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In this paper we study the Benjamin-Ono equation on the torus T := R/2πZ,
∂ u = ∂ ∂ u u2 , (1) t x | x| − T R β β−1 R where u u(x, t), x , t , is real valued and ∂x : Hc Hc , β , is the Fourier≡ multiplier∈ ∈ | | → ∈
inx inx ∂x : v(n)e n v(n)e | | Z 7→ Z | | nX∈ nX∈ Z b b β β β T C where v(n), n , are the Fourier coefficients of v Hc and Hc H ( , ) is the Sobolev space∈ of complex valued distributions on∈ the torus T.≡ The equation (1) wasb introduced in 1967 by Benjamin [1] and Davis & Acrivos [5] as a model for a special regime of internal gravity waves at the interface of two fluids. It is well known that (1) admits a Lax pair representation (cf. [19]) that leads to an infinite sequence of conserved quantities (cf. [19], [4]) and that it can be written in Hamiltonian form with Hamiltonian 2π 1 1 2 1 (u) := ∂ 1/2u u3 dx (2) H 2π 2 | x| − 3 Z0 by the use of the Gardner bracket
1 2π F, G (u) := ∂ F G dx (3) { } 2π x∇u ∇u Z0 2 1 s R s where uF and uG are the L -gradients of F, G C (Hr , ) at u Hr where s ∇s T R ∇ ∈ ∈T Hr H ( , ) is the Sobolev space of real valued distributions on . By the ≡ 1/2 3 Sobolev embedding Hr ֒ L (T, R), the Hamiltonian (2) is well defined and 1/2 → real analytic on Hr , the energy space of (1). The problem of the existence and the uniqueness of the solutions of the Benjamin-Ono equation is well studied – see [7], [21] and references therein. We refer to [21] for an excellent survey and a derivation of (1). By using the Hamiltonian formalism for (1), it was recently proven in [6, 7] that for any s> 1/2, the Benjamin-Ono equation has a Birkhoff map − 1 +s Φ: Hs h 2 , u (Φ (u)) , (Φ (u)) , (4) r,0 → r,0 7→ −n n≤−1 n n≥1 where Hβ := Hβ Hβ , Hβ := u Hβ u(0) = 0 , r,0 r ∩ c,0 c,0 ∈ c and hβ := z hβ z = z n 1b (5) r,0 ∈ c,0 −n n ∀ ≥ is a real subspace in the complex Hilbert space
β 2β 2 h := (z ) Z z = 0 and n z < , n := max 1, n . (6) c,0 n n∈ 0 h i | n| ∞ h i { | |} n∈Z X
2 By [6, Theorem 1] and [7, Theorem 6], the Birkhoff map (4) is a homeomorphism that transforms the trajectories of the Benjamin-Ono equation (1) into straight lines, which are winding around the underlying invariant torus, defined by the action variables. Furthermore, these trajectories evolve on the isospectral sets of potentials of the corresponding Lax operator (see (15) below). In this sense, the Birkhoff map can be considered as a non-linear Fourier transform that sig- nificantly simplifies the construction of solutions of (1). This fact allows us to 0 s prove that for any 1/2 ∞ ∞ ∞ 2 Φ−1(z)= n2I I , (7) H◦ n − k n=1 n=1 X X kX=n where I (z) := z 2/2 are action variables for any n 1 (cf. (5)). In addition, n | n| ≥ by [6, Corollary 7.1], the Birkhoff coordinate functions ζn(u) Φn(u), n 1, satisfy the (well defined) canonical relations ≡ ≥ ζ , ζ =0, ζ , ζ = iδ , n,m 1, (8) { n m} { n m} − nm ∀ ≥ s with respect to the Gardner bracket (3) on Hr,0 for s 0. In the present paper we prove that for any s > ≥1/2 the Benjamin-Ono equation admits an analytic Birkhoff normal form in− an open neighborhood s of zero in Hr,0 (see Theorem 1.1 below). Let us first recall the notion of an analytic Birkhoff normal form in the case of a Hamiltonian system on R2N with coordinates (x1,y1, ..., xN ,yN ) and canonical symplectic structure Ω = N n=1 dxn dyn. Assume that the Hamiltonian H H(x, y) is real analytic and ∧ ≡ P N x2 + y2 H(x, y)= ω n n + O(r3), r := x 2 + y 2, (9) n 2 | | | | n=1 X p where O(r3) stands for terms of order 3 in the Taylor expansion of H at zero ≥ and ωn R, 1 n N. Note that then the Hamiltonian vector field XH of ∈ ≤ ≤ 2N 2N H has a singular point at zero and its linearization d0XH : T0R T0R at zero has imaginary eigenvalues, Spec d X = iω , ..., iω . By→ definition, 0 H {± 1 ± N } the Hamiltonian system corresponding to XH has an analytic Birkhoff normal form in an open neighborhood of zero if there exist open neighborhoods U and V of zero in R2N and a real analytic canonical diffeomorphism : U V, (x ,y , ..., x ,y ) (q ,p , ..., q ,p ), F → 1 1 N N 7→ 1 1 N N 3 such that the Hamiltonian H −1 : V R has the form ◦F → p2 + q2 H −1(q ,p , ...q ,p )= (I , ..., I ) I := n n , 1 n N. ◦F 1 1 N N H 1 N n 2 ≤ ≤ In this case, the quantities I1,...,IN are Poisson commuting integrals and the Hamiltonian equations can be easily solved in terms of explicit formulas. A classical theorem of Birkhoff [3] states that in the case when the coefficients ω1, ..., ωN in (9) are rationally independent then : U V can be constructed in terms of a formal power series. However, this formalF → power series generically diverges because of small denominators. In the case when the Hamiltonian sys- tem is completely integrable in an open neighborhood of zero and the integrals satisfy a non-degeneracy condition at zero, then the Hamiltonian system ad- mits an analytic Birkhoff normal form (Vey [22]). Vey’s result was subsequently improved – see [23] and the references therein. We remark that typically, the Hamiltonians of infinite dimensional Hamiltonian systems appearing in applica- tions such as the Benjamin-Ono equation, the Korteweg-de Vries equation, or the NLS equation, are only defined on a continuously embedded proper subspace of the corresponding phase space. The following theorem can be considered as an instance of a Hamiltonian system of infinite dimension that admits an ana- lytic Birkhoff normal form in an open neighborhood of zero. Theorem 1.1. For any s> 1/2 there exists an open neighborhood U U s of s − ≡ zero in Hc,0 such that the Birkhoff map of the Benjamin-Ono equation 1 +s Φ: Hs h 2 , (10) r,0 → r,0 1 2 +s introduced in [6, 7], extends to an analytic map Φ: U hc,0 . The restriction 1 +s → s h 2 Φ U∩Hs : U Hr,0 r,0 is a real analytic diffeomorphism onto its image. r,0 ∩ → In particular, in view of (7) and (8), for s> 1/2 the Benjamin-Ono equation − s (1) has an analytic Birkhoff normal form in the open neighborhood U Hr,0 of s −1 ∩ zero in Hr,0 such that the Hamiltonian H Φ given by (7) is well defined and 1 ◦ real analytic on hr,0. Remark 1.1. Since, prior to this work, the Benjamin-Ono equation was not known to be integrable in terms of an appropriate infinite dimensional real an- alytic setup, Theorem 1.1 should not be viewed as an instance of Vey’s result for the infinite dimensional Hamiltonian system (1). For the same reason, the techniques developed by Kuksin &Perelman in [16] do not apply in this case. We apply Theorem 1.1 to further analyze regularity properties of the solution s map of the Benjamin-Ono equation. Assume that s > 1/2. For u0 Hr,0, denote by t u(t) u(t,u ) the solution of the Benjamin-Ono− equation∈ (1) 7→ ≡ 0 with initial data u0, constructed in [7]. For given t R and T > 0 consider the flow map ∈ t : Hs Hs , u u(t,u ), S0 r,0 → r,0 0 7→ 0 4 as well as the solution map : Hs C [ T,T ],Hs , u u , S0,T r,0 → − r,0 0 7→ |[−T,T ] of the Benjamin-Ono equation. To state our results, we need to introduce one more definition. A continuous map F : X Y between two Banach spaces X and Y is called nowhere locally uniformly continuous→ in an open neighborhood U in X if the restriction F V : V Y of F to any open neighborhood V U is not uniformly continuous. In a similar→ way one defines the notion of a nowhere⊆ locally Lipschitz map in an open neighborhood U X. ⊆ t Theorem 1.2. (i) For any 1/2 k : Hs Cj [ T,T ],Hs−2j S0,T r,0 → − r,0 j=0 \ s is well defined and real analytic in an open neighborhood of zero in Hr,0. Remark 1.2. Item (i) of Theorem 1.2 improves on the result by Molinet in [17, R t Theorem 1.2], saying that for any s< 0, t , the flow map 0 (if it exists at all) is not of class C1,α for any α > 0. Item∈ (ii) of TheoremS1.2 improves on R t the result by Molinet, saying that for any s 0, t , the flow map 0 is real analytic near zero ([17, Theorem 1.2]). ≥ ∈ S s Remark 1.3. Any solution u of the Benjamin-Ono equation in Hr,0, s> 1/2, constructed in [7], has the property that for any c R, u(t, x 2ct)+ c is− again a solution with constant mean value c. It is straightforward∈ to− see that for any c R, Theorem 1.2 holds on the affine space u Hs u(0) = c . ∈ ∈ r Remark 1.4. Note that in Theorem 1.2 we restrict our attention to solutions with mean value zero. In case when the mean value of solutionsb is not prescribed, s the flow map is no longer locally uniformly continuous on Hr with s 0 as can be seen by considering families of solutions c+u(t, x 2ct) of the Benjamin-Ono≥ equation, parametrized by c R, where c tends to zero− and u is oscillating. 1 Based on this observation, Koch∈ &Tzvetkov succeeded in proving that the solu- tion map of the Benjamin-Ono equation on the line is not uniformly continuous 1This fact can also be proved in a very transparent way using Birkhoff coordinates (cf. [13, Appendix A]). 5 on bounded subsets of initial data in Hs(R, R) with s > 0 (cf. [14]). We point out that Theorem 1.2(i) does not follow from the lack of uniform continuity on bounded subsets of initial data of the solution map of the Benjamin-Ono equa- s tion on Hr , s > 1/2, and its proof requires new arguments. We remark that the Birkhoff coordinates− allow to explain in clear terms why on any neighbor- s hood of zero in Hr,0 with 1/2 1 +s Ideas of the proofs. The Birkhoff map Φ : Hs h 2 , u (Φ (u)) , is r,0 → r,0 7→ n |n|≥1 defined in terms of the Lax operator Lu = i∂x Tu where Tu denotes the s − − Toeplitz operator with potential u Hr,0, s> 1/2. We refer to Section 2 for a review of terminology and results∈ concerning− this operator, established in the previous papers [6], [7], and [8]. At this point we only mention that the spectrum of Lu is discrete and consists of a sequence of simple, real eigenvalues, bounded from below, λ (u) < λ (u) < . Furthermore, there exist L2 normalized 0 1 ··· − eigenfunctions fn(u) of Lu, corresponding to the eigenvalues λn(u), which are uniquely determined by the normalization conditions f (u) 1 > 0, eixf (u) f (u) > 0, n 1. (11) h 0 | i h n−1 | n i ∀ ≥ The components of the Birkhoff map Φ are then defined as 1 fn(u) Φn(u)= h | i, n 1, (12) + κn(u) ∀ ≥ where κn(u) > 0 are scaling factorsp (cf. Section 4). We point out that the normalization conditions (11) are defined inductively. It is this fact which makes it more difficult to prove that Φ extends to an analytic map. For u = 0, one has λ (0) = n, f (0) = einx, κ (0) = 1, n 0. n n n ∀ ≥ We then use perturbation arguments to show that for any s> 1/2, Φ extends s − to an analytic map on a neighborhood of zero in Hc,0. Let us outline the main steps of the proof of this result. In [8] we proved that there exists a neighborhood U U s of zero in Hs ≡ c,0 so that for any u U, the spectrum of Lu consists of simple eigenvalues λn(u), n 0. These eigenvalues∈ are analytic maps on U and so are the Riesz projectors ≥ Pn(u) onto the one dimensional eigenspaces, corresponding to these eigenvalues. See Proposition 2.1 and Proposition 2.2 in Section 2 for a review of these results. inx It then follows that for any n 0, hn(u) = Pn(u)e is an analytic function U H1+s (see (13) below). ≥ → + 6 In Section 3 we introduce the pre-Birkhoff map Ψ(u)=(Ψ (u)) , Ψ (u)= h (u) 1 , n 1. n n≥1 n h n | i ∀ ≥ With the help of the Taylor expansion of Pn(u), n 1, we show that for any 1+s 1+s ≥ u U, Ψ(u) h+ and that Ψ : U h+ is analytic (see (14) below). The arguments∈ developed∈ allow to prove that→ these results hold for any s > 1/2, not just for 1/2 In this section we review results from [8] on the Lax operator Lu of the Benjamin- s Ono equation with potentials u in Hc with s > 1/2, used in this paper. Throughout this section we assume that 1/2 8 For u Hs with 1/2 Vert (̺) := λ C λ n ̺, Re(λ) n 1/2 , n 0, (16) n ∈ | − |≥ | − |≤ ≥ Dn(̺) := λ C λ n <̺ , n 0, (17) ∈ | − | ≥ and let ∂Dn(̺) be the counterclockwise oriented boundary of Dn(̺) in C. De- 0 C note by Vertn(̺) the interior of Vertn(̺) in . Proposition 2.2. For any 1/2 < s 0, there exists an open neighborhood s s − ≤ U U of zero in Hc,0 so that for any u U the operator Lu is a closed ≡ s 1+s ∈ operator in H+ with domain H+ . The operator has a compact resolvent and all its eigenvalues are simple. When listed appropriately, they satisfy λ D (1/4), n 0. n ∈ n ≥ 9 Moreover, for any n 0 the resolvent map ≥ U Vert0 (1/4) Hs ,H1+s , (u, λ) (L λ)−1, × n → L + + 7→ u − −1 −1 −1 −1 is analytic. In addition, (Lu λ) = (D λ) I Tu(D λ) and for any n 0, the Neumann series− − − − ≥ −1 I T (D λ)−1 = [T (D λ)−1]m (18) − u − u − m≥0 X converges in Hs absolutely and uniformly for (u, λ) U Vert0 (1/4). L + ∈ × n Remark 2.1. By [8, Corollary 2] the neighborhood U in Proposition 2.2 can be chosen so that for any n 0, ≥ −1 Tu(D λ) L(Hs;n) < 1/2, (u, λ) U Vertn(1/4), − + ∀ ∈ × s; n s where H+ stands for the space H+ equipped with the equivalent “shifted” norm 2s 2 1/2 f s;n := k≥0 n k f(k) (cf. [8, Lemma 3]). In particular, for any kn k 0 the series inh (−18)i converges| | uniformly on U Vert (1/4) with respect to ≥ × n an operatorP norm which isb equivalent to the operator norm in Hs . L + s Remark 2.2. For any u Hr,0, 1/2 3 Analytic extension of the pre-Birkhoff map In this section we introduce the pre-Birkhoff map and study its properties. Throughout this section, we assume that s> 1/2 and define − σ σ(s) := min(s, 0). (21) ≡ It follows from Proposition 2.2 (cf. (20)) that for any s> 1/2, there exists σ σ − σ an open neighborhood U of zero in Hc,0 so that for any u U and n 0 the Riesz projector ∈ ≥ 1 P (u)= (L λ)−1 dλ Hσ ,H1+σ , n −2πi u − ∈ L + + I ∂Dn 10 is well defined and the map U Hσ ,H1+σ , u P (u), is analytic. Here → L + + 7→ n ∂Dn is the counterclockwise oriented boundary of Dn Dn(1/3) (see (17)). Hence, for any n 0 the map ≡ ≥ U σ H1+σ, u h (u), (22) → + 7→ n is analytic, where h (u) := P (u)e H1+σ, e := einx. (23) n n n ∈ + n The main objective in this section is to prove the following Proposition 3.1. For any s > 1/2 there exists an open neighborhood U s of s − zero in Hc,0 so that the following holds: (i) The map, Ψ: U s h1+s, u 1,h (u) , (24) → + 7→ h n i n≥1 is analytic. We refer to Ψ as the pre-Birkhoff map (near zero). (ii) For any n 0, the map U s C, u h (u) e , is analytic and ≥ → 7→ h n | ni h (u) e 1/2, u U s. h n | ni ≥ ∀ ∈ Proof of Proposition 3.1. Take s> 1/2 and let U σ be an open neighborhood σ − of zero in Hc,0 so that the statement of Proposition 2.2 holds. Then, by the discussion above, the map (22) is analytic for any n 0. In particular, the ≥ σ components 1,hn(u) , n 1, of the map (24) are analytic on U . Hence, item (i) will follow from [11, Theorem≥ A.5] once we prove that the restriction of the s σ s s map (24) to a sufficiently small neighborhood U U Hc,0 of zero in Hc,0 is bounded. Let us show this. It follows from Proposition⊆ ∩2.2 that for any n 1 and for any u U σ we have ≥ ∈ 1 Ψ (u) := 1,h (u) = (L λ)−1e , 1 dλ n n −2πi u − n I∂Dn 1 dλ = [T (D λ)−1]me , 1 2πi u − n λ m≥1 I∂Dn X (1) (m+1) =Ψn (u)+ Ψn (u) (25) mX≥1 where u( n) Ψ(1)(u) := − , (26) n − n Ψ(m+1)(u) := Ψ (kb, ..., k ), m 1, (27) n n,u 1 m ≥ kj ≥0 1≤Xj≤m and 1 u( k ) u(k k ) u(k n) dλ Ψ (k , ..., k ) := − m m − m−1 1 − . (28) n,u 1 m 2πi k λ k λ ··· n λ λ I∂Dn m − m−1 − − b b b 11 By Proposition 2.2 and Remark 2.1, the series in (25) and (27) converge abso- lutely and uniformly on U σ. Moreover, by Remark 2.1, for any n 1 and for (m+1) σ C ≥ any m 0, Ψn : H+ is a bounded polynomial map of order m + 1. ≥ → σ Hence, for any n 1 the series (25) is the Taylor’s expansion of Ψn : U C at zero u = 0. For≥ m = 1, one obtains from (27), (28), and Cauchy’s formula→ that for any n 1, ≥ (2) 1 u(k1 n) 1 u(l1) Ψn (u)= u( k1) − = u( n l1) . −n − k1 n −n − − l1 k1≥0,k16=n − l1≥−n,l16=0 X b X b Let us now consider the generalb case m 1. By passing to theb variable µ := λ n ≥ − in the contour integral (28) and then setting lj := kj n,1 j m, we obtain from (27) and (28) that − ≤ ≤ Ψ(m+1)(u)= H (l , ..., l ), m 2, (29) n u,n 1 m ≥ lj ≥−n 1≤Xj≤m where Hu,n(l1,...,lm) is given by 1 1 u( n l ) u(l l ) u(l l ) dµ − − m m − m−1 2 − 1 u(l ) . (30) − 2πi n + µ l µ l µ ··· l µ 1 µ I m − m−1 − 1 − ∂D0 b b b b Note that the latter integral and hence Hu,n is defined for any l1,...,lm in Z. To estimate the term Hu,n(l1,...,lm), note that for any µ ∂D0, n 1, and ℓ ,...,ℓ Z ∈ ≥ 1 m ∈ n ℓ +1 n + µ , ℓ µ | j| , 1 j m , (31) | |≥ 2 | j − |≥ 5 ≤ ≤ implying that 2 u( n l ) u(l l ) u(l l ) H (l ,...,l ) 5m | − − m | | m − m−1 | | 2 − 1 | u(l ) | u,n 1 m |≤ n l +1 l +1 ··· l +1 | 1 | | m| | m−1| | 1| By Lemma 3.1 below, it thenb follows thatb for a y u U σ Hb s , ∈ ∩ c,0 b 2 n 2(1+s) H (l , ..., l ) | | u,n 1 m n≥1 lj ≥−n X 1≤Xj≤m 2 m+1 2 2s u( n lm) u(l2 l1) (5 ) n | − − | | − | u(l1) ≤ | | Z lm +1 ··· l1 +1 | | n≥1 lj ∈ | | | | X 1≤Xj≤m b b 2 2(m+1) b 5m+1Cm u m+1 5C u ≤ s k ks ≤ sk ks σ s where Cs 1 is the constant given by Lemma 3.1. Hence, by shrinking U Hc,0 to a (bounded)≥ neighborhood U s Hs , the latter estimate implies that∩ ⊆ c,0 (m+1) s Ψ(u) h1+s Ψ (u) h1+s 1, u U . + ≤ + ≤ ∀ ∈ m≥0 X This proves item (i). The proof of item (ii) is similar and hence omitted. 12 The following estimate follows directly from [8, Lemma 1]. Lemma 3.1. For any z = (z ) Z hs with s> 1/2, the linear operator n n∈ ∈ c − s s zℓ−k Q(z): h h , y = (yk)k∈Z Q(z)[y] := yk , c c Z → 7→ Z k +1 ℓ∈ Xk∈ | | is well defined and there exists a constant C 1 so that s ≥ Q(z)[y] C z y , z,y hs. k ks ≤ sk ksk ks ∀ ∈ c 4 Analytic extension of the Birkhoff map in a neighborhood of zero The goal of this section is to extend for any s> 1/2 the Birkhoff map, defined s − on Hr,0 cf. [6] (s = 0), [7] (s > 1/2) , to an analytic map on an open neighborhood of zero in Hs . − c,0 Recall from [7] that for any s> 1/2, the Birkhoff map is given by − 1 1 f (u) s 2 +s n Φ: H h , u Φn(u) , Φn(u) ζn(u) := | , (32) r,0 + n≥1 + → 7→ ≡ κn(u) 1 +s p where the sequence space h 2 is defined in (14), the eigenfunctions f f (u), + n ≡ n n 0, of the Lax operator Lu, corresponding to the eigenvalues λn λn(u), are≥ normalized so that f = 1 and (cf. [6, Definition 2.1], [7, Lemma≡ 6]), k nk 1 f > 0, eixf f > 0, n 1, (33) h | 0i h n−1| ni ∀ ≥ and the norming constants κn κn(u) > 0, n 0, are given by (cf. [6, Corollary 3.1], [7, (29)]) ≡ ≥ γk 1 γk κ0 := 1 , κn := 1 , n 1, (34) − λk λ0 λn λ0 − λk λn ≥ kY≥1 − − n6=Yk≥1 − with γk γk(u) defined as in Proposition 2.1. To construct the analytic ex- tension of≡ Φ, we express the normalisation conditions (33) of the eigenfunctions (fn)n≥0 in terms of (κn)n≥0, and the norming constants µn µn(u) > 0, n 1, defined for u Hs , s> 1/2, by cf. [6, Remark 4.1], [8, formula≡ (28)] (s =≥ 0), ∈ r,0 − [7, Section 3]( 1/2 Hs Hs+1/2, f f f f . + → + 7→ h | ni n 13 Expressed in terms of Pn, the normalisation conditions (33) of fn read cf. [6, Remark 4.1, Corollary 3.1] (s = 0), [7, Lemma 6] ( 1/2 1 + fn(u)= Pn Sfn−1(u) , n 1, f0(u), 1 = κ0(u), (36) + µn(u) ≥ h i p p 1+s 1+s ix where S denotes the shift operator S : H+ H+ , u(x) u(x)e . We note s → 7→ that for any u Hr,0, 1/2 s Remark 4.2. We record that for u Hr,0, s > 1/2, κn(u) > 0, n 0, and µ (u) > 0, n 1. If u = 0 then∈ f (u) = h−(u) = e (cf. (23))≥ and n ≥ n n n λn(u)= n for any n 0, implying that γn(u)=0 for any n 1, κ0(u)=1 and nκ (u)=1, µ (u)=1≥ for any n 1. ≥ n n ≥ s s For any 1/2 P Sh (u) = ν (u) h (u), n 1, (39) n n−1 n n ≥ where νn : U C is analytic. It follows from [8, Theorem 3] and Proposition 2.2 that for any→ n 1, the infinite product in (35) converges absolutely for ≥ u U. Hence µn, n 1, extend as analytic functions to the neighborhood U of ∈ s ≥ zero in Hc,0. By shrinking the neighborhood U if needed, we then obtain from [8, Proposition 5] that 1 µ (u) 1 < , u U, n 1, (40) n − 2 ∈ ≥ 1 +s 2 2 We ignore the complex structure of h+ and consider the space as a real Banach space. 14 implying that for any n 1, the map ≥ √+ µ : U C (41) n → is well defined and analytic. By shrinking the neighborhood U once more if necessary, we obtain from [8, Corollary 6] that κn, n 0, in (34) extend as analytic functions to the neighborhood U. Similarly, by≥ shrinking U further if necessary, we obtain from [8, Proposition 4] and the fact that κ0(0) = 1 that for any u U, ∈ 1 9 κ (u) 1 < and nκ (u) 1 < , n 1, 0 − 2 n − 10 ≥ and, by [8, Proposition 3], we have the following lemma. Lemma 4.1. For any 1/2 < s 0 there exists an open neighborhood U s of s − ≤ zero in Hc,0 so that the map κ : U s ℓ∞, u (κ (u)) , κ (u) := 1/ + nκ (u), (42) → + 7→ n n≥1 n n is analytic. p Proposition 2.2, (40), and (41), then allow us to extend fn to a neighbor- s hood of zero in Hc,0, 1/2 < s 0, so that the extension is analytic and the normalization conditions− (36) are≤ satisfied. We have the following Lemma 4.2. For any 1/2 < s 0, there exists an open neighborhood U s of s − ≤ s s 1+s zero in Hc,0 so that for any n 0, the map fn : U Hr,0 H+ extends to an analytic map ≥ ∩ → U s H1+s, u f (u). (43) → + 7→ n Proof of Lemma 4.2. Let 1/2 < s 0 and choose the neighborhood U U s s − ≤ ≡ of zero in Hc,0 so that Proposition 2.2 and Proposition 3.1 hold and so that the properties, discussed above, are satisfied. First we extend the eigenfunction f0 1+s to U. Since by (22), h0 : U H+ is analytic and by Proposition 3.1, h0, 1 does not vanish on U, the map→ h i + κ (u) U C, u a (u) := 0 , (44) → 7→ 0 h (u), 1 hp0 i 1+s and hence U H+ , u f0(u) := a0(u)h0(u), are well defined and analytic. → 7→ s 1+s Using that for any n 1, U H+,H+ , u Pn(u) (cf. (20)) is analytic and µ : U C satisfies≥ (40→) and L (41) one then7→ concludes by induction that n for any n →1, ≥ 1+s 1 U H , u fn(u) := Pn(Sfn−1(u)) + + → 7→ µn(u) is analytic as well. We thus have provedp that for any u U, f : U H1+s, ∈ n → + n 0, are analytic, satisfying Lufn(u) = λn(u)fn(u) and the normalisation conditions≥ (36). 15 Let us now study for any u U and n 0 the relation between fn(u) and h (u) where U U s is the neighborhood∈ of≥ zero in Hs of Lemma 4.2. Recall n ≡ c,0 that by the proof of Lemma 4.2, f0(u) = a0(u)h0(u) where a0 : U C is the analytic map given by (44). Let us now turn to the case n 1. Since→f (u) and ≥ n hn(u) belong to the one dimensional (complex) eigenspace of Lu corresponding to the simple eigenvalue λn(u) and both do not vanish we have that f (u)= a (u)h (u), n 1, (45) n n n ≥ where the map a : U C (46) n → is analytic. It now follows from (45), (39), and the normalization conditions (36), extended to complex valued potentials u U as explained above, that for any u U and n 1, ∈ ∈ ≥ + + √µnanhn = √µn fn = Pn Sfn−1 = Pn S(an−1hn−1) = an−1Pn Shn− 1 = an−1νnhn . In view of (40), (45), and the fact that h0(u) 1 = 0 (cf. Proposition 3.1 (ii)), one then infers that h | i 6 νn(u) an(u)= an−1(u), n 1. (47) + µn(u) ∀ ≥ Hence, for any u U and n p1 we have that ∈ ≥ n n n νk(u) 1 an(u)= a0(u) = a0(u) νk(u) . (48) + + µk(u) µk(u) kY=1 kY=1 kY=1 It follows from (39) thatp for any u U and n 1, p ∈ ≥ Pn Shn−1(u) en νn(u)= . hn(u) en h | i Hence, δn(u) νn(u)=1+ , αn(u) := Pnen en (49) αn(u) h | i where δ (u) := β (u) α (u), β (u) := P SP e e . (50) n n − n n n n−1 n−1 n Recall from Proposition 3.1 (ii) that α (u) 1/2 (51) | n |≥ for any u U and n 1. In Section 5 we prove the following important proposition,∈ which will be≥ a key ingredient into the proof of Theorem 1.1. 16 Proposition 4.1. For any 1/2 < s 0, there exists an open neighborhood s s − ≤ U of zero in Hc,0 so that the map δ : U s ℓ1 , u δ (u) , (52) → + 7→ n n≥1 with δn defined in (50), is analytic and bounded. Remark 4.3. By Remark 4.2, δ (0) = 0 for any n 1. n ≥ Proposition 4.1 allows us to prove the following lemma. Lemma 4.3. For any 1/2 < s 0 there exists an open neighborhood U s of s − ≤ zero in Hc,0 so that the map a : U s ℓ∞, u a (u) , (53) → + 7→ n n≥1 with an defined by (45), is analytic. Proof of Lemma 4.3. For any given 1/2 This together with (49) and (51) then implies that there exists C > 0 such that for any u U and n 1, ∈ ≥ n n δk(u) log νk(u) = log 1+ C, (54) αk(u) ≤ k=1 k=1 X X where log λ denotes the standard branch of the (natural) logarithm on C ( , 0], defined by log λ := log λ + i arg λ and π < arg λ < π. Hence, in\ view−∞ of (54), for any u U and n| | 1 we obtain − ∈ ≥ n n νk(u) = exp log νk(u) k=1 k=1 Y Xn exp log ν (u) exp C. ≤ k ≤ k=1 X Let us now prove the boundedness of the second product on the right side of (48). It follows from (40) that log µ (u) = log 1 (1 µ (u)) is well defined n − − n 17 for any u U and n 1. By shrinking the neighborhood U if necessary, we obtain from∈ [8, Theorem≥ 3 and Remark 4] that there exists C > 0 such that for any u U and n 1 we have that ∈ ≥ n log µ (u) C. k ≤ k=1 X This implies that for any u U and n 1 we obtain ∈ ≥ n 1 1 n = exp log µk(u) + µk(u) − 2 k=1 k=1 Y n X p 1 exp log µ (u) exp C/2 , ≤ 2 k ≤ k=1 X which completes the proof of the lemma. As a consequence from Lemma 4.1, Proposition 4.1, and Lemma 4.3, we obtain the following more general statement. Corollary 4.1. For any s > 1/2 there exists an open neighborhood U s of s − zero in Hc,0 such that the maps (42), (52), and (53) are well defined and ana- lytic. The neighborhood U s can be chosen invariant with respect to the complex conjugation of functions. Proof of Corollary 4.1. Take s> 1/2 and denote σ σ(s) := min(s, 0). Then, it follows from Lemma 4.1, Proposition− 4.1, and Lemma≡ 4.3 that there exists an σ σ open neighborhood U of zero in Hc,0 such that the maps (42), (52), and (53) (with U s replaced by U σ) are analytic. Denote by U s the open neighborhood σ s s U Hc,0 of zero in Hc,0. The analyticity of the maps (42), (52), and (53) then follow∩ from the boundedness of the inclusion U s ֒ U σ. Finally, by taking U σ σ → above to be an open ball in Hc,0, centered at zero, we obtain the last statement in Corollary 4.1. Our last step is to rewrite formula (37), which defines the Birkhoff map 1 s 2 +s s Φ: Hr,0 hr,0 for real valued u Hr,0, s > 1/2, in a form that will allow us to extend→ Φ to an analytic map∈ in a (complex)− neighborhood U s of zero in s Hc,0. To this end, we use the symmetries of the Lax operator Lu established in Corollary A.2 in Appendix A and argue as follows: First, we note that if the map F : U Y , where U X is an open neighborhood and X and Y are (complex) Banach→ spaces, is⊆ analytic then so is the map F ∗ : U Y, x F ∗(x) := F (x), (55) → 7→ where ( ) denotes complex conjugation. With this in mind, we note that for any u Hs · with s> 1/2 and n 0 we have that ∈ r,0 − ≥ 1, fn(u) = 1, fn(u) . (56) 18 For any u Hs with s > 1/2, we obtain from Corollary A.2 that λ (u) = ∈ r,0 − n λn(u), n 0. This, together with the definition (34) of κn implies that for any u Hs ,≥s> 1/2, ∈ r,0 − κ (u)= κ (u), n 0. (57) n n ≥ s s We choose the neighborhood U of zero in Hc,0, s > 1/2, so that Corollary 4.1 and Proposition 3.1 hold and so that it is invariant− with respect to complex conjugation. It then follows from (32), (45), (56), and (57), that for any u U s Hs , s> 1/2, and n 1, ∈ ∩ r,0 − ≥ an(u) Φn(u)= √n 1, hn(u) + nκn(u) a∗ (u) = √n p n 1,h∗ (u) (58) + n nκn(u) and p an(u) Φn(u)= √n 1,hn(u) , (59) + nκn(u) where we used the notation, introducedp in (55). Hence, the Birkhoff map (37) satisfies for any u U s Hs , s> 1/2, ∈ ∩ r,0 − 1 +s Φ(u)= Φ∗ (u) , Φ (u) h 2 . (60) −n n≤−1 n n≥1 ∈ r,0 Note that by the discussion above, the right hand sides of the identities in (58) and (59) are well defined and analytic (cf. Corollary 4.1) for any u U s, s> 1/2, and n 1. The main result of this section is the following proposition.∈ − ≥ Proposition 4.2. For any s> 1/2, there exists an invariant with respect to − s s the complex conjugation open neighborhood U of zero in Hc,0 so that the right hand side of the identity in (60) is well defined for any u U s and the map ∈ 1 +s Φ: U s h 2 , u Φ∗ (u) , Φ (u) , (61) → c,0 7→ −n n≤−1 n n≥1 is analytic. s s Remark 4.4. Since Hr,0 is a real space in the complex space Hc,0 and, similarly, 1 1 2 +s 2 +s hr,0 is a real space in the complex space hc,0 , we conclude from Proposition 4.2 that the Birkhoff map (37) is real analytic on U s Hs . ∩ r,0 Proof of Proposition 4.2. For a given s > 1/2 we choose the open neighbor- s s − hood U of zero in Hc,0 as in Corollary 4.1 and Proposition 3.1 and assume that U s is invariant with respect to the complex conjugation. Then, the map (61) 1 s 2 +s is well defined on U and takes values in hc,0 . The map (61) is analytic if the map 1 +s Φ(2) : U s h 2 , u Φ (u) , (62) → + 7→ n n≥1 19 1/2 with Φn(u) given by (58), is analytic. Let Λ denote the linear multiplier, s− 1 Λ1/2 : hs h 2 , (x ) √n x . + → + n n≥1 7→ n n≥1 The map Φ(2) is then represented by the commutative diagram Φ(2) U s (a∗,κ,Ψ∗) 1/2 1 +s ℓ∞ ℓ∞ h1+s M h1+s Λ h 2 + × + × + + + where (a∗, κ, Ψ∗): U s ℓ∞ ℓ∞ h1+s, u a∗(u), κ(u), Ψ∗(u) , (63) → + × + × + 7→ M : ℓ∞ ℓ∞ h1+s h1+s, (x ) , (y ) , (z ) x y z , + × + × + → + n n≥1 n n≥1 n n≥1 7→ n n n n≥1 (64) σ ∞ κ σ ∞ s 1+s and a : U ℓ+ , : U ℓ+ , and Ψ : U h+ are given respectively in (53), (42),→ and (24). By→ Corollary 4.1 and→ Proposition 3.1 the map (63) is analytic, whereas the map (64) is analytic since it is a bounded (complex) trilinear map. This, together with the analyticity of the multiplier operator Λ1/2 and the commutative diagram above implies that (62) is analytic. This completes the proof of the proposition. 5 Analyticity of the delta map In this section we prove Proposition 4.1 of Section 4, which plays a significant role in the proof of Proposition 4.2. Our proof of Proposition 4.1 is based on a vanishing lemma – see Lemma 5.1 below. Throughout this section we assume that 1/2 20 converges uniformly in Hs for (u, λ) U Vert0 (1/4). By the defini- L + ∈ × n≥0 n tion of Pn−1 in (20), this implies that βn βn(u), n 1, satisfies ≡ S ≥ 1 m β = P S(D λ)−1 T (D λ)−1 e e dλ, (67) n − 2πi n − u − n−1 n m≥0 I X ∂Dn−1 where the series converges absolutely and where Dn−1 = Dn−1(1/3) and ∂Dn−1 is counterclockwise oriented. The first term in the latter series is 1 1 dλ P S(D λ)−1e e dλ = P Se e −2πi n − n−1 n n n−1 n 2πi λ (n 1) I I − − ∂Dn−1 ∂Dn−1 = Pnen en = αn, (68) where we used that Sen−1 = en. By combining (65) with (67) and (68) we obtain that for n 1, ≥ 1 m δ δ (u) = P S(D λ)−1 T (D λ)−1 e e dλ n ≡ n − 2πi n − u − n−1 n m≥1 I X ∂Dn−1 = C (k , ..., k ) P Se e (69) − u 1 m h n km | ni m≥1 kj ≥0 X 1≤Xj≤m where 1 u(k (n 1)) u(k k ) u(k k ) dλ C (k , ..., k ) := 1 − − 2 − 1 m − m−1 . u 1 m 2πi (n 1) λ k λ ··· k λ k λ I − − 1 − m−1 − m − ∂Dn−1 b b b By passing to the variable µ := λ (n 1) in the contour integral above and then setting l := k (n 1), 1 −j −m, we obtain from (69) that j j − − ≤ ≤ δ = A(l , ..., l )B (l , ..., l ) P e e (70) n 1 m u 1 m h n lm+n| ni m≥1 lj ≥−n+1 X 1≤Xj≤m where m 1 1 1 A(l , ..., l ) := dµ, (71) 1 m 2πi µ l µ I j=1 j ∂D0 Y − and u(l ) if m =1, B (l , ..., l ) := 1 (72) u 1 m u(l )u(l l ) u(l l ) if m 2. 1 2 − 1 ··· m − m−1 ≥ b Consider the term P e e that appears in (70). It follows from (18) and n lmb+n bn b the formula for theh Riesz projector| i that 1 r P e e = (D λ)−1 T (D λ)−1 e e dλ. h n lm+n| ni − 2πi − u − lm+n n r≥0 I X ∂Dn 21 For the first term in the latter series we have 1 1 dλ (D λ)−1e e dλ = e e = δ . − 2πi − lm+n n h lm+n| ni2πi λ (l + n) lm0 I I − m ∂Dn ∂Dn (73) Hence P e e = δ + C (k , ..., k ) (74) h n lm+n| ni lm0 u 1 r−1 r≥1 kj ≥0 X 1≤Xj≤r−1 where b 1 u(−lm) dλ if r =1, 2πi (lm+n)−λ n−λ − ∂Dn Cu(k1, ..., kr−1) := b b b 1 H u(k1−(lm+n)) u(k2−k1) u(n−kr−1) dλ if r 2. 2πi (lm+n)−λ k1−λ kr−1−λ n−λ − ∂Dn ··· ≥ H By passing to the variable µ := λ n in the contour integral and then setting l := k n, 1 j r 1, we obtain− from (74) that m+j j − ≤ ≤ − P e e = δ + A(l , ..., l )B′ (l , ..., l ) h n lm+n| ni lm0 m m+r−1 u m m+r−1 r≥1 lj ≥−n X m+1≤Xj≤m+r−1 ′ = δlm0 + A(lm, ..., lm+r)Bu(lm, ..., lm+r) (75) r≥0 lj ≥−n X m+1X≤j≤m+r where u(l l ) u(l l )u( l ) if r 1, B′ (l , ..., l ) := m+1 m m+r m+r−1 m+r u m m+r u( l )− if r···=0. − − ≥ − m b b b (76) In view of the expansion (75), we split P e e , with one of the terms b h n lm+n| ni being δlm0, and then split δn accordingly. To this end, it turns out to be useful to introduce u(l )u( l ) if d =1, E (l , ..., l ) := 1 1 (77) u 1 d u(l )u(l− l ) u(l l )u( l ) if d 2 1 2 − 1 ··· d − d−1 − d ≥ By (72) and (76) one hasb b b b b b ′ Bu(l1, ..., lm)Bu(lm, ..., lm+r)= Eu(l1, ..., lm+r). By (70) and (75), we then have (1) (2) δn = δn + δn (78) where (2) δn := A(l1, ..., lm−1, 0)Bu(l1, ..., lm−1, 0) m≥1 lj ≥−n+1 X 1≤jX≤m−1 = A(l , ..., l , 0)E (l , ..., l ) (79) − 1 d u 1 d d≥1 lj ≥−n+1 X 1X≤j≤d 22 and (1) δn := A(l1, ..., lm)A(lm, ..., lm+r)Eu(l1, ..., lm+r) m≥1 r≥0 lj ≥−n+1 lj ≥−n X X 1≤Xj≤m m+1X≤j≤m+r = A(l1, ..., lm)A(lm, ..., ld)Eu(l1, ..., ld). d≥1 lj ≥−n+1 lj ≥−n 1≤Xm≤d 1≤Xj≤m m+1X≤j≤d Since the range of lj ,1 j m, and the one of lj , m +1 j d, are different, we split the latter sum≤ into≤ two parts ≤ ≤ (1) δn = A(l1, ..., lm)A(lm, ..., ld) Eu(l1, ..., ld) d≥1 lj ≥−n+1 1≤m≤d X 1X≤j≤d X + A(l1, ..., lm)A(lm, ..., ld)Eu(l1, ..., ld). d≥2 lj ≥−n+1 m+1≤k≤d lk=−n 1≤mX≤d−1 1≤Xj≤m X lj ≥−n+1X,m+1≤j δ = (l , ..., l )E (l , ..., l )+ (u) (80) n D 1 d u 1 d Rn d≥1 lj ≥−n+1 X 1X≤j≤d where by (71), (l , ..., l ) := A(l , ..., l )A(l , ..., l ) A(l , ..., l , 0) D 1 d 1 m m d − 1 d 1≤m≤d X 1 1 m 1 1 1 d 1 = dµ dµ 2πi µ l µ 2πi µ l µ ∂D0 j=1 j ∂D0 j=m j 1≤Xm≤d I Y − I Y − d 1 1 1 dµ, (81) − 2πi µ2 l µ ∂D0 j=1 j I Y − and where the remainder (u) equals Rn A(l1, ..., lm)A(lm, ..., ld)Eu(l1, ..., ld). d≥2 lj ≥−n+1 m+1≤k≤d lk=−n 1≤mX≤d−1 1≤Xj≤m X lj ≥−n+1X,m+1≤j 23 For any given (ℓ ,...ℓ ) (C D ) 0 , set 1 d ∈ \ 0 ∪{ } J := j 1,...,d ℓ =0 ,K := k 1,...,d ℓ =0 . { ∈{ } j } { ∈{ } k 6 } In the case where K = , I := A(l , ..., l )A(l , ..., l ) and II := ∅ 1≤m≤d 1 m m d A(2)(l , ..., l ) both vanish by the residue theorem and hence by the definition 1 d P (81), the claimed identity (83) holds. For the remaining part of the proof we assume that K = . Notice that the terms I and II are holomophic functions of the variable (6 ℓ ∅) (C D )K , so we may assume that the ℓ , k K, k k∈K ∈ \ 0 k ∈ are pairwise distinct complex numbers in C D0. By Cauchy’s theorem and (2) \ Leibniz’s rule, A (ℓ1,...,ℓd) can be computed as |J| ( 1) |J|+1 1 |J| 1 − ∂µ = ( 1) q +1 , ( J + 1)! ℓ µ |µ=0 k k − ℓk | | h kY∈K − i q∈Q(XK,|J|+1) kY∈K where for any integer p 1, we set ≥ Q(K,p) := q = (q ) ZK q = p . k k∈K ∈ ≥0 k n k∈K o X Given m 1,...,d , define ∈{ } J := J [1,m], J ′ := J [m, d],K := K [1,m],K′ := K [m, d]. m ∩ m ∩ m ∩ m ∩ One then obtains, in a similar way, |Jm| 1 A(ℓ1,...,ℓm) = ( 1) , rk+1 − ℓk r∈Q(KXm,|Jm|) k∈YKm |J′ | 1 A(ℓm,...,ℓd) = ( 1) m . − sk+1 s ′ ′ k∈K′ ℓk ∈Q(KXm,|Jm|) Ym Since ′ J + 1 if m J Jm + J = | | ∈ (84) | | | m| J if m K (| | ∈ d one infers that m=1 A(ℓ1,...,ℓm)A(ℓm,...,ℓd) equals P 1 1 ( 1)|J| − rk+1 sk+1 m∈K r k∈K ℓk s ′ ′ k∈K′ ℓk X ∈Q(KXm,|Jm|) Ym ∈Q(KXm,|Jm|) Ym 1 1 ( 1)|J| . − − rk+1 sk+1 m∈J r k∈K ℓk s ′ ′ k∈K′ ℓk X ∈Q(KXm,|Jm|) Ym ∈Q(KXm,|Jm|) Ym ′ ′ For any m 1,...,d , r Q(Km, Jm ), and s Q(Km, Jm ), the correspond- ∈{ } ∈ | | ∈ 1| | ing term in the latter expression is of the form k∈K qk+1 , where ℓk Q qk := rk (k 24 and in case m K, qm := rm + sm +1. It then follows from (84) that (qk)k∈K ∈ d belongs to Q(K, J +1). In order to describe m=1 A(ℓ1,...,ℓm)A(ℓm,...,ℓd) in more detail, we| | define for q in Q(K, J + 1), | | P J (q) := m J q = J , ad ∈ k | m| n k∈Km o X ′ Kad(q) := m K qk Jm , qk Jm . ∈ ≤ | | ′ ≤ | | n k∈Km\{m} k∈Km\{m} o X X We remark that Jad(q) might be empty and that by (84), for any m Jad(q) ′ ∈ and q Q(K, J + 1), the identity ′ qk = J is automatically satisfied. k∈Km m In view∈ of these| | definitions, one has | | P d A(ℓ1,...,ℓm)A(ℓm,...,ℓd) m=1 X |J| 1 = ( 1) ( Kad(q) Jad(q) ) qk+1 − | | − | | ℓk q∈Q(XK,|J|+1) kY∈K and (83) follows from the following combinatorial statement, K (q) = J (q) +1, q Q(K, J + 1). (85) | ad | | ad | ∀ ∈ | | (Recall that we consider the case (ℓ1,...ℓd) (C D0) 0 where K = .) To prove identity (85), let us fix q Q(K, J +∈ 1). For\ notational∪{ } convenience,6 ∅ we ∈ | | write Jad, Kad instead of Jad(q), Kad(q). Furthermore, define S(E) := q , E K. k ⊆ kX∈E In a first step we prove that Kad = . (Recall that we assume K = .) Denote by m is the smallest element of K6 . Then∅ S(K m )=0 J . Hence6 ∅ m\{ } ≤ | m| the largest number among all m in K, satisfying S(Km m ) Jm , exists. \{ ′} ≤ | | ′ We denote it by m and claim that m Kad, i.e., that S(Km m ) Jm . Indeed, this clearly holds if m is the largest∈ element of K. Otherwise,\{ } let≤ |j be| the successor of m in K. Then S(K )= S(K j ) J +1, m j \{ } ≥ | j | and S(K′ m ) = ( J + 1) S(K ) can be estimated by (84) as m \{ } | | − m S(K′ m ) ( J + 1) ( J +1)= J ′ J ′ . m \{ } ≤ | | − | j | | j | ≤ | m| We thus have proved that m Kad. Using the same type of arguments, one verifies that the following properties∈ are satisfied. (P1) n J , m ,m K with m 25 (P2) n ,n J with n ′ ′ ′ J +1= S(Kn )+ S(K )= S(Kn )+ S(K )= Jn + J , | | 1 n1 2 n1 | 2 | | n1 | ′ implying that J +1 Jn +1+ J = J + 2, which is a contradiction. | | ≥ | 1 | | n1 | | | Denote by m− the smallest element in (n1,n2) K. Since S(Kn1 ) = Jn1 it then follows that ∩ | | S(K m )= S(K )= J J . m− \{ −} n1 | n1 | ≤ | m− | Hence the largest number among all m in K, satisfying S(Km m ) Jm and m 26 and hence m+ Kad. Otherwise, let j be the successor of m+ in Km2 . Then by the definition∈ of m , S(K j ) J + 1. Therefore + j \{ } ≥ | j | ′ S(K m+ ) = ( J + 1) S(Km ) ( J + 1) S(Kj j ) m+ \{ } | | − + ≤ | | − \{ } yields ′ ′ ′ S(K m+ ) ( J + 1) ( Jj +1)= J J . m+ \{ } ≤ | | − | | | j | ≤ | m+ | Hence also in this case, m K . + ∈ ad It remains to prove (P3). Let m1,m2 Kad with m1 and thus n Jad. In the case (n,m2) J = , denote by j the smallest ∈ ∩ 6 ∅ ′ ′ element of J (n,m2). Since by the definition of n, S(Kj) Jj + 1 and J +1 J ′ ∩= J = J +1 one has ≥ | | | | − | j | | j | | n| S(K ) S(K ) ( J + 1) ( J ′ +1)= J 1= J , n ≤ j ≤ | | − | j | | j |− | n| and we conclude again that n J . ∈ ad (A3) S(K ) J 1. We claim that this case does not occur. Indeed, n ≤ | n|− S(K′ ) S(K′ m ) J ′ = J ′ . n ≤ m1 \{ 1} ≤ | m1 | | n| Since J + J ′ = J + 1 (cf. (84)) it then follows that | n| | n| | | J +1= S(K )+ S(K′ ) J 1+ J ′ = J , | | n n ≤ | n|− | n| | | which is a contradiction. This proves (P3). Finally, properties (P1), (P2), (P3) above together with the fact that Kad is not empty, imply identity (85). 27 We are now ready to prove Proposition 4.1. Proof of Proposition 4.1. For any given 1/2 s s Proof of Lemma 5.2. Let U U be an open neighborhood of zero in Hc,0 so that the statement of Proposition≡ 2.2 holds. Let d 2 and any 1 m d 1. By (31), one has ≥ ≤ ≤ − m d 1 1 A(l , ..., l 5m , A(l , ..., l 5d−m+1 . | 1 m|≤ l +1 | m d|≤ l +1 j=1 j j=m j Y | | Y | | Hence A(l , ..., l ) A(l , ..., l ) E 5d+1F (l , ..., l )F ′ (l , ..., l ) where | 1 m || m d || u|≤ u 1 m u m d 1 u(l l ) u(l l ) F (l , ..., l ) := | m − m−1 | | 2 − 1| u(l ) , u 1 m l +1 l +1 ··· l +1 | 1 | | m| | m−1| | 1| b b 1 u( (l l )) u( (l l b)) F ′ (l , ..., l ) := | − m − m+1 | − d−1 − d u( l ) . u m d l +1 l +1 ··· l +1 | − d | | m| | m+1| | d| By Lemma 3.1 it then followsb that for any u U b ∈ b F (l , ..., l ) Cm−1 u m, u 1 m s+1 ≤ s k ks −∞ F ′ (l , ..., l ) Cd−m u d−m+1 u m d s+1 ≤ s k ks −∞ 28 where Cs 1 is the constant of Lemma 3.1. One then concludes that the left hand side≥ of the inequality in (87) is bounded by 5d+1d2Cd−1 u d+1 (C′ u )d+1 s k ks ≤ sk ks Xd≥2 Xd≥2 ′ d+1 2 d−1 ′ d+1 where the constant Cs satisfies 5 d Cs (Cs) . By shrinking U, if needed, the claimed estimate (87) follows. ≤ Lemma 5.3. Let U be an open neighborhood in a complex Banach space X. Assume that the map f : U ℓ1 , x f (x) is locally bounded and for → + 7→ n n≥1 C 1 any n 1 the “coordinate” function fn : X is analytic. Then f : X ℓ+ is analytic.≥ → → Proof of Lemma 5.3. The lemma follows if we prove that f : U ℓ1 is weakly → + analytic (cf. [11, Appendix A]). To this end, fix x0 U, h X, h X = 1, and ∞ 1 ∗ C ∈ ∈ k k c = (cn)n≥1 ℓ+ ℓ+ . For any λ with λ < r and r > 0 sufficiently small, consider∈ the≡ complex valued function∈ | | g(λ) := cnfn(x0 + λh) nX≥1 N 1 and the partial sums gN (λ) := n=1 cnfn(x0 + λh), N 1. Since f : U ℓ+ is locally bounded, there exist M > 0, r> 0 so that for any≥ λ < r and N→ 1, P | | ≥ gN (λ) c ℓ∞ g(x0 + λh) ℓ1 M. | |≤k k + k k + ≤ Hence, the holomorphic functions (gN )N≥1 are uniformly bounded in the disk D0(r) = λ C λ < r . By Montel’s theorem, there exists a subsequence (g ) of{ (g∈ ) | |, converging} to g uniformly on any compact subset of D (r). Nk k≥1 N N ≥1 0 Hence, by the Weierstrass theorem, g is holomorphic. Since x0 U, h X with h =1, and (c ) ℓ∞ are arbitrary, f is weakly analytic.∈ ∈ k kX n n≥1 ∈ + 6 Proof of Theorem 1.1 and Theorem 1.2 In this section we prove Theorem 1.1 and Theorem 1.2, stated in Section 1. Proof of Theorem 1.1. Recall from Proposition 4.2 that for any s> 1/2, there s s − exists a neighborhood U U of zero in Hc,0 so that the Birkhoff map extends from U Hs to an analytic≡ map ∩ r,0 1 +s Φ: U h 2 , u Φ∗ (u) , Φ (u) (88) → c,0 7→ −n n≤−1 n n≥1 where, by (58), ∗ an(u) ∗ Φn(u)= √n Ψ (u), n 1, (89) + n nκn(u) ≥ p 29 with Ψ : U h1+s being the pre-Birkhoff map (24), studied in Section 3. → + Let us compute the differential d0Φ of Φ at u = 0. To this end note that the differential of Ψ at u = 0 is given by the weighted Fourier transform (cf. (26)), s 1+s ub(−n) d0Ψ: Hc,0 h+ , u n . Hence, in view of (55), → 7→ − n≥1 u(n) d Ψ∗ : Hs h1+s, u . (90) 0 c,0 → + 7→ − n n≥1 b Recall from Remark 4.2 that for u = 0 one has κ (0) = 1, nκ (0) = 1, n 1, f (0) = e , h (0) = e , n 0. (91) 0 n ≥ n n n n ≥ The definition (45) of an(u) then implies that for u = 0, a∗ (0) = a (0) = 1, n 1. (92) n n ≥ From (88)–(92), the fact that Ψn(u) = 0, n 1, and Leibniz’s rule one then infers that ≥ 1 s 2 +s u(n) d0Φ: Hc,0 hc,0 , u . (93) → 7→ − n ! | | n∈Z\{0} b 1 p s 2 +s Hence, being a weighted Fourier transform, d0Φ: Hc,0 hc,0 is a linear → 1 +s s h 2 isomorphism of complex linear spaces. Moreover, d0Φ Hs : Hr,0 r,0 is r,0 → a linear isomorphism of the corresponding real subspaces. Theorem 1.1 now follows from the inverse function theorem in Banach spaces together with the 1 +s s h 2 fact that Φ Hs : Hr,0 r,0 is a homeomorphism (cf. [7, Theorem 6]). r,0 → Let us now turn to the proof of Theorem 1.2. To prove item (i), we first need to make some preliminary considerations. Recall that the Benjamin-Ono s equation is globally well-posed in Hr,0 for any s> 1/2 (cf. [7] for details and R t − references). For any given t , denote by 0 the flow map of the Benjamin- s t ∈ s s S t 1/2+s 1/2+s Ono equation on Hr,0, 0 : Hr,0 Hr,0 and by B : h+ h+ the t S → S → version of 0 obtained, when expressed in the Birkhoff coordinates (ζn)n≥1 (cf. S t Remark 4.1). To describe B more explicitly, recall that the nth frequency ωn, n 1, of the Benjamin-OnoS equation is the real valued, affine function defined ≥ s+1/2 on h+ (cf. [6], [7]), n ∞ ω (ζ)= n2 2 k ζ 2 2n ζ 2 . (94) n − | k| − | k| Xk=1 k=Xn+1 For any initial data ζ(0) h1/2+s, s> 1/2, t (ζ(0)) is given by ∈ + − SB t (ζ(0)) := ζ (0)eitωn(ζ(0)) . (95) SB n n≥1 The key ingredient into the proof of Theorem 1.2 (i) is a corresponding result for the flow map t . More precisely, one has the following SB 30 t 1/2+s 1/2+s Lemma 6.1. For any t = 0 and any 1/2 (m,δ) (0) (m,δ) (0) √ s ζ ζ h1/2+s = δ , ξ ζ h1/2+s = δ 1+ m . − + − + (m,δ) (m,δ) Choose δ0 > 0 so small that ζ and ξ are elements of U for any m>N and 0 <δ δ . Furthermore one has for any m > N, ≤ 0 (m,δ) (m,δ) s/2 ζ ξ h1/2+s = δm , (96) − + and by (94), δ2ms 1 ω (ζ(m,δ)) ω (ξ(m,δ)) =2m = 2δ2 . m − m m1+2s ms For any given t = 0, choose an integer k 1 so large that 6 ≥ πks 1/2 δ δ(t) := <δ ≡ 2 t 0 | | and hence 2 t δ2 = πks. It then follows that | | k s ω (ζ(m,δ))t ω (ξ(m,δ))t = π. m − m m Since 0 < s < 1/2 and | | m+1 1 (m + 1)|s| m|s| s dx s , − ≤ | | x1−|s| ≤ | | Zm there exists a subsequence (mj )j≥ 1 of the sequence (kn)n≥N+1 so that (with N = 1, 2, 3,... ), { } k s dist , 2N 1 < 1/2, j 1 . m − ∀ ≥ j 31 Since for any x R with dist(x, 2N 1) < 1/2 one has exp( ixπ) 1 > 1, one concludes that∈ − ± − exp itω (ζ(mj ,δ)) itω (ξ(mj ,δ)) 1 > 1 mj − mj − and hence 1/2+s m ξ(mj ,δ) exp itω (ζ(mj ,δ)) exp itω (ξ(mj ,δ)) j | mj | · mj − mj 1/2+s (mj ,δ) (mj ,δ) (mj ,δ) = m ξ exp it ωm (ζ ) ω m (ξ ) 1 j | mj | · j − j − 1+ ms δ . (97) ≥ j q In view of the estimate (cf. (96)) 1/2+s s/2 m ζ(mj ,δ) ξ(mj ,δ) exp itω (ζ(mj ,δ)) δm , (98) j mj − mj · mj ≤ j one then concludes, by comparing only the m th component of t (ζ(mj ,δ)) with j SB the one of t (ξ(mj ,δ)) (cf. (95)) that SB t (mj ,δ) t (mj ,δ) 1/2+s (mj ,δ) (mj ,δ) (ζ ) (ξ ) 1/2+s m ζ (t) ξ (t) B B h j mj mj S − S + ≥ − 1/2+s (mj ,δ) (mj ,δ) (mj ,δ) m ξm exp itωmj (ζ ) exp itωmj (ξ ) ≥ j | j | · − 1/2+s m ζ(mj ,δ) ξ(mj ,δ ) . − j mj − mj This together with (97) and (98) yields t (mj ,δ) t (mj ,δ) s 1/2 s/2 B (ζ ) B (ξ ) h1/2+s (1 + mj ) mj δ , S − S + ≥ − with the latter expression converging to δ > 0 as j , whereas by (96), → ∞ (mj ,δ) (mj ,δ) s/2 ζ ξ h1/2+s = δmj , − + which converges to 0 as j . This completes the proof of Lemma 6.1. → ∞ Proof of Theorem 1.2(i). The claimed result follows directly from Lemma 6.1 and Theorem 1.1. To prove Theorem 1.2(ii), we first need to make some preliminary consid- erations. Denote by ℓ∞ the space ℓ∞(Z 0 , C) of bounded complex valued c,0 \{ } sequences z = (zn)n6=0, endowed with the supremum norm. It is a Banach ∞ space and can be identified with the subspace of ℓc , consisting of sequences ∞ (zn)n∈Z with z0 = 0. Note that ℓc,0 is a Banach algebra with respect to the multiplication ℓ∞ ℓ∞ ℓ∞ , (z, w) z · w := (z w ) . (99) c,0 × c,0 → c,0 7→ n n n6=0 32 Similarly, for any s R, the bilinear map ∈ hs ℓ∞ hs , (z, w) z · w, (100) c,0 × c,0 → c,0 7→ is bounded. For any given T > 0 and s R introduce the C-Banach spaces ∈ := C [ T,T ], hs and := C [ T,T ],ℓ∞ , CT,s − c,0 CT,∞ − c,0 endowed with supremum norm. Elements of these spaces are denot ed by ξ, or more explicitly, ξ(t) = (ξn(t))n6=0. The multiplication in (99) and (100) induces in a natural way the following bounded bilinear maps , , ℓ∞ . (101) CT,∞ ×CT,∞ →CT,∞ CT,s ×CT,∞ →CT,s CT,s × c,0 →CT,s For example, the boundedness of the first map in (101) follows from the Banach ∞ algebra property of the multiplication in ℓc,0 and the estimate (1) (2) (1) (2) (1) (2) · · ∞ ∞ ∞ ξ ξ CT,∞ = sup ξ (t) ξ (t) ℓc sup ξ (t) ℓc ξ (t) ℓc k k t∈[−T,T ] k k ≤ t∈[−T,T ] k k k k ξ(1) ξ(2) , ξ(1), ξ(2) . ≤k kCT,∞ k kCT,∞ ∈CT,∞ Hence, T,∞, · is a Banach algebra. The following lemma can be shown in a straightforwardC way and hence we omit its proof. Lemma 6.2. The map , ξ eξ := eξn , is analytic. CT,∞ →CT,∞ 7→ n6=0 For any s> 1/2 and any n 1, the nth frequencies (94) of the Benjamin- − 1 ≥ 2 +s Ono equation extends from hr,0 (cf. Remark 4.1) to an analytic function on 1 2 +s hc,0 given by 2 ωn(ζ)= n +Ωn(ζ), (102) where n ∞ 1 +s Ω (ζ) := 2 k ζ ζ 2n ζ ζ , ζ h 2 . (103) n − −k k − −k k ∈ c,0 Xk=1 k=Xn+1 For n 1 we set ≤− 1 +s Ω (ζ) := Ω (ζ), ζ h 2 . (104) n − −n ∈ c,0 We have the following Lemma 6.3. For any s 0 the map ≥ 1 +s Ω: h 2 ℓ∞ , ζ Ω (ζ) , (105) c,0 → c,0 7→ n n6=0 is analytic. 33 1 2 +s Proof of Lemma 6.3. Let s 0. Then for any n 1, ζ hc,0 , one has by (103) ≥ ≥ ∈ n ∞ ∞ 2 Ωn(ζ) 2 k ζ−kζk +2 n ζ−kζk 2 k ζ−k ζk 2 ζ 1/2+s . ≤ ≤ | || || |≤ k khc k=1 k=n+1 k=1 X X X This, together with (104) then implies that for any s 0, the map (105) is well defined and locally bounded. The estimate above also≥ shows that for any n =0 1 6 2 +s the nth component of (105) is a (continuous) quadratic form in ζ hc,0 and hence analytic. The lemma then follows from [11, Theorem A.3]. ∈ We now consider the curves ξ(1), ξ(2) , defined by ∈CT,∞ 2 ξ(1) :[ T,T ] ℓ∞ , ξ(1)(t) := ei sign(n) n t, n =0, (106) − → c,0 n 6 and, respectively, ξ(2) :[ T,T ] ℓ∞ , ξ(2)(t) := t, n =0. (107) − → c,0 n 6 The following corollary follows from Lemma 6.2, Lemma 6.3, and the continuity of the bilinear maps in (101). Corollary 6.1. For any s 0, the map (cf. (95)) ≥ 1 +s · (2) 2 · (1) · i Ω(ζ) ξ B,T : hc,0 T, 1 +s, ζ ζ ξ e , (108) S →C 2 7→ is analytic. Proof of Theorem 1.2(ii). Let s 0 and let U s be the neighborhood of zero ≥ 1 s s s 2 +s in Hc,0 of Theorem 1.1, so that Φ : U Φ(U ) hc,0 is a diffeomorphism. → ⊆ 1 2 +s According to Corollary 6.1 there exists a neighborhood W of zero in hc,0 , which s t s is contained in Φ(U ), so that for any ζ W , one has B(ζ) Φ(U ) for any t [ T,T ]. Let V s := Φ−1(W ). By Corollary∈ 6.1, TheoremS ∈1.1, Lemma 6.4 ∈ − t −1 t s below, and the fact 0(u0)=(Φ SB Φ)(u0) for any u0 V and t [ T,T ], it then follows that S extends to◦ an analytic◦ map V s ∈C [ T,T ],H∈ s− . S0,T → − c,0 The following lemma is used in the proof of Theorem 1.2 (ii). To state it, we first need to introduce some more notation. Let X be a complex Banach space with norm X . For any T > 0, denote by C [ T,T ],X the Banach space of continuousk·k≡k·k functions x :[ T,T ] X, endowed− with the supremum norm x := sup x(t) .− Furthermore,→ for any open neighborhood k kT,X t∈[−T,T ] k k U of X, denote by C [ T,T ],U the subset of C [ T,T ],X , consisting of continuous functions [ T,T− ] X with values in U. − − → Lemma 6.4. Let f : U Y be an analytic map from an open neighborhood U in X where X and Y are→ complex Banach spaces. Then for any T > 0, the associated push forward map f : C [ T,T ],U C [ T,T ], Y , [t x(t)] t f(x(t)) , ⋆ − → − 7→ 7→ 7→ is analytic. 34 Proof of Lemma 6.4. Let x0 C [ T,T ],U be given. Since x0 [ T,T ] is compact in U and f : U Y∈is locally− bounded there exist M > 0− and r > 0 so that for any x C [ T,T→ ],U with x x < 2r one has ∈ − k − 0kT,X f∗x T,Y := sup f(x(t)) Y M, (109) k k t∈[−T,T ] k k ≤ where Y denotes the norm of Y . For any given t [ T,T ] we obtain from the analyticityk·k of the map f : U Y and Cauchy’s∈ formula− that for any x, h C [ T,T ],U such that → ∈ − x x < r/2, h < r, (110) k − 0kT,X k kT,X we have that for any 0 < µ < 1/2, | | f x(t)+ µh(t) f(x(t)) 1 f x(t)+ λh(t) − = dλ µ 2πi λ(λ µ) I|λ|=1 − and 1 f x(t)+ λh(t) d f h(t) = dλ x(t) 2πi λ2 I|λ|=1 These identities imply that for any x, h C [ T,T ],U satisfying (110), and for any t [ T,T ], 0 < µ < 1/2, one has∈ − ∈ − | | f x(t)+ µh(t) f(x(t)) µ f x(t)+ µh(t) − dx(t)f h(t) = 2 dλ µ − 2πi |λ|=1 λ (λ µ) Y I − Y 2M µ ≤ | | where we used (109). Expressed in terms of the push forward map f∗, it means that for any x, h C [ T,T ],U satisfying (110), ∈ − f∗ x + µh f∗(x) − dxf∗(h) 2M µ , 0 < µ < 1/2. µ − ≤ | | | | T,Y 1 Therefore, f∗ : C [ T,T ],U C [ T,T ], Y is a C -map between C-Banach spaces (see e.g. [20−, Problem 3])→ and− hence analytic. It remains to prove the Addendum to Theorem 1.2 (ii), stated in Section 1. Proof of Addendum to Theorem 1.2(ii). Corollary 6.1 continues to hold when 1 k 2 +s−2k 1 is replaced by the Banach space C [ T,T ], h , k 1, of k T, 2 +s c,0 C − 1 ≥ 2 +s−2k times continuously differentiable functions ξ :[ T,T ] hc,0 . For any k 1, this follows from the proof of Theorem 1.2(ii)− and the→ fact that the curve ≥ (2) ζ · ξ(1) [resp. ei Ω(ζ)·ξ ], which appears as a factor on the right side of (108), 1 k 2 +s−2k k ∞ belongs to C [ T,T ], hc,0 [resp. C [ T,T ],ℓc,0 ]. By combining this with Theorem 1.1− one obtains the claimed result.− 35 A Symmetries of the Lax operator In this appendix we study symmetries of the Lax operator Lu. Recall that in 1+s s [8], we defined and studied the Lax operator Lu = D Tu : H+ H+ for u Hs, 1/2 (Luf)∗ = Lu∗ f∗. This completes the proof of the lemma. By combining Lemma A.1 with Proposition 2.1 we obtain Corollary A.1. For any 1/2 < s 0, there exists an open neighborhood s s s − ≤ W W of Hr,0 in Hc,0, invariant under the involution (112) and the complex ≡ − conjugation of functions, so that for any u W , the operator Lu given by (111) s ∈ 1+s is a closed operator in H− with domain H− . The operator has a compact − resolvent and all its eigenvalues are simple. When appropriately listed, λn , n 0, satisfy Re(λ−) < Re(λ− ) for n 0 and λ−(u) n 0 as n . ≥ n n+1 ≥ | n − | → → ∞ For u Hs , 1/2 < s 0, let (f (u)) , be the eigenfunctions of L , ∈ r,0 − ≤ n n≥0 u corresponding to the eigenvalues (λn(u))n≥0, normalized as f (u) =1, n 0, 1 f (u) > 0, f (u) Sf (u) > 0, n 1 (114) k n k ≥ h | 0 i n | n−1 ≥ − (cf. [6, Definition 2.1])). Similarly, denote by (fn (u))n≥0, the eigenfunctions of − − Lu , corresponding to the eigenvalues (λn (u))n≥0, normalized as f −(u) =1, n 0, 1 f −(u) > 0, Sf −(u) f − (u) > 0, n 1. (115) k n k ≥ h | 0 i n | n−1 ≥ It follows from Proposition 2.2 (cf. (20)) and Lemma A.1 that for any 1/2 < s s − s 0, there exists an open neighborhood U of zero in Hc,0 so that for any u ≤U s and n 0, the Riesz projector ∈ ≥ 1 P −(u)= (L− λ)−1 dλ Hs ,H1+s , n −2πi u − ∈ L − − I ∂Dn s s 1+s − is well defined and the map U H−,H− , u Pn (u), is analytic. Here ∂D is the counterclockwise oriented→ L boundary of 7→D D (1/3) (see (17)). n n n Hence for any n 0, the map ≡ ≥ U s H1+s, u h−(u) := P −(u)e H1+s. (116) → − 7→ n n −n ∈ − is analytic. Since the map ( )∗ and the complex conjugation are isometries, without of loss of generality, we· can choose U s to be invariant under these two maps. With this notation established, we can state the following Corollary A.2. Let 1/2 37 Remark A.1. In a straightforward way it follows from Proposition 2.1, Propo- sition 2.2, and Remark 2.2 that for any 1/2 fn(u) and fn(u∗) ∗ (120) − − are eigenfunctions of Lu with eigenvalue λn (u) follow directly from the com- mutative diagrams in Lemma A.1. Since for any f,g H1+s, ∈ c,0 f = f = f , 1 f = 1 f , 1 f = 1 f , k k k k k ∗k h | i h | i h | i | ∗ and Sf g = f Sg , Sf g = f Sg , h | i h | i h | i h ∗| ∗i the normalization conditions (114) then imply that the eigenfunctions in (120) satisfy the normalization condition (115). By the simplicity of the eigenvalue − − − λn (u) we then conclude that fn (u)= fn(u) and fn (u)= fn(u∗) ∗. The last statement of the corollary follows from the definition of h (u) in n (23) and Lemma A.1. Indeed, for any u Hs and any λ in the resolvent set ∈ r,0 of Lu, Lemma A.1 implies that −1 −1 − −1 −1 −1 − −1 (Lu λ) = (L λ¯) and (Lu λ) = (L λ) , − C u − C − I u∗ − I s s where : H+ H− denotes the restriction of the complex conjugation of C s→ s s functions to H+ and : H+ H− the one of the map ( )∗. This, together with (23) then impliesI that for→ any u Hs U s and n 0· we have ∈ r,0 ∩ ≥ 1 h (u) = −1 (L− λ¯)−1 e dλ n −2πi C u − −n I ∂Dn −1 1 − −1 − = (L λ) e dλ = hn (u) , (121) C − 2πi u − −n ∂DIn − where we used the definition (116) of hn . By similar arguments one shows that for any u Hs U s and n 0 we have ∈ r,0 ∩ ≥ 1 −1 − −1 hn(u) = (L λ) e−n dλ −2πi I u∗ − I ∂Dn 1 = −1 (L− λ)−1 e dλ = h−(u ) , (122) I − 2πi u∗ − −n n ∗ ∗ I ∂Dn − s where we used λn (u)= λn(u∗) (cf. (117)), the assumption that U is invariant − under , and the definition (116) of hn . The formulas in (119) now follow from (121) andI (122). 38 References [1] T. Benjamin, Internal waves of permanent form in fluids of great depth,J. Fluid Mech., 29(1967), 559-592 [2] M. Berti, T. Kappeler, R. Montalto, Large KAM tori for perturbations of the defocusing NLS equation, Ast´erisque 403, 2018 [3] G. Birkhoff, Dynamical Systems, AMS, 1927 [4] T. Bock, M. 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Kappeler, The defocusing NLS and its normal form, EMS Series of Lectures in Mathematics, EMS, Z¨urich, 2014 [11] T. Kappeler, J. P¨oschel, KdV& KAM, 45, Springer-Verlag, Berlin, Heidel- berg, 2003 [12] T. Kappeler, J. Molnar, On the wellposedness of the KdV/KdV2 equa- tions and their frequency maps, Ann. Inst. H. Poincar´eAnal. Non Lin´eaire, 35(2018), no. 1, 101-160 [13] T. Kappeler, P. Topalov, Global well-posedness of KdV in H−1(T, R), Duke Journal of Mathematics, 135(2006), no. 2, 327–360 [14] H. Koch, N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, IMRN, 2005, no. 30, 1833–1847 [15] S. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Math- ematics and its Applications, 19, Oxford University Press, Oxford, 2000 [16] S. Kuksin, G. Perelman, Vey theorem in infinite dimensions and its appli- cation to KdV, Discrete Contin. Dyn. Syst., 27(2010), no. 1, 1–24 39 [17] L. Molinet, Global well-posednes in L2 for the periodic Benjamin-Ono equa- tion, American J. Math, 130(3)(2008), 2793-2798 [18] L. Molinet, D. Pilod, The Cauchy problem for the Benjamin-Ono equation in L2 revisited, Anal. and PDE, 5(2012), no. 2, 365-395 [19] A. Nakamura, A direct method of calculating periodic wave solutions to non- linear evolution equations. I. Exact two-periodic wave solutions, J. Phys. Soc. Japan, 47(1979), 1701-1705 [20] J. P¨oschel , E. Trubowitz, Inverse Spectral Theory, Academic Press, 1978 [21] J.-C. Saut, Benjamin-Ono and intermediate long wave equations: modeling, IST, and PDE, Fields Institute Communications, 83, Springer, 2019 [22] J. Vey, Sur certains systemes dynamiques separables, American Journal of Mathematics, 100(1978), no. 3, 591-614 [23] N. Zung, Convergence versus integrability in Birkhoff normal forms, Ann. of Math., 161 (2005), 141-156 40 1/2, the Birkhoff map Φ : Hr,0 hr,0 is a real analytic diffeomorphism. The− proof of this result uses, in broad→ terms, the same strategy developed to prove Theorem 1.1, but it is more technical, relying on the approximation of s arbitrary elements in Hr,0 by finite gap potentials and on properties of the spectrum of the Lax operator, associated to such potentials. As in the proof of Theorem 1.1, one of the main ingredients of the proof of the analytic extension is Lemma 5.1 (Vanishing Lemma) of Section 5. 1 s 2 +s The result saying that the Birkhoff map Φ : Hr,0 hr,0 is a real analytic diffeomorphism for the appropriate range of s, shows→ that similarly as for the Korteweg-de Vries (KdV) equation (cf. [11], [13]) and the defocusing nonlinear Schr¨odinger (NLS) equation (cf. [10]), the Benjamin-Ono equation on the torus is integrable in the strongest possible sense. We point out that the proof of the analyticity of the Birkhoff map in the case of the Benjamin-Ono equation significantly differs from the one in the case of the KdV and NLS equations due to the fact the Benjamin-Ono equation is not a differential equation. The above result on the Birkhoff map Φ can be applied to prove in a straight- s forward way that Theorem 1.2 extends to all of Hr,0. We remark that a result of this type has been first proved for the KdV equation in [12, Theorem 3.10]. It turns out that the analysis of the solution map of the KdV equation, expressed in Birkhoff coordinates, can also be used to prove Theorem 1.2 and its extension. Similarly as in the case of the KdV equation (cf. [11], [15]) and the NLS equation (cf. [2] and references therein), a major application of the result, saying 1 s 2 +s that Φ : Hr,0 hr,0 is a real analytic diffeomorphism for any s > 1/2, concerns its use→ to study (Hamiltonian) perturbations of the Benjamin-Ono− equation by KAM methods near finite gap solutions of arbitrary large amplitude. Notation. In this paragraph we summarize the most frequently used notations in the paper. For any β R, Hβ denotes the Sobolev space Hβ T, C of complex ∈ c 7 valued functions on the torus T = R/2πZ with regularity exponent β. The norm β in Hc is given by 1/2 2β 2 u β := n u(n) k k Zh i | | nX∈ Z β where u(n), n , are the Fourier coefficientsb of u Hc . For β = 0 and 0 2 ∈T C β ∈ u Hc L ( , ) we set u u 0. By Hc,0 we denote the complex ∈ ≡ β k k≡k k subspaceb in Hc of functions with mean value zero, Hβ = u Hβ u(0) = 0 . c,0 ∈ c For f Hβ and g H−β, define the sesquilinear and bilinear pairing, ∈ c ∈ c b f g := f(n)g(n), f,g := f(n)g( n). h | i Z h i Z − nX∈ nX∈ b β b b b The positive Hardy space H with regularity exponent β R is defined as + ∈ Hβ := f Hs f(n)=0 n< 0 . (13) + ∈ c ∀ p p Z C For 1 p< denote by ℓ+ ℓ ≥1 ,b the Banach space of complex valued sequences≤ z =∞ (z ) with finite≡ norm n n≥1 1/p p z ℓp := zn < . k k + | | ∞ nX≥1 Similarly, denote by ℓ∞ ℓ∞ Z , C the Banach space of complex valued + ≡ ≥1 ∞ sequences with finite supremum norm z ℓ+ := supn≥1 zn . More generally, for 1 p and m Z, we introduce thekk Banach space | | ≤ ≤ ∞ ∈ ℓp ℓp Z , C , Z := n Z n m , ≥m ≡ ≥m ≥m { ∈ | ≥ } as well as the space ℓp ℓp Z, C , defined in a similar way. Furthermore, denote c ≡ by hβ , β R, the Hilbert space of complex valued sequences z = (z ) with + ∈ n n≥1 1/2 2β 2 z hβ := n zn < . (14) k k + | | | | ∞ nX≥1 β In a similar way, we define the space of complex valued sequences h≥m for any m Z and β R. Finally, for z C ( , 0], we denote by √+ z the principal ∈ ∈ ∈ \ −∞ branch of the square root of z, defined by Re √+ z > 0. 2 The Lax Operator 0 consider the sets in C, 1/2, the inclusion − 1 +s 1 +s ı : h 2 h 2 , (z ) ( z ) , (z ) , + → r,0 n n≥1 7→ −n n≤−1 n n≥1 is an R-linear isomorphism2, it follows from [6, Theorem 1] (s = 0) and [7, Theorem 6] ( 1/2 0 such that for any u U and n 1 we have that ∈ ≥ n δn(u) 1 δk(u) C and . | |≤ αn(u) ≤ 2 k=1 X 0≤so that for any u U s c,0 ∈ (d m) A(l , ..., l ) A(l , ..., l ) E C u 3. (87) − | 1 m || m d | | u|≤ k ks d≥2 −∞ 0 . Indeed, it − 1+∈s s ∀ follows from Lemma 1 in [8] that Tu : H− H− is a well defined bounded − → s 1+s linear operator and hence Lu defines an operator on H− with domain H− so − 1+s s R β that the map Lu : H− H− is bounded. For β , denote by ( )∗ : Hc Hβ the involution, defined→ for v Hβ by ∈ · → c ∈ c v (x) := v( x), x T. (112) ∗ − ∈ Clearly, ( ) is a C-linear isometry. The Fourier coefficients of v satisfy · ∗ ∗ v (k)= v( k), k Z. (113) ∗ − ∈ Lemma A.1. For any u Hs, 1/2 < s 1/2, we have the commutative b c b diagrams ∈ − ≤ 1+s Lu s 1+s Lu s H+ H+ H+ H+ and (·)∗ (·)∗ (·) (·) L− L− 1+s u∗ s 1+s u s H− H− H− H− β β R where ( )∗ denotes the involution defined by (112) and ( ): H+ H−, β , is the complex· conjugation of functions. · → ∈ Proof of Lemma A.1. Since the proof of the commutativity of the second dia- gram is similar to the proof of the one of the first diagram we will prove only the first one. For any u Hs and f H1+s we have ∈ c ∈ + Π(uf) = u(n k)fˆ(k) e−inx = u(k n)fˆ( k) einx ∗ − − − n≥0 k≥0 n≤0 k≤0 X X X X = ub (n k)fˆ (k) einx = Π−(u f )b. ∗ − ∗ ∗ ∗ nX≤0 kX≤0 b 36 By combining this with the fact that (Df) = D(f ) we conclude that ∗ − ∗