Gevrey Normal Form and Effective Stability of Lagrangian Tori

A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.


Introduction
The aim of this paper is to obtain a Birkhoff Normal Form (shortly BNF) in Gevrey classes of a Gevrey smooth Hamiltonian near a Kronecker torus Λ with a Diophantine vector of rotation. Such a normal form implies "effective stability" of the quasi-periodic motion near the invariant torus, that is stability in a finite but exponentially long time interval. As in [17,19,20] it can be used to obtain a microlocal Quantum Birkhoff Normal Form for the Schrödinger operator P h = −h 2 ∆ + V (x) near Λ and to describe the semi-classical behavior of the corresponding eigenvalues (resonances).
A Kronecker torus of a smooth Hamiltonian H in a symplectic manifold of dimension 2n is a smooth embedded Lagrangian submanifold Λ, diffeomorphic to the flat torus T n := R n /2πZ n , which is invariant with respect to the flow Φ t of H, and such that the restriction of Φ t to Λ is smoothly conjugated to the linear flow g t ω (ϕ) := ϕ + tω (mod 2π) on T n for some ω ∈ R n . Hereafter, we suppose that ω satisfies the usual Diophantine condition (2.4). Then there is a symplectic mapping χ from a neighborhood of the zero section T n 0 := T n × {0} to a neighborhood of Λ in X sending T n 0 to Λ and such that the Hamiltonian H 0 := χ * H becomes H 0 (ϕ, I) = H 0 (I) + R 0 (ϕ, I), where ∇H 0 (0) = ω, and the Taylor series of R 0 at I = 0 vanishes (cf. [10], Proposition 9.13). In particular, T n 0 is an invariant torus of H 0 , the restriction of the flow of H 0 to T n 0 is given by g t ω (ϕ) = ϕ + tω (mod 2π), and for any α, β ∈ N, and any N ≥ 1, we have ∂ α ϕ ∂ β I R 0 (ϕ, I) = O α,β,N (|I| N ). Our aim is to replace these polynomial estimates by exponential estimates of ∂ α ϕ ∂ β I R 0 of the form O α,β (exp(−c/|I| a )), a > 0, c > 0, in the case when both the Hamiltonian H and the Kronecker torus are Gevrey smooth. In the case when the Hamiltonian and the torus are analytic a similar BNF has been obtained by Morbidelli and Giorgilli. They have proved as well effective stability of the action near analytic KAM tori and even a super exponential stability of the action [4,13,14] for convex Hamiltonians using Nekhoroshev's theory. A simultaneous normal form for a family of Gevrey KAM tori has been obtained in [16,18]. The existence of large family of Kronecker tori of Diophantine vectors of rotation is given by the classical KAM theorem in the case of real-analytic Hamiltonians satisfying the Kolmogorov non-degeneracy condition and for Gevrey smooth Hamiltonians it has been proved by one of the authors in [18]. Similar results for analytic (Gevrey-smooth) Hamiltonians satisfying the Rüssmann non-degeneracy conditions have been obtained in [22].

Main results
Let X be a bounded domain in R n . Fix ρ ≥ 1 and a positive constant L, and denote by G ρ L (X) the set of all C ∞ -smooth real-valued functions H in X such that where N is the set of non negative integers, α! = α 1 ! · · · α n ! and |α| = α 1 + · · · + α n is the "length" of α = (α 1 , . . . , α n ) ∈ N n . A function H is said to be G ρ -smooth on X if it satisfies (2.1) with some L > 0. In the same way, using local coordinates, we define G ρ -smooth functions on a G ρ -smooth manifold X of dimension n. Note that the G 1 -smooth functions in a bounded domain (real-analytic manifold) X are just the analytic functions in X. On the other hand, the class of G ρ -smooth function is not quasi-analytic for ρ > 1; there exist functions of a compact support which are G ρ -smooth. For more properties of Gevrey smooth functions we refer to [9,11] and [18,Appendix], where the implicit function theorem and the composition of Gevrey functions is discussed. When dealing with the KAM theory in Gevrey classes one looses Gevrey regularity in frequencies, and there naturally arise anisotropic Gevrey classes. They are defined as follows. Let ρ, µ ≥ 1 and L 1 , L 2 be positive constants. Given a bounded domain D ⊂ R n , we consider A := T n × D provided with the canonical symplectic structure, and denote by G ρ,µ L 1 ,L 2 (A) the set of all C ∞ -smooth real valued Hamiltonians H in A such that
Let ρ ≥ 1 and let X be a G ρ -smooth symplectic manifold of dimension 2n. Let H be a G ρ -smooth Hamiltonian in X. A G ρ -smooth Kronecker torus of H of a vector of rotation ω ∈ R n is given by a G ρsmooth embedding f : T n → X, such that Λ = f (T n ) is a Lagrangian submanifold of X which is invariant with respect to the Hamiltonian is commutative for any t ∈ R. Recall that g t ω (ϕ) = ϕ+tω (mod 2π). We will suppose that ω is (κ, τ )-Diophantine for some κ > 0 and τ > n − 1, which means the following: where |k| = n j=1 |k j |. Note that if X is exact symplectic and Λ ⊂ X is an embedded submanifold satisfying (2.3) with a Diophantine vector ω then Λ is Lagrangian (see [8], Sect. I.3.2). The existence of such tori in A := T n × D with vectors of rotation ω satisfying (2.4) is provided by the KAM theorem. It follows from [18, Theorems 1.1 and 3.12] and [16], that if H ∈ G ρ (A) is a "small" (in terms of κ) real-valued perturbation of a completely integrable Hamiltonian satisfying the Kolmogorov nondegeneracy conditions, then there is a Cantor set Ω κ ∈ R n of frequencies satisfying (2.4) and of a positive Lebesgue measure such that for any ω ∈ Ω κ there is a G ρ -smooth Kronecker torus Λ ω with frequency ω. In the analytic case (ρ = 1) this follows from the classical KAM theorem. Moreover, the family Λ ω , ω ∈ Ω κ , is G µ -smooth in Whitney sense, where µ = ρ(τ + 1) + 1 when ρ > 1 and µ could be any number greater than τ + 2 when ρ = 1 (see [16,18,21]). The main result in this paper is concerned with a Gevrey smooth Birkhoff Normal Form of H near any Kronecker torus with a Diophantine frequency. Theorem 1. Let ω ∈ R n satisfy the (κ, τ )-Diophantine condition (2.4) with some κ > 0 and τ > n − 1. Fix ρ ≥ 1 and set µ = ρ(τ + 1) + 1. Let H ∈ G ρ (X, R) be a real-valued Hamiltonian and let Λ be a G ρsmooth Kronecker torus of H of a vector of rotation ω. Then there is a neighborhood D of 0 in R n and a symplectic mapping χ ∈ G ρ,µ (A, X), where A = T n × D, such that χ(T n 0 ) = Λ, and and ∂ α I R 0 (ϕ, 0) = 0 for any ϕ ∈ T n and α ∈ N n .
In the analytic case (ρ = 1), a similar BNF near an elliptic equilibrium point of the Hamiltonian has been obtained by Giorgilli, Delshams, Fontich, Galgani and Simó in [3]. Moreover, effective stability of the action, that is stability of the action in a finite but exponentially long time interval has been proved in [3]. Effective stability near an analytic KAM torus has been investigated by Morbidelli and Giorgilli in [13], [14] and [4]. Combining it with the Nekhoroshev theorem they obtained a super-exponential effective stability of the action near the torus. The Nekhoroshev theory for Gevrey smooth Hamiltonians has been developed by J.-P. Marco and D. Sauzin [12]. As it was mentioned above, if H ∈ G ρ (A) is a "small" (in terms of κ) real-valued perturbation of a completely integrable G ρ -smooth Hamiltonian satisfying the Kolmogorov non-degeneracy conditions, then there is a Cantor set Ω κ ⊂ R n of frequencies satisfying (2.4) of positive Lebesgue measure such that for any ω ∈ Ω κ there is a G ρ -smooth Kronecker torus Λ ω with frequency ω. The family Λ ω , ω ∈ Ω κ , is G µ -smooth in Whitney sense, where µ = ρ(τ + 1) + 1 if ρ > 1 and µ > τ + 2 if ρ = 1 (see [16,18,21]). This implies a simultaneous G ρ,µ -smooth BNF of the corresponding Hamiltonian at a family of KAM tori Λ ω , ω ∈ Ω κ , where Ω κ ⊂ Ω κ is the set of points of positive Lebesgue density in Ω κ [18, Corollary 1.2]. Normal forms for reversible analytic vector fields with an exponentially small error term have been obtained by Iooss and Lombardi [6,7].
Here we obtain a BNF of any single G ρ -smooth Kronecker torus Λ of the Hamiltonian. This normal form implies effective stability not only of the action but of the quasi-periodic motion near Λ as well (cf. [18,Corollary 1.3]). Moreover, our method allows us to keep track on the corresponding Gevrey constants. In the case of KAM tori [16,18] this yields an uniform bound on the corresponding Gevrey constants with respect to ω ∈ Ω κ . It could be applied as in [13], [14] and [4] to obtain a super-exponential effective stability of the action near the torus in the case of convex Hamiltonians using the Nekhoroshev theory for Gevrey Hamiltonians developed by J.-P. Marco and D. Sauzin [12]. It seems that this method could be applied to obtain a Gevrey normal form in the case of elliptic tori and near an elliptic equilibrium point of Gevrey smooth Hamiltonians as well as in the case of hyperbolic tori and reversible systems.
The method we use relies on an explicit construction of the generating function of the canonical transformation putting the Hamiltonian in a normal form which allows us to obtain an explicit form of the corresponding homological equation (see Sect. 5). It is different from those used in [3] and [13] which is based on the formalism of the Lie transform.
It is an interesting question if the exponent µ = ρ(τ + 1) + 1 is optimal. As it was mentioned above the same exponent appears in the KAM theorem in Gevrey classes when ρ > 1 and our exponent µ is smaller when ρ = 1, in particular we obtain the same exponent as in the simultaneous BNF of the family of KAM tori Λ ω , ω ∈ Ω κ in [18, Corollary 1.2] when ρ > 1. In the analytic case (ρ = 1) there is an heuristic argument of Morbidelli and Giorgilli [14,§3. Discussion] showing that µ = τ + 2 should be optimal.
Theorem 1 can be used as in [17,19,20] to obtain a microlocal Quantum Birkhoff Normal Form in Gevrey classes for the Schrödinger operator P h = −h 2 ∆ + V (x) near a Gevrey smooth Kronecker torus Λ of the Hamiltonian H(x, ξ) = ξ 2 + V (x).

Birkhoff Normal Form in Gevrey classes and Effective Stability
We are going to reduce the problem to the case of a Gevrey smooth (real-analytic) Hamiltonian in A = T n × D having a Kronecker torus T n 0 = T n × {0}, where D is a connected neighborhood of 0 in R n and A is provided with the canonical symplectic two-form. By a result of Weinstein there is a symplectic transformation χ 0 : A → X such that χ 0 (T n 0 ) = Λ and χ 0 • ı = f , where ı(θ) = (θ, 0) ∈ T n 0 for any θ ∈ T n . To construct χ 0 we first find a tubular neighborhood U of Λ in T * Λ and a G ρ -smooth symplectic transformation F : U → X which maps the zero section of Λ in T * Λ to Λ. If ρ > 0 one just follows the proof of Weinstein. In the real-analytic case (ρ = 1), we first take a C ∞ -smooth symplectic map F 0 with this property, which exists by the Weinstein theorem, next we approximate it with a real-analytic one, and then we use a deformation argument of Moser to get F . Set f = F −1 •f . Arguing as in the proof of Proposition 9.13 [10], we obtain a G ρ -smooth symplectic mapping χ 1 from a bounded neighborhood A = T n × D of the torus T n 0 in T * T n to a tubular neighborhood of the zero section of Λ in T * Λ such that χ 1 • ı =f , and we set χ 0 = F • χ 1 . In particular, χ 0 (T n 0 ) = Λ. Moreover, the pull-back of the Hamiltonian vector field to A is globally Hamiltonian and we denote by H ∈ G ρ (A, R) its Hamiltonian in A. It follows from (2.3) that the restriction of the flow of the Hamiltonian vector field of H to T n 0 is just g t ω . Moreover, H(θ, 0) is constant since the flow is transitive in T n 0 , and we take it to be zero. Hence, Denote by Γ(t), t > 0, the Gamma function (7.1). Using Remark 7.2, we write the corresponding Gevrey estimates as follows for any (θ, r) ∈ A and α, β ∈ N n , where L 0 , L 1 and L 2 are positive constants, and we suppose that L 0 ≥ 1, L 1 ≥ 1 and L 2 ≥ 1.
A smooth function g(θ, I) in A ′ = T n × D ′ is said to be a generating function of a canonical transformation χ : Without loss of generality we can suppose that κ ≤ 1 in (2.4). Theorem 1 follows from the following , and ∂ α I R 0 (θ, 0) = 0 for any α ∈ N n .
where c 1 and c 2 are positive constant depending only on ρ, τ and n, while κ is the constant in (2.4).
Theorem 2 and Remark 3.1 will be proved in Sect. 6. By the Taylor formula of order m applied to R 0 (ϕ, I) at I = 0 we obtain for any α, β ∈ N n , m ∈ N, and (ϕ, I) ∈ T n × D ′ the estimate where A > 0 and the positive constants C 1 and C 2 are as in (3.6). Using Stirling's formula we minimize the right-hand side with respect to m ∈ N. An optimal choice for m will be which leads to for any α, β ∈ N n uniformly with respect to (ϕ, I) ∈ T n × D ′ , where C 1 and C 2 are of the form (3.6). This estimate yields effective stability of the quasi-periodic motion near the invariant tori as in [ It follows from (3.2) that the coefficients b α satisfy the following Gevrey type estimates for any θ ∈ T n and any multi-indices α, β ∈ N n , |α| ≥ 2.
We are looking for a function g ∈ G ρ,µ (A ′ ), where A ′ = T n × D ′ and D ′ ⊂ R n is a neighborhood of 0, such that g(θ, 0) = 0, ∇ I g(θ, 0) = 0, and and ∂ α I R(θ, 0) = 0 for any α ∈ N n . (3.10) If such a function g exists, and if D ′ is sufficiently small, we get by means of the implicit function theorem in anisotropic Gevrey classes [9], [17, Proposition A.2], a function θ(ϕ, I) in G ρ,µ (A ′ ) which solves the equation with respect to θ ∈ T n , and we denote by χ the canonical transformation defined by g by means of (3.3). Hence, Setting R 0 (ϕ, I) = R(θ(ϕ, I), I) we obtain R 0 ∈ G ρ,µ (A ′ ) by the theorem of composition in anisotropic Gevrey classes [17,Proposition A.4], as well as the identities ∂ α I R 0 (θ, 0) = 0 . for any α ∈ N n and θ ∈ T n . Theorem 2 follows from the following where C 1 and C 2 are given by (3.5).

Weighted Wiener norms
To obtain sharp estimates in Gevrey classes we will use weighted Wiener norms. These norms are well adapted to solve the so called homological equation and they provide a sharp estimate for the product of two functions. Given u ∈ C(T n ), we denote by u k , k ∈ Z n , the corresponding Fourier coefficients, and by the mean value of u on T n . For any s ∈ R + := [0, +∞) we define the corresponding weighted Wiener norm of u by . Moreover, the following relations between Wiener spaces and Hölder spaces hold for any s ≥ 0 and q > s + n/2, and the corresponding inclusion maps are continuous. The first relation is a special case of a theorem of Bernstein (n = 1) and its generalizations for n ≥ 2 [1, Chap. 3, § 3.2]. For more properties of these spaces see [20].
Weighted Wiener spaces are perfectly adapted for solving the homological equation where L ω := ω, ∂ ∂θ . We have the following Let ω satisfy the (κ, τ )-Diophantine condition (2.4) and let s ≥ 0. Then for any f ∈ A s+τ (T n ) such that f = 0 the homological equation has an unique solution u ∈ A s (T n ), and it satisfies the estimate Proof. Comparing the Fourier coefficients u k and f k , k ∈ Z n , of u and f respectively, we get and set u 0 = 0. Then using (2.4) we obtain Since f 0 = f = 0, taking the sum with respect to k = 0 we get the function u and the corresponding estimate of S s (u). In this way we obtain an unique solution u of (4.1) normalized by u = 0.  Proof. For any k ∈ Z n we set k := 1+|k|. Obviously, k < l + k−l for any k, l ∈ Z n , and we obtain We have used the inequality a+b a, b ∈ N and x ≥ 0. Summing with respect to k ∈ Z n we prove the claim. 2 A similar inequality can be obtained for the Sobolev s-norm of uv, but there appears an additional factor 2 s/2 coming from the inequality (a+b) 2 ≤ 2(a 2 +b 2 ), which makes it useless for the estimates in Sect. 6, because it changes the Gevrey constant at any step of the construction.
To get rid of the sum in Lemma 4.2, we consider the modified norms P s (u) = (s + 1) 2 S s (u) , s ≥ 0 , u ∈ A s (T n ) .
If f ∈ A s+τ (T n ) and f = 0, and if u ∈ A s (T n ) is a solution of the homological equation (4.1) such that u = 0, then by Lemma 4.1 we obtain Moreover, for any u, v ∈ A s (T n ) we obtain from Lemma 4.2 the following estimate where C = 16 ∞ q=1 q −2 = 8π 2 /3. Another usefull property of the norm P s (·), s ≥ 0, is that P s (∂ α u) ≤ P s+|α| (u) for any α ∈ N n and u ∈ C ∞ (T n ). For any p ∈ N and u ∈ C ∞ (T n ) we set There is a positive constant C 0 = C 0 (n) depending only on the dimension n such that for any u ∈ C ∞ (T n ) and s ≥ 0.
Proof. We have Integrating by parts we get for any p ∈ N and any k = 0 the inequality Moreover, (1+s) 2 ≤ 2e 1+s , and we obtain the second inequality in (4.4) with C 0 = 2e 2 (2n) n+2 C ′ 0 . The proof of the first one is straightforward. 2 Consider now the functions b α given by (3.8).
for any s ≥ 0 and any α ∈ N n with a length |α| ≥ 2, where the Gevrey constants L 1 ≥ 1 and L 2 ≥ 1 are equivalent to the corresponding Gevrey constants in (3.9) andL 0 is equivalent to L 0 L n+2 Recall that the positive constantL is equivalent to L if there is c(n, ρ, τ ) > 0 such thatL = c(n, ρ, τ )L.

Gevrey estimates
We are going to show that there are positive constants C 1 and C 2 depending on the constantsL 0 , L 1 and L 2 in (4.5) such that for any α ∈ N n with length m = |α| ≥ 2 and for any β ∈ N n we have where µ = ρ(τ + 1) + 1 (see the statement of Theorem 2). Consider for any m ≥ 2 the solution (g m , R m ) of the homological equation (5.5), where ω is (κ, τ )-Diophantine, 0 < κ ≤ 1 and τ > n − 1. Denote the unit poly-disc in C n by D n , i.e. I = (I 1 , . . . , I n ) ∈ C n belongs to D n if |I j | ≤ 1 for any 1 ≤ j ≤ n.
Proposition 6.1. There is A 0 = A 0 (n, ρ, τ ) ≥ 1 depending only on n, ρ and τ , such that for C 1 = e ρ L 1 and for any the following estimates hold for m ≥ 3 and any s ∈ R + , and for m ≥ 2 and any s ∈ R + , whereL 0 , L 1 and L 2 are the corresponding Gevrey constants in (4.5) and B 0 = B 0 (n, ρ, τ ) ≥ 1.
Proof of Lemma 6.2. We are going to prove (6.5) by induction with respect to p ≥ 1. For p = 1, we have in view of (6.4). Set Now take p = 2 and 2 ≤ m 1 , m 2 ≤ m − 1, and fix I ∈ D n . Using (4.3) and (6.6) we obtain On the other hand, where B(x, y), x, y > 0, is the Beta function (7.3). Recall that B(x, y) is decreasing with respect to both variables x > 0 and y > 0. Then using (7.4) we get for any δ ∈ (0, µ − 1) the inequalities Moreover, Lemma 7.1 implies as well as .

(6.8)
Hence, for any non-negative integer 0 ≤ q ≤ [s] we have In the same way we estimate the quantity 1)).
Proof. To simplify the notations we will write below M instead of M p .

Using (7.4) we get
Moreover, using Lemma 7.1 we get as above .
Next using the theorem of composition of Gevrey functions [18, Proposition A.4], we prove that R 0 (ϕ, I) := R(θ(ϕ, I), I) belongs to the class G ρ,µ C 1 ,C 2 (A ′ , A), where C 1 and C 2 are given by Remark 3.1. By the same argument, the canonical transformation χ generated by g belongs to the class G ρ,µ C 1 ,C 2 (A ′ , A). This completes the proof of Theorem 2 and of Remark 3.1.
More generally, we have the following Lemma 7.1. For any ν ≥ 1 and δ > 0 there is a constant C ′ (ν, δ) ≥ 1 such that for any x, y ≥ 0 the following inequality holds
Since B(x, y) is a decreasing function with respect to both variables x and y, we obtain where C ′ = 2ν δ (ν+1)/2 5 ν C. This completes the proof of the assertion since the inequality in Lemma 7.1 is symmetric with respect to x, y. 2 Remark 7.2. As in (7.6) one proves that for any ρ > 0 there is a constant C(ρ) > 1 such that C(ρ) −m Γ(ρm + 1) ≤ m! ρ ≤ C(ρ) m Γ(ρm + 1) for any m ∈ N.