Invariant curves of smooth quasi-periodic mappings

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.


Introduction
In this paper we are concerned with the existence of invariant curves of the following planar quasi-periodic mappings M : θ 1 = θ + r + f (θ, r), where the perturbations f (θ, r) and g(θ, r) are quasi-periodic in θ with the frequency ω = (ω 1 , ω 2 , · · · , ω n ), C p smooth in θ and r.
He obtained the existence of invariant closed curves of M 0 which is of class C 333 . About M 0 , an analytic version of the invariant curve theorem was presented in [13], a version in class C 5 in Rüssmann [10] and a optimal version in class C p with p > 3 in Herman [2,3].
When the perturbations f (θ, r), g(θ, r) in (1.1) are quasi-periodic in θ, there are some results about the existence of invariant curves of the following planar quasi-periodic mappings M 1 : θ 1 = θ + β + r + f (θ, r), where the functions f (θ, r) and g(θ, r) are quasi-periodic in θ with the frequency ω = (ω 1 , ω 2 ,· · · , ω n ), real analytic in θ and r, and β is a constant. When the map M 1 in (1.2) is an exact symplectic map, ω 1 , ω 2 , · · · , ω n , 2πβ −1 are sufficiently incommensurable, Zharnitsky [14] proved the existence of invariant curves of the map M 1 and applied this result to present the boundedness of all solutions of Fermi-Ulam problem. His proof is based on the Lagrangian approach introduced by Moser [9] and used by Levi and Moser in [5] to show a proof of the twist theorem.
In this paper, motivated by the above references, especially by Rüssmann [12], instead of the exact symplecticity or reversibility assumption on M, we assume that this mapping satisfies the intersection property, and obtain the invariant curve theorem for the quasi-periodic mapping M in the smooth case, other than analytic case.
Incidently, in [4] we use this theorem to establish the existence of invariant curves of the planar quasi-periodic mapping M δ : θ 1 = θ + β + δl(θ, r) + δf (θ, r, δ), r 1 = r + δm(θ, r) + δg(θ, r, δ), where the functions l, m, f, g are quasi-periodic in θ with the frequency ω=(ω 1 ,ω 2 , · · · , ω n ), f (θ, r, 0) = g(θ, r, 0) = 0, β is a constant, 0 < δ < 1 is a small parameter. As an application, we also use them to study the existence of quasi-periodic solutions and the boundedness of all solutions for an asymmetric oscillation where a, b are two different positive constants, x + = max{x, 0}, x − = max{−x, 0}, f (t) is a smooth quasi-periodic function with the frequency ω = (ω 1 , ω 2 , · · · , ω n ). Finally, we must point out that in order to obtain the existence of invariant curves for the quasi-periodic mapping M, we need to assume that this mapping belongs to C p with p > 2n + 1 and n is the number of the frequency ω = (ω 1 , ω 2 , · · · , ω n ). Meanwhile we note that when n = 1, quasi-periodic mappings are periodic mappings, and the optimal smoothness assumption is C p with p > 3. Hence our smoothness assumption for quasi-periodic mappings agrees with that for periodic mappings, and is optimal in this sense.
Our efforts in this paper are same as Rüssmann [12], we are more interested in weak conditions for the perturbations f, g than in high differentiability properties of the constructed invariant curves, and the main line of the proofs is also similar to that of Rüssmann [12].
The rest of the paper is organized as follows. In Section 2, we list some properties of quasi-periodic functions, and then state the main invariant curve theorem (Theorem 2.8) for the quasi-periodic mapping M which is given by (1.1). The proofs of Theorem 2.8 are given in Sections 3,4,5. In this section 6, we formulate the detail proofs of the Lemma 2.11 which have been used in the previous sections.
Moreover, f (t) is called a C p /real analytic quasi-periodic function, if F is C p /real analytic, meanwhile we say that F is a shell function of f .
Denote by Q(ω) the space of real analytic quasi-periodic functions with the frequency ω = (ω 1 , ω 2 , · · · , ω n ). Given f (t) ∈ Q(ω), suppose that the corresponding shell function F has the following Fourier expansion which is 2π-periodic in each variable, real analytic and bounded in a complex neighborhood Π n r = {(θ 1 , θ 2 , · · · , θ n ) ∈ C n : |Im θ j | ≤ r, j = 1, 2, · · · , n} of R n for some r > 0. The function f (t) is obtained from F (θ) by replacing θ by ωt, and has the following expansion Definition 2.2. For r > 0, let Q r (ω) ⊆ Q(ω) be the set of real analytic quasiperiodic functions f such that the corresponding shell functions F are bounded on the subset Π n r with the supremum norm Also we define the norm of f as f r = F r .
The following properties of quasi-periodic functions can be found in [13, chapter 3].
Throughout this paper, we assume that the frequency ω = (ω 1 , ω 2 , · · · , ω n ) satisfies the Diophantine condition for all integer vectors k = 0. It is not difficult to show that for σ 0 > n, the Lebesgue measure of the set of ω satisfying the above inequalities is positive for a suitably small c.

The main result
First we give the following definitions.
Definition 2.5. Let M be a mapping given by (1.1). We say that M : R × [a, b] → R 2 is an exact symplectic if M is symplectic with respect to the usual symplectic structure dr∧dθ and for every curve Γ : θ = ξ+ϕ(ξ), r = ψ(ξ), where the continuous functions ϕ and ψ are quasi-periodic in ξ with the frequency ω = (ω 1 , ω 2 , · · · , ω n ), we have We claim that if the mapping M is an exact symplectic map, then it has intersection property. In order to prove this result, we first give an useful lemma, and its proof is simple. Lemma 2.6. If r = r(θ) is quasi-periodic in θ and F (θ, r) is quasi-periodic in θ with the same frequency, then F (θ, r(θ)) is also quasi-periodic in θ with the same frequency. Now we are going to prove the following lemma.
Lemma 2.7. If the mapping M is an exact symplectic map, then it has intersection property.
Proof. Since the mapping M is exact symplectic and it is also a monotonic twist map, according to the paper by Zharnitsky [14], there is a function H such that the mapping M can be written by where H is quasi-periodic in the second variable. Now we prove the intersection property of the mapping M, that is, given any continuous quasi-periodic curve Γ : r = r(θ), we need to prove that M(Γ) ∩ Γ = ∅. Define two sets B and B 1 : the set B is bounded by four curves (θ, r) : θ = t , (θ, r) : θ = T , (θ, r) : r = r * and (θ, r) : r = r(θ) , the set B 1 is bounded by four curves (θ, r) : θ = t , (θ, r) : θ = T , (θ, r) : r = r * and the image of Γ under M. Here we choose r * < min r(θ). It is easy to show that the difference of the areas of B 1 and B is From the definition of M and Lemma 2.6, we know that θ 1 (T ) − T = r(T ) + f (T, r(T )) is quasi-periodic in T and θ 1 (t) − t = r(t) + f (t, r(t)) is quasi-periodic in t. Hence using Lemma 2.6 again, it follows that ∆(t, T ) is quasi-periodic in t and T .
Hence there are at least two pairs of (t 1 , T 1 ) and (t 2 , T 2 ) such that ∆(t 1 , T 1 ) < 0, ∆(t 2 , T 2 ) > 0. The intersection property of M follows from this fact, which proves the lemma.
For the quasi-periodic mapping M we assume that f, g : R 2 → R are of class C p , and define

is an integer, and
We choose a rotation number α satisfying the inequalities with some constants γ, τ satisfying Now we are in a position to state our main result.
Theorem 2.8. Suppose that the quasi-periodic mapping M given by (1.1) is of class C p (p > 2τ + 1), and satisfies the intersection property, the functions f (θ, r), g(θ, r) are quasi-periodic in θ with the frequency ω = (ω 1 , ω 2 , · · · , ω n ), and satisfy the following smallness conditions (2.4) where Γ is the Gamma function, γ, τ satisfy (2.3), c 0 , c 1 , c 2 are positive constants depending only on p and ω, and q is a number satisfying Then for any number α satisfying the inequalities (2.2), the quasi-periodic mapping M has an invariant curve Γ 0 with the form where ϕ, ψ are quasi-periodic with the frequency ω = (ω 1 , ω 2 , · · · , ω n ), and the invariant curve Γ 0 is continuous and quasi-periodic with the frequency ω = (ω 1 , ω 2 , · · · , ω n ). Moreover, the restriction of M onto Γ 0 is Remark 2.9. Here we assume that the mapping M is of class C p with p > 2τ + 1 > 2n + 1. n = 1 corresponds to the periodic case, in which p > 3 is the optimal smoothness condition. Hence our smoothness assumption for quasiperiodic mappings is optimal in this sense.
Remark 2.10. If all conditions of Theorem 2.8 hold, then the mapping M has many invariant curves Γ 0 , which can be labeled by the form of the restriction of M onto Γ 0 . In fact, given any α satisfying the inequalities (2.2), there exists an invariant curve Γ 0 of M which is quasi-periodic with the frequency ω = (ω 1 , ω 2 , · · · , ω n ), and the restriction of M onto Γ 0 has the form The existence of such α can be found in Lemma 2.12.
The constants c 0 , c 1 , c 2 in the main result depend on how well functions of class C p can be approximated by analytic ones.
The detail proof of Lemma 2.11 is given in the Appendix. The proof of Lemma 2.11 is similar to the periodic case. When h ∈ C p is a periodic function, there are some detail proofs of Lemma  Proof : Choose some n-dimensional frequency vector ω = (ω 1 , ω 2 , · · · , ω n ) satisfying (2.1) and let D ω γ,τ denote the set of all α ∈ R satisfying (2.2) with the fixed γ and τ . Then D ω γ,τ is the complement of the open dense set R ω γ,τ , where Since 1 ≤ |k| kmax ≤ n, then we have the following measure estimate Next we estimate the measure of the set R ω γ,τ . Since for α ∈ R k,j ω,γ,τ , Hence, for any τ > n, This completes the proof.

The iteration process
In this section we present an iteration process leading to the proof of Theorem 2.8.
Firstly, we introduce new variables by the linear transformation where α is the chosen rotation number satisfying (2.2), ε 0 is defined by In the new coordinates the given mapping (1.1) having the intersection property in the strip S = {(θ, r) ∈ R 2 : a < r < b} gets the form A : Clearly the intersection property is preserved and holds in the strip where we have used (3.1) and Γ(τ + 1) ≥ 1 for τ > n.
(iii) For D = D(r, s), denote by T (D) = T (r, s) the set of all holomorphic functions F : D → C 2 satisfying the identity (iv) Define the mappings Ω k (k = 0, 1, · · · ) by Ω k : Now we are going back to the quasi-periodic mappings A k : E k → C 2 defined above. We try to fix domains , and Z k − id, H k − Ω k are quasiperiodic with the frequency ω = (ω 1 , ω 2 , · · · , ω n ) in the first variable, such that the diagrams (3.6) Then obviously D k ⊆ D ′ k (k = 0, 1, · · · ). About the mappings Z k , H k the following relations are needed , then by means of this iteration process, if it exists, the assertion of Theorem 2.8 can easily be proved.
In fact, from (3.6), (3.8) k and (3.9) k , the sequence Z 0 , Z 1 , · · · converges uniformly on R × {0} and the limit Z ∞ (ξ) = lim Now the commutativity of (3.5) k yields and by virtue of (3.8) k , (3.10) k consequently Passing to the limit we get Therefore, we can obtain the existence of invariant curves of the mapping A, and from the relationship between the mappings A and M, one can also get the existence of invariant curves of the mapping M.
From the above analysis, firstly we need to prove the assertion The diagram (3.5) k exists and commutes and the estimate (3.9) k for k = 0, 1, · · · . The proofs of (3.9) k and (3.11) k (k = 0, 1, · · · ) are done by the complete induction. Let us first consider the case (3.11) 0 . As a consequence of the definition of Z 0 , the relations (3.7) 0 and (3.8) 0 are obvious. Moreover if we define H 0 = A 0 D0 , by virtue of (2.4), (3.1), (3.4), (3.6), then which is the wanted estimate (3.10) 0 . From the definition of ε 0 and q, we have Hence From the definitions of D 0 and E 0 , D 0 ⊆ E 0 holds. Therefore Thus, the diagram (3.5) 0 exists and commutes. Now let us suppose that (3.11) k is true for some k ≥ 0. We have to show (3.11) k+1 and (3.9) k . On this way the crucial result is the construction of the commuting diagram D k+1 − Ω k+1 are quasi-periodic with the frequency ω = (ω 1 , ω 2 , · · · , ω n ) in the first variable.
The existence of the commuting diagram (3.12) is guaranteed by the inductive theorem (Theorem 5.3), which we will prove in Section 5. This theorem also gives the following estimates where Q k is a polynomial of degree 2 in the second variable only With these assertions of the inductive theorem we can show (3.11) k+1 and (3.9) k . From the diagrams (3.5) k and (3.12) we see that D k+1 In fact, where Z k+1 is real for real arguments as an element of T ′ (D k+1 ), we get Using (3.3), (3.6), (3.8) k we obtain With these assertions we obtain the following commuting diagram D k+1 Comparing the diagrams (3.5) k+1 and (3.17), it remains to prove that we can replace Φ k+1 by H k+1 if we replace A k E k+1 by A k+1 . Moreover we have to show (3.10) k+1 , which is not possible without going back to the original quasiperiodic mapping A in order to use the intersection property, and to estimate the polynomial (3.16) well enough such that (3.10) k+1 follows from (3.15).
In the following, we will prove these assertions. First of all we give some useful definitions and lemmas. F stands for C or R, we call a function analytic if it is holomorphic in the case F = C and R-analytic in the case F = R. Moreover, for D ⊆ F m and d > 0, define the set   holds for all ζ, ζ ′ ∈ D ′ with some b > 0. Then for any continuous mapping A ′′ : E → F m of class Σ satisfying the estimate there exists a continuous mapping Φ ′′ of a class Σ such that the diagram exists and commutes, and the estimate We apply Lemma 3.2 to the diagram (3.17) in order to obtain (3.5) k+1 . Set Moreover, ∆ is the set of all subsets of C 2 which are invariant under Λ = σ such that Σ is the class of all functions F : D → C 2 with D ∈ ∆ and σ • F = F • σ.
We also need to prove H k+1 satisfies (3.10) k+1 . By (2.5), (3.1), (3.3), (3.4), we obtain The necessary condition which we have to require is for then we get the estimate By the definitions of s 0 , b k+1 , M k+1 in (3.6), (3.8) k+1 and (3.10) k+1 , we have According to Lemma 3.2, we have then we have the estimate with a polynomial Q k defined in (3.16).
In order to obtain a proper estimate for Q k , we apply Lemma 3.2 once more to the diagram (3.17), where this time we restrict D k+1 to D = R 2 ∩ D k+1 such that we consider (3.18) with Here we put Λ = σ R 2 such that D, D ′ , E are open subsets of R 2 belonging to ∆, and A ′ , Φ, Z are analytic functions of class Σ. Also the original quasiperiodic mapping A defined at the beginning of Section 3 is of class Σ, and it is continuous. Therefore using (3.7) k+1 , Lemma 3.2 is again applicable and we obtain a continuous function Ψ k+1 = Φ ′′ of class Σ such that the diagram exists and commutes provided (3.19) can be satisfied. Furthermore by means of (3.4), (3.20) and (3.23) we get the estimate which leads with (3.15) to We recall that the quasi-periodic mapping A has the intersection property at least in the strip (3.2). We apply this property to the family of curves where it is clear that these curves lie in S ⋆ . Moreover these curves satisfy the conditions of Definition 2.4. For each η with −s k+1 < η < s k+1 , there are real numbers ξ 0 , ξ 1 such that On the other hand from the commuting diagram (3.24) we have The mapping Z k+1 is analytic, and as a consequence of (3.8) k+1 it is injective. Thus the injectivity of Z k+1 yields (ξ 1 , η) = Ψ k+1 (ξ 0 , η), where (2) indicates the second component of a vector. This equation leads to a reasonable estimate for the polynomial (3.16). Since we use the maximum norm we get for −s k+1 < η < s k+1 with the notation a = a 0k , b = a 1k , c = a 2k , the estimate holds by virtue of (3.25), where we put Then we have |a + bη + cη 2 | ≤ N.
Let η = ±σ, 0 <σ < s k+1 , we get Lettingσ → s k+1 , we have In the previous setting, we have obtained This inequality obviously gives the wanted estimate (3.10) k+1 . The proof by induction for justifying the iteration process has finished. It remains to find a better form for condition (3.22). Equivalently we may write As a consequence is sufficient for (3.22). This is one of the conditions for q appearing in (2.6).

Linear difference equations
In this section we will solve the difference equations (4. 2) The functions f (·, y), g(·, y) ∈ Q r (ω) are given holomorphic functions of the complex variables x, y, and u, v are wanted holomorphic functions of the complex variables x, y. ε is a positive constant to be determined in such a way that the functions u, v will be of the same size.
Passing to the limit s → r yields Adding these inequalities and by (4.6), we have The proof of this lemma is completed.
(ii) For a function f ∈ P m (D), denote its mean value over the variable x by Theorem 4.4. Let α be a real number satisfying (4.2), and f (·, y) ∈ Q r (ω) be a function belonging to P 1 (r, s) for some positive constants r, s. Then the difference equation has a unique solution u ∈ Q(ω), u ∈ P 1 (r, s) with [u] = 0. For this solution the estimate holds, where ε is defined by Proof : Since the restriction of f (x, y) onto R 2 is a continuously differentiable and quasi-periodic function in x, it can be expanded into its Fourier series After straightforward calculations we obtain the relation between Fourier coefficients f k (y) and u k (y) as follows which is the uniquely determined Fourier expansion of the wanted solution u satisfying u ∈ Q(ω) with [u] = 0. Firstly, we estimate the sum By Cauchy-Schwarz inequality, we have k∈Z n 0<|k|≤m e i k,ω α − 1 For the last series we get the estimate which completes the proof of the lemma. Now we are ready to solve equation (4.1).
Theorem 4.5. Let α be a real number satisfying (4.2), and f (·, y), g(·, y) ∈ Q r (ω) be functions belonging to P 1 (r, s) and satisfying the estimates f r,s ≤ M, g r,s ≤ M (4.10) with some positive constants r, s, M. Then the difference equations (4.1) with ε defined in (4.8) have a unique solution u, v ∈ Q(ω), u, v ∈ P 1 (r, s) with [u] = 0. For this solution the estimates are valid for 0 < 2ρ < r.
Proof : In the first equation of (4.1) the mean value must vanish on both sides. Hence we get the condition for the mean value of v. As a consequence, we have [v] ∈ P 1 (r, s) and [v] r,s ≤ ε −1 M (4.14) in view of (4.10). Theorem 4.4 gives a unique solution v =ṽ ∈ P 1 (r, s) of the second equation of (4.1) with [ṽ] = 0. This solution has the estimate because of (4.10). Define v =ṽ+[v], we obtain the uniquely determined solution v ∈ P 1 (r, s) of the second equation of (4.1) satisfying (4.13). This solution has the estimate (4.12) as a consequence of (4.14) and (4.15). Define h = εṽ + f , note thatṽ is defined in D(r − ρ, s), then h is well defined in D(r − ρ, s). As a consequence we have h r−ρ,s = εṽ + f r−ρ,s ≤ 2M (4.16) and = εv + f.
Hence, the first equation of (4.1) can be rewritten in the form Thus Theorem 4.4 gives a uniquely determined solution u ∈ P 1 (r, s) of (4.17) with [u] = 0. For an estimate of u we apply Theorem 4.4 to (4.17) restricted to D(r − ρ, s) such that in (4.7) we have to replace f by h and r by r − ρ. Then (4.11) follows by means of (4.16). The proof is finished.

The inductive theorem
First of all we give together constants, domains, and mappings appearing in the formulation of the inductive theorem. (

I) Constants and their relations
We introduce the constants ω, γ, τ, M, q, ε, ε + , r, r + , s, s + , r ′ , r ′ + , s, s ′ + (5.1) and the auxiliary constants θ, ρ satisfying the relations for all (x, y) ∈ C 2 , where we use the same symbol for the mappings Ω, Ω + , Θ as well as for their restrictions to subsets of C 2 .
For the proof of the inductive theorem we need two useful lemmas.
Lemma 5.2 (Lemma 6 in [12]). For some r > 0, let f : z ∈ C : |z| < r → C be a holomorphic function with power series expansion Then for the polynomial of degree m − 1 depending on q (0 < q < 1), we have the estimate there are mappings W ∈ T (D ′ + ), Φ + ∈ T (D + ) such that the diagram exists and commutes. Moreover W − Θ, Φ + − Ω + − Q are quasi-periodic with the frequency ω in the first variable and the following estimates are satisfied with some constants a 0 , a 1 , a 2 ∈ R. Let d Ω be the differential of Ω and define With these definitions and notations, we have .
Hence, the difference equations (5.9) can be written in the more compact form we ought to show w ∈ P 2 (D), φ ∈ P 2 (D + ). After having obtained a solution w ∈ P 2 (D ′ + ) of (5.10), we try to determine φ from the equation which holds because the diagram in the inductive theorem can commute. Now (5.11) can be rewritten in the form First, Since ε+ ε = θ, then Ω • Θ = x + α + εθy θy = x + α + ε + y θy and Thus (5.11) is changed into the form Now (5.11) gets by (5.10) the form Careful estimates will lead to a solution z ∈ P 2 (D ′ + ) and z is quasi-periodic with the frequency ω in the first variable , which implies that φ is quasi-periodic with the frequency ω in the first variable and φ ∈ P 2 (D ′ + ) can be determined. 2) Properties of W .
Since H − Ω is quasi-periodic with the frequency ω in the first variable and |H − Ω| D ≤ M, using Theorem 4.5 with ρ = r 6 , we get a solution w which is quasi-periodic with the frequency ω in the first variable of (5.10) with Since D ′ + ⊆ D(4ρ, t), we define W = Θ + w| D ′ + and obtain W − Θ is quasiperiodic with the frequency ω in the first variable with Now we look for the range of W . Since

3) Estimate for
First ε < 1 because Γ(τ + 1) ≥ 1 for τ > n. Hence we get Since Ω ⋆ (D + ) = D + , s + < t = θ −1 s, we have Moreover applying Lemma 5.1 to ξ → w(ξ, η) with d = R, D = ξ ∈ C : |Im ξ| < 4ρ , we obtain Using (5.15) and the definitions of Ω ⋆ , Ω + yields By (5.15) and (I) we have Since h ≤ M , we apply Lemma 5.1 with d = 2 3 s to obtain In the previous setting we have proved Since q ≤ 10 −2 θ 2 , by (I) we get In the previous setting we define By the definition of h ⋆ we get h ⋆ • Θ D+ ≤ M , and therefore it is easy to see that Furthermore an application of Lemma 5.1 with d = 2s 3θ yields As a consequence we have by (5.19). Since q ≤ 10 −2 θ 2 , we have The set of all z satisfying z D+ ≤ 1 24 θ 2 M is a complete metric space. Using Hence is a mapping of this metric space into itself. Furthermore letting we have as above for all z, z ′ satisfying (5.18), where 0 < q < 1. Hence F is a contraction, there is a fixed point F (z) = z, which leads to the existence of a mapping Φ such that the diagram in the inductive theorem exists and commutes. If the unknown function z is represented by Z(θ, η), where z(ξ, η) = Z(ωξ, η). From the above, we know Z is well defined in D + , and by (5.21), Z have period 2π in each of variable θ i (1 ≤ i ≤ n). Hence, z is quasi-periodic with the frequency ω in the first variable which means φ is quasi-periodic with the frequency ω in the first variable.

Appendix
In this appendix we give the detail proof of Lemma 2.11 which have been used in the previous sections. For this purpose we need a well known and fundamental approximation result. Lemma 6.1 (Lemma 2.1 in [1]). Let f ∈ C p (R ℓ ) for some p > 0 with finite C p norm over R ℓ . Let φ be a radial-symmetric, C ∞ function, having as support the closure of the unit ball centered at the origin, also φ is completely flat and takes value 1, let K =φ be its Fourier transform and for all δ > 0 define x − y δ f (y)dy.
Then there exists a constant c ≥ 1 depending only on p and ℓ such that for any δ > 0, the function f δ (x) is real analytic on C ℓ , and for all β ∈ N ℓ with |β| ≤ p, one has and, for all 0 < δ < δ ′ , Moreover, the Hölder norms of f δ satisfy, for all 0 ≤ s ≤ p ≤ r, f δ − f s ≤ c f p δ p−s , f δ r ≤ c f p δ p−r .
The function f δ preserves periodicity, this is, if f is T-periodic in any of its variable x j , so is f δ .
Now we are in a position to prove Lemma 2.11.
An application of Lemma 6.1 to the function F (θ, y) ∈ C p (R n+1 ), there exists a function F δ , which is an analytic approximation of F , and for any β ∈ N n+1 with |β| ≤ p, λ! (iIm z) λ ≤ c 3 F p δ p−|β| and for any 0 < δ < δ ′ , where c 3 ≥ 1 is a positive constant depending only on p and n. Specially, if β = 0, one has where 0 < δ < δ ′ . Hence, where c 4 is a positive constant depending only on p, n. Now we define the analytic approximation of h(x, y) in E δ as follows h δ (x, y) = F δ (ω 1 x, ω 2 x, · · · , ω n x, y).
By (6.1), (6.2), we have Similarly, for any 0 < δ < δ ′ , one can obtain where c 2 > 0 is a constant depending only on p, n, ω. Hence, The proof of this lemma is completed.