t : ( a , a ) → ( a , a ), t ∈ ( e a , e a ), such that: supp ( f t ) ⊂ [ e a , e a ] ,f t ( a + a ) = t,f a a = 1l ( a ,a ) We extend the function c ( y ) to a smooth function on the interval ( e α , e β ), such that c ( y ) = a + a , for y close enough to the points e α , e β . Next, define a diffeomorphismΨ : ( a , a ) × ( e α , e β ) → ( a , a ) × ( e α , e β )by the requirement: Ψ ( x, y ) = ( f c ( y ) ( x ) , y ) . It is not hard to check that the diffeomorphism Ψ is the identity near the boundaryof the rectangle ( a , a ) × ( e α , e β ), and therefore one can extend it by the identity,allowing ourselves a slight abuse of notation, to a diffeomorphism Ψ : V → V .Denote G = Ψ ∗ G . It follows from the definition of Ψ that for y ∈ ( α , β ), one has: ( ∂∂x G ( x, y ) > , for a < x < a + a , ∂∂x G ( x, y ) < , for a + a < x < a , (3.1.25)21nd moreover that supp ( G ) = [ a , a ] × [ α , β ], and G ( x, y ) = u ( x ) v ( y ) for x ∈ [ a , a ] and y being near α or β , where u ( x ) = ( f c ( α ) ) ∗ ( u ( x ) φ ( x − aǫ )), v ( x ) = v ( x ). Step II:
Let 0 < ǫ < a − a , and consider three families of smooth positive functions χ ǫj :[ a , a ] → [0 , j = 1 , ,
3, such that the following holds: χ ǫ ( x ) = ( , for x ∈ [ a , a + ǫ ] ∪ [ a + a − ǫ, a + a + ǫ ] ∪ [ a − ǫ, a ] , , for x ∈ [ a + 2 ǫ, a + a − ǫ ] ∪ [ a + a + 2 ǫ, a − ǫ ] ,χ ǫ ( x ) = ( , for x ∈ [ a , a + ǫ ] ∪ [ a + a − ǫ, a ] , , for x ∈ [ a + 2 ǫ, a + a − ǫ ] ,χ ǫ ( x ) = ( , for x ∈ [ a , a + a + ǫ ] ∪ [ a − ǫ, a ] , , for x ∈ [ a + a + 2 ǫ, a − ǫ ] , and moreover, ( χ ǫ ( x ) > , for x ∈ ( a + ǫ, a + a − ǫ ) ,χ ǫ ( x ) > , for x ∈ ( a + a + ǫ, a − ǫ ) . Next, denote by C ∞ ([ a , a ]) the set of smooth functions [ a , a ] → R , such thatthe derivatives of any order (including zero) vanish at the boundary points a and a .Fix g ∈ C ∞ ([ a , a ]), and define h ǫ ( x ) by: h ǫ ( x ) = g ′ ( x ) χ ǫ ( x ) + Aχ ǫ ( x ) − Bχ ǫ ( x ) , where A and B are two constants given by: A = g ( a + a ) − R a a a g ′ ( x ) χ ǫ ( x ) dx R a a a χ ǫ ( x ) dx , and B = g ( a + a ) + R a a a g ′ ( x ) χ ǫ ( x ) dx R a a a χ ǫ ( x ) dx . Note that one has: Z a a a h ǫ ( x ) dx = g ( a + a , and Z a a a h ǫ ( x ) dx = − g ( a + a . g ǫ : [ a , a ] → R be the unique function such that g ′ ǫ ( x ) = h ǫ ( x ), and g ǫ ( a ) = 0.It follows from the definition that g ǫ ( x ) = g ( x ) , for x ∈ [ a , a + ǫ ] ∪ [ a + a − ǫ, a + a ǫ ] ∪ [ a − ǫ, a ] , and in particular, g ǫ ∈ C ∞ ([ a , a ]). Note moreover that if g ( x ) satisfies g ′ ( x ) > x ∈ ( a , a + a ) and g ′ ( x ) < x ∈ ( a + a , a ), then so is g ǫ ( x ) i.e., g ′ ǫ ( x ) > x ∈ ( a , a + a ) and g ′ ǫ ( x ) < x ∈ ( a + a , a ).Next, we define a family of operators L ǫ : C ∞ ([ a , a ]) → C ∞ ([ a , a ]), by therequirement that L ǫ g = g ǫ . It is not hard to check that L ǫ is linear, and continuousin the C ∞ -topology. Moreover, let I ǫ := [ a , a + 2 ǫ ] ∪ [ a + a − ǫ, a + a ǫ ] ∪ [ a − ǫ, a ]Then, from the definition of g ǫ , and the fact that χ ǫ and χ ǫ has disjoint support, onehas the following estimate:max [ a ,a ] | g ′ ǫ ( x ) | max x ∈I ǫ | g ′ ( x ) | + max {| A | , | B |} . Furthermore, from the definition of A and B one has: | A | , | B | | g ( a + a ) | + 4 ǫ max x ∈I ǫ | g ′ ( x ) | a − a − ǫ . Therefore, we conclude thatmax [ a ,a ] | g ′ ǫ ( x ) | | g ( a + a ) | a − a − ǫ + (cid:18) ǫ a − a − ǫ (cid:19) max x ∈I ǫ | g ′ ( x ) | . (3.1.26)Next, define H ǫ : [ a , a ] × [ α , β ] → R by H ǫ ( · , y ) = L ǫ G ( · , y ) for every y ∈ [ α , β ].Note that H ǫ | [ a ,a ] × [ α ,β ] is a smooth function. Moreover, if ǫ > | ∂∂x H ǫ ( x, y ) | k G k ∞ a − a = 3 k G k ∞ a − a , for every ( x, y ) ∈ [ a , a ] × [ α , β ]We fix such an ǫ , and set H := H ǫ . From the definition of H and (3.1.25 ) one has: ( ∂∂x H ( x, y ) > , for a < x < a + a , ∂∂x H ( x, y ) < , for a + a < x < a , (3.1.27)for any y ∈ ( α , β ). Furthermore, H ( x, y ) = G ( x, y ) (3.1.28)23or x ∈ [ a , a + ǫ ] ∪ [ a + a − ǫ, a + a + ǫ ] ∪ [ a − ǫ, a ], and y ∈ ( α , β ). Note moreoverthat since the operator L ǫ is linear, one has that H ( x, y ) = ( L ǫ u )( x ) v ( y ) for any x ∈ [ a , a ] and y being near the boundary points α or β .It follows from (3.1.27 ) and (3.1.28 ) above, that there is a unique diffeomorphismΨ : ( a , a ) × ( α , β ) → ( a , a ) × ( α , β ), of the form Ψ ( x, y ) = ( w ( x, y ) , y ), suchthat H | ( a ,a ) × ( α ,β ) = Ψ ∗ G | ( a ,a ) × ( α ,β ) , and supp (Ψ) ⊂ (cid:0) [ a + ǫ, a + a − ǫ ] ∪ [ a + a + ǫ, a − ǫ ] (cid:1) × [ α , β ]. Moreover, we have G ( x, y ) = u ( x ) v ( y ), H ( x, y ) = ( L ǫ u )( x ) v ( y ) for x ∈ [ a , a ] and y being near α or β . From this we conclude that w ( x, y ) is independent of y , for y being close to α , β . From Step I, we have e α < α < β < e β , such that [ a , a ] × [ e α , e β ] ⊂ V .One can easily extend the diffeomorphism Ψ toΨ : ( a , a ) × ( e α , e β ) → ( a , a ) × ( e α , e β ) , such that Ψ is the identity diffeomorphism near the boundary of ( a , a ) × ( e α , e β ).Then we can extend Ψ by the identity to be a diffeomorphism Ψ : V → V . We have H = Ψ ∗ G .Finally, denote Φ = Ψ Ψ : V → V . The diffeomorphism Φ is compactly sup-ported inside V , and H = Φ ∗ G satisfies | ∂∂x H | k G k ∞ a − a , and supp ( H ) = supp ( G )This completes the proof of the lemma.We are finally in a position to prove Lemma 3.8. Proof of Lemma 3.8.
Let f : R → R be a smooth function with k f k ∞
1, and supp ( f ) ⊂ int ( R ). We fix some parameters α i , α ′ i , β i , β ′ i , where i = 1 ,
2, such that0 < α i < α ′ i < β ′ i < β i < L i , for i = 1 , β − α > L ; and supp ( f ) ⊂ int ([ α ′ , β ′ ] × [ α ′ , β ′ ]) ⊂ int ([ α , β ] × [ α , β ]) ⊂ int ( R )Moreover, we choose a smooth function u : [0 , L ] → R , such that u ( x ) = e − x − α near α , u ( x ) = e − β − x near β , u ( x ) = 1 on [ α ′ , β ′ ], and k u k ∞ = 1. Similarly, we take v : [0 , L ] → R , with v ( y ) = e − y − α near α , v ( y ) = e − β − y near β , v ( y ) = 2 on [ α ′ , β ′ ],and k v k ∞ = 2. Next, we consider the decomposition f = F − F , where F ( x, y ) = f ( x, y ) + u ( x ) v ( y ) , and F ( x, y ) = u ( x ) v ( y )We have k F ς ( x, y ) k ∞ ς ∈ { , } . From Lemma 3.13 it follows that there is ǫ > < ǫ < ǫ , and any a ∈ R , the following holds: let G ς ( x, y ) =24 ς ( x, y ) φ ( x − aǫ ), where ς ∈ { , } (we may and shall assume in what follows that G ς = 0). Take V ς to be any open neighborhood of U ς := supp ( G ς ) = [ a ς , a ς ] × [ α , β ].Then, there is a compactly supported diffeomorphism Φ ς : V ς → V ς , such that H ς = (Φ ς ) ∗ G ς satisfies (cid:12)(cid:12) ∂∂x H ς (cid:12)(cid:12) a ς − a ς , and supp ( H ς ) = supp ( G ς ) (3.1.29)Fix 0 < ǫ < ǫ as above. For n ∈ Z and ς ∈ { , } denote G ς,n = F ς ( x, y ) φ ( x − nǫǫ ).Note that F ς = P n ∈ Z G ς,n , and that only finitely many summands are not identicallyzero. For i = 1 , , ,
4, let K ς,i = P j ∈ Z G ς,i +4 j . Note moreover that the supports of allthe non-zero summands of K ς,i are pairwise disjoint, and F ς = P i =1 K ς,i , and thus f = P ς =1 P i =1 K ς,i . Next, we fix 1 i
4. Consider K ς,i = P j ∈ Z G ς,i +4 j , and choosepairwise disjoint open neighborhoods V ςi ,j ⊃ supp ( G ς,i +4 j ) of those summands whichare not identically zero. Now, apply Lemma 3.13 to each element in the decomposition K ς,i = P j ∈ Z G ς,i +4 j . We obtain that for any non-zero summand G ς,i +4 j , there isa compactly supported diffeomorphism Φ ςi ,j : V ςi ,j → V ςi ,j , such that the function H ςi ,j = (Φ ςi ,j ) ∗ G ς,i +4 j satisfies (cid:12)(cid:12) ∂∂x H ςi ,j (cid:12)(cid:12) µ ( π x ( supp ( G ς,i +4 j ))) , and supp ( H ςi ,j ) = supp ( G ς,i +4 j ) (3.1.30)Here π x denotes the projection to the interval [0 , L ], and µ is the Lebesgue measure.Note that the supports { supp (Φ ςi ,j ) } are mutually disjoint. We shall denote by e Φ ςi the composition of all the Φ ςi ,j ’s for which G ς,i +4 j = 0. Moreover, we denote by Π ςi ,k , k = 1 , , ..., M i all the non-empty supports among { supp ( G ς,i +4 j ) } . Note that eachΠ ςi ,k is a rectangle contained in [ α , β ] × [ α , β ]. Consider a sequence of rectangles e Π ςi ,k := [ α , β ] × [ α + (2 k − β − α M i , α + 2 k β − α M i ]It is not hard to check that there exists a diffeomorphism Ψ ςi : R → R , such thatΨ ςi ( e Π ςi ,k ) = Π ςi ,k , and moreover that on each e Π ςi ,k it coincides with a linear con-traction on the directions of the axes, composed with a translation. As a result, for k ς,i := (Ψ ςi ) ∗ ( e Φ ςi ) ∗ K ς,i , one has | ∂∂x k ς,i | β − α < L . The proof of Lemma 3.8 isnow complete. The proof of Theorem 3.4 for arbitrary dimension relies on the 2-dimensional case,and on the following proposition, the proof of which we postpone to Subsection 3.2.1.
Proposition 3.14.
There is a finite family of functions
F ⊂ C ∞ c ( W ) , such that: i) Any f ∈ C ∞ c ( W ) that can be represented as a product f ( q, p ) = Q ni =1 f i ( q i , p i ) ,for some f i ∈ C ∞ c ( I ) , satisfies that k f k F , max C k f k ∞ , for some constant C .(ii) For any f ∈ C ∞ c ( W ) , one has k f k F , max C k f k C n +1 , for some constant C . Remark 3.15.
In what follows, we fix F to be the collection of functions given byProposition 3.14 above. Moreover, in order to simplify the presentation, we shall use x = q , x = p , ..., x n − = q n , x n = p n , as another notation for the coordinates of apoint x = ( q , p , ..., q n , p n ) in the 2 n -dimensional cube W = ( − L, L ) n . Proof of Theorem 3.4 ( the higher dimensional case).
For simplicity, the proof ofthe theorem is divided into two steps:
Step I (
Decomposing the function):
We consider a smooth function r : [ − , → R ,satisfying: r ( t ) = ( t ∈ [ − , ] , t ∈ [ − , − ] ∪ [ , , and such that P i ∈ Z r ( t + i ) = 1, and k r k ∞ = 1. For any ǫ >
0, we denote R ǫ ( x ) = R ǫ ( x , x , ..., x n ) = n Y i =1 r (cid:16) x i ǫ (cid:17) Clearly, one has P v ∈ ǫ Z n R ǫ ( x − v ) = 1l( x ). Moreover, for a sufficiently small ǫ > w ∈ X := { , , , } n , we consider a finite grid Γ ǫw ⊂ W given byΓ ǫw = ǫw + 4 ǫ Z n ∩ ( − L + 3 ǫ, L − ǫ ) n (3.2.31)Furthermore, we define a partition function R ǫw ( x ) by: R ǫw ( x ) = X v ∈ Γ ǫw R ǫ ( x − v )Note that P w ∈ X R ǫw ( x ) = 1l( x ) for any x ∈ ( − L + 4 ǫ, L − ǫ ) n . Next, consider anarbitrary function f ∈ C ∞ c ( W ). Take ǫ > supp ( f ) ⊂ ( − L + 4 ǫ , L − ǫ ) n ,and fix ǫ < ǫ . For any w ∈ X , denote f w ( x ) = R ǫw ( x ) f ( x ). Note that f ( x ) = X w ∈ X f w ( x )Moreover, for a fix w ∈ X one has f w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) f ( x ) , (3.2.32)where the support of each summand satisfies supp (cid:0) R ǫ ( x − v ) f ( x ) (cid:1) ⊂ v + h − ǫ , ǫ i n , for v ∈ Γ ǫw tep II ( Estimating the norm k f k F , max ): Fix v ∈ Γ ǫw , and consider the decomposi-tion of f ∈ C ∞ c ( W ) to a Taylor polynomial of order 2 n + 1 and a remainder, aroundthe point v : f ( x ) = P v n +1 ( x − v ) + R v n +1 ( x − v )It follows from (3.2.32 ) above that f w ( x ) = g w ( x ) + h w ( x ), where g w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) P v n +1 ( x − v ) , and h w ( x ) = X v ∈ Γ ǫw R ǫ ( x − v ) R v n +1 ( x − v ) Lemma 3.16.
With the above notations, there is a constant C = C ( n ) such that k h w k C n +1 Cǫ k f k C n +2 Proof of Lemma 3.16.
From the fact that the family {R ǫ ( x − v ) R v n +1 ( x − v ) } v ∈ Γ ǫw has mutually disjoint support, and the definition of the norm k · k C n +1 , it follows thatthere is a constant C (depending on the dimension) such that k h w ( x ) k C n +1 ≤ max v ∈ Γ ǫw kR ǫ ( x − v ) R v n +1 ( x − v ) k C n +1 ≤ C ( n ) max v ∈ Γ ǫw (cid:16) max ≤ k ≤ n +1 kR ǫ ( x − v ) k C k k R v n +1 ( x − v ) k C n +1 − k (cid:17) Note that from the definition of R ǫ it follows that for every 0 ≤ k ≤ n + 1, one has kR ǫ ( x − v ) k C k C ′ ǫ − k , for some constant C ′ (independent of k ). Note moreover, that for 0 ≤ k ≤ n + 1 , k R v n +1 ( x − v ) k C n +1 − k C ′′ k f k C n +2 ǫ k , (3.2.33)for some constant C ′′ . Indeed, let α be a multiindex with | α | = 2 n + 1 − k , andconsider the order- k Taylor’s expension of ∂ α f near the point v . The remainderequals to ∂ α R v n +1 ( x − v ), and the estimate (3.2.33 ) follows from the standard boundon the size of the remainder. This completes the proof of the lemma. Corollary 3.17.
From Proposition 3.14 (ii), and Lemma 3.16, we conclude that: k h w k F , max Cǫ k f k C n +2 , for some constant C = C ( n ) (3.2.34)To complete the proof of the theorem we shall need the following proposition: Proposition 3.18.
There is a constant C = C ( n ) such that k g w k F , max C (cid:0) n +1 X i =0 k f k C i ǫ i (cid:1) (3.2.35)Postponing the proof of Proposition 3.18 to Subsection 3.2.2, we first completethe proof of Theorem 3.4. From (3.2.34 ) and (3.2.35 ), letting ǫ →
0, we concludethat k f k F , max ≤ C k f k ∞ , for some absolute constant C , and the proof is complete.27 .2.1 Proof of Proposition 3.14Part (i): Let W = Q ni =1 W i , where W i = ( − L, L ) ⊂ R ( q i , p i ), and denote by F = { f , f , f } the collection of functions constructed in the proof of Theorem 3.4in the 2-dimensional case. For any multi-index β = ( l , . . . , l n ) ∈ X ′ := { , , } n , weset f β ( q, p ) = Q nk =1 f l k ( q k , p k ). In what follows we denote by F the set { f β ; β ∈ X ′ } .Consider f ∈ C ∞ c ( W n ) of the form f ( q, p ) = Q ni =1 f i ( q i , p i ), where f i ∈ C ∞ c ( W i ).Let ǫ >
0. From the proof of Theorem 3.4 in the 2-dimensional case it follows thatthere exists functions f i,k ∈ L F , i = 1 , , ..., n ; k ∈ N , such that f i,k k →∞ −−−→ f i in the C ∞ -topology, and such that k f i,k k L F < k f i k F , max + ǫ . Next, for every 1 ≤ i ≤ n and k ∈ N , we decompose f i,k = X j,l c j,li,k (Φ j,li,k ) ∗ f l , (3.2.36)where Φ j,li,k ∈ Ham c ( W i , ω ); l ∈ { , , } , and, X j,l | c j,li,k | < k f i,k k L F + ǫ (3.2.37)Denote f k ( q, p ) = Q ni =1 f i,k ( q i , p i ). Clearly, f k k →∞ −−−→ f ∈ C ∞ c ( W ) in the C ∞ -topology.Moreover, from (3.2.36 ) it follows that f k = X β =( l ,...,l n ) γ =( j ,...,j n ) c γ,βk (Φ γ,βk ) ∗ f β , where c γ,βk = n Y i =1 c j i ,l i i,k , and Φ γ,βk ( q , p , ..., q n , p n ) = (cid:16) Φ j ,l ,k ( q , p ) , ..., Φ j n ,l n n,k ( q n , p n ) (cid:17) This shows that f k ∈ L F , and moreover that k f k k L F X β =( l ,...,l n ) γ =( j ,...,j n ) | c γ,βk | = n Y i =1 X j i ,l i | c j i ,l i i,k | ! < n Y i =1 (cid:0) k f i,k k L F + ǫ (cid:1) n Y i =1 ( k f i k F , max + 2 ǫ ) (3.2.38)Recall, that from the proof of Theorem 3.4 in the 2-dimensional case one has k f i k F , max C k f i k ∞ , for some absolute constant C . Combining this with (3.2.38 ) we conclude that k f k k L F n Y i =1 ( C k f i k ∞ + 2 ǫ ) , k f k F , max lim inf k →∞ k f k k L F n Y i =1 ( C k f i k ∞ + 2 ǫ )In particular, for any ǫ >
0, one has k f k F , max n Y i =1 ( C k f i k ∞ + 2 ǫ )Taking ǫ →
0, we obtain k f k F , max C n n Y i =1 k f i k ∞ = C n k f k ∞ This completes the proof of part (i) of Proposition 3.14.For the proof of the second part of Proposition 3.14 we shall need the followingpreliminaries. Let f be an integrable function on the m -dimensional torus T m , anddenote its Fourier coefficients byˆ f r = 1(2 π ) m Z T m f ( t ) e ir · t dt, where r = ( r , . . . , r m ) ∈ Z m , and t = ( t , . . . , t m ) ∈ T m . We denote the j th -partialsum of the Fourier series of f by S j ( f, t ) = X max | r l |≤ j ˆ f r e ir · t The next lemma is a well known result in Fourier analysis.
Lemma 3.19.
Let f ∈ C ∞ ( T m ) . Then S j ( f ) j →∞ −−−→ f in the C ∞ -topology and X r ∈ Z m | ˆ f r | ≤ A k f k C n +1 , (3.2.39) for some universal constant A . Proof of Lemma 3.19.
The fact that S j ( f ) j →∞ −−−→ f in the C ∞ -topology follows,e.g., from Theorem 33.7 in Section 79 of [9], and the fact that ∂ α S j ( f ) = S j ( ∂ α f )for every multi-index α and j ≥
0. For the estimate (3.2.39 ), we use Lemma 9.5 inSection 79 of [9] to obtain the following upper bound for the Fourier coefficients: | ˆ f r | ≤ A k f k C n +1 k r k n +1 for all r = 0 , (3.2.40)for some constant A . From this we conclude that X r ∈ Z m | ˆ f r | ≤ A k f k C n +1 Z S n − Z ∞ ρ − n − ρ n − dρ dθ ≤ A k f k C n +1 , where A = A is a constant which depends solely on the dimension.29 emark 3.20. We remark that Lemma 3.19 holds (with different constants) forany torus of the form T m = ( R /a Z ) m , where a >
0. Moreover, the lemma holds ifinstead of the basis { e πia rt } , we choose the trigonometric basis consists of productsof { cos( πa r i t i ) } or { sin( πa r i t i ) } for i = 1 , . . . , m .We now turn to complete the proof of the second part of Proposition 3.14: Proposition 3.14, Part (ii):
Let f ∈ C ∞ c ( W ). By gluing together the boundaryof the cube W in an appropriate way, we obtain a well defined smooth function onthe torus T n = ( R / L Z ) n , which by abuse of notation we still denote by f . Weapply Lemma 3.19 to the function f (note the comment regarding the trigonometricbasis in Remark 3.20). We order the trigonometric basis in Remark 3.20 by { e k } ∞ k =1 .Note that each e k is a product function with k e k k ∞ = 1. Denoting the correspondingFourier sums of f by S k = P ki =1 c i e i . We have S k → f in the C ∞ -topology and P ∞ k =1 | c k | A k f k C n +1 for some A = A ( n ). We turn back to the situation where weconsider f defined on W . Take any smooth cutoff function ρ : W → R , which equals1 on supp ( f ), equals 0 near the boundary ∂W , and which has k ρ k ∞ = 1 (one caneasily find such ρ , since supp ( f ) ⊂ W ). Then we have ρS k = P ki =1 c i ρe i → ρf = f in C ∞ c ( W ), in the C ∞ topology as well. Moreover, the functions { ρe k } are productfunctions with k ρe k k ∞
1. From part (i) or Proposition 3.14, and Lemma 3.19, itfollows that for a suitable collection F , one has k S k k F , max k X i =1 | c i |k ρe i k F , max C k X i =1 | c i | CA k f k C n +1 . Hence, from Remark 3.2 we conclude that k f k F , max CA k f k C n +1 . The proof of the second part of the proposition is now complete.
For any multi-index α = ( i , i , ..., i n ), where | α | n + 1, denote g αw ( x ) = X v =( v ,v ,...,v n ) ∈ Γ ǫw i ! i ! ...i n ! ∂f | α | ∂x i ∂x i ...∂x i n n ( v ) (cid:16) n Y j =1 ( x j − v j ) i j (cid:17) R ǫ ( x − v )Note that the function g w is the sum of g αw , for α = ( i , i , ..., i n ) with | α | n + 1.Note moreover that each summand of g αw is a constant multiple of the function Ξ α ( x − v ) := (cid:16) n Y j =1 ( x j − v j ) i j (cid:17) R ǫ ( x − v ) , Ξ α ( x ) = x i x i ...x i n n R ǫ ( x ) = n Y l =1 q i l − l p i l l r (cid:16) q l ǫ (cid:17) r (cid:16) p l ǫ (cid:17) We shall need the following lemma which will be proven in Subsection 3.2.3
Lemma 3.21.
Let ξ ∈ C ∞ c (( − ǫ, ǫ ) n ) be a compactly supported smooth function whichcan be represented as a product ξ = Q nj =1 ξ j ( q j , p j ) , where ξ j ∈ C ∞ c (( − ǫ, ǫ ) ) . Then,for every function H ( x ) = P v ∈ Γ ǫw a v ξ ( x − v ) , where a v are real coefficients and Γ ǫw isthe grid defined in ( ) , one has k H k F , max C k H k ∞ , for some absolute constant C Applying Lemma 3.21, with ξ = Ξ α , to the function H = g αw , we conclude that k g αw k F , max C k g αw k ∞ Ci ! i ! . . . i n ! k Ξ α k ∞ max v ∈ Γ ǫw ∂f | α | ∂x i ∂x i . . . ∂x i n n ( v ) C k Ξ α k ∞ k f k C | α | Since k r k ∞ = 1, and supp ( r ) ⊂ ( − ǫ, ǫ ), it follows that k Ξ α k ∞ ≤ ǫ | α | . Thus, we obtain k g αw k F , max C ǫ | α | k f k C | α | , and hence k g w k F , max X | α | n +1 C ǫ | α | k f k C | α | C ′ n +1 X k =0 ǫ k k f k C k This completes the proof of Proposition 3.18.
Note first that the grid Γ ǫw = ǫw + 4 ǫ Z n ∩ ( − L + 3 ǫ, L − ǫ ) n admits a decompositioninto the product Γ ǫw = Q ni =1 γ i , where γ i = γ ǫ,wi ⊂ ( − L + 3 ǫ, L − ǫ ) ⊂ ( − L, L ) aregrids on the plane. Next, let H be as in Lemma 3.21. Given a bijection τ : Γ ǫw → Γ ǫw ,we denote H τ ( x ) = X v ∈ Γ ǫw a τ ( v ) ξ ( x − v ) Lemma 3.22.
For any bijection τ : Γ ǫw → Γ ǫw , one has k H τ k F , max = k H k F , max . Proof of Lemma 3.22.
It is not hard to check that every bijection τ : Γ ǫw → Γ ǫw ,can be written as a product of transpositions that interchange two neighboring pointsof Γ ǫw (here, by neighboring points we mean v ′ , v ′′ ∈ Γ ǫw , such that | v ′ − v ′′ | = 4 ǫ ).Therefore it is enough to prove the lemma for the case of such a transposition.31et v ′ = ( z ′ , ..., z ′ n ) , v ′′ = ( z ′′ , ..., z ′′ n ) ∈ Γ ǫw be a pair of neighboring points, where z ′ i , z ′′ i ∈ γ i for i = 1 , , ..., n . There exists 1 k n , such that z ′ i = z ′′ i for i = k ,and moreover z ′′ k = z ′ k ± ǫ or z ′′ k = z ′ k ± ǫi . The union of the neighboring squares Q ′ := z ′ k + [ − ǫ, ǫ ] , and Q ′′ := z ′′ k + [ − ǫ, ǫ ] is a rectangle S = Q ′ ∪ Q ′′ . Since thesupport supp ( ξ k ) ⊂ ( − ǫ, ǫ ) , there exists 0 < ǫ < ǫ , such that supp ( ξ k ) ⊂ [ − ǫ , ǫ ] .Looking on Q ′ = z ′ k + [ − ǫ , ǫ ] , Q ′′ = z ′′ k + [ − ǫ , ǫ ] ⊂ int ( S ), one can clearlymove Q ′ to Q ′ and Q ′ to Q ′ simultaneously, using affine translations, such that atevery moment the images of Q ′ , Q ′ will not intersect, and are contained in int ( S ).Moreover, this can be done by a smooth Hamiltonian isotopy Φ tK k , supported in S ,where K k ( t, z k ) : [0 , × W k → R is the Hamiltonian that generates this isotopy,and such that we have supp ( K k ( t, · )) ⊂ int ( S ) for all t ∈ [0 , j = k ,1 j n consider a smooth function K j ( z j ) : W j → R such that K j ( z j ) = 1for z j ∈ z ′ j + [ − ǫ, ǫ ] and K j ( z j ) = 0 for z j ∈ W j \ ( z ′ j + [ − ǫ, ǫ ] ). Now define aHamiltonian K : [0 , × W → R by K ( t ; z , z , ..., z n ) = K k ( t, z k ) Y j nj = k K j ( z j )Note that K ( t ; z , z , ..., z n ) = K k ( t, z k ) for z = ( z , ..., z n ) ∈ U := k − Y j =1 ( z ′ j + [ − ǫ, ǫ ] ) × S × n Y j = k +1 ( z ′ j + [ − ǫ, ǫ ] ) . Moreover, U is invariant under the flow Φ tK , andΦ tK ( z , ..., z n ) = ( z , ..., z k − , Φ tK k ( z k ) , z k +1 , ..., z n ) , for any z = ( z , ..., z n ) ∈ U . In particular, Φ K ( z ) = z + v ′′ − v ′ for z ∈ v ′ + [ − ǫ, ǫ ] n ,and Φ K ( z ) = z + v ′ − v ′′ for z ∈ v ′′ + [ − ǫ, ǫ ] n . Furthermore, for U := k − Y j =1 ( z ′ j + [ − ǫ, ǫ ] ) × S × n Y j = k +1 ( z ′ j + [ − ǫ, ǫ ] )we have that supp ( K ( t, · )) ⊂ U for all t ∈ [0 , v +[ − ǫ, ǫ ] n ) ∩ U = ∅ for all v ∈ Γ ǫw \ { v ′ , v ′′ } , we conclude that Φ K ( z ) = z for z ∈ v + [ − ǫ, ǫ ] n for any v ∈ Γ ǫw \ { v ′ , v ′′ } . Hence if τ : Γ ǫw → Γ ǫw is a transposition that interchanges v ′ with v ′′ , we conclude that H τ = (Φ K ) ∗ H . Therefore we conclude k H τ k F , max = k H k F , max . Proof of Lemma 3.21.
Consider the decomposition Γ ǫw = Q ni =1 γ i , and write each γ i explicitly as γ i = { z i, , ..., z i,N i } ⊂ ( − L, L ) . We order each set γ i by setting32 i, < z i, < ... < z i,N i , for each i , and consider the lexicographic order ≺ on Γ ǫw induced by these orders. We can arrange all the elements of Γ ǫw by increasing order v ≺ v ≺ ... ≺ v N , where N = Q ni =1 N i . Take a bijection τ : Γ ǫw → Γ ǫw such that a τ ( v ′′ ) a τ ( v ′ ) if and only if v ′ (cid:22) v ′′ , where v ′ , v ′′ ∈ Γ ǫw , and rewrite H τ ( x ) = P v ∈ Γ ǫw a τ ( v ) ξ ( x − v ) as H τ ( x ) = N X j =1 b j ξ ( x − v j ) , and b b ... b N (3.2.41)By Lemma 3.22, one has k H τ k F , max = k H k F , max . Next, write H τ ( x ) = b N K N ( x ) + N − X j =1 ( b j − b j +1 ) K j ( x ) , (3.2.42)where K j ( x ) = P jl =1 ξ ( x − v j ). Also set K ( x ) = 0. k H τ k F , max | b N |k K N ( x ) k F , max + N − X j =1 | b j − b j +1 |k K j k F , max | b N |k K N ( x ) k F , max + N − X j =0 ( b j +1 − b j ) max j N k K j k F , max = | b N |k K N ( x ) k F , max + ( b N − b ) max j N k K j k F , max (cid:16) max v ∈ Γ ǫw | a v | (cid:17) max j N k K j k F , max (3.2.43)Next, consider some K j , where 1 j N . There exist a unique sequence j = 0 j j ... j n − j n = j, such that for any 1 m n we have Q nl = m +1 N l | j m − j m − , and we have k l := j m − j m − Q nl = m +1 N l < N m . Here we mean Q nl = n +1 N l = 1. Take any 1 m n . Then provided j m − < j m , wecan write ξ m ( z ) := K j m − K j m − = n Y l =1 ξ ml ( z l ) , ξ ml ( z l ) = ξ l ( z l − z l,k l ) , for l = 1 , ..., m − ,ξ mm ( z m ) = k m X i m =1 ξ m ( z m − z m,i m ) ,ξ ml ( z l ) = N l X i l =1 ξ l ( z l − z l,i l ) , for l = m + 1 , ..., n. Moreover, for any 1 m n we have k ξ m k ∞ = n Y l =1 k ξ ml k ∞ = n Y l =1 k ξ l k ∞ = k ξ k ∞ . From this, and from Proposition 3.14 (i), we conclude that k ξ m k F , max C k ξ m k ∞ = C k ξ k ∞ , for some C = C ( n ). We have K j = n X m =1 ξ m , hence k K j k F , max n X m =1 k ξ m k F , max nC k ξ k ∞ , and this holds for any 1 j N . Therefore we conclude k H k F , max = k H τ k F , max (cid:18) max v ∈ Γ ǫw | a v | (cid:19) max j N k K j k F , max nC (cid:18) max v ∈ Γ ǫw | a v | (cid:19) k ξ k ∞ = 3 nC k H k ∞ . The proof of the lemma is now complete.
The proof of Theorem 1.3 follows from Theorem 3.4 by a standard partition of unityargument. For the sake of completeness, we provide the details below.As explained in Section 2, it is enough to prove Theorem 1.3 for Ham(
M, ω )-invariant pseudo norms on C ∞ ( M ). Indeed, any Ham( M, ω )-invariant pseudo norm k · k on A that is continuous in the C ∞ -topology, can be naturally extended to a34am( M, ω )-invariant pseudo-norm k · k ′ on C ∞ ( M ), which is again continuous in the C ∞ -topology, by setting k f k ′ = k f − M f k , where M f = V ol ( M ) R M f ω n Consider a Darboux chart i : U ֒ → M , where U ⊂ ( R n , ω std ) is an open set.Without loss of generality we assume that the origin of R n lies inside U . Choosesome L >
0, such that W = ( − L, L ) n ⊂ U . Since i ( W ) ⊂ M , we have a naturalembedding C ∞ c ( i ( W )) ֒ → C ∞ ( M ), and therefore any Ham( M, ω )-invariant pseudonorm k · k on C ∞ ( M ) restricts to C ∞ c ( i ( W )). From Lemma 3.3 and Theorem 3.4,we conclude that (when the norm is continuous in the C ∞ -topology) there exists aconstant C > k f k C k f k ∞ , for every function f ∈ C ∞ c ( i ( W ))Next, for any point x ∈ M there exists an open neighborhood V x ⊂ M , and a smoothHamiltonian diffeomorphism Φ x ∈ Ham(
M, ω ), such that Φ x ( V x ) ⊂ W . Consider theopen covering S x ∈ M V x = M . The compactness of M allows us to pass to a finitesubcover S Ni =1 V x i = M . Moreover, one can find a partition of unity { ρ , ρ , ..., ρ N } ,such that for every i = 1 , , ..., N , ρ i : M → R is a smooth positive function supportedin V x i , and ρ + ρ + ... + ρ N = 1l M Finally, let f ∈ C ∞ ( M ), and consider the decomposition f = ρ f + ρ f + ... + ρ N f Since k · k is a Ham(
M, ω )-invariant norm, it follows that k f k N X i =1 k ρ i f k = N X i =1 k (Φ − x i ) ∗ ( ρ i f ) k Moreover, it follows from the above that supp (cid:0) (Φ − x i ) ∗ ( ρ i f ) (cid:1) ⊂ W , and hence k (Φ − x i ) ∗ ( ρ i f ) k C k (Φ − x i ) ∗ ( ρ i f ) k ∞ = C k ρ i f k ∞ C k f k ∞ . Therefore we conclude that k f k C ′ k f k ∞ , where C ′ = N C . The proof of the theorem is now complete.
Here we prove the claim mentioned in Remark 1.1. More precisely:35 roposition 5.1.
Let M be a closed symplectic manifold, and let k · k be a normon the Lie algebra A of Ham ( M, ω ) . Then, smooth paths [0 , → Ham ( M, ω ) havefinite length if and only if the norm k · k is continuous in the C ∞ -topology.Proof. The “if” part of the statement is clear. Let us show the “only if” part.Throughout, we equip M with a Riemmanian metric, and denote k · k ∞ = k · k C k · k C k · k C ... the corresponding C , C , C , ... -norms on C ∞ ( M ).Let k · k be an invariant pseudo-norm on C ∞ ( M ) which is not continuous in the C ∞ -topology. Consider two sequences { a k } , { b k } in the interval [0 , < a < b < a < b < ... < c : [0 , → [0 ,
1] be a smooth function such that c ( t ) = 0 for t ∈ [0 , ] ∪ [ , c ( t ) = 1 for t ∈ [ , ]. For a sequence of smooth functions H k : M → R , wedefine a function H : M × [0 , → R in the following way: H ( x, t ) = t ∈ [0 , a ] ∪ [ b , a ] ∪ [ b , a ] ∪ ...,c ( t − a k b k − a k ) H k ( x ) for t ∈ [ a k , b k ] , t = 1 . (5.1)Note that H is smooth on M × [0 , H k ∈ C ∞ ( M ), one has H ( x, t ) ∈ C ∞ ( M × [0 , Z k H ( · , t ) k dt = + ∞ (5.2)Thus, the Hamiltonian flow of H has infinite length with respect to the Finsler metric d k·k . Indeed, note that Z k H ( · , t ) k dt = ∞ X k =1 ( b k − a k ) (cid:18)Z | c ( t ) | dt (cid:19) k H k k > ∞ X k =1 ( b k − a k ) k H k k . Hence, for the estimate (5.2 ), it is enough to choose H k such that k H k k > b k − a k .Moreover, to ensure that H ( x, t ) is smooth in M × [0 , t → k ∂ j ∂t j H ( t, · ) k C m = 0 , for any j, m > t ∈ ( a k , b k ). Note that in that case k ∂ j ∂t j H ( t, · ) k C m = (cid:18) b k − a k (cid:19) j (cid:12)(cid:12)(cid:12) c ( j ) ( t − a k b k − a k ) (cid:12)(cid:12)(cid:12) k H k k C m (cid:18) b k − a k (cid:19) j k c k C j k H k k C m . Therefore, to show (5.3 ) it is enough to choose H k such thatlim k →∞ (cid:18) b k − a k (cid:19) j k H k k C m = 0 , for any j, m > H k ∈ C ∞ ( M ), that for every k > ( k H k k > b k − a k , k H k k C k ( b k − a k ) k , (5.4)would give rise (via definition (5.1 )) to a smooth function H : M × [0 , → R , suchthat R k H ( · , t ) k dt = + ∞ .Since the norm k · k is assumed to be non-continuous in the C ∞ -topology, one canalways find a sequence { H k } which satisfy (5.4 ). References [1] Arnold, V. I.
Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimensioninfinie et ses applications `a l’hydrodynamique des fluides parfaits, (French) Ann.Inst. Fourier (Grenoble) 16 1966 fasc. 1 319-361.[2] Arnold, V. I., Khesin, B. A.
Topological methods in hydrodynamics,
AppliedMathematical Sciences, 125. Springer-Verlag, New York, 1998.[3] Banyaga, A.
Sur la structure du groupe des diff´eomorphisms qui pr´eservent uneforme symplectique.
Comment. Math. Helv. (1978), no.2, 174-227.[4] Eliashberg, Y. Symplectic topology in the nineties,
Symplectic geometry. Differ-ential Geom. Appl. 9 (1998), no. 1-2, 59-88.[5] Eliashberg, Y., Polterovich, L.
Bi-invariant metrics on the group of Hamiltoniandiffeomorphisms,
Internat. J. Math. (1993), 727-738.[6] Hofer, H. On the topological properties of symplectic maps.
Proceedings of theRoyal Society of Edinburgh, (1990), 25-38.[7] Hofer, H., Zehnder, E.
Symplectic Invariants and Hamiltonian Dynamics ,Birkh¨auser Advanced Texts, Birkh¨auser, Basel (1994).[8] Khesin, B. A., Wendt, R.
The geometry of infinite-dimensional groups,
Ergeb-nisse der Mathematik, vol. 51. Springer, New York (2008).[9] K¨orner, T. W.
Fourier analysis,
Cambridge University Press, Cambridge, 1988.[10] Lalonde, F., McDuff, D.
The geometry of symplectic energy , Ann. of Math 141(1995), 349-371. Math 141 (1995), 349-371.3711] McDuff, D., Salamon, D.
Introduction to Symplectic Topology,
Spectral invariants, analysis of the Floer moduli space, and geometryof the Hamiltonian diffeomorphism group,
Duke Math. J. 130 (2005), 199-295.[13] Ostrover, Y., Wagner, R.
On the extremality of Hofer’s metric on the group ofHamiltonian diffeomorphisms , Inter. Math. Res. Notices 35, 2123-2142 (2005).[14] Polterovich, L.
Symplectic displacement energy for Lagrangian submanifolds,
Er-godic Theory and Dynamical Systems, 13 (1993), 357-67.[15] Polterovich, L.
The geometry of the group of symplectic diffeomorphisms , Lec-tures in Mathematics ETH Z¨urich. Birkh¨auser Verlag, Basel, 2001.[16] F. Schlenk,
Embedding problems in symplectic geometry , De Gruyter Expositionsin Mathematics, , Walter de Gruyter Verlag, Berlin, 2005.[17] Schwarz, M. On the action spectrum for closed symplectically aspherical mani-folds,
Pacific J. Math. 193 (2000), 1046-1095.[18] Viterbo, C.
Symplectic topology as the geometry of generating functions,
Math.Ann. 292 (1992) 685-710.Lev BuhovskyThe Mathematical Sciences Research Institute, Berkeley, CA 94720-5070 e-mail : [email protected] OstroverSchool of Mathematics, Institute for Advanced Study, Princeton NJ 08540 e-maile-mail