Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schrödinger equations
aa r X i v : . [ m a t h . D S ] F e b Almost global solutions to two classes of 1-dHamiltonian Derivative Nonlinear Schr¨odingerequations
Jing Zhang † School of Mathematical sciences,Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice,East China Normal University, Shanghai, 200241, China
Abstract
Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schr¨odinger (DNLS)equations with respect to different symplectic forms under periodic boundary con-ditions. The nonlinearities of these equations depend not only on ( x, ψ, ¯ ψ ) butalso on ( ψ x , ¯ ψ x ), which means the nonlinearities of these equations are unbounded.Suppose that the nonlinearities depend on the space-variable x periodically. Undersome assumptions, for most potentials of these two kinds of Hamiltonian DNLSequations, if the initial value is smaller than ε ≪ p -Sobolev norm, then thecorresponding solution to these equations is also smaller than 2 ε during a time in-terval ( − cε − r ∗ , cε − r ∗ )(for any given positive r ∗ ). The main methods are constructingBirkhoff normal forms to two kinds of Hamiltonian systems which have unboundednonlinearities and using the special symmetry of the unbounded nonlinearities ofHamiltonian functions to obtain a long time estimate of the solution in p -Sobolevnorm. Keyword.
Derivative Nonlinear Schr¨odinger (DNLS) equations, Hamiltonian systems,unbounded, long time stability, momentum, Birkhoff normal form
AMS subject classifications.
It is very interesting to research the behavior of the solution in high-index Sobolevnorm to nonlinear evolution equations with derivative in nonlinearities during a long timeinterval.Consider a nonlinear Schr¨odinger equation i ψ t = ∂ xx ψ + F ( x, ψ, ¯ ψ, ∂ x ψ, ∂ x ¯ ψ ) , x ∈ [0 , π ] (1.1) † Email: [email protected] by National Natural Science Foundation of China 11871023 ψ ( t, x ) = ψ ( t, x + 2 π ) . Suppose that F satisfies F ( x + 2 π, ψ, ¯ ψ, ∂ x ψ, ∂ x ¯ ψ ) = F ( x, ψ, ¯ ψ, ∂ x ψ, ∂ x ¯ ψ ) and F ( x, , , ,
0) = 0 . Under this assumption ψ = 0 is an equilibrium solution to equation (1.1). I am interestedin the behavior of solutions around ψ = 0 during a long time interval.If F only depends on ( x, ψ, ¯ ψ ) and vanishes at order n + 1 about ( ψ, ¯ ψ ) at the origin ( n is a positive integer), local existence theory implies that when initial value k ψ ( x, k H p ≤ ε ≪ ψ ( x, t ) exists at least over an interval ( − cε − n , cε − n ) and k ψ ( x, t ) k H p stays bounded on such an interval. The problem that I am interested in is thatconstruct almost global solutions when F depends on ( ψ x , ¯ ψ x ). An almost global solutionmeans that for any given k >
0, when the initial value ψ ( x,
0) is smaller than 0 < ε ≪ ψ ( x, t ) is also small in a high index Sobolev norm for any t ∈ ( − cε − k , cε − k ) (refer[31]).When investigation concerns equation (1.1) on a compact manifold, no dispersionis available. Nevertheless, two ways may be used to obtain solutions, defined on time-intervals larger than the one given by local existence theory. The first one is using KAMtheory to get small amplitude periodic or quasi-periodic (hence global) solutions. A lotof work have been devoted to these questions and readers refer [5, 6, 8, 9, 11, 13, 21, 39,24, 28, 29, 30, 32, 33, 34, 35].The second approach concerns the construction of almost global H p -small solutionsfor (1.1) on compact manifold. Use Birkhoff normal form method to improve the orderof normal form and then exploit integral principle to get almost global solutions. Whennonlinearity F to equation (1.1) depends only on ( ψ, ¯ ψ ), small initial data give rise toglobal solutions and keep uniform control of the p -Sobolev norm of solutions ( p largeenough), over time-intervals of length ε − k , for any given positive k . This has been initiatedby Bourgain [10], who stated results of almost global existence and uniform control toequation i ψ t − ψ xx + V ( x ) ψ + ∂H∂ ¯ ψ ( ψ, ¯ ψ ) = 0 , (1.2)for any typical (with large probability) V ( x ). Bourgain in [11] stated that for any smalltypical initial value to equation i ψ t − ψ xx + v | ψ | ψ + ∂H∂ ¯ ψ ( ψ, ¯ ψ ) = 0 , (1.3)the solution ψ will satisfy k ψ ( t ) k H p < Cε, for any | t | ≤ ε − B . More results may be found in the book of Bourgain [12]. Almost global solutions forHamiltonian Semi-linear Klein-Gordon Equations (without derivative in nonlinearity) onspheres and Zoll manifolds have been obtained by Bambusi, Delort, Gr´ebert and Szeftel in[3]. Berti and Delort in [7] give almost global existence of solutions for capillarity-gravitywater waves equations with periodic spatial boundary conditions.Bambusi and Gr´ebert [4] (see also Bambusi [1] and Gr´ebert [25]) prove an abstractBirkhoff normal form theorem for Hamiltonian partial differential equations and apply2his theorem to semi-linear equations: nonlinear wave equation, nonlinear Schr¨odingerequation on the d -dimensional ( d ≥
1) torus with nonlinearities satisfying a property-tame modulus. In a non-resonant case they deduce that any small amplitude solutionremains very close to a torus for a very long time. In [2] Bambusi researches the NLWequation with nonlinearity function depending on x periodically.Faou and Gr´ebert in [26] consider a general class of infinite dimensional reversibledifferential systems and prove that if the p -Sobolev norm of initial data is smaller than ε ( 0 < ε small enough) then the solution is bounded by 2 ε during time of order ε − r with r arbitrary. This theorem applies to a class of reversible semi-linear PDEs includingnonlinear Schr¨odinger equation on the d -dimensional torus and a class of coupled NLSequations which is reversible but not Hamiltonian. Feola and Iandoli in [23] give thelong time existence for a large class of fully nonlinear, reversible and parity preservingSchr¨odinger equations on the one dimensional torus.Delort and Szeftel in [17], [18], Delort in [19], [20] research semi-linear Klein-Gordonequation (with derivative in nonlinearity) on spheres and Zoll manifold, quasi-linear Klein-Gordon equation on tori and S , and obtain that when the initial value is small than ε > | t | ≤ ε − r . Given a DNLS equation i ψ t = ∂ xx ψ + V ∗ ψ + ( ∂f ( ψ, ¯ ψ ) ∂ ¯ ψ ) x , x ∈ [0 , π ] , (1.4)Yuan and Zhang in [37] obtain that for most V the solution to (1.4) is still smaller than2 ε among time | t | ≤ ε − r (for any given positive r ), if the initial value is smaller than ε ≪
1. The nonlinearity in (1.4) does not directly depend on space variable x . In [38]Yuan and Zhang research the long time behavior of the solution to the perturbed KdVequation the nonlinearity of which is trigonometric polynomial about x .In this paper, I focus on the behavior of solutions during a long time interval to twotypes of Hamiltonian Derivative Nonlinear Schr¨odinger (DNLS) equations which dependon x periodically. One type is of the following form i ψ t = ∂ xx ψ + V ∗ ψ + 12 ∂ x ∂ ¯ ψ F ( x, ψ, ¯ ψ ) + ∂ ψ ¯ ψ F ( x, ψ, ¯ ψ ) ψ x , (1.5)where V belongs to Θ m := (cid:26) V ( x ) ∈ L ([0 , π ] , R ) (cid:12)(cid:12)(cid:12)(cid:12) b V j · max { , | j | m } ∈ [ − ,
12 ] , b V j = b V − j , ∀ j ∈ Z (cid:27) and the other type is as follow i ψ t = ∂ xx ψ + V ∗ ψ + i ∂ x (cid:0) ∂ ¯ ψ F ( x, ψ, ¯ ψ ) (cid:1) , (1.6)where V belongs to Θ m := (cid:26) V ∈ L ([0 , π ] , C ) (cid:12)(cid:12)(cid:12)(cid:12) b V j · max { , | j | m } ∈ [ − ,
12 ] , ∀ j ∈ Z \ { } , b V = 0 (cid:27) . Under some assumptions, (1.5) becomes into a Hamiltonian equation with respect to asymplectic form w := J dψ ∧ d ¯ ψ, J − := (cid:0) − ii (cid:1) and (1.6) is Hamiltonian under asymplectic form w := J dψ ∧ d ¯ ψ, ( J ) − := (cid:0) ∂ x ∂ x (cid:1) .3hen ∂ ¯ ψ F ( x, ,
0) = 0 and ∂ ψψ F ( x, ,
0) = 0 , ψ = 0 is an equilibrium point ofequations (1.5) and (1.6). In order to get the almost global solution around the origin to(1.5) and (1.6), it is required to research the behavior of solutions around ψ = 0 during along time interval.The result in [37] holds ture for Hamiltonian DNLS equation with nonlinearity inde-pendent of x . In other words the momentum of the corresponding Hamiltonian functionequals to zero. This property is important in proving the long time stability result. In[38] one researches an unbounded perturbed KdV equation the nonlinearity of which is atrigonometric polynomial about sin kx and cos kx ( | k | ≤ M ), i.e., the momentum of thecorresponding Hamiltonian function are bounded. But generally, the sets of the momen-tum of Hamiltonian functions to equation (1.5) and (1.6) under Fourier transformationmay be unbounded. Even if assume that for any ( ψ, ¯ ψ ) around origin F ∈ H β ([0 , π ] , R )( β big enough), the corresponding nonlinear vector field of equations (1.5) and (1.6) aresitll unbounded. Denote the part of F , the momentum of which is bigger than δ >
0, as F . Even if δ is very large, the Hamiltonian vecotr field of F in equations (1.5) and (1.6)are still unbounded. The results and methods in [37] and [38] do not work to equations(1.5) and (1.6), directly. In [14] one consider quasi-linear Klein-Gordon equation on S .The nonlinearities are polynomials and smooth depend on x . Their methods are not suit-able to DNLS equations (1.5) and (1.6). In [23] they consider the reversible and paritypreserving Schr¨odinger equation. It is necessary to construct a long time stability theoryto solutions of Hamiltonian DNLS equations (1.5) and (1.6) around the origin.Under Fourier transformation, equations (1.5) and (1.6) are transformed into two typesof Hamiltonian systems θ ∈ { , } ˙ u j = − i sgn θ ( j ) · ∂ ¯ u j H w θ ( u, ¯ u ) , ˙¯ u j = i sgn θ ( j ) · ∂ u j H w θ ( u, ¯ u ) , j ∈ Z ∗ := Z or Z \{ } when θ = 0 Z \{ } when θ = 1 (1.7)with Hamiltonian function H w θ ( u, ¯ u ) = H w θ + P w θ ( u, ¯ u ) , ( u, ¯ u ) ∈ H p , θ ∈ { , } (1.8)under symplectic form w θ := i P j ∈ Z ∗ d u j ∧ d¯ u j θ = 0 , i P j ∈ Z ∗ sgn( j )d u j ∧ d¯ u j θ = 1 , (1.9)where H w θ := P j ∈ Z θ ω w θ j | u j | and ω w θ j := (cid:26) ( − j + ˆ V j ) θ = 0sgn( j )( − j + ˆ V j ) θ = 1 . P w θ ( u, ¯ u ) is apower series having ( β, θ )-type symmetric coefficients semi-bounded by C θ > P w θ ( u, ¯ u ) are notbounded. This leads to the Hamiltonian vector field of P w θ ( u, ¯ u ) being unbounded. SeeProposition 4.1 in section 4.The problem of finding almost global solutions around the origin to equations (1.5)and (1.6) is changed into considering a long time stability of solutions around equilibriumpoint ( u, ¯ u ) = 0 of (1.7).In section 3 Theorem 3 states that under some assumptions, the solution to the twotype of Hamiltonian systems which have ( β, θ )-type symmetric coefficients ( θ ∈ { , } ) is4till smaller than 2 ε during a time interval ( − ε − r ∗ , ε − r ∗ ), if its initial value is smaller than ε ≪ β, θ )-type symmetric coefficients used to obtain energy inequalities.Let us introduce the important steps in proving Theorem 3.First step: construct a coordination transformation T ( r ) w θ under which the Hamiltonianfunction H w θ ( u, ¯ u ) in (1.8) can be transformed into a new Hamiltonian function H ( r,w θ ) = H w θ ◦ T ( r ) w θ = H w θ + Z ( r,w θ ) + R N ( r,w θ ) + R T ( r,w θ ) | {z } P ( r,wθ ) with a high degree ( θ, γ, α, N )-normal form Z ( r,w θ ) (see definition 5.1). Because the system(1.7) is in an infinite dimension, one can only get a partial normal form. R N ( r,w θ ) is atleast 3 order about ( u j , ¯ u j ) | j | >N ( N is large enough) and R T ( r,w θ ) has a zero of high orderabout ( u, ¯ u ). The Hamiltonian vector field of P ( r,w θ ) is still unbounded. The constructionof T ( r ) w θ is from solving Homological equation (refer Lemma 5.2). Because the perturbationin equation (1.7) is unbounded, a strong non resonant condition to frequencies { ω w θ j ( V ) } isneeded to keep the transformation T ( r ) w θ bounded. This condition will effort the estimateof sets of potential V ( x ) and the expression of ( θ, γ, α, N )-normal form. Moreover, if P w θ ( u, ¯ u ) has ( β, θ )-type symmetric coefficients, then P ( r,w θ ) ( u, ¯ u ) is still of ( β, θ )-typesymmetric coefficients.Second step: The solution to the new Hamiltonian system satisfies d k u k p dt = {k u k p , H ( r,w θ ) ( u, ¯ u ) } w θ . (1.10)From above equation, it is obvious that estimating {k u k p , H ( r,w θ ) ( u, ¯ u ) } w θ is the key toget a long time behavior of the solution. For a general function f ( u, ¯ u ) with unboundedcoefficients, {k u k p , f ( u, ¯ u ) } w θ is not bounded even if k u k p is small enough. Fortunately, P ( r,w θ ) ( u, ¯ u ) has ( β, θ )-type symmetric coefficients semi-bounded by C θ >
0. Studyingthe Possion bracket of k u k p and P ( r,w θ ) ( u, ¯ u ) with ( β, θ )-type symmetric coefficients is animportant problem in this paper. Proposition 4.2 in section 4 and Lemma 5.1 in section5 state that |{k u k p , R N ( r,w θ ) ( u, ¯ u ) + R T ( r,w θ ) ( u, ¯ u ) } w θ | ≺ R r +1 (1.11)and |{k u k p , Z ( r,w θ ) ( u, ¯ u ) } w θ | ≺ R r +1 (1.12)hold true for any k u k p ≤ R ≪ N . With the help of (1.11), (1.12) and(1.10), the long time behavior of solution to the new Hamiltonian system can be obtained.Since the two Hamiltonian DNLS equations have some difference, there still are somedifferences in the results of existence of almost global solution. The main difference is thesets of the potentials. From Lemma 7.1, there exist positive measure subsets ˜Θ θm ⊂ Θ θm ( θ ∈ { , } ) such that when V ∈ ˜Θ θm , frequencies { ω w θ j ( V ) } are ( θ, γ, α, N )-non resonant(see definition 5.1). When V ∈ Θ m , its Fourier coefficients satisfy ˆ V j = ˆ V − j ∈ R , for any j ∈ Z , which makes the corresponding frequencies satisfying ω w j = ω w − j ; while V ∈ Θ m , ˆ V j does not always equal to ˆ V − j . Thus ω w j is not related to ω w − j for any j ∈ N . The potentialsets to equations (1.5) and (1.6) are different, because (1.5) and (1.6) have different5ymplectic forms and nonlinearities. To be specific, from the definitions of symplecticstructures, the following equations hold true for any j ∈ Z \ { }{ u j ¯ u − j + ¯ u j u − j , k u k p } w = 0 (1.13)and { u j ¯ u − j + ¯ u j u − j , k u k p } w = 0 . (1.14)In order to make |{ H w θ ( u, ¯ u ) , k u k p } w θ | being high order small as k u k p is small, when θ = 0, from (1.13) the terms depending on (cid:0) ( u j ¯ u − j + ¯ u j u − j ) (cid:1) j ∈ Z in H w θ ( u, ¯ u ) will notneed to be eliminated by symplectic transformations; when θ = 1, from (1.14) it needs toeliminate the terms depending on (cid:0) ( u j ¯ u − j + ¯ u j u − j ) (cid:1) j ∈ Z in H w θ ( u, ¯ u ). Therefore, it needsmore parameters in the case θ = 1 than in the case θ = 0 and the sets of potential V ( x )are different to equations (1.5) and (1.6).The paper is organized as follows: The section 2 of this paper is devoted to introductionof two types of Hamiltonian DNLS equations with respect to different symplectic forms.There are many differences between these two types of equations (see Remark 2.1). ThenI give the main results in this paper, the existence of global solutions with small initialvalues to these two types of DNLS equations (See Theorem 1 and Theorem 2).In the third section I present a definition of ( β, θ )-type symmetric( θ ∈ { , } ). Usingthis definition, one can describe the coefficients of nonlinearities of two types of Hamil-tonian DNLS equations under Fourier transformation. The long time stability result toinfinite dimensional Hamiltonian systems owing ( β, θ )-type symmetric coefficients is givenin Theorem 3. Theorem 1 and Theorem 2 follow from Theorem 3.In the fourth section I give two main estimates. One is the estimate of the H p − norm of Hamiltonian vector field of a polynomial f w θ ( u, ¯ u ) with ( β, θ )-type symmetriccoefficients under symplectic form w θ ( θ ∈ { , } ). This estimate is given in Proposition4.1. It is easy to found that the Hamiltonian vector field of f w θ ( u, ¯ u ) is unbounded.Proposition 4.2 states that |{ f w θ ( u, ¯ u ) , k u k p } w θ | is small when k u k p is small enough and f w θ ( u, ¯ u ) has ( β, θ )-type symmetric coefficients. Even if the set of momentum of f w θ ( u, ¯ u )is unbounded, the result still holds true. The property of having ( β, θ )-type symmetriccoefficients is invariant under some operators, such as truncated operators Γ N ≤ and Γ N> defined in (4.4) and (4.5). See Corollary 1.In the fifth section, in order to improve the order of Birkhoff normal forms of Hamilto-nian systems under two different symplectic forms, I will find suitable bounded symplec-tic transformations (See Theorem 4). These transformations are constructed by solvingHomological equations. Since the nonlinear vector fields of Hamiltonian systems are un-bounded (see Proposition 4.1), a stronger non-resonant condition (see definition 5.1) isneeded. Under these transformations, new Hamiltonian systems are obtained. The non-linearities of the new Hamiltonian functions still have ( β, θ )-type symmetric coefficients(See Lemma 5.3). Although the high order normal forms Z ( r,w θ ) ( u, ¯ u ) ( θ ∈ { , } ) inthe new Hamiltonian functions are not standard Birkhoff normal forms, from Lemma 5.1 {k u k p , Z ( r,w θ ) ( u, ¯ u ) } w θ is high order small when k u k p is small enough. The detail of theproof of Theorem 4 is listed in Appendix.In the sixth section Theorem 3 is proved by applying Theorem 4, Proposition 4.2,Corollary 1 and Lemma 5.1, .In the seventh section the proofs of Theorem 1 and Theorem 2 are given. Using Lemma7.1, there exists a positive measure subset ˜Θ θm ⊂ Θ θm ( θ ∈ { , } ) such that when V ∈ ˜Θ θm the eigenvalues of linear operator ∂ xx + V ( x ) ∗ are stronger non resonant.6 Hamiltonian DNLS equations and main results
Let H p ([0 , π ] , C ) := (cid:26) ψ ∈ L ([0 , π ] , C ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ r ψ∂x r ∈ L ([0 , π ] , C ) , ∀ ≤ r ≤ p (cid:27) be a p-Sobolev space. The inner product of the space L ([0 , π ] , C ) is defined as h ζ , η i := Re Z π ζ · ¯ η d x, for any ζ , η ∈ L ([0 , π ] , C ) . The important definition of Hamiltonian PDEs is introduced in [32]. I list it as following.Consider an evolution equation ˙ ξ = Aξ + f ( ξ ) (2.1)defined in symplectic Hilbert scales ( { H p ([0 , π ] , C ) × H p ([0 , π ] , C ) } , α ), where α is anon-degenerate closed 2-form. Equation (2.1) is called a Hamiltonian equation , ifthere exists a Hamiltonian function H ( ξ ) defined in a domain O p ⊂ H p ([0 , π ] , C ) × H p ([0 , π ] , C ) making α ( Aξ + f ( ξ ) , η ) = −h dH ( ξ ) , η i for any ξ ∈ O p , η ∈ T O p ( T O p is the tangent space of O p ) . The dual space and the tangent space of H p ([0 , π ] , C ) × H p ([0 , π ] , C ) are isometry to H p ([0 , π ] , C ) × H p ([0 , π ] , C ), without confusion I denote them in the same signal in thefollowing content.Denote d A as the order of the linear operator A : H p ([0 , π ] , C ) × H p ([0 , π ] , C ) → H p − d A ([0 , π ] , C ) × H p − d A ([0 , π ] , C )and d f as the order of the mapping f : H p ([0 , π ] , C ) × H p ([0 , π ] , C ) → H p − d f ([0 , π ] , C ) × H p − d f ([0 , π ] , C ) . When the nonlinearity of a partial differential equation includes derivative, the cor-responding order of the nonlinear vector field is positive. Otherwise, the order is non-positive. The following notations “bounded” and “unbounded” are given by the signs ofthe order of the vector field, and readers can refer [28], [37], [38]. For the sake of referenceI list the definitions again.
Definition 2.1. If d f ≤ , call f in (2.1) bounded ; If d f > , f is called unbounded .Moreover, If d A − d f > , call f critical unbounded . In this paper, I focus on two kinds of Hamiltonian Derivative Nonlinear Schr¨odinger(DNLS) equations.
Type I –DNLS equation has the following form i ψ t = ∂ xx ψ + V ∗ ψ + i f ( x, ψ, ¯ ψ, ψ x , ¯ ψ x ) , ψ ∈ H p ([0 , π ] , C ) (2.2)under periodic boundary condition ψ ( x, t ) = ψ ( x + 2 π, t ) , (2.3)7here V belongs to Θ m := V ( x ) = X j ∈ Z b V j e i jx ∈ L ([0 , π ] , R ) (cid:12)(cid:12)(cid:12)(cid:12) v w j := b V j h j i m ∈ [ − ,
12 ] , v w j = v w − j , ∀ j ∈ Z (2.4) with m > / h j i := max { , | j |} and ¯ ψ is the complex conjugate of ψ. Suppose that there exists a function F ( x, ψ, ¯ ψ ) such that f ( x, ψ, ¯ ψ, ψ x , ¯ ψ x ) = 12 ∂ x ∂ ¯ ψ F ( x, ψ, ¯ ψ ) + ∂ ψ ¯ ψ F ( x, ψ, ¯ ψ ) ψ x . (2.5)Moreover, F satisfies assumptions as follows. A : F ( x, ξ, η ) is analytic about ( ξ, η ) in a neighborhood of the origin and satisfies F ( x, ψ, ¯ ψ ) = F ( x, ψ, ¯ ψ ) (2.6)and F ( x, ψ, ¯ ψ ) vanishes at least at order 2 in ( ψ, ¯ ψ ) at the origin. A For any fixed ( ψ, ¯ ψ ) a neighborhood of the origin, F ∈ H β +1 ([0 , π ] , C ) ( β is a bigenough positive real number) satisfies F ( x + 2 π, ψ, ¯ ψ ) = F ( x, ψ, ¯ ψ ) . Then (2.2) becomes a Hamiltonian PDE with a real value Hamiltonian function ∗ H ( . ) ( ψ, ¯ ψ ) = Z π −| ∂ x ψ | + ( V ∗ ψ ) ¯ ψ + i ∂ x F ( x, ψ, ψ ) + i ∂ ψ F ( x, ψ, ψ ) ψ x dx (2.8)on symplectic space ( H p ([0 , π ] , C ) × H p ([0 , π ] , C ) , w ), where w = J dψ ∧ d ¯ ψ, J − = (cid:18) − ii (cid:19) . (2.9)The corresponding Hamiltonian vector of H ( . ) ( ψ, ¯ ψ ) under symplectic form w is X w H ( . ) := (cid:16) − i ∂ ¯ ψ H ( . ) , i ∂ ψ H ( . ) (cid:17) T ∗ Since F ( x, ψ, ¯ ψ ) satisfies assumptions A - A , the following equation holds true for any ψ ∈ H p ([0 , π ] , C ) fulfilling ψ ( x + 2 π, t ) = ψ ( x, t )0 = Z π dFdx ( x, ψ, ¯ ψ ) dx = Z π ∂ x F ( x, ψ, ψ ) + ∂ ψ F ( x, ψ, ψ ) ψ x + ∂ ¯ ψ F ( x, ψ, ψ ) ¯ ψ x dx, i.e., Z π ∂ x F ( x, ψ, ψ ) + ∂ ψ F ( x, ψ, ψ ) ψ x dx = − Z π ∂ x F ( x, ψ, ψ ) − ∂ ¯ ψ F ( x, ψ, ψ ) ¯ ψ x dx. (2.7)From (2.7) and assumptions A - A , it follows Z π i ∂ x F ( x, ψ, ψ ) + i ∂ ψ F ( x, ψ, ψ ) ψ x dx = Z π ( − i ) 12 ∂ x F ( x, ψ, ψ ) − i ∂ ¯ ψ F ( x, ψ, ψ ) ¯ ψ x dx = Z π i ∂ x F ( x, ψ, ψ ) + i ∂ ψ F ( x, ψ, ψ ) ψ x dx, which means that the Hamiltonian function H ( . ) ( ψ, ¯ ψ ) is real. ˙ ψ = − i ∂ ¯ ψ H ( . ) ( ψ, ¯ ψ ) , ˙¯ ψ = i ∂ ψ H ( . ) ( ψ, ¯ ψ ) . (2.10) Type II –DNLS equation has the form as following i ψ t = ∂ xx ψ + V ∗ ψ + i ∂ x (cid:0) ∂F ( x, ψ, ¯ ψ ) ∂ ¯ ψ (cid:1) (2.11)defined on H p ([0 , π ] , C ) := (cid:26) ψ ∈ H p ([0 , π ] , C ) (cid:12)(cid:12)(cid:12)(cid:12) Z π ψ ( x, t ) dx = 0 (cid:27) (2.12)under periodic boundary condition ψ ( x, t ) = ψ ( x + 2 π, t ) . (2.13)The potential V belongs toΘ m := (cid:26) V ∈ L ([0 , π ] , C ) (cid:12)(cid:12)(cid:12)(cid:12) v w j := b V j h j i m ∈ [ − ,
12 ] , ∀ j ∈ Z \ { } , v w = b V = 0 (cid:27) (2.14)with m > / . If equation (2.11) satisfies the following assumptions: B F ( x, ξ, η ) is analytic at the origin about ( ξ, η ) ∈ H p ([0 , π ] , C ) × H p ([0 , π ] , C ) andvanishes at least at order 2 in ( ψ, ¯ ψ ) at origin. For any ψ ∈ H p ([0 , π ] , C ), it holds F ( x, ψ, ¯ ψ ) = F ( x, ψ, ¯ ψ ) . B For any fixed ( ψ, ¯ ψ ) in a neighborhood of the origin, F ∈ H β +1 ([0 , π ] , C ) ( β is a bigenough positive real number) satisfies F ( x + 2 π, ψ, ¯ ψ ) = F ( x, ψ, ¯ ψ );then equation (2.11) becomes into a Hamiltonian PDE with a real Hamiltonian H ( . ) ( ψ, ¯ ψ ) = Z π − i ∂ x ψ ¯ ψ − i ( ∂ x ) − ( V ( x ) ∗ ψ ) · ¯ ψ + F ( x, ψ, ¯ ψ )d x under symplectic space ( H p ([0 , π ] , C ) × H p ([0 , π ] , C ) , w ), where w := J dψ ∧ d ¯ ψ, ( J ) − := (cid:18) ∂ x ∂ x (cid:19) (2.15)is a symplectic from ( w is a non-degenerate closed two form in space H p ([0 , π ] , C ) × H p ([0 , π ] , C )). 9he Hamiltonian vector X w H ( . ) of H ( . ) ( ψ, ¯ ψ ) equals to (cid:16) ∂ x ( ∂ ¯ ψ H ( . ) ) , ∂ x ( ∂ ψ H ( . ) ) (cid:17) T . Equation (2.11) can be written as follow ˙ ψ = ∂ x ∂ ¯ ψ H ( . ) ( ψ, ¯ ψ ) , ˙¯ ψ = ∂ x ∂ ψ H ( . ) ( ψ, ¯ ψ ) . The DNLS equation researched in [37] is a special case of equation (2.11), i.e., F ( x, ψ, ¯ ψ )is independent of x . Remark 2.1.
There are some differences between type I-DNLS equations and type II-DNLS equations: • Nonlinearities of these two kinds of DNLS equations are different. It is an essentialdifference. • Symplectic spaces are different. Type I-DNLS equation is defined in ( H p ([0 , π ] , C ) × H p ([0 , π ] , C ) , w ) and type II-DNLS equation is defined in ( H p ([0 , π ] , C ) × H p ([0 , π ] , C ) ,w ) . w and w are also different. • The Potential V in type I-DNLS equation belongs to Θ m and the one in type II-DNLSequation belongs to Θ m . Θ m is different from Θ m . When V ∈ Θ m it fulfills ˆ V j =ˆ V − j ∈ R which means V ( x ) = V ( x ) ; while V ∈ Θ m , V is a complex valued potential.The potential V will directly determine the eigenvalues of the linear operator − ∂ xx + V ( x ) ∗ . It is clear that when V ∈ Θ m the corresponding eigenvalues ω j and ω − j ofthe linear operator − ∂ xx + V ( x ) ∗ are resonant, when V ∈ Θ m , they are independent.The measures of Θ m and Θ m are defined as followsmeas (Θ m ) := Y j ∈ N ∪{ } meas (cid:8) v j ∈ [ − ,
12 ] (cid:12)(cid:12) V ∈ Θ m , b V j h j i m = v j (cid:9) and meas (Θ m ) := Y j ∈ Z \{ } meas (cid:8) v j ∈ [ − ,
12 ] | V ∈ Θ m , b V j | j | m = v j (cid:9) , where “meas” means the Lebesgue measure. Remark 2.2. • If type II-DNLS equation (2.11) is defined in space H p ([0 , π ] , C ) , it is easy toverify that for any solution ψ ( x, t ) to equation (2.11) the quantity R π ψ ( x, t ) dx is a constant for any t ∈ R . Set φ := ψ − c, c := Z π ψ ( x, t ) dx = Z π ψ ( x, dx. If ψ ∈ H p ([0 , π ] , C ) , then φ ∈ H p ([0 , π ] , C ) . When G ( x, φ, ¯ φ ) := F ( x, φ + c, φ + c ) satisfies B and B , then equation (2.11) becomes into a Hamiltonian equationunder symplectic form w about ( φ, ¯ φ ) . • From Proposition 4.1 in section 4, the nonlinearities of type I-DNLS equation (2.2)and type II-DNLS equation (2.11) are unbounded. .2 Main result The long time behavior of the solutions around equilibrium point to type I and typeII-DNLS Hamiltonian equations are given in this subsection.
Theorem 1.
Suppose that the equation (2.2) satisfies assumptions A - A . For anyinteger r ∗ > , there exist an almost full measure set e Θ m ⊂ Θ m and p ∗ > such that forany fixed V ∈ e Θ m and any p fulfilling ( β − / > p > p ∗ , if the initial data of the solutionto (2.2) satisfies k ψ ( x, k H p ([0 , π ] , C ) ≤ ε < ε ∗ , then one has k ψ ( x, t ) k H p ([0 , π ] , C ) < ε, ∀ | t | ≺ ε − r ∗ − . Theorem 2.
Suppose that equation (2.11) fulfills assumptions B - B . For any integer r ∗ > , there exist a positive p ∗ and an almost full measure set e Θ m ⊂ Θ m ( m > ) suchthat for any fixed V ∈ e Θ m and any p fulfilling ( β − / > p > p ∗ , the solution to (2.11)satisfies k ψ ( x, t ) k H p +1 / ([0 , π ] , C ) < ε, for any | t | ≺ ε − r ∗ − , if the initial value fulfills k ψ ( x, k H p +1 / ([0 , π ] , C ) < ε ≪ . Remark 2.3.
As type I and type II-DNLS equations have many differences (Readers canrefer Remark 2.1), the proofs of Theorem 1 and Theorem 2 still have some differences.
Remark 2.4.
The DNLS equations researched in our paper are not always invariantunder gauge transformation. For example, take F ( ψ, ¯ ψ ) = ψ ¯ ψ + ψ ¯ ψ (2.16) in equation (2.11), which fulfills assumption B . It is easy to check that equation (2.11)with F in (2.16) is not invariant under the transformation φ = e i θ ψ , θ ∈ R . ( β, θ ) -type symmetric coeffi-cients ( θ ∈ { , } ) ( β, θ ) -type symmetric coefficients ( θ ∈ { , } ) Under Fourier transformation, Hamiltonian DNLS equations with respect to periodicboundary condition can be transformed into two classes of infinite dimension Hamiltoniansystems with “unbounded” nonlinearities. In this section, I will introduce long time sta-bility results to two classes of infinite dimension Hamiltonian systems with “unbounded”nonlinearities. First, give some notations and annotations. In this paper, Z ∗ means Z or Z \ { } . Denote weighted Hilbert spaces ℓ p ( Z ∗ , C ) := (cid:26) u ∈ ℓ ( Z ∗ , C ) (cid:12)(cid:12)(cid:12)(cid:12) k u k p := X j ∈ Z ∗ h j i p · | u j | < + ∞ , h j i := max {| j | , } (cid:27) , H p ( Z ∗ , C ) := { ( u, v ) ∈ ℓ p ( Z ∗ , C ) × ℓ p ( Z ∗ , C ) (cid:12)(cid:12) v = ¯ u } with norm k ( u, ¯ u ) k p := q k u k p + k ¯ u k p . Let the neighborhood of the origin with a radius R be noted by B p ( R ) := (cid:8) ( u, ¯ u ) ∈ H p ( Z ∗ , C ) (cid:12)(cid:12) k ( u, ¯ u ) k p < R (cid:9) . Definition 3.1.
For any fixed l, k ∈ N Z ∗ , call the integer X j ∈ Z ∗ j ( l j − k j ) be the momentum of the ordered vector ( l, k ) and denote it as M ( l, k ) . Readers can refer this definition in [37] and [38].
Remark 3.1. If M ( l, k ) = i , from the definition of momentum, it holds that M ( k, l ) = − i. Definition 3.2.
Call a power series f ( u, ¯ u ) = X t ≥ X | k + l | = t,l,k ∈ NZ ∗M ( l,k )= i ∈ Mft ⊂ Z f it,lk u l ¯ u k , ( u, ¯ u ) ∈ H p ( Z ∗ , C ) have symmetric coefficients , if for any l, k fulfilling | l + k | = t and M ( l, k ) = i , thecoefficient holds f it,lk = f − it,kl . Moreover, fixed β > , call f ( u, ¯ u ) has β - bounded coefficients bounded by C > , if | f it,lk | ≤ C t − h i i β , for any l, k satisfying | l + k | = t and M ( l, k ) = i . Remark 3.2.
A power series f : H p ( Z ∗ , C ) → C is of symmetric coefficients, if and onlyif f satisfies f ( u, ¯ u ) = f ( u, ¯ u ) , for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) . Hence, a real-value Hamiltonian function has symmetric coefficients.
Now define two kinds of power series with “unbounded” special symmetric coefficients.
Definition 3.3.
Given β > and C f > , call a power series f ( u, ¯ u ) = X t ≥ X | k + l | = t,l,k ∈ NZ ∗M ( l,k )= i ∈ Mft ⊂ Z f it,lk u l ¯ u k , ( u, ¯ u ) ∈ H p ( Z ∗ , C ) have ( β, -type symmetric coefficients , if its coefficients have the following form f it,lk := X ( l ,k ,i ) ⊂A fit,lk f i ( l ,k ,i ) t,lk (cid:0) M ( l , k ) − i (cid:1) , here A f it,lk ⊂ { (˜ l, ˜ k, ˜ i ) | ≤ ˜ l j ≤ l j , ≤ ˜ k j ≤ k j , for any j ∈ Z ∗ , (˜ l, ˜ k ) ∈ N Z ∗ × N Z ∗ , ˜ i ∈ Z } , and for any ( l , k , i ) ∈ A f it,lk , the followings hold true ( k − k , l − l , i − i ) ∈ A f − it,kl , f i ( l ,k ,i ) t,lk = f − i ( k − k ,l − l ,i − i ) t,kl . Moreover, call f ( u, ¯ u ) have ( β, -type symmetric coefficients semi-bounded by C f ,if f ( u, ¯ u ) have ( β, -type symmetric coefficients and there exists a constant C f > suchthat for any l, k ∈ N Z ∗ with | l + k | = t and M ( l, k ) = i , the following inequality holds true X ( l ,k ,i ) ∈A fit,lk | f i ( l ,k ,i ) t,lk | · max {h i i , h i − i i} ≤ C t − f h i i β . (3.1)Suppose that Type I-DNLS equation satisfies assumption A - A . Under Fourier trans-formation, there exists a constant C > β, -type symmetric coefficients semi-bounded by C . Seesection 7 for details. This symmetric property is invariant under a symplectic transfor-mation. Refer Lemma 5.3 in section 5. Definition 3.4.
Given β > and C g > , call a power series g ( u, ¯ u ) = X t ≥ X | k + l | = t, l,k ∈ NZ ∗M ( l,k )= i ∈ Mgt ⊂ Z g it,lk u l ¯ u k have ( β, -type symmetric coefficients , if for any l, k ∈ N Z ∗ with | l + k | = t and M ( l, k ) = i ∈ M g t ⊂ Z , its coefficient has the following form g it,lk := ˜ g it,lk Y j ∈ Z ∗ h j i ( l j + k j ) and satisfies ˜ g it,lk = ˜ g − it,kl . Moreover, call g ( u, ¯ u ) have ( β, -type symmetric coefficients semi-bounded by C g ,if g ( u, ¯ u ) have ( β, -type symmetric coefficients and there exists a constant G g > suchthat | ˜ g it,lk | ≤ C t − g h i i β (3.2) for any l, k fulfilling | l + k | = t and M ( l, k ) = i. Remark 3.3.
Suppose a power series f w θ ( u, ¯ u ) is of ( β, θ ) -type symmetric coefficientssemi-bounded by C θ > ( θ ∈ { , } ). Thence, • the “semi-bounded” does not means the coefficients of f w θ ( u, ¯ u ) are bounded, evenif f w θ ( u, ¯ u ) is an r -degree polynomial; the momentum set M f wθt is symmetric, i.e., if i ∈ M f wθt , then − i ∈ M f wθt . And thenumber of elements in M f wθt satisfies ♯M f wθt ≤ M f wθt + 1 . (3.3) So does the power series having β -bounded symmetric coefficients. • when θ = 0 for any given ( l, k ) ∈ N Z ∗ × N Z ∗ and i ∈ Z , ( f w ) it,lk = X ( l ,k ,i ) ⊂A fit,lk ( f w ) i ( l ,k ,i ) t,lk (cid:0) M ( l , k ) − i (cid:1) = X ( l ,k ,i ) ⊂A ( fw it,lk ( f w ) − i ( k − k ,l − l ,i − i ) t,kl (cid:0) M ( l , k ) − i (cid:1) = X ( k − k ,l − l ,i − i ) ⊂A ( fw − it,kl ( f w ) − i ( k − k ,l − l ,i − i ) t,kl (cid:0) M ( k − k , l − l ) − ( i − i ) (cid:1) = ( f w ) − it,kl , the last second equation follows from M ( l , k ) − i M ( k − k , l − l ) − ( i − i ) and ( k − k , l − l , i − i ) ∈ A f − it,kl ; when θ = 1 for any l, k ∈ N Z ∗ and any i ∈ Z , ( f w ) it,lk = ( ˜ f w ) it,lk Y j ∈ Z ∗ h j i ( l j + k j ) = ( ˜ f w ) − it,kl Y j ∈ Z ∗ h j i ( k j + l j ) = ( f w ) − it,kl . (3.4) Hence, the coefficients of f w θ ( u, ¯ u ) are symmetric. Let ( H p ( Z ∗ , C ) , w θ ) ( θ ∈ { , } ) be a symplectic space endowed with symplectic form w θ := i P j ∈ Z ∗ d u j ∧ d¯ u j θ = 0 , i P j ∈ Z ∗ sgn( j )d u j ∧ d¯ u j θ = 1 . (3.5)When θ = 0, Z ∗ can be either Z or Z \{ } . When θ = 1, Z ∗ means Z \{ } only.The possion bracket of differential functions f and f defined in the domain of H p ( Z ∗ , C ) under the symplectic form w θ ( θ ∈ { , } ) has the following form { f , f } w θ = w θ ( ∇ f , ∇ f ) . (3.6)Given a differential function f , its corresponding Hamiltonian vector field under the sym-plectic form w θ is defined as X w θ f := J θ ∇ f, J θ := (cid:18) − i I θ i I θ (cid:19) , θ ∈ { , } , (3.7)where I is an identity operator on space ℓ p ( Z ∗ , C ), and for any u = ( u j ) j ∈ Z ∗ ∈ ℓ p ( Z ∗ , C ), I u = (cid:0) ( I u ) j (cid:1) j ∈ Z ∗ , ( I u ) j := sgn( j ) · u j , j ∈ Z ∗ . .2 Result of Hamiltonian system with ( β, θ )-Type symmetriccoefficients In order to use an uniformly formula to describe two kinds of Hamiltonian equations, inthis paper, denote 0 = 1.Let θ ∈ { , } . Consider Hamiltonian systems defined in ( H p ( Z ∗ , C ) , w θ ), for any j ∈ Z ∗ ˙ u j = − i sgn θ ( j ) · ∂ ¯ u j H w θ ( u, ¯ u ) , ˙¯ u j = i sgn θ ( j ) · ∂ u j H w θ ( u, ¯ u ) , (3.8)with a Hamiltonian function H w θ ( u, ¯ u ) = H w θ + P w θ ( u, ¯ u ) , (3.9)where H w θ := P j ∈ Z ∗ ω w θ j | u j | . Theorem 3.
Suppose that equation (3.8) satisfies the following assumptions: A θ : ω w θ := ( ω w θ j ) j ∈ Z ∗ , ω w θ j ∈ R . ω w θ satisfies strong non resonant condition. † B θ : P w θ ( u, ¯ u ) is a power series beginning with at least at order 2 in ( u, ¯ u ) and has ( β, θ ) -type symmetric coefficients semi-bounded by C θ > ( β is big enough positivenumber).Given integer r ∗ > , there exists an integer p r ∗ > , for any p fulfilling ( β − / > p > p r ∗ there exists ε r ∗ ,p > such that the solution to (3.8) satisfies k ( u ( t ) , ¯ u ( t )) k p < ε, for any | t | (cid:22) ε − r ∗ − , if the initial data fulfills k ( u (0) , ¯ u (0)) k p < ε < ε r ∗ ,p . Let us give the basic procedure of proving Theorem 3 which consists of the followingsteps.The first step is to construct a bounded symplectic transformation around the originunder which the nonlinearity of Hamiltonian function (3.9) becomes into the sum ofthe following three parts: one is a high order ( θ, γ, α, N )-normal form Z ( r ∗ ,w θ ) ( u, ¯ u ) ‡ ,one of the others is R T ( r ∗ ,w θ ) ( u, ¯ u ) which vanishes at r ∗ + 3 order of ( u, ¯ u ) at originand the last one denoted as R N ( r ∗ ,w θ ) ( u, ¯ u ) has zero at least order 3 about high indexvariable ( u j , ¯ u j ) | j | >N ( N is big enough). Moreover, when P w θ ( u, ¯ u ) in (3.9) has ( β, θ )-typesymmetric coefficients semi-bounded by C θ >
0, the new Hamiltonian function is stillof ( β, θ )-type symmetric coefficients semi-bounded by C ( θ, r ) > C ( θ, r ) is defined inTheorem 4). In order to guarantee the boundedness of the symplectic transformation, astrong non-resonant condition is presented in Definition 5.1. See Theorem 4 in section 5for details.Since (3.8) is a Hamiltonian system, the following equation holds true k u ( t ) k p − k u (0) k p = Z t ddτ k u ( τ ) k p dτ = Z t {k u k p , H w θ ( u, ¯ u ) } w θ dτ. † See Definition 5.1 in section 5. ‡ See Definition 5.1. H w θ ( u, ¯ u ) and k u k p under thecorresponding symplectic form w θ is important. The second step is to estimate the Pos-sion bracket of function f w θ ( u, ¯ u ) and k u k p . Suppose that f w θ ( u, ¯ u ) has ( β, θ )-typesymmetric coefficients semi-bounded by C θ >
0. If the momentum of f w θ ( u, ¯ u ) arebounded, partial result can be found in [37] and [38]. When the set of the momen-tum of f w θ ( u, ¯ u ) is unbounded, the corresponding Hamiltonian vector field of f w θ ( u, ¯ u )is small under H p − norm but not H p norm (see Proposition 4.1 and Remark 4.1). Inorder to deal with it, I will make use of the Hamiltonian structure and ( β, θ )-type sym-metric coefficients semi-bounded by C θ > H w θ ( u, ¯ u ) and k u k p under the corresponding symplectic form w θ .See Proposition 4.2 and Corollary 1 for details. By Proposition 4.2 and Corollary 1, |{R T ( r ∗ ,w θ ) ( u, ¯ u ) + R N ( r ∗ ,w θ ) ( u, ¯ u ) , k u k p } w θ | ≺ R r ∗ +1 holds true for k ( u, ¯ u ) k p ≤ R . FromRemark 5.2, ( θ, γ, α, N )-normal form Z ( r ∗ ,w θ ) ( u, ¯ u ) is not a standard Birkhoff normal form.By Lemma 5.1 when Z ( r ∗ ,w θ ) ( u, ¯ u ) has ( β, θ )-type symmetric coefficients semi-bounded by C ( θ, r ∗ ), for any k ( u, ¯ u ) k p < R ≪ N satisfying (5.28), |{ Z ( r ∗ ,w θ ) ( u, ¯ u ) , k u k p }| ≺ R r ∗ +1 . { f w θ r ( u, ¯ u ) , k u k p } w θ and Hamiltonian vectorfield X f wθr ( f w θ r ( u, ¯ u ) has ( β, θ ) -type symmetric coef-ficients θ ∈ { , } ) Suppose that an r -degree homogeneous power series f w θ r ( u, ¯ u ) defined on H p ( Z ∗ , C ) isof ( β, θ )-type symmetric coefficients semi-bounded by C θ > f w θ r ( u, ¯ u ) under symplecticform w θ is unbounded with order 1. See Proposition 4.1.Next, the estimate of the possion bracket of the power series f w θ r ( u, ¯ u ) and k u k p isgiven in Proposition 4.2.Last but not least, I introduce truncated operators Γ N ≤ and Γ N> , and estimate theHamiltonian vector fields of the functions Γ N ≤ f w θ r ( u, ¯ u ), Γ N> f w θ r ( u, ¯ u ) in H p − ( Z ∗ , C )-norm, |{ Γ N ≤ f w θ r ( u, ¯ u ) , k u k p }| and |{ Γ N> f w θ r ( u, ¯ u ) , k u k p }| for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) in Corollary1. These results will be used in proving Theorem 3. Proposition 4.1.
Suppose that an r -degree ( r ≥ ) homogeneous polynomial f w θ r ( u, ¯ u ) ( θ ∈ { , } ) defined on H p ( Z ∗ , C ) has ( β, θ ) -type symmetric coefficients semi-bounded by C f wθ > , and β − p ≥ . Then for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) and any θ ∈ { , } , k X w θ f wθr ( u, ¯ u ) k p − ≤ C r − f wθ r p +1 c r − k u k r − k u k p . (4.1) If an r -degree homogeneous polynomial f r ( u, ¯ u ) has β -bounded symmetric coefficientsbounded by C f > , then the following inequality holds true for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) and any θ ∈ { , } k X w θ f r ( u, ¯ u ) k p ≤ C r − f r r p +1 c r − k u k r − k u k p . (4.2) Remark 4.1.
If an r -degree homogeneous polynomial f w θ r ( u, ¯ u ) : B p ( R ∗ ) → C ( θ ∈{ , } , R ∗ > ) has ( β, θ ) -type symmetric coefficients semi-bounded by C f wθ > ( θ ∈{ , } ), from Proposition 4.1 it holds that The Hamiltonian vector field X w θ f wθr is from B p ( R ∗ ) to H p − ( Z ∗ , C ) , but not to H p ( Z ∗ , C ) .It means that X w θ f wθr is unbounded with order 1. • There exists a constant δ ∈ (0 , such that | f w θ r ( u, ¯ u ) | = |h∇ ( u, ¯ u ) f w θ r ( δu, δ ¯ u ) , ( u, ¯ u ) i| . (4.3) Together with Cauchy estimate and (4.1), one has that the function f w θ r ( u, ¯ u ) ( θ ∈{ , } ) is analytic about ( u, ¯ u ) on some B p ( R ) ⊂ H p ( Z ∗ , C ) ( p > ). (In this paper,when I mention the “analyticity” of functions or vector fields, I take u and ¯ u asindependent variables). Proposition 4.2.
Suppose that an r -degree ( r ≥ ) homogeneous polynomial f w θ r ( u, ¯ u )( θ ∈ { , } ) has ( β, θ ) -type symmetric coefficients semi-bounded by C f wθr > . Then thefollowing inequality holds true for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) ( p ≥ , β − p ≥ ) (cid:12)(cid:12) { f w θ r ( u, ¯ u ) , k u k p } w θ (cid:12)(cid:12) ≤ C r − f wθr p +1 pr p − c r − k u k p k u k r − . Given an integer
N >
0, two projection operators Γ >N and Γ ≤ N on N Z ∗ and ℓ p ( Z ∗ , C )are defined as follows. For any l = ( l j ) j ∈ Z ∗ ∈ N Z ∗ , Γ >N l and Γ ≤ N l ∈ N Z ∗ with(Γ >N l ) j := (cid:26) l j , | j | > N, , | j | ≤ N, (Γ ≤ N l ) j := (cid:26) , | j | > N,l j , | j | ≤ N. For any u = ( u j ) j ∈ Z ∗ ∈ ℓ p ( Z ∗ , C ), Γ >N u , Γ ≤ N u ∈ ℓ p ( Z ∗ , C ) with(Γ >N u ) j := (cid:26) u j , | j | > N, , | j | ≤ N, (Γ ≤ N u ) j := (cid:26) , | j | > N,u j , | j | ≤ N. Now I will introduce two truncated operators Γ N ≤ and Γ N> defined as follows. For anypower series f ( u, ¯ u ) := X r ≥ X | l + k | = r M ( l,k )= i f ir,lk u l ¯ u k denote Γ N ≤ f ( u, ¯ u ) := X r ≥ X | l + k | = r, M ( l,k )= i | Γ >N ( l + k ) |≤ , | i |≤ N f ir,lk u l ¯ u k , (4.4)Γ N> f ( u, ¯ u ) := f ( u, ¯ u ) − Γ N ≤ f ( u, ¯ u ) . (4.5) Remark 4.2.
Fix a positive integer N . Suppose that a power series f w θ ( u, ¯ u ) has ( β, θ ) -type symmetric coefficients semi-bounded by C f wθ > θ ∈ { , } ) . Then Γ N ≤ f w θ ( u, ¯ u ) , Γ N> f w θ ( u, ¯ u ) also have ( β, θ ) -type symmetric coefficients semi-bounded by C f wθ > . Corollary 1.
Suppose that an r -degree ( r ≥ ) homogeneous polynomials f w θ r ( u, ¯ u ) ( θ ∈ { , } ) defined on H p ( Z ∗ , C ) has ( β, θ ) -type symmetric coefficients semi-boundedby C f wθ > , and β − p ≥ . Given an integer N > , then k X w θ Γ N ≤ f wθr ( u, ¯ u ) k p − ≤ C r − f wθ r p +1 c r − k u k r − k u k p , X w θ Γ N> f wθr ( u, ¯ u ) k p − ≤ C r − f wθ r p +1 c r − k u k r − k Γ >N u k k u k p +16 N − ( β − p − ) C r − f wθ r p +1 c r − k u k r − k u k p , (cid:12)(cid:12) { Γ N ≤ f w θ r ( u, ¯ u ) , k u k p } w θ (cid:12)(cid:12) ≤ C r − f wθr p +1 pr p − c r − k u k p k u k r − , (cid:12)(cid:12) { Γ N> f w θ r ( u, ¯ u ) , k u k p } w θ (cid:12)(cid:12) ≤ C r − f wθr p +1 pr p − c r − k u k p k u k r − k Γ >N u k + N − ( β − p − ) C r − f wθr p +2 pr p − c r − k u k p k u k r − . (4.6)The proof of Corollary 4.5 is similar with Proposition 4.1-4.2 and I omit it. In orderto give the proof of Proposition 4.1-4.2, the following Lemmas are needed. Lemma 4.1.
If power series f w θ ( u, ¯ u ) and g w θ ( u, ¯ u ) have ( β, θ ) -type symmetric coeffi-cients ( θ ∈ { , } ) semi-bounded by C f wθ > and C g wθ > , respectively, then for any a, b ∈ R , ( af w θ + bg w θ )( u, ¯ u ) also has ( β, θ ) -type symmetric coefficients semi-bounded by C af wθ + bg wθ := max {| a | C f wθ + | b | C g wθ , | a | C f wθ + C g wθ , C f wθ + | b | C g wθ , C f wθ + C g wθ } > . (4.7) Proof.
I only give the proof in the case θ = 0, while in the case θ = 1 the proof is similarto the case θ = 0. For any a , b ∈ R ,( af w + bg w )( u, ¯ u ) = X t ≥ X | l + k | = t M ( l,k )= i ∈ M ( afw bgw t ( af w + bg w ) it,lk u l ¯ u k , (4.8)where M ( af w + bg w ) t := M f w t ∪ M g w t . For any l, k ∈ N Z ∗ with M ( l, k ) = i ∈ M ( af w + bg w ) t ,the corresponding coefficient of af w + bg w has the following form( af w + bg w ) it,lk = X ( l ,k ,i ) ∈A ( afw bgw it,lk ( af w + bg w ) i ( l ,k ,i ) t,lk ( M ( l , k ) − i , where A ( af w + bg w ) it,lk := A ( f w ) it,lk ∪ A ( g w ) it,lk and( af w + bg w ) i ( l ,k ,i ) t,lk := af w i ( l ,k ,i ) t,lk + bg w i ( l ,k ,i ) t,lk , ( l , k , i ) ∈ A ( f w ) it,lk ∩ A ( g w ) it,lk ,af w i ( l ,k ,i ) t,lk , ( l , k , i ) ∈ A ( f w ) it,lk ∩ A C ( g w ) it,lk ,bg w i ( l ,k ,i ) t,lk , ( l , k , i ) ∈ A C ( f w ) it,lk ∩ A ( g w ) it,lk . It is easy to check that( af w + bg w ) i ( l ,k ,i ) t,lk = ( af w + bg w ) − i ( k − k ,l − l ,i − i ) t,kl and X ( l ,k ,i ) ∈A ( afw bgw it,lk | ( af w + bg w ) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i}≤ X ( l ,k ,i ) ∈A ( fw it,lk | a ( f w ) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i} X ( l ,k ,i ) ∈A ( gw it,lk | b ( g w ) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i}≤ ( C af w + bg w ) t − h i i β , where C af w + bg w is defined in (4.7). Lemma 4.2.
Given real numbers q i ≤ p ( ≤ i ≤ r ), suppose that F = ( F j ) j ∈ Z ∗ is an r -multiple linear vector field defined as following F j ( u (1) , · · · , u ( r ) ) := X j = j (1) ± j (2) ···± j ( r ) F j,j (1) ··· j ( r ) u (1) j (1) · · · u ( r ) j ( r ) , j ∈ Z ∗ where u (1) := (cid:0) u (1) j (1) (cid:1) j (1) ∈ Z ∗ , · · · u ( r ) := (cid:0) u ( r ) j ( r ) (cid:1) j ( r ) ∈ Z ∗ ∈ ℓ p ( Z ∗ , C ) . If there exist a positiveconstant C and an integer n ∈ { , · · · , r } such that | F j,j (1) ··· j ( r ) | ≤ C h j ( n ) i q n · r Y t =1 ,t = n h j ( t ) i q t − , (4.9) then k F ( u (1) , · · · , u ( r ) ) k ℓ ( Z ∗ , C ) ≤ Cc r − (cid:13)(cid:13) u ( n ) (cid:13)(cid:13) q n · r Y i =1 ,i = n (cid:13)(cid:13) u ( i ) (cid:13)(cid:13) q t , where c := qP j ∈ Z h j i − . Proof.
Using Young’s inequality § the following inequality holds true for any a ∈ ℓ ( Z ∗ , C )and b ∈ ℓ ( Z ∗ , C ) (cid:13)(cid:13)(cid:0) X k ∈ Z ∗ a j ± k b k (cid:1) j ∈ Z ∗ (cid:13)(cid:13) ℓ ( Z ∗ , C ) ≤ k a k ℓ ( Z ∗ , C ) · k b k ℓ ( Z ∗ , C ) . (4.10)Together with (4.9), using (4.10) repeatedly, one has k F ( u (1) , · · · , u ( r ) ) k ℓ ( Z ∗ , C ) ≤ C (cid:13)(cid:13)(cid:0) X j = j (1) ± j (2) ···± j ( r ) h j (1) i q − | u (1) j (1) | · · · h j ( n − i q n − − | u ( n − j ( n − | · h j ( n ) i q n | u ( n ) j ( n ) | ·h j ( n +1) i q n +1 − | u ( n +1) j ( n +1) | · · · h j ( r ) i q r − | u ( r ) j ( r ) | (cid:1) j ∈ Z ∗ (cid:13)(cid:13) ℓ ( Z ∗ , C ) ≤ C k u ( n ) k q n · Y ≤ i ≤ r,i = n (cid:13)(cid:13) ( h j ( i ) i q i − · | u ( i ) j ( i ) | ) j ( i ) ∈ Z ∗ (cid:13)(cid:13) ℓ ( Z ∗ , C ) . (4.11)Since q i ≤ p , the following inequality holds true for any u = ( u j ) j ∈ Z ∗ ∈ ℓ p ( Z ∗ , C ), k ( h j i q i − | u j | ) j ∈ Z ∗ k ℓ ( Z ∗ , C ) = X j ∈ Z ∗ (cid:0) h j i q i | u j | · h j i (cid:1) ≤ sX j ∈ Z h j i − · k u k q i = c k u k q i . (4.12) § Suppose a ∈ ℓ p ( Z ∗ , C ), b ∈ ℓ q ( Z ∗ , C ) and p + q = r + 1 , with 1 ≤ p, q, r ≤ ∞ . Then k f ∗ g k ℓ r ( Z ∗ , C ) ≤k f k ℓ p ( Z ∗ , C ) · k g k ℓ q ( Z ∗ , C ) .
19n view of (4.12) and (4.11), one has k F ( u (1) , · · · , u ( r ) ) k ℓ ( Z ∗ , C ) ≤ Cc r − (cid:13)(cid:13) u ( n ) (cid:13)(cid:13) q n · r Y i =1 ,i = n (cid:13)(cid:13) u ( i ) (cid:13)(cid:13) q i . Corollary 2.
Given integers p > q ≥ and real number ρ ≥ , suppose that there exists apositive number C f > such that the coefficients of an r -degree homogeneous polynomial f ( u, ¯ u ) = X l,k ∈ NZ ∗ , i ∈ Mfr ⊆ Z | l + k | = r, M ( l,k )= i f ir,lk u l ¯ u k satisfy that for any l, k ∈ N Z ∗ with M ( l, k ) = i ∈ M f r | f ir,lk | ≤ C f h i i ρ l j h j i p − q +1 (cid:0) X t ∈ Z ∗ ( l + k − e j ) t h t i ( p − q +1) (cid:1) Y m ∈ Z ∗ h m i ( q − l m + k m ) . (4.13) Then for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) , it satisfies that | f ( u, ¯ u ) | ≤ C f c r − r k u k p · k u k r − q . Remark 4.3.
The result of Corollary 2 still holds true for ρ > . To simplify the processof proof, assume ρ ≥ .Proof. By Cauchy estimate, | f ( u, ¯ u ) | ≤ |h F, G i| ≤ k F k ℓ · k G k ℓ , (4.14)where G := ( h j i p | u j | ) j ∈ Z ∗ and F ( u, ¯ u ) = ( F j ( u, ¯ u )) j ∈ Z ∗ F j ( u, ¯ u ) := X | l − ej + k | = r − , M ( l,k )= i ∈ Mfr ⊆ Z f ir,lk h j i p u l − e j ¯ u k . For any n ∈ { , , · · · , r − } , there exist an ( r − e F n ( u (1) , · · · u ( r − ) := (cid:0) X | l − ej | = n, | k | = r − − n,i ∈ Mfr ⊆ Z , j = M ( l − ej,k ) − i f ir,lk h j i p u (1) · · · u ( n ) | {z } n u ( n +1) · · · u ( r − | {z } r − − n (cid:1) j ∈ Z ∗ such that F j ( u, ¯ u ) = r − X n =0 e F nj ( u, · · · , u | {z } n , ¯ u, · · · , ¯ u | {z } r − − n ) , for any j ∈ Z ∗ , . By condition (4.13), the coefficients of each ( r − e F n satisfythe condition (4.9) of Lemma 4.2 with q n = p and q i = q ≤ p ( i = n ). Hence, by Lemma4.2, for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ), one has k F ( u, ¯ u ) k ℓ ≤ r − X n =0 k e F n ( u, · · · , u | {z } n , ¯ u, · · · , ¯ u | {z } r − − n ) k ℓ ( Z ∗ , C ) C f c r − r k u k p · k u k r − q X i ∈ M fr ⊂ Z h i i − ρ ≤ C f c r − r k u k p · k u k r − q . (4.15)In view of (4.14) and (4.15), the following inequality holds true | f ( u, ¯ u ) | ≤ k F ( u, ¯ u ) k ℓ · k G ( u, ¯ u ) k ℓ ≤ k F ( u, ¯ u ) k ℓ · k u k p ≤ C f c r − r k u k p · k u k r − q . Now give the proof of Proposition 4.1.
Proof.
In the case θ = 0, (cid:13)(cid:13) X w f w r ( u, ¯ u ) (cid:13)(cid:13) p − = q k∇ ¯ u f w r ( u, ¯ u ) k p − + k∇ u f w r ( u, ¯ u ) k p − . (4.16)Note that k∇ ¯ u f w r ( u, ¯ u ) k p − equals to equals to ℓ ( Z ∗ , C ) norm of the following vectorfield (cid:18) X j = M ( l,k − ej ) − il,k ∈ NZ ∗ , i ∈ Mfw r ⊆ Z k j h j i p − · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k − e j (cid:19) j ∈ Z ∗ . (4.17) Accordingly, k∇ u f w r ( u, ¯ u ) k p − equals to equals to ℓ ( Z ∗ , C ) norm of the following vectorfield (cid:18) X j = i −M ( l − ej,k ) l,k ∈ NZ ∗ , i ∈ Mfw r ⊆ Z l j h j i p − · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l − e j ¯ u k (cid:19) j ∈ Z ∗ . (4.18) In order to use Lemma 4.2 to give the ℓ -norm of (4.17) and (4.18), it is required toestimate the coefficients of vector fields (4.17) and (4.18). Note that for any fixed l, k ∈ N Z ∗ and j ∈ Z ∗ with ( f w ) ir,lk = 0 and k j = 0, the indices satisfysgn( k j ) · j = M ( l, k − e j ) − i. (4.19)Since p >
2, it follows | j | p − ≤ r p − ( X t ∈ Z ∗ l t | t | p − + X t ∈ Z ∗ ,t = j k t | t | p − + ( k j − · | j | p − + | i | p − ) ≤ r p − ( X t ∈ Z ∗ l t | t | p − + X t ∈ Z ∗ ,t = j k t | t | p − + ( k j − · | j | p − ) · h i i p − , (4.20)the last inequality holds true by the fact that for any integer a, b ≥ a + b ≤ ab + 1 ≤ ab. (4.21)Furthermore, for any ( l , k , i ) ∈ A ( f w ) ir,lk , it holds that0 ≤ l j ≤ l j , ≤ k j ≤ k j , for any j ∈ Z ∗ (4.22)21nd the momentum of ( l , k ) equals to M ( l , k ) = M (cid:0) l , k − sgn( k j ) e j (cid:1) − sgn( k j ) · j, k j = 0 , M ( l , k ) , k j = 0 . (4.23)Together with (4.19), (4.21), (4.22) and (4.23), one has |M ( l , k ) − i | ≤ (cid:0) X n ∈ Z ∗ l n | n | + X n ∈ Z ∗ n = j k n | n | + ( k j − | j | (cid:1) · max {h i i , h i − i i} . (4.24)In view of (4.19), (4.20), (4.24) and f w ( u, ¯ u ) having the ( β, C f w >
0, one has h j i p − · (cid:12)(cid:12) X ( l ,k ,i ) ∈A ( fw ir,lk k j ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1)(cid:12)(cid:12) ≤ C r − f w r p h i i β − p +1 (cid:0) X m ∈ Z ∗ , l m + k m > h m i sgn ( l m +( k − e j ) m ) Y t ∈ Z ∗ h t i ( p − sgn ( l t +( k − e j ) t ) (cid:1) . (4.25)Therefor, using (4.25) and Lemma 4.2 by taking q n = p and q i = 2 ( i = n ), the followinginequality holds true k∇ ¯ u f w r ( u, ¯ u ) k p − ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:0) X j = M ( l,k − ej ) − il,k ∈ NZ ∗ , i ∈ Mfw r ⊆ Z , X ( l ,k ,i ) ∈A ( fw ir,lk (cid:12)(cid:12) k j j p − ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k − e j (cid:12)(cid:12)(cid:1) j ∈ Z ∗ (cid:13)(cid:13)(cid:13)(cid:13) ℓ ≤ r p +1 c r − C r − f w k u k r − k u k p . Similarly, one has k∇ u f w r ( u, ¯ u ) k p − ≤ r p +1 c r − C r − f w k u k r − k u k p . Thus, (cid:13)(cid:13) X w f w ( u, ¯ u ) (cid:13)(cid:13) p − ≤ r p +1 c r − C r − f w k u k r − k u k p . In the case θ = 1, (cid:13)(cid:13) X w f w r ( u, ¯ u ) (cid:13)(cid:13) p − = q k∇ ¯ u f w r ( u, ¯ u ) k p − + k∇ u f w r ( u, ¯ u ) k p − . (4.26)Similarly, k∇ ¯ u f w r ( u, ¯ u ) k p − and k∇ u f w r ( u, ¯ u ) k p − equal to ℓ ( Z ∗ , C )-norm of the followingvector fields respectively (cid:0) k j · h j i p − h j i X j = M ( l,k − ej ) − il,k ∈ NZ ∗ , i ∈ Mfw r ⊆ Z ( ˜ f w ) ir,lk Y n ∈ Z ∗ h n i l n Y n ∈ Z ∗ n = j h n i k n u l ¯ u k − e j (cid:1) j ∈ Z ∗ (4.27)and (cid:0) l j · h j i p − h j i X j = i −M ( l − ej,k ) l,k ∈ NZ ∗ , i ∈ Mfw r ⊆ Z ( ˜ f w ) ir,lk Y n ∈ Z ∗ h n i l n Y n ∈ Z ∗ n = j h n i k n u l − e j ¯ u k (cid:1) j ∈ Z ∗ . (4.28)22or any fixed l, k ∈ N Z ∗ and j ∈ Z ∗ with ( f w ) ir,lk = 0. M ( l, k ) = i and k j = 0, (4.19) stillholds true. By (4.19) and (4.20), the coefficients of the vector field in (4.27) are boundedby h j i p − · | k j ( ˜ f w ) ir,lk | Y n ∈ Z ∗ h n i l n Y n ∈ Z ∗ n = j h n i k n ≤ C r − f w r p − h i i β − p + (cid:0) X m ∈ Z ∗ l m h m i p − + X m ∈ Z ∗ m = j k m h m i p − + ( k j − h j i p − (cid:1) Y t ∈ Z ∗ h t i ( l t +( k − e j ) t ) . (4.29)Using Lemma 4.2 and (4.29), one has k∇ ¯ u f w r ( u, ¯ u ) k p − ≤ C r − f w r p +1 c r − k u k r − k u k p . Similarly, k∇ u f w r ( u, ¯ u ) k p − ≤ C r − f w r p +1 c r − k u k r − k u k p . Thus, (cid:13)(cid:13) X w f w ( u, ¯ u ) (cid:13)(cid:13) p − ≤ r p +1 c r − C r − f w k u k r − k u k p . By the same approach, when f has β -bounded symmetric coefficients bounded by C f > k X w θ f k p ≤ C r − f r p +1 c r − k u k r − k u k p . Next the proof of Proposition 4.2 is given.
Proof.
Step 1: (delete unbounded part)In the case θ = 0, since f w r ( u, ¯ u ) has ( β, f w r ( u, ¯ u ) has the following form f w r ( u, ¯ u ) = X | l + k | = r M ( l,k )= i ∈ Mfw r ⊆ Z X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k . Under the definition of Possion bracket, it holds that { f w r ( u, ¯ u ) , k u k p } w = X j ∈ Z ∗ (cid:18) ∂f w r ∂u j i ∂ k u k p ∂ ¯ u j (cid:19) − X j ∈ Z ∗ (cid:18) ∂f w r ∂ ¯ u j i ∂ k u k p ∂u j (cid:19) = i X | l + k | = r M ( l,k )= i ∈ Mfw r ( l j − k j ) h j i p X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k . For the sake of convenience, rewrite { f w r ( u, ¯ u ) , k u k p } w as the sum of the following twoparts O + ( u, ¯ u ) := i X | l + k | = r M ( l,k )= ii ∈ Mfw r ⊆ Z ( l j − l j − k j + k j ) h j i p X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k , − ( u, ¯ u ) := i X | l + k | = r M ( l,k )= ii ∈ Mfw r ⊆ Z ( l j − k j ) h j i p X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k . Since the coefficients of f w r ( u, ¯ u ) are ( β, O − ( u, ¯ u ) and obtain O − ( u, ¯ u )= i X i ∈ M fw r X | l + k | = r M ( l,k )= i ( l j − k j ) h j i p X ( l ,k ,i ) ∈A ( fw it,lk ( f w ) i ( l ,k ,i ) t,lk (cid:0) M ( l , k ) − i (cid:1) u l ¯ u k = i X − i ∈ M fw r X | l + k | = r M ( k,l )= − i ( k j − l j ) h j i p X ( k − k ,l − l ,i − i ) ∈A ( fw − it,kl ( f w ) − i ( k − k ,l − l ,i − i ) t,kl · (cid:0) M ( k − k , l − l ) − ( i − i ) (cid:1) u k ¯ u l = O + ( u, ¯ u ) . (4.30)Together with (4.30), rewrite { f w r ( u, ¯ u ) , k u k p } w as the following { f w r ( u, ¯ u ) , k u k p } w = 2 Re O + ( u, ¯ u )= A ( u, ¯ u ) + A ( u, ¯ u ) + A ( u, ¯ u ) , where A ( u, ¯ u ) := − Re X | l + k | = r M ( l,k )= ii ∈ Mfw r ⊆ Z i ( l j − l j − k j + k j ) h j i p X ( l ,k ,i ) ∈A ( fw ir,lk i ( f w ) i ( l ,k ,i ) t,lk u l ¯ u k ; (4.31) A ( u, ¯ u ) : = 2 Re X | l + k | = r M ( l,k )= i ∈ Mfw r ⊆ Z i ( l j − l j − k j + k j ) h j i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ ( l t − k t ) t ( h j i p − h t i p ) u l ¯ u k (4.32)and A ( u, ¯ u ) := 2 Re X | l + k | = r M ( l,k )= i ∈ Mfw r ⊆ Z i ( l j − l j − k j + k j ) h j i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ ( l t − k t ) · t · h t i p u l ¯ u k . (4.33)The estimate of { f w r ( u, ¯ u ) , k u k p } w follows the estimates of A ( u, ¯ u ), A ( u, ¯ u ) and A ( u, ¯ u ).In fact, I can not estimate A ( u, ¯ u ) by Corollary 2 directly, because the coefficients of A ( u, ¯ u ) are not satisfy condition (4.13). Fortunately the bad unbounded part (not sat-isfy the condition (4.13)) in A ( u, ¯ u ) can be handled by ( β, w r ( u, ¯ u ). Then A ( u, ¯ u ) is transformed into a new form, the coefficients of which satisfy(4.13). Thus, the estimate of A ( u, ¯ u ) can be obtained by Corollary 2.Now the details of deleting the unbounded terms in A ( u, ¯ u ) are given in the follows. Forany ( l , k , i ) ∈ A ( f w ) ir,lk , it holds that( l t − k t ) · t = X n = j (cid:0) ( k − k ) n − ( l − l ) n (cid:1) · n + i − X n = t ( l n − k n ) · n | {z } R w ( l,k,t,j,i ) + (cid:0) ( k − k ) j − ( l − l ) j (cid:1) · j. (4.34)Using (4.34) A ( u, ¯ u )= 2 Re X i ∈ M fw r ⊆ Z , X | l + k | = r M ( l,k )= i i ( l j − l j − k j + k j ) h j i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ R w ( l, k, t, j, i ) h t i p u l ¯ u k + 2 Re X i ∈ M fw r ⊆ Z , X | l + k | = r M ( l,k )= i i ( l t − k t ) h t i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ (cid:0) ( k − k ) j − ( l − l ) j (cid:1) · j h j i p u l ¯ u k . (4.35)Since the two parts in the right side of (4.35) are real value functions, they are invariantunder complex conjugation. Taking complex conjugation to the second part of the rightside of (4.35) and using that fact that f w r ( u, ¯ u ) has ( β, Re X i ∈ M fw r ⊆ Z X | l + k | = r M ( l,k )= i i ( l t − k t ) h t i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ ( k j − k j − l j + l j ) · j h j i p u l ¯ u k = − Re X − i ∈ M fw r ⊆ Z X | l + k | = r M ( k,l )= − i i ( l t − k t ) h t i p · X ( k − k ,l − l ,i − i ) ∈A ( fw − ir,kl ( f w ) − i ( k − k ,l − l ,i − i ) r,kl X t ∈ Z ∗ ( k j − k j − l j + l j ) · j h j i p u k ¯ u l = − A ( u, ¯ u ) . (4.36)Together with (4.35) and (4.36), it follows A ( u, ¯ u ) = Re X i ∈ M fw r ⊆ Z , X | l + k | = r M ( l,k )= i i ( l j − l j − k j + k j ) h j i p · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z ∗ R w ( l, k, t, j, i ) h t i p u l ¯ u k . (4.37)25n the case θ = 1, { f w r ( u, ¯ u ) , k u k p } w = Q + ( u, ¯ u ) + Q − ( u, ¯ u ) , (4.38)where Q + ( u, ¯ u ) := X | l + k | = r M ( l,k )= i ∈ Mfw r i sgn( j ) l j · h j i p Y n ∈ Z ∗ h n i ( l n + k n ) ( ˜ f w ) ir,lk u l ¯ u k ,Q − ( u, ¯ u ) := − X | l + k | = r M ( l,k )= i ∈ Mfw r i sgn( j ) k j · h j i p Y n ∈ Z ∗ h n i ( l n + k n ) ( ˜ f w ) ir,lk u l ¯ u k . Since the coefficients of f w r ( u, ¯ u ) are ( β, Q − ( u, ¯ u ) = Q + ( u, ¯ u ) . (4.39)From (4.38) and (4.39), one has { f w r , k u k p } w = 2 Re Q + ( u, ¯ u )= 2 Re X i ∈ M fw r X | l + k | = r M ( l,k )= i i sgn( j ) l j · h j i p Y n ∈ Z ∗ h n i ( l n + k n ) ( ˜ f w ) ir,lk u l ¯ u k = 2 Re X i ∈ M fw r i X | l + k | = r M ( l,k )= i ( ˜ f w ) ir,lk h j i p − l j · j Y n ∈ Z ∗ h n i ln + kn u l ¯ u k , (4.40)the last equation is obtained by sgn( j ) | j | = j . For any l, k ∈ N Z ∗ and any j ∈ Z ∗ with M ( l, k ) = i ∈ M f w r and l j = 0, one has that l j · j = i − M ( l − l j e j , k ) = i − X n ∈ Z ∗ n = j l n · n + X n ∈ Z ∗ k n · n. (4.41)Together with (4.40) and (4.41), { f w r ( u, ¯ u ) , k u k p } w = E ( u, ¯ u ) + B ( u, ¯ u ) + D ( u, ¯ u ) , (4.42)where E ( u, ¯ u ) := 2 Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − ( X n ∈ Z ∗ k n · n ) Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k , (4.43) B ( u, ¯ u ) := − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − ( X n ∈ Z ∗ n = j l n · n ) Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k , (4.44) D ( u, ¯ u ) := 2 Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − i Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k . (4.45) In order to estimate of E ( u, ¯ u ) and B ( u, ¯ u ), E ( u, ¯ u ) is rewritten as the sum of E ( u, ¯ u )and E ( u, ¯ u ), where E ( u, ¯ u ) := 2 Re X | l + k | = r M ( l,k )= ii ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − ( h j i p − − h n i p − ) X n nk n Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k , (4.46) ( u, ¯ u ) := 2 Re X | l + k | = r M ( l,k )= ii ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − ( X n k n · n h n i p − ) Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k (4.47) and B ( u, ¯ u ) is rewritten as the sum of B ( u, ¯ u ) and B ( u, ¯ u ), where B ( u, ¯ u ) := − Re X | l + k | = r M ( l,k )= ii ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − X n = j l n · n ( h j i p − − h n i p − ) Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k , (4.48) B ( u, ¯ u ) := − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk sgn( l j ) h j i p − ( X n = j l n · n h n i p − ) Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k . (4.49) Using the ( β, f w r ( u, ¯ u ), delete the bad unbounded parts(not satisfy the condition (4.13)) in B ( u, ¯ u ) and E ( u, ¯ u ) in the followings. For any l, k ∈ N Z ∗ and any j, n ∈ Z ∗ with k n = 0, l j = 0 and M ( l, k ) = i , the following equationholds true k n · n = i + X t ∈ Z ∗ t = j l t · t − X t ∈ Z ∗ ,t = n k t · t | {z } R w ( l,k,n,j,i ) + l j · j. (4.50)Take (4.50) into E ( u, ¯ u ), one obtains that E ( u, ¯ u )= − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ R w + ( l, k, n, j, i ) h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ l j · j h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k . (4.51)Take complex conjugation to the second part of the right side of (4.51) and get E ( u, ¯ u )= − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ R w + ( l, k, n, j, i ) | n | p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ l j · j h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k = − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ R w + ( l, k, n, j, i ) h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k +2 Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) − ir,kl h j i p − X n ∈ Z ∗ l j · j h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u k ¯ u l = − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ R w + ( l, k, n, j, i ) h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k − E ( u, ¯ u ) , (4.52)27he last equation is obtained by the coefficients of f w r ( u, ¯ u ) being ( β, E ( u, ¯ u )= − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n ∈ Z ∗ R w + ( l, k, n, j, i ) h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k . (4.53) Similarly, by the fact l n · n = i + X t ∈ Z ∗ k t · t − X t = j,n l t · t | {z } R w − ( l,k,n,j,i ) − l j · j, the following equation holds true B ( u, ¯ u ) = − Re X | l + k | = r M ( l,k )= i ∈ Mfw r i ( ˜ f w ) ir,lk h j i p − X n = j R w − ( l, k, n, j, i ) h n i p − Y m ∈ Z ∗ h m i ( l m + k m ) u l ¯ u k . (4.54) Summarize this step, it satisfies that { f w r , k u k p } w m Re O + ( u, ¯ u ) = A ( u, ¯ u ) defined in (4 . A ( u, ¯ u ) defined in (4 . A ( u, ¯ u ) in (4 . ( β, − type symmetrical −−−−−−−−−−−−−−−−−→ A ( u, ¯ u ) in (4 . . and { f w r , k u k p } w m Re Q + ( u, ¯ u ) = B ( u, ¯ u ) in (4 .
44) = B ( u, ¯ u ) in (4 . B ( u, ¯ u ) in (4 . ( β, −−→ B ( u, ¯ u ) in (4 . E ( u, ¯ u ) in (4 .
43) = E ( u, ¯ u ) in (4 . E ( u, ¯ u ) in (4 . ( β, −−→ E ( u, ¯ u ) in (4 . D ( u, ¯ u ) in (4 . . Step 2:Estimate A ( u, ¯ u )- A ( u, ¯ u ), B ( u, ¯ u ), B ( u, ¯ u ), E ( u, ¯ u ), E ( u, ¯ u ) and D ( u, ¯ u ).It is clear that A ( u, ¯ u ) can be written as an inner product of the vector fields G :=( h j i p ¯ u j ) j ∈ Z and F := ( F j ) j ∈ Z , where F j ( u, ¯ u ) := X | l + k − ej | = r − M ( l,k )= i ∈ Mfw r ⊆ Z F ij,lk − e j u l ¯ u k − e j , (4.55)and F ij,lk − e j := (cid:0) l j − l j − k j + k j (cid:1) h j i p X ( l ,k ,i ) ∈A ( fw ir,lk i · ( f w ) i ( l ,k ,i ) r,lk . (4.56)28oting the fact the momentum of ( f w ) ir,lk u l ¯ u k being i and 0 ≤ l j ≤ l j , 0 ≤ k j ≤ k j , thefollowing equation holds true( l j − l j − k j + k j ) · j = i − ( l j − k j ) · j − X t = j ( l t − k t ) · t. (4.57)Using the fact that | x | p ( p ≥
2) is convex function and (4.21), it follows that | i − ( l j − k j ) · j − X t = j ( l t − k t ) · t | p ≤ r p − (cid:0) X t = j ( l t + k t ) | t | p + ( l j + k j ) | j | p (cid:1) · h i i p . (4.58)In view of (4.57) and (4.58), it holds that | l j − l j − k j + k j |h j i p ≤ r p − ( X t = j ( l t + k t ) h t i p + ( l j + k j ) h j i p (cid:1) · h i i p . (4.59)Since f w r has ( β, C f w , together with(4.59), the coefficients of vector field F in (4.55) are bounded by the following | F ij,lk − e j | ≤ r p − C r − f w (cid:0) X t = j ( l t + k t ) h t i p + ( l j + k j ) h j i p (cid:1) · h i i β − p . Then by Corollary 2, the following inequality holds true | A ( u, ¯ u ) | ≤ |h F, G i| ≤ k F k ℓ · k G k ℓ = k F k ℓ · k u k p ≤ C r − f w r r p c r − k u k p k u k r − . (4.60)Using the same method, one has | D ( u, ¯ u ) | ≤ C r − f w r r p − c r − k u k p k u k r − . (4.61)In order to estimate A ( u, ¯ u ), the following inequality is given for any j, m ∈ Z ∗ (cid:12)(cid:12) | j | a − | m | a (cid:12)(cid:12) ≤ (cid:12)(cid:12) Z d | m + θ ( j − m ) | a dθ dθ (cid:12)(cid:12) ≤ Z a | j − m | · | m + θ ( j − m ) | a − dθ ≤ a − a ( | m | a − | j − m | + | j − m | a ) , (4.62)with a ≥ a = p into (4.62). Given l, k ∈ N Z ∗ fulfilling M ( l, k ) = i with k j = 0 and l m = 0,together with (4.21), it holds that (cid:12)(cid:12) | j | p − | m | p (cid:12)(cid:12) ≤ p − p (cid:18) | m | p − (cid:0) X n = j k n | n | + X n = m l n | n | (cid:1) · h i i +( r − p − ( X n = j k n | n | p + X n = m l n | n | p ) · h i i p ) (cid:19) . (4.63)The similar inequality holds in the case k j = 0, k m = 0 ( m = j ).Since | A ( u, ¯ u ) | ≤ (cid:12)(cid:12) X i ∈ M fw r X | l + k | = ri = M ( l,k ) ( l j − l j − k j + k j ) · h j i p X ( l ,k ,i ∈A ( fw ir,lkm ∈ Z ∗ ( f w ) i ( l ,k ,i ) r,lk (cid:0) l m m ( h j i p − h m i p ) − k m m ( h j i p − h m i p ) (cid:1) u l ¯ u k (cid:12)(cid:12) , (4.64) take the right side of (4.64) as an inner product of vectors F and G , where F := (2 h j i p ¯ u j ) j ∈ Z ∗ , G := ( X i ∈ M fw r X | l + k − ej | = r − j = M ( l,k − ej ) − i ( G j ) ir − ,lk − e j u l ¯ u k − e j ) j ∈ Z ∗ and ( G j ) ir − ,lk − e j :=( l j − l j − k j + k j ) X ( l ,k ,i ∈A ( fw ir,lkm ∈ Z ∗ ( f w ) i ( l ,k ,i ) r,lk (cid:0) l m m ( h j i p − h m i p ) − k m m ( h j i p − h m i p ) (cid:1) . By (4.63) and f w r ( u, ¯ u ) having ( β, C f w ,the coefficients of G j are bounded by | ( G j ) ir − ,lk − e j |≤ C r − f w r h i i β − p p − p ( r − p − ( Y t ∈ Z ∗ h t i sgn (cid:0) l t +( k − e j ) t (cid:1) X m ∈ Z ∗ h m i ( p − sgn (cid:0) ( k − e j ) m + l m (cid:1) ) . Using Corollary 2, one has | A ( u, ¯ u ) | ≤ k G k ℓ · k F k ℓ ≤ C r − f w r p p ( r − p − c r − k u k p k u k r − . By the same method, E ( u, ¯ u ) and B ( u, ¯ u ) satisfy the following inequalities | B ( u, ¯ u ) | ≤ C r − f w r p p ( r − p − c r − k u k p k u k r − and | E ( u, ¯ u ) | ≤ C r − f w r p p ( r − p − c r − k u k p k u k r − . Since A ( u, ¯ u ), B ( u, ¯ u ) and E ( u, ¯ u ) can be estimated by the same method, I only givethe details of estimate of A ( u, ¯ u ) in (4.37). | A ( u, ¯ u ) | ≤ k F k ℓ · k G k ℓ , where F = ( F j ) j ∈ Z ∗ and G = ( G j ) j ∈ Z ∗ with F j = h j i p u j and G j ( u, ¯ u ) := X i ∈ M fw r X | l + k | = rj = M ( l,k − ej ) − i i (cid:0) ( l − l ) j − ( k − k ) j (cid:1) · X ( l ,k ,i ) ∈A ( fw ir,lk ( f w ) i ( l ,k ,i ) r,lk X t ∈ Z R w ( l, k, t, j, i ) h t i p u l ¯ u k − e j . From (4.21) and (4.50), one has | R w ( l, k, t, j, i ) | ≤ (cid:0) X n = j (cid:12)(cid:12) ( k − k ) n − ( l − l ) n (cid:12)(cid:12) · h n i + X n = t | l n − k n | · h n i (cid:1) · h i i . G are bounded by | ( G j ) ir,lk − e j |≤ C r − f w r h i i β − (cid:0) X n = j (cid:12)(cid:12) ( k − k ) n − ( l − l ) n (cid:12)(cid:12) · h n i + X n = t | l n − k n | · h n i (cid:1) h t i p . Using Corollary 2, it holds that | A ( u, ¯ u ) | ≤ C r − f w r c r − r k u k p k u k r − . Similarly, | B ( u, ¯ u ) | , | E ( u, ¯ u ) | ≤ C r − f w r c r − r k u k p k u k r − . Thus, the following inequalities hold true |{ f w r ( u, ¯ u ) , k u k p } w | ≤ | A ( u, ¯ u ) | + | A ( u, ¯ u ) | + | A ( u, ¯ u ) |≤ C r − f w r p +1 pr p − c r − k u k p k u k r − and |{ f w r ( u, ¯ u ) , k u k p } w | ≤ | B ( u, ¯ u ) | + | B ( u, ¯ u ) | + | D ( u, ¯ u ) | + | E ( u, ¯ u ) | + | E ( u, ¯ u ) |≤ C r − f w r p +1 pr p − c r − k u k p k u k r − . ( θ, γ, α, N ) -normal form In order to guarantee the boundedness of the symplectic transformation, it is required astrong non resonant condition. Given integers
N > r ≥
3, let E r,N := (cid:8) ( l, k ) | l, k ∈ N Z ∗ , ≤ | l + k | = t ≤ r, | Γ >N ( l + k ) | ≤ (cid:9) . (5.1) Definition 5.1.
Given γ > , α > , θ ∈ { , } and N, r ∈ N , frequencies ω = ( ω j ) j ∈ Z ∗ is said to be r -degree ( θ, γ, α, N ) -non resonant , if for any ( l, k ) belongs to O w θ r,N := ( l, k ) ∈ E r,N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) when | Γ >N ( l + k ) | < ,θ P j ∈ Z ∗ | l j − k j | + (1 − θ ) P j ∈ Z ∗ | l j + l − j − k j − k − j | 6 = 0 when | Γ >N ( l + k ) | = 2 , P | j | >N | l j + l − j − k j − k − j | 6 = 0 , (5.2) it satisfies |h ω, I θ ( l − k ) i| > γM l,k N α , where M l,k := max (cid:0) { | j | (cid:12)(cid:12) k j = 0 or l j = 0 , k, l ∈ N Z ∗ } ∪ { N } (cid:1) . (5.3)31ith the r -degree ( θ, γ, α, N )-non resonant condition, a symplectic transformation will beobtained. Under this transformation the Hamiltonian function H ( u, ¯ u ) are transformedinto the sum of an r -degree normal form and a remainder term. However this r -degreenormal form is not a standard Hamiltonian Birkhoff normal form (A standard 2 r -degreeHamiltonian Birkhoff normal form in variables ( u, ¯ u ) is an r -degree polynomial which onlydepends on variables ( | u j | ) j ∈ Z ∗ ). Now I introduce a definition to describe this normalform. Definition 5.2.
Given γ > , α > and an integer N > , call an r -degree polynomial f ( u, ¯ u ) = X r ≥ t ≥ X | l + k | = t M ( l,k )= i ∈ Mft f it,lk u l ¯ u k ( θ, γ, α, N ) -normal form with respect to ω ∈ R Z ∗ , if for any ( l, k ) ∈ E r,N ( E r,N isdefined in (5.1)) with M ( l, k ) = i ∈ M f t and f it,lk = 0 , it satisfies that |h ω, I θ ( k − l ) i| ≤ γM k,l N α , where h , i is the inner product of space ℓ ( Z ∗ , C ) , I θ and M l,k are defined in (3.7) and(5.3). Remark 5.1.
Let f ( u, ¯ u ) be an r -degree polynomial. For any given γ > , α > , θ ∈ { , } , integers N > and ω ∈ R Z ∗ , denote Γ ω ( θ,γ,α,N ) f ( u, ¯ u ) := X r ≥ t ≥ , X | l + k | = t, M ( l,k )= i (Γ ( θ,γ,α,N ) f ) it,lk u l ¯ u k with (Γ ω ( θ,γ,α,N ) f ) it,lk := f it,lk , if l, k fulfills |h ω, I θ ( l − k ) i| ≤ γM l,k N α , the other casesas ( θ, γ, α, N ) -normal form of f ( u, ¯ u ) with respect to ω . Moreover, suppose that f w θ ( u, ¯ u ) has ( β, θ ) -type symmetric coefficients semi-bounded by C θ > ( θ ∈ { , } ). Sodoes Γ ω ( θ,γ,α,N ) f w θ ( u, ¯ u ) . Remark 5.2.
Assume that ω = ( ω j ) j ∈ Z ∗ is an r ∗ -degree ( θ, γ, α, N ) -non resonant fre-quencies and f w θ ( u, ¯ u ) is an r ∗ -degree ( θ, γ, α, N ) -normal form with respect to ω . Then f w θ ( u, ¯ u ) has the following form f w θ ( u, ¯ u ) := A w θ ( u, ¯ u ) + B w θ ( u, ¯ u ) + C w θ ( u, ¯ u ) , where A w θ ( u, ¯ u ) := X ≤ r ≤ r ∗ X ( l,k ) ∈ Ω θ A Nr ,i = M ( l,k ) ( f w θ ) ir,lk u l ¯ u k , (5.4) B w θ ( u, ¯ u ) := X ≤ r ≤ r ∗ X ( l,k ) ∈ Ω θ B Nr ,i = M ( l,k ) ( f w θ ) ir,lk u l ¯ u k , (5.5) w θ ( u, ¯ u ) := X ≤ r ≤ r ∗ X ( l,k ) ∈ Ω θ C Nr ,i = M ( l,k ) ( f w θ ) ir,lk u l ¯ u k , (5.6) and Ω θ A Nr := { ( l, k ) ∈ E r,N (cid:12)(cid:12) | Γ >N ( l + k ) | < , P j ∈ Z ∗ | l j + l − j − k j − k − j | = 0 } , θ = 0 { ( l, k ) ∈ E r,N (cid:12)(cid:12) | Γ >N ( l + k ) | < , P j ∈ Z ∗ | l j − k j | = 0 } , θ = 1Ω θ B Nr := (cid:8) ( l, k ) ∈ E r,N (cid:12)(cid:12) | Γ >N ( l + k ) | = 2 , there exists | j | > N such that l j = k j = 1 (cid:9) , Ω θ C Nr := (cid:8) ( l, k ) ∈ E r,N (cid:12)(cid:12) | Γ >N ( l + k ) | = 2 , there exists | j | > N such that l − j = k j = 1 (cid:9) . Lemma 5.1.
Let ω = ( ω j ) j ∈ Z ∗ be an r -degree ( θ, γ, α, N ) non-resonant frequency. Sup-pose that f w θ r ( u, ¯ u ) is an r -degree ( r ≥ ) homogeneous ( θ, γ, α, N ) -normal form with re-spect to ω and has ( β, θ ) -type symmetric coefficients semi-bounded by C f wθr > ( β − p ≥ ).Then for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) it has |{ f w θ r ( u, ¯ u ) , k u k p } w θ | ≤ r p +1 c r − C r − f wθr N k Γ >N u k · k u k r − · k u k p (5.7) Remark 5.3.
Although f w θ r ( u, ¯ u ) in Lemma 5.1 is at most 2 degree about (Γ >N u, Γ >N ¯ u ) , itstill satisfies an inequality similar to (4.6) in Corollary 1. But for the general polynomialsbeing at most 2 degree about (Γ >N u, Γ >N ¯ u ) , this inequality dose not always hold.Proof of Lemma 5.1. From Remark 5.2, f w θ r ( u, ¯ u ) = A w θ r ( u, ¯ u ) + B w θ r ( u, ¯ u ) + C w θ r ( u, ¯ u ) , (5.8)where A w θ r ( u, ¯ u ), B w θ r ( u, ¯ u ) and C w θ r ( u, ¯ u ) are defined in (5.4)-(5.6) in Remark 5.2.Step 1: Calculate {A w θ r ( u, ¯ u ) , k u k p } w θ .It is easy to verify that {A w r ( u, ¯ u ) , k u k p } w = X j ∈ Z ∗ X ( l,k ) ∈ Ω A Nr i ( l j + l − j − k j − k − j ) h j i p ( f w ) ir,lk u l ¯ u k = 0; (5.9)and {A w r ( u, ¯ u ) , k u k p } w = X j ∈ Z ∗ X ( l,k ) ∈ Ω A Nr i sgn( j ) · ( l j − k j ) h j i p ( f w ) ir,lk u l ¯ u k = 0 . (5.10)Step 2: Estimate {B w θ r ( u, ¯ u ) , k u k p } w θ .Since the function B w θ r ( u, ¯ u ) depends on ( u j , ¯ u j ) | j |≤ N and ( | u j | ) | j | >N , the following equa-tion holds true {B w θ r ( u, ¯ u ) , X | j | >N h j i p | u j | } w θ = 0 . Thus, {B w θ r ( u, ¯ u ) , k u k p } w θ = {B w θ r ( u, ¯ u ) , X | j |≤ N h j i p | u j | } w θ . (5.11)33rom the definition of { , } w θ and the structure of B w θ r ( u, ¯ u ), {B w θ r ( u, ¯ u ) , P | j |≤ N h j i p | u j | } w θ still depends on ( | u j | ) | j | >N and (Γ ≤ N u, Γ ≤ N ¯ u ). To be more specific, {B w θ r ( u, ¯ u ) , X | j |≤ N h j i p | u j | } w θ = 2 Re X ( l,k ) ∈ Ω θ B Nr , | j |≤ N i sgn θ ( j ) l j h j i p (cid:0) f w θ (cid:1) ir,lk u l ¯ u k . (5.12)For any no zero term u l ¯ u k of the right side of (5.12), its index ( l, k ) satisfies M ( l, k ) = M (Γ ≤ N l, Γ ≤ N k ) = X | j |≤ N ( l j − k j ) · j = i. (5.13)From (5.13), for any | j | ≤ N with l j = 0, it satisfies that j = X | t |≤ N,t = j ( l t − k t ) · t + ( l j − − k j ) · j − i. (5.14)Moreover, given p ≥
2, by (4.21) and (5.14), the following inequalities hold true | j | p ≤ r p (cid:0) X | t |≤ N,t = j ( l t + k t ) · | t | p + ( l j − k j ) · | j | p (cid:1) · h i i p (5.15)and | j | p ≤ | j | · | j | p − · | j | p − ≤ N r p − | j | p − (cid:0) X | t |≤ N,t = j ( l t + k t ) · | t | p − + ( l j − k j ) · | j | p − (cid:1) · h i i p − . (5.16)I will estimate the coefficients of {B w θ r ( u, ¯ u ) , k u k p } w θ . When θ = 0, for any ( l , k , i ) ∈ A ( f w ) ir,lk with ( l, k ) ∈ Ω θ B Nr , it holds that |M ( l , k ) − i | ≤ X | j |≤ N ( l j + k j ) · | j | + X | j | >N | sgn( l j + k j ) j | + | i |≤ ( r − N + X | j | >N | sgn( l j + k j ) j | + | i |≤ r − N · (cid:0) X | j | >N | sgn( l j + k j ) j | (cid:1) · h i i (5.17)the last inequality hold by (4.21). From (5.15) and (5.17), the coefficients of {B w r ( u, ¯ u ) , k u k p } w are bounded by8 C r − f w h i i β − p r p +1 N l j h j i p (cid:0) X | t |≤ N,t = j ( l t + k t ) ·| t | p + ( l j − k j ) · | j | p (cid:1) · (cid:0) X | j | >N | sgn( l j + k j ) j | (cid:1) . (5.18)Similarly, in the case θ = 1, using (5.16), the coefficients of {B w r ( u, ¯ u ) , k u k p } w are bounded by8 C r − f w r h i i β − p + r p N l j | j | p − (cid:0) X | t |≤ N,t = j ( l t + k t ) · | t | p − + ( l j − k j ) · | j | p − (cid:1) Y t ∈ Z ∗ | t | lt + kt . (5.19)By Corollary 2, it holds that |{B w θ r ( u, ¯ u ) , k u k p } w θ | ≤ r p +1 N c r − C r − f wθr k Γ >N u k k u k r − p . (5.20) tep 3: Estimate {C w θ r ( u, ¯ u ) , k u k p } w θ .When θ = 0, |{C w r ( u, ¯ u ) , k u k p } w | = {C w r ( u, ¯ u ) , X | t |≤ N | u t | h t i p } w + {C w r ( u, ¯ u ) , X | t | >N | u t | h t i p } w = {C w r ( u, ¯ u ) , X | t |≤ N | u t | h t i p } w + 0 , (5.21)the last equation holds by the fact that { u j ¯ u − j , | j | p ( | u j | + | u − j | ) } w = 0 . It is easy to verify that {C w r ( u, ¯ u ) , P | t |≤ N | u t | h t i p } w is still dependent on ( u j ¯ u − j ) | j | >N and(Γ ≤ N u, Γ ≤ N ¯ u ). Using the method of estimate {B ( w θ ) r ( u, ¯ u ) , P | t |≤ N | u t | h t i p } w θ , the estimate of {C w r ( u, ¯ u ) , P | t |≤ N | u t | h t i p } w is obtained.When θ = 1, |{C w r ( u, ¯ u ) , k u k p } w | = {C w r ( u, ¯ u ) , X | t |≤ N | u t | h t i p } w + {C w r ( u, ¯ u ) , X | t | >N | u t | h t i p } w . Using the method of estimate {B w θ r ( u, ¯ u ) , P | t |≤ N | u t | h t i p } w θ in step 2, the estimate of {C w r ( u, ¯ u ) , P | t |≤ N | u t | h t i p } w can be obtained. The estimate of {C w r ( u, ¯ u ) , P | t | >N | u t | h t i p } w will be obtained by the following. For any nonzero term of C w r ( u, ¯ u ) with index ( l, k ), thereexists | j | > N with l j = 1, k − j = 1 (or l − j = 1, k j = 1) such that2 | j | = | X | t |≤ N ( l t − k t ) · t − i | , (5.22)which follows from M ( l, k ) = i . From the relation (5.22), using (4.21) it holds that | j | ≤ (cid:0) X | t |≤ N ( l t + k t ) · | t | + | i | (cid:1) ≤ h i i (cid:0) X | t |≤ N ( l t + k t ) · | t | (cid:1) ≤ rN h i i (5.23)and h j i p − ≤ r p − (cid:0) X | t |≤ N ( l t + k t ) · h t i p − (cid:1) · h i i p − . (5.24)By (5.23) and (5.24), the coefficients of |{C w r ( u, ¯ u ) , X | t | >N | u t | h t i p } w | = | Re X ( l,k ) ∈ Ω C Nr i h j i p ( ˜ f w ) ir,lk Y t ∈ Z ∗ | t | lt + kt u l ¯ u k | (5.25)are smaller than 2 h j i p − r p + N C r − f w r h i i β − p + (cid:0) X | n |≤ N ( l n + k n ) h n i p − (cid:1) Y t ∈ Z ∗ | t | lt + kt . (5.26)Using Corollary 2, it holds that |{C w θ r ( u, ¯ u ) , k u k p } w θ | ≤ r p +1 c r − C r − f wθr N k Γ >N u k · k u k r − · k u k p . (5.27)Summing (5.9), (5.10), (5.20) and (5.27), inequality (5.7) is obtained. .2 Birkhoff normal form theorem In this subsection, construct a coordinate transformation under which the Hamiltoniansystem (3.8) will have an r ∗ + 3 degree ( θ, γ, α, N )-normal form, for any given positive r ∗ . Theorem 4 (Birkhoff normal form theorem) . Suppose that system (3.8) satisfies as-sumptions A θ - B θ and P w θ ( u, ¯ u ) in H w θ ( u, ¯ u ) defined in (3.9) has ( β, θ ) -type symmetriccoefficients semi-bounded by C θ > ( β is big enough). Given α > , < γ ≪ andinteger r ∗ > , take p satisfying ( β − / > p > α ( r ∗ + 1) r ∗ . There exist a positivereal number e R > and a Lie-transformation T ( r ∗ ) w θ : B p ( e R/ → B p ( e R ) such that:For any R < e R and any integer N fulfilling ( R r ∗ − γ r ∗ +1 ) − p − − α ( r ∗ +1) ≤ N ≤ ( γR − r ∗ +1)( r ∗ +2) ) α , (5.28) the transformation T ( r ∗ ) w θ puts Hamiltonian H w θ into H ( r ∗ ,w θ ) := H w θ ◦ T ( r ∗ ,w θ ) = H w θ + Z ( r ∗ ,w θ ) + R N ( r ∗ ,w θ ) + R T ( r ∗ ,w θ ) which satisfies that1) Both Z ( r ∗ ,w θ ) and R N ( r ∗ ,w θ ) are ( r ∗ + 3) -degree polynomials and R T ( r ∗ ,w θ ) is a powerseries which starts with r ∗ + 4 degree polynomial. All of them have ( β, θ ) -type sym-metric coefficients semi-bounded by C ( θ, r ∗ ) := C θ (cid:0) β N α γ (cid:1) ( r ∗ +1) .2) The polynomial Z ( r ∗ ,w θ ) ( u, ¯ u ) is r ∗ + 3 -degree ( θ, γ, α, N ) -normal form with respectto ω w θ .3) The polynomial R N ( r ∗ ,w θ ) ( u, ¯ u ) := P r ∗ +3 r =3 Γ N> g ( r,w θ ) r +4 , where g ( r,w θ ) r +4 is an ( r + 4) -degreehomogeneous polynomial with ( β, θ ) -type symmetric coefficients semi-bounded by C ( θ, r ) ;4) The canonical Lie-transformation T ( r ∗ ) w θ satisfies sup ( u, ¯ u ) ∈ B p ( R/ kT ( r ∗ ) w θ ( u, ¯ u ) − ( u, ¯ u ) k p ≤ C ( θ, p, r ∗ ) R − r ∗ +1)2 , (5.29) where C ( θ, p, r ∗ ) is a constant dependent on θ, p and r ∗ . In order to prove Theorem 4, it need not only to construct a bounded canonical trans-formation under which the Hamiltonian H w θ in (3.9) has an r ∗ + 3-degree ( θ, γ, α, N )-normal form, but also to show that the new Hamiltonian function has ( β, θ )-type symmet-ric coefficients semi-bounded by C ( θ, r ∗ ). First, let us review the definition of canonicaltransformation. Definition 5.3.
Call a map φ : H p ( Z ∗ , C ) ∋ ( u, ¯ u ) → ( ξ, ¯ ξ ) ∈ H p ( Z ∗ , C ) canonicaltransformation under a symplectic form w θ (or a symplectic change of coordiantes), if φ is a diffeomorphism and preserves the Poisson bracket, i.e. { f, g } w θ ◦ φ = { f ◦ φ, g ◦ φ } w θ .
36 convenient way of constructing canonical transformations is as followings. Let Φ tS wθ be the flow generated by a regular function S w θ ( u, ¯ u ) defined in H p ( Z ∗ , C ) with respectto the symplectic structure w θ . Φ tS wθ | t =0 = id . If Φ tS wθ is well defined up to t = 1, thenthe map Φ tS wθ | t =1 is called a Lie transformation associated to S w θ ( u, ¯ u ) under symplecticform w θ . Φ S wθ is canonical.Given a regular function g , the new function g ◦ Φ tS wθ satisfies d n dt n ( g ◦ Φ tS wθ ) = {{ g, S w θ } w θ , · · · } w θ | {z } n times ◦ Φ tS wθ . Thus the Taylor expansion of g ◦ Φ tS wθ in the variable t is g ◦ Φ tS wθ = ∞ X ν =0 g ( ν,S wθ ) ◦ Φ tS wθ (cid:12)(cid:12) t =0 t ν = ∞ X ν =0 g ( ν,S wθ ) t ν , where g (0 ,S wθ ) := g, g ( ν,S wθ ) := 1 ν { g ( ν − ,S wθ ) , S w θ } w θ , ν ≥ . (5.30)Take t = 1 and it follows that g ◦ Φ tS wθ | t =1 = ∞ X ν =0 g ( ν,S wθ ) . In this paper, denote ≺ as ≤ ˜ C · , where ˜ C > R . To improve theorder of the ( θ, γ, α, N )-normal form of H w θ , it needs to solve a linear equation to find asuitable generated function S w θ under symplectic form w θ . The following lemma is to dothis with respect to w θ -Possion bracket. Lemma 5.2. (Homological Equation)
Given an integer
N > , real numbers γ > and α > , suppose that an r -degree homogeneous polynomial g w θ ( u, ¯ u ) has ( β, θ ) -typesymmetric coefficients semi-bounded by C g wθ > ( θ ∈ { , } ). Then there exists anunique S w θ ( u, ¯ u ) such that { H w θ , S w θ ( u, ¯ u ) } w θ + Γ ω θ ( θ,γ,α,N ) Γ N ≤ g w θ ( u, ¯ u ) = Γ N ≤ g w θ ( u, ¯ u ) , (5.31) where H w θ := P j ∈ Z ∗ ω w θ j | u j | with ω w θ j ∈ R . Moreover for any ( u, ¯ u ) ∈ H p ( Z ∗ , C ) theHamiltonian vector of S w θ ( u, ¯ u ) holds k X w θ S wθ ( u, ¯ u ) k p ≤ r p +1 C r − g wθ c r − N α γ k u k p k u k r − . Proof.
By the definition of Possion bracket { , } w θ , the solution S w θ ( u, ¯ u ) of (5.31) is stillan r -degree homogeneous polynomial and has the following form S w θ ( u, ¯ u ) = X i ∈ M Swθr , X | l + k | = r, M ( l,k )= i ( S w θ ) ir,lk u l ¯ u k (5.32)with undetermined coefficients. Since g w θ ( u, ¯ u ) has ( β, θ )-type symmetric coefficientssemi-bounded by C g wθ , by Remark 4.2 and Remark 5.1, Γ ω θ ( θ,γ,α,N ) Γ N ≤ g w θ ( u, ¯ u ) is an r -degree ( θ, γ, α, N )-normal form of Γ N ≤ g w θ ( u, ¯ u ) with ( β, θ )-type symmetric coefficientssemi-bounded by C g wθ , and its coefficients have the following form(Γ ω θ ( θ,γ,α,N ) Γ N ≤ g w θ ) ir,lk := (Γ N ≤ g w θ ) ir,lk , if |h ω w θ , I θ ( l − k ) i| ≤ γM l,k N α , , if |h ω w θ , I θ ( l − k ) i| > γM l,k N α , (5.33)37here M l,k is defined in Definition 5.1. Take (5.32) into equation (5.31) and get that forany i ∈ M g wθ and any l, k ∈ N Z ∗ with | l + k | = r and M ( l, k ) = i , − i h ω w θ , I θ ( l − k ) i ( S w θ ) ir,lk + (Γ ω θ ( θ,γ,α,N ) Γ N ≤ g w θ ) ir,lk = (Γ N ≤ g w θ ) ir,lk , (5.34)which means that the coefficients of S w θ ( u, ¯ u ) has the following form( S w θ ) ir,lk = − (Γ N ≤ g wθ ) ir,lk i h ω wθ , I θ ( l − k ) i , when |h ω w θ , I θ ( l − k ) i| > γM l,k N α , when |h ω w θ , I θ ( l − k ) i| ≤ γM l,k N α (5.35)and satisfy that( S w θ ) ir,lk = − (Γ N ≤ g w θ ) ir,lk i h ω w θ , I θ ( l − k ) i = − (Γ N ≤ g w θ ) − ir,kl i h ω w θ , I θ ( k − l ) i = ( S w θ ) − ir,kl , (5.36)the second equality holds by Γ N ≤ g w θ ( u, ¯ u ) having symmetric coefficients from Remark 4.2and ω w θ j ∈ R ( j ∈ Z ∗ ).The norm of Hamiltonian vector field X w θ S wθ (cid:13)(cid:13) X w θ S wθ ( u, ¯ u ) (cid:13)(cid:13) p = q k∇ ¯ u S w θ ( u, ¯ u ) k p + k∇ u S w θ ( u, ¯ u ) k p ≤ (cid:13)(cid:13)(cid:0) X j = M ( l,k − ej ) − ii ∈ MSwθ | l + k − ej | = r − k j ( S w θ ) ir,lk u l ¯ u k − e j (cid:1) j ∈ Z ∗ (cid:13)(cid:13) p + (cid:13)(cid:13)(cid:0) X j = −M ( l − ej,k ) − i | l − ej + k | = r − i ∈ MSwθ l j ( S w θ ) ir,lk u l − e j ¯ u k (cid:1) j ∈ Z ∗ (cid:13)(cid:13) p equals to the ℓ norm of the vector fields Q w θ := (cid:18) X i ∈ M Swθ X j = M ( l,k − ej ) − i | l + k − ej | = r − h j i p k j · ( S w θ ) ir,lk u l ¯ u k − e j (cid:19) j ∈ Z ∗ and Q w θ := (cid:18) X i ∈ M Swθ X j = −M ( l − ej,k ) − i | l − ej + k | = r − h j i p l j · ( S w θ ) ir,lk u l − e j ¯ u k (cid:19) j ∈ Z ∗ . When θ = 0, for any l, k ∈ N Z ∗ with M ( l, k ) = i and k j = 0, by (4.21), it holds | j | p ≤ r p − (cid:0) X t ∈ Z ∗ l t | t | p + X t ∈ Z ∗ , t = j k t | t | p + ( k j − | j | p (cid:1) · h i i p . (5.37)For any ( l , k , i ) ⊂ A f ir,lk , by (4.21) the following inequality holds |M ( l , k ) − i | ≤ rM l,k · h i i . (5.38)By (5.35) and (5.38), the coefficients of S w satisfy that P ( l ,k ,i ) ∈A ( gw ir,lk (cid:12)(cid:12) (Γ N ≤ g w ) i ( l ,k ,i ) r,lk ( M ( l , k ) − i ) (cid:12)(cid:12) |h ω w , I ( l − k ) i| N α γM l,k X ( l ,k ,i ) ∈A ( gw ir,lk | (Γ N ≤ g w ) i ( l ,k ,i ) r,lk | · rM l,k · h i i ≤ r N α C r − g w γ h i i β . (5.39)From (5.37) and (5.39), the coefficients of vector fields Q w ( u, ¯ u ) and Q w ( u, ¯ u ) fulfill h j i p · | k j ( S w ) ir,lk | , h j i p · | l j ( S w ) ir,lk |≤ r p +1 (cid:0) X t ∈ Z ∗ l t h t i p + X t ∈ Z ∗ t = j k t h t i p + ( k j − h j i p (cid:1) N α C r − g w γ h i i β − p . (5.40)By (5.40), using Corollary 2, it holds that (cid:13)(cid:13) X w S w ( u, ¯ u ) (cid:13)(cid:13) p ≤ k Q w k ℓ + k Q w k ℓ ≤ r p +1 c r − N α C r − g w γ k u k p k u k r − . (5.41)When θ = 1, in order to estimate the ℓ -norm of Q w ( u, ¯ u ) and Q w ( u, ¯ u ), let us considerthe coefficients of Q w ( u, ¯ u ) and Q w ( u, ¯ u ) firstly. For any i ∈ M S w and any l, k ∈ N Z ∗ satisfying | l + k | = r, M ( l, k ) = i and k j = 0 (or l j = 0), using (5.35), the coefficients of u l ¯ u k − e j in Q w are bounded by the following2 r p + C r − g w r N α γ h i i β − p + ( X t l t | t | p − + X t = j ( k − e j ) t | t | p − ) Y t ∈ Z ∗ | t | ( l t +( k − e j ) t ) (5.42)and the coefficients of u l − e j ¯ u k in Q w ( u, ¯ u ) are bounded by2 r p + C r − g w r N α γ h i i β − p + ( X t = j ( l − e j ) t | t | p − + X t k t | t | p − ) Y t ∈ Z ∗ | t | (( l − e j ) t + k t ) . (5.43)By Corollary 2 and (5.42)-(5.43), the following estimate is obtained k X w S w ( u, ¯ u ) k p ≤ k Q w k ℓ + k Q w k ℓ ≤ r p + C r − g w r N α γ c r − k u k p k u k r − . The following Lemma shows that the Possion bracket of an ˜ r -degree homogeneouspolynomial f w θ ( u, ¯ u ) with ( β, θ )-type symmetric coefficients semi-bounded by C f wθ ˜ r > S ω θ ( u, ¯ u ) to equation (5.31) is still of ( β, θ )-type symmetric coefficients.Moreover, its coefficients satisfy some inequalities. Lemma 5.3.
Let an ˜ r -degree homogeneous polynomial f w θ ( u, ¯ u ) ( θ ∈ { , } ) have ( β, θ ) -type symmetric coefficients semi-bounded by C f wθ ˜ r > . Then the possion bracket of f w θ ( u, ¯ u ) and the solution S w θ ( u, ¯ u ) to equation (5.31) under the symplectic form w θ is an (˜ r + r − -degree homogeneous polynomial with ( β, θ ) -type symmetric coefficientsand it holds that • when θ = 0 , for any l ′ , k ′ , i fulfilling | l ′ + k ′ | = ˜ r + r − and M ( l ′ , k ′ ) = i , it holdsthat X ( l ′ ,k ′ ,i ) ∈A ( fw ,Sw i ˜ r + r − ,l ′ k ′ | ( f w (1 ,S w ) ) i ( l ′ ,k ′ ,i )˜ r + r − ,l ′ k ′ | · max {h i i , h i − i i}≤ β +2 r c (˜ r + 1) C ˜ r − f w r C r − g w r (2 N + 1) r − N α +1 γ h i i β . when θ = 1 the following inequality holds true | ( f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ | ≤ β +1 cr (˜ r + 1) C ˜ r − f w r C r − g w r (2 N + 1) r − N α γ h i i β Y t ∈ Z ∗ | t | l ′ t + k ′ t . Remark 5.4.
Under the same assumptions of Lemma 5.3, for any integer ν ≥ , f w θ ( ν,S wθ ) is an (cid:0) ˜ r + ν ( r − (cid:1) -degree homogeneous polynomial with ( β, θ ) -type symmetric coefficients. • When θ = 0 , the following inequality holds X ( l ,k ,i ) ∈A ( fw ν,Sw i ˜ r +( r − ν,lk | ( f w ( ν,S w ) ) i ( l ,k ,i )˜ r +( r − ν,lk | · max {h i i , h i − i i}≤ C ˜ r − f w (cid:0) β +2 r c (2 N + 1) ( r − N α +1 γ C r − g w (cid:1) ν Q ν − n =0 (cid:0) ˜ r + n ( r −
2) + 1 (cid:1) h i i β ν ! ; • When θ = 1 , it holds that | ( f w ( ν,S w ) ) i ˜ r +( r − ν,lk | ≤ C ˜ r − f w r ν ! h i i β (cid:0) β +1 rc (2 N ) ( r − N α γ C r − g w (cid:1) ν ν − Y n =0 (cid:0) ˜ r + n ( r −
2) + 1 (cid:1) Y t ∈ Z ∗ | t | l ′ t + k ′ t . Before proving Lemma 5.3, I denote a set of indexes and give a Lemma to count thenumber of this set. This Lemma is used to prove Lemma 5.3.For any ( l ′ , k ′ ) ∈ N Z ∗ × N Z ∗ and any i ′ ∈ Z , letΩ( l ′ , k ′ , i ′ ) := (cid:26) (cid:0) ( l, k, i ) , ( L, K, i ) , j (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) l, k, L, K ∈ N Z ∗ , i , i ∈ Z , j ∈ Z ∗ ;satisfying A , B , D1 or A , B , D2 (cid:27) , where A: | l + k | = ˜ r, | L + K | = r, | Γ >N ( L + K ) | ≤ B: M ( l, k ) = i , M ( L, K ) = i , i ′ = i + i ; D1: ( l − e j ) + L = l ′ , k + ( K − e j ) = k ′ , with l j > K j > D2: l + ( L − e j ) = l ′ , ( k − e j ) + K = k ′ , with L j > k j > . From the definition of set Ω( l ′ , k ′ , i ′ ), if element (( l, k, i ) , ( L, K, i ) , j ) ∈ Ω( l ′ , k ′ , i ′ ),then (( k, l, − i ) , ( K, L, − i ) , j ) ∈ Ω( k ′ , l ′ , − i ′ ). Lemma 5.4.
Fix β ≥ . For any given l ′ , k ′ ∈ N Z ∗ with | l ′ + k ′ | = r + ˜ r − and M ( l ′ , k ′ ) = i ′ , it holds X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ′ ) K j l j h i − i ′ i β · h i i β , X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ′ ) k j L j h i − i ′ i β · h i i β ≤ β +1 h i ′ i β cr (˜ r + 1)(2 N + 1) r − . roof. Consider the non-zero components of vectors k ′ and l ′ . For example, e j has onlyone non-zero component with index j , being 1; Taking multiplicity into account, regardthat ke j ( k is a positive integer) has k non-zero components whose values are 1 and theirindexes are j . So ( l ′ , k ′ ) with | l ′ + k ′ | = r + ˜ r − r + ˜ r − j,i ( l ′ , k ′ , i ′ ) := (cid:8)(cid:0) ( l, k, i ) , ( L, K, ˜ i ) , t (cid:1) ∈ Ω( l ′ , k ′ , i ′ ) | t = j and ˜ i = i (cid:9) . It follows Ω( l ′ , k ′ , i ′ ) = [ j ∈ Z ∗ , [ i ∈ Z Ω j,i ( l ′ , k ′ , i ′ ) . The element in Ω j,i ( l ′ , k ′ , i ′ ) is unique determined, if ( l − e j , k ) is fixed. The estimate of˜ ♯ Ω j,i ( l ′ , k ′ , i ′ ) is obtained as follows.In the case | j | ≤ N , since | Γ >N ( L + K ) | ≤
2, there are at least r − l ′ , k ′ ) coming from ( L, K ) with the indexes being bounded by N and the choices ofthat is smaller than (2 N + 1) r − . As for the remaining three components of ( L, K ) whosevalues are 1, one of their positions is j with | j | ≤ N ; One position among the other twocan be selected from the rest non-zero components of ( l ′ , k ′ ) and the choices is ˜ r + 1; Thelast one may be determined by the fact that M ( L, K ) = i . It holds X (( l,k,i ) , ( L,K,i ) ,j ) ∈ S | j |≤ N S i ∈ Z Ω j,i ( l ′ ,k ′ ,i ′ ) , K j l j h i i β · h i i β ≤ r h i ′ i β X | j |≤ N, X Ω j,i ( l ′ ,k ′ ,i ′ ) , X i ∈ Z l j h i ′ i β h i i β · h i i β ≤ r h i ′ i β X | j |≤ N, X Ω j,i ( l ′ ,k ′ ,i ′ ) , X i ∈ Z l j β − ( h i i β + h i i β ) h i i β · h i i β ≤ rc β h i ′ i β (˜ r + 1)(2 N + 1) r − . (5.44)In the case | j | > N , there are at least r − l ′ , k ′ ) coming from( L, K ) whose indexes are bounded by N , and there are at most (2 N + 1) r − choices; Oneposition of the last two value-1 components of ( L, K ) is chosen from the rest ˜ r non-zerocomponents of ( l ′ , k ′ ) and the choices is ˜ r ; The position of the last component of ( L, K )is determined by the momentum of (
L, K ) being i . It holds X (( l,k,i ) , ( L,K,i ) ,j ) ∈ S | j | >N S i ∈ Z Ω j,i ( l ′ ,k ′ ,i ′ ) , K j l j h i i β · h i i β ≤ r h i ′ i β X | j | >N, X i ∈ Z , X Ω j,i ( l ′ ,k ′ ,i ′ ) , l j h i ′ i β h i i β · h i i β ≤ r h i ′ i β X | j | >N, X i ∈ Z , X Ω j,i ( l ′ ,k ′ ,i ′ ) , l j β − ( h i i β + h i i β ) h i i β · h i i β ≤ rc β h i ′ i β ˜ r (2 N + 1) r − . (5.45)The result is obtained from (5.44) and (5.45).In the following, the proof of the Lemma 5.3 is given.41 roof. By the definition of {· , ·} w θ , the following equation holds f w θ (1 ,S wθ ) = { f w θ , S w θ } w θ = X | l ′ + k ′| = r +˜ r − M ( l ′ ,k ′ )= i ′∈ Mfwθ (1 ,Swθ ) ( f w θ (1 ,S wθ ) ) i ′ r +˜ r − ,l ′ k ′ u l ′ ¯ u k ′ with M f wθ (1 ,Swθ ) := { i = i + i | i ∈ M f wθ ⊂ Z , i ∈ M S wθ ⊂ Z } and( f w θ (1 ,S wθ ) ) i ′ ˜ r + r − ,l ′ ,k ′ : = X j ∈ Z ∗ i ′ = i i i sgn θ ( j ) X ( l − ej )+ L = l ′ k +( K − ej )= k ′M ( l,k )= i , M ( L,K )= i l j K j ( f w θ ) i ˜ r,lk ( S w θ ) i r,LK − X j ∈ Z ∗ i ′ = i i i sgn θ ( j ) X l +( L − ej )= l ′ ( k − ej )+ K = k ′M ( l,k )= i , M ( L,K )= i L j k j ( f w θ ) i ˜ r,lk ( S w θ ) i r,LK . In the case θ = 0, I will give the exact definition of A ( f w ,Sw ) i ′ ˜ r + r − ,l ′ k ′ and ( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ and prove that the coefficients ( f w (1 ,S w ) ) i ′ ˜ r + r − ,l ′ k ′ can be rewritten as the following form( f w (1 ,S w ) ) i ′ ˜ r + r − ,l ′ k ′ := X ( l ′ ,k ′ ,i ′ ) ∈A ( fw ,Sw i ′ ˜ r + r − ,l ′ k ′ ( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ (cid:0) M ( l ′ , k ′ ) − i ′ (cid:1) and satisfy ( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ = ( f w (1 ,S w ) ) − i ′ ( k ′ − k ′ ,l ′ − l ′ ,i ′ − i ′ )˜ r + r − ,k ′ l ′ . In order to describe the set A ( f w ,Sw ) i ′ ˜ r + r − ,l ′ k ′ clearly, for any fixed (cid:0) ( l, k, i ) , ( L, K, i ) , j (cid:1) ∈ Ω w ( l ′ , k ′ , i ′ ), define a mapping D on set A ( f w ) i r,lk , for any ( l , k , i ) ∈ A ( f w ) i r,lk , D ( l , k , i ) := ( l , k , i ) when ( l ′ , k ′ ) = (cid:0) ( l − e j ) + L, k + ( K − e j ) (cid:1) and l j = 0 . ( l − e j + L, k + K − e j , i + 2 i ) when ( l ′ , k ′ ) = (cid:0) ( l − e j ) + L, k + ( K − e j ) (cid:1) and l j ≥ l j > . ( l , k , i ) when ( l ′ , k ′ ) = (cid:0) l + ( L − e j ) , ( k − e j ) + K (cid:1) and k j > k j ≥ . ( l − e j + L, k + K − e j , i + 2 i ) when ( l ′ , k ′ ) = (cid:0) l + ( L − e j ) , ( k − e j ) + K (cid:1) and k j = k j > . (5.46) Base on the set Ω( l ′ , k ′ , i ′ ) and the map D , denote A ( f w ,Sw ) i ′ ˜ r + r − ,l ′ k ′ := [ (( l,k,i ) , ( L,K,i ) , j ) ∈ Ω( l ′ ,k ′ ,i ′ ) D A ( f w ) i r,lk and A ( f w ,Sw ) − i ′ ˜ r + r − ,k ′ l ′ := [ (( k,l, − i ) , ( K,L, − i ) , j ) ∈ Ω( k ′ ,l ′ , − i ′ ) D A ( f w ) − i r,kl , where D A ( f w ) i r,lk := { D ( l , k , i ) | ( l , k , i ) ∈ A ( f w ) i r,lk } .
42t is easy to check that D is not an inverse mapping from A ( f w ) i r,lk to D A ( f w ) i r,lk . Denote ( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ := X ( l ,k ,i ) ∈ D − ( l ′ ,k ′ ,i ′ ) i ( l j K j − L j k j )( f w ) i ( l ,k ,i )˜ r,lk ( S w ) i r,LK , (5.47) where D − ( l ′ , k ′ , i ′ ) := { ( l , k , i ) ∈ A ( f w ) i r,lk | D ( l , k , i ) = ( l ′ , k ′ , i ′ ) } . For any( l ′ , k ′ , i ′ ) ∈ A ( f w ,Sw ) i ′ ˜ r + r − ,l ′ k ′ , it is easy to verify that ( k ′ − k ′ , l ′ − l ′ , i ′ − i ′ ) ∈A ( f w ,Sw ) − i ′ ˜ r + r − ,k ′ l ′ . Moreover, by (5.47) and the facts that f w having ( β, S w having symmetric coefficients, it holds( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ = i X ( l ,k ,i ) ∈ D − ( l ′ ,k ′ ,i ′ ) ( l j K j − k j L j )( f w ) i ( l ,k ,i )˜ r,lk ( S w ) i r,LK = i X ( k − k ,l − l ,i − i ) ∈ D − ( k ′ − k ′ ,l ′ − l ′ ,i ′ − i ′ ) ( L j k j − l j K j )( f w ) − i ( k − k ,l − l ,i − i )˜ r,kl ( S w ) − i r,KL = ( f w (1 ,S w ) ) − i ′ ( k ′ − k ′ ,l ′ − l ′ ,i ′ − i ′ )˜ r + r − ,k ′ l ′ , the last second equation is holding by the definition of D in (5.46) and M ( l ′ , k ′ ) − i ′ M ( k ′ − k ′ , l ′ − l ′ ) − ( i ′ − i ′ ) . So f w (1 ,S w ) has ( β, C g w r . By (5.47), (5.39)in Lemma 5.2, it holds X ( l ′ ,k ′ ,i ′ ) ∈A ( fw ,Sw i ′ ˜ r + r − ,l ′ k ′ | ( f w (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ | · max {h i ′ i , h i ′ − i ′ i}≤ X ( l ′ ,k ′ ,i ′ ) ∈A ( fw ,Sw i ′ ˜ r + r − ,l ′ k ′ | X ( l ,k ,i ) ∈ D − ( l ′ ,k ′ ,i ′ ) ( k j L j − K j l j )( f w ) i ( l ,k ,i )˜ r,lk ( S w ) i r,LK |· max {h i ′ i , h i ′ − i ′ i}≤ rC r − g w r N α γ X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ′ ) (cid:12)(cid:12) k j L j − K j l j (cid:12)(cid:12) h i i β · X ( l ,k ,i ) ∈A ( fw i r,lk (cid:12)(cid:12) ( f w ) i ( l ,k ,i )˜ r,lk (cid:12)(cid:12) · (cid:0) max {h i i , h i − i i} + N (cid:1) ≤ r N α +1 C r − g w r C ˜ r − f w r γ X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ′ ) , X i ,i ∈ Z | k j L j − K j l j |h i i β · h i i β . (5.48) By (5.48) and Lemma 5.4, it follows that X ( l ′ ,k ′ ,i ′ ) ∈A ( fw ,Sw i ′ ˜ r + r − ,l ′ k ′ | ( f (1 ,S w ) ) i ′ ( l ′ ,k ′ ,i ′ )˜ r + r − ,l ′ k ′ | · max {h i ′ i , h i ′ − i ′ i}≤ β +2 r c (˜ r + 1) N α +1 C r − g w r C ˜ r − f w r (2 N + 1) r − γ h i ′ i β . (5.49)43hen θ = 1, the coefficients of f w (1 ,S w ) have the following form( f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ := ( ˜ f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ Y t ∈ Z ∗ | t | l ′ t + k ′ t , (5.50)where( ˜ f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ := X (( l,k,i , ( L,K,i ,j ) ∈ Ω( l ′ ,k ′ ,i ) i sgn( j )( l j K j − L j k j )( ˜ f w ) i ˜ r,lk | j | (˜ g w ) i r,LK i h ω w , I ( l − k ) i . (5.51)Since f w ( u, ¯ u ) and g w ( u, ¯ u ) have ( β, g w ( u, ¯ u ) is givenin equation (5.31)), from (5.51) the coefficients of f w (1 ,S w ) satisfy that( ˜ f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ = X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ) i sgn( j )( l j K j − L j k j )( ˜ f w ) i ˜ r,lk | j | (˜ g w ) i r,LK i h ω w , I ( l − k ) i = X ( k,l, − i ) , ( K,L, − i ) ,j ) ∈ Ω( k ′ ,l ′ , − i ) i sgn( j )( L j k j − l j K j )( ˜ f w ) − i ˜ r,kl | j | (˜ g w ) − i r,KL i h ω w , I ( k − l ) i = ( ˜ f w (1 ,S w ) ) − i ˜ r + r − ,k ′ l ′ . From (5.35), for any ( l, k ) with nonzero ( S w ) ir,lk , |h ω w , I ( l − k ) i| > γM l,k N α , which implies that | ( ˜ f w (1 ,S w ) ) i ˜ r + r − ,l ′ k ′ |≤ X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ) (cid:12)(cid:12) ( l j K j − L j k j ) · ( ˜ f w ) i ˜ r,lk j (˜ g w ) i r,LK i h ω w , I ( l − k ) i (cid:12)(cid:12) ≤ N α C r − g w r C ˜ r − f w r γ X (( l,k,i ) , ( L,K,i ) ,j ) ∈ Ω( l ′ ,k ′ ,i ) | l j K j − L j k j |h i i β h i i β ≤ (˜ r + 1) cr (2 N ) r − N α β +1 C r − f w r C ˜ r − g w r γ h i i β , the last inequality holds by Lemma 5.4.The proof of Theorem 4 is a purely technical matter and is relegated to Appendix. For any given integer r ∗ ≥
0, using Theorem 4, there exists a transformation T ( r ∗ ) w θ changing the system (3.8) into ˙˜ u j = − i sgn θ ( j ) · ∂ ¯˜ u j H ( r ∗ ,w θ ) (˜ u, ¯˜ u ) , ˙˜ u j = i sgn θ ( j ) · ∂ ˜ u j H ( r ∗ ,w θ ) (˜ u, ¯˜ u ) , (6.1)44ith Hamiltonian H ( r ∗ ,w θ ) (˜ u, ¯˜ u ) = H w θ + Z ( r ∗ ,w θ ) (˜ u, ¯˜ u ) + R N ( r ∗ ,w θ ) (˜ u, ¯˜ u ) + R T ( r ∗ ,w θ ) (˜ u, ¯˜ u ) . (6.2)The solution (˜ u, ¯˜ u ) to (6.1) satisfies ddt k ˜ u ( t ) k p = {k ˜ u k p , H ( r ∗ ,w θ ) (˜ u, ¯˜ u ) } w θ = {k ˜ u k p , H w θ + Z ( r ∗ ,w θ ) (˜ u, ¯˜ u ) + R N ( r ∗ ,w θ ) (˜ u, ¯˜ u ) + R T ( r ∗ ,w θ ) (˜ u, ¯˜ u ) } w θ . (6.3)It is easy to get that {k ˜ u k p , H w θ (˜ u, ¯˜ u ) } w θ = 0 . (6.4)Using Theorem 4, Proposition 4.2 and Corollary 1, when N satisfies (5.28), it holds thatsup (˜ u, ¯˜ u ) ∈ B p ( R/ |{k ˜ u k p , R N ( r ∗ ,w θ ) (˜ u, ¯˜ u ) + R T ( r ∗ ,w θ ) (˜ u, ¯˜ u ) } w θ | ≤ C ( θ, p, r ∗ ) R r ∗ +1 , (6.5)the inequality is holding by the fact that for any (˜ u, ¯˜ u ) ∈ B p ( R ) k Γ >N ˜ u k ≤ k Γ >N ˜ u k p N p − and X | i | >N h i i β ≤ N p ( X | i | >N h i i ) , as β > p + 4 . (6.6)By Lemma 5.1 and (6.6), when N satisfies (5.28), it follows thatsup (˜ u, ¯˜ u ) ∈ B p ( R/ |{k ˜ u k p , Z ( r ∗ ,w θ ) (˜ u, ¯˜ u ) } w θ | ≤ C ( θ, p, r ∗ ) R r ∗ +1 . (6.7)Suppose that the initial value to (3.8) satisfies ( u (0) , ¯ u (0)) ∈ B p ( R/ R is smallenough, the initial value ( u (0) , ¯ u (0)) ∈ B p ( R/
6) is transformed into(˜ u (0) , ¯˜ u (0)) ∈ B p ( R/ . (6.8)Together with (6.3)-(6.5) and (6.7)-(6.8), the following inequality holds true (cid:12)(cid:12) k ˜ u ( t ) k p − k ˜ u (0) k p | ≤ | Z T d k ˜ u ( τ ) k p dτ dτ | ≤ C ( θ, p, r ∗ ) R r ∗ +1 T, (6.9)where T := min { | t | | k (cid:0) ˜ u ( t ) , ¯˜ u ( t ) (cid:1) k p = R/ } , which means that for any | t | ≤ T := C ( θ,p,r ∗ ) R r ∗− , k ˜ u ( t ) k p ≤ R/ , k (cid:0) ˜ u ( t ) , ¯˜ u ( t ) (cid:1) k p ≤ R/ . (6.10)From Theorem 4, when R ≪ T ( r ∗ ) w θ is an inverse transformation from B p ( R/
2) to B p ( R ). Then by (6.10), the solution ( u ( t ) , ¯ u ( t )) to systems (3.8) with ( u (0) , ¯ u (0)) ∈ B p ( R/
6) satisfies k ( u ( t ) , ¯ u ( t )) k p ≤ R, for any | t | ≺ R r ∗ − . Proof of Theorem 1 and Theorem 2
It is common knowledge that ( j ) j ∈ Z are the eigenvalues of − ∂ xx under periodic bound-ary condition ψ ( x, t ) = ψ ( x + 2 π, t ) with the corresponding eigenfunctions { φ j ( x ) := e i jx √ π } j ∈ Z . Take ψ ( x, t ) = X j ∈ Z u j ( t ) φ j ( x ) , u j := Z π ψ ( x, t ) φ − j ( x ) dx (7.1)into equation (2.2) and obtain a Hamiltonian system, ˙ u j = − i ∂H w ∂ ¯ u j ( u, ¯ u ) , ˙¯ u j = i ∂H w ∂u j ( u, ¯ u ) , for any j ∈ Z (7.2)with respect to 2-form w in (3.5), and the Hamiltonian function has the form H w ( u, ¯ u ) := H w + P w ( u, ¯ u ) , (7.3)where H w := X j ∈ Z ω j | u j | , ω j := − j + b V j = − j + v w j h j i m . (7.4)Under assumptions A and A in section 2.1, the power series P w ( u, ¯ u ) has the followingform P w ( u, ¯ u ) = X r ≥ X | k + l | = r M ( l,k )= i ∈ MPw r X ( l ′ ,k ′ , − i ) ∈A ( Pw ir,lk ( P w ) i ( l ′ ,k ′ , − i ) r,lk · ( M ( l ′ , k ′ ) − i u l ¯ u k , where A ( P w ) ir,lk := { ( l, , − i ) | i ∈ M P w r ⊂ Z } , M P w r is a symmetric set and( P w ) i ( l, , − i ) r,lk := ∂ | l | ψ | l | ∂ | k | ¯ ψ | k | b F | (0 , ( − i )( − π ) r l ! k ! . Moreover, the following equation holds true for any l, k with | l + k | = r and M ( l, k ) = i ( P w ) i ( l, , − i ) r,lk = ∂ | l | ψ | l | ∂ | k | ¯ ψ | k | b F | (0 , ( − i )( − π ) r l ! k ! = ∂ | k | ψ | k | ∂ | l | ¯ ψ | l | b F | (0 , ( i )( − π ) r l ! k ! = ( P w ) − i ( k, , − i ) r,kl and there exists a constant C > X ( l ,k ,i ) ∈A ( Pw ir,lk max {h i i , h i − i i}| ( P w ) i ( l ,k ,i ) r,lk | ≤ C r − h i i β , which means that P w ( u, ¯ u ) has ( β, C >
0. 46 emma 7.1.
For any given integers r ∗ , N > and real numbers α > m + r ∗ + 8 , ≫ γ > , there exists an open subset e Θ θm ⊂ Θ θm ( Θ θm defined in (2.4) and (2.14),respectively) such that for any V ∈ e Θ θm and any ( l, k ) belongs to O w θ r ∗ +3 ,N defined in (5.2),it satisfies |h ω w θ ( V ) , I θ ( l − k ) i| > γM l,k N α , where ω w θ ( V ) = ( ω w θ j ) j ∈ Z θ , ω w θ j := sgn θ ( j ) · (cid:0) − j + v w θ j h j i m (cid:1) , v w θ j ∈ [ − / , / . (7.5) Moreover, meas (Θ θm / e Θ θm ) ≤ r ∗ +4+ m r m +3 ∗ γN α − m − r ∗ − . Remark 7.1. If γ > is small enough, the set e Θ θm will have a positive measure. Inparticular, if γ approaches to 0, then the measure of e Θ θm will approach to the measure of Θ θm . Now give the proof of Lemma 7.1.
Proof.
Denote Θ θm / e Θ θm := [ ≤ r ≤ r ∗ +3 (cid:0) [ ( l,k ) ∈ O wθ,rr ∗ +3 ,N, ∪ O wθ,rr ∗ +3 ,N, ∪ O wθ,rr ∗ +3 ,N, Θ θr,lk (cid:1) , where Θ θr,lk := (cid:26) V ∈ Θ θm (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) h ω w θ ( V ) , I θ ( l − k ) i (cid:12)(cid:12) ≤ γM l,k N α (cid:27) and O w θ ,rr ∗ +3 ,N,n := (cid:26) ( l, k ) ∈ O w θ r ∗ +3 ,N (cid:12)(cid:12)(cid:12)(cid:12) | Γ >N ( l + k ) | = n, | l + k | = r (cid:27) , for n ∈ { , , } and 3 ≤ r ≤ r ∗ + 3.I only give the estimate of the measure of Θ θr,lk in the case ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, , which ismore complex than the case ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, ∪ O w θ ,rr ∗ +3 ,N, .When the multi-index ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, , estimate the measure of Θ θr,lk in two cases.(1)The first case( l, k ) ∈ O w θ ,rr ∗ +3 ,N, a := { ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, | | Γ >N l | = 2 or | Γ >N k | = 2 } . In this case, there exists | j | , | j | > N such that l j = l j = 1 or k j = k j = 1.Without loss of generality, assume | j | ≥ | j | > N with l j = l j = 1. So M l,k = | j | and | ω w θ j ( V ) | > j − , | ω w θ j ( V ) | > j − . The other frequencies ω w θ j ( V )( | j | ≤ N ) arebounded by | ω w θ j | ≤ N + 1.If | j | > √ rN , it follows that |h ω w θ ( V ) , I θ ( l − k ) i| = | l j ω w θ j + l j ω w θ j + X | j |≤ N ω w θ j ( l j − k j ) |≥ | ω w θ j + ω w θ j | − | X j = j ,j , | j |≤ N ( l j − k j ) ω w θ j | ≥ j − − ( N + 1)( r − γ | j | N α = γM l,k N α . That means when | j | > √ rN , the set Θ θr,lk is empty. So it is only need to calculate themeasure of Θ θr,lk whose multi-index ( l, k ) being in the following set, ^ O w θ ,rr ∗ +3 ,N, a := (cid:8) ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, a (cid:12)(cid:12) N ≤ M l,k ≤ √ rN (cid:9) ⊂ O w θ ,rr ∗ +3 ,N, a , (7.6)the number of which are bounded by ♯ ^ O w θ ,rr ∗ +3 ,N, a ≤ √ r (4 N ) r . (7.7)For any fixed ( l, k ) ∈ ^ O w θ ,rr ∗ +3 ,N, a , there exists 4 √ rN ≥ | j | > N fulfilling l j + l − j − k j − k − j = 0 , θ = 0 ,l j − k j = 0 or l − j − k − j = 0 , θ = 1 . such that (cid:12)(cid:12) ∂g w ∂v w j (cid:12)(cid:12) = | l j + l − j − k j − k − j || j | m ≥ | j | m = 0 , θ = 0 , (cid:12)(cid:12) ∂g w ∂v w j (cid:12)(cid:12) = | l j − k j |h j i m ≥ h j i m = 0 , (or (cid:12)(cid:12) ∂g w ∂v w − j (cid:12)(cid:12) = | l − j − k − j |h j i m ≥ | j | m = 0) , θ = 1 . (7.8)The measure of Θ θr,lk has the following estimate by (7.8)meas(Θ θr,lk ) ≤ M l,k γN α (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∂g w θ ∂v w θ j (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ | j | m +1 N α ≤ γ (4 √ r ) m +1 N α − m − . (7.9)From (7.7) and (7.9), it holds thatmeas( [ ( l,k ) ∈ O wθ,rr ∗ +3 ,N, a Θ θr,lk ) = meas( [ ( l,k ) ∈ ^ O wθ,rr ∗ +3 ,N, a Θ θr,lk ) ≤ m +2 r +3 √ r m +2 γN α − m − r − . (7.10)(2)The second case ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, b := { ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, | there exist | j | 6 = | j | > N, l j = k j = 1 or l j = k j = 1 } . Without loss of generality, assume | j | > | j | and M l,k = | j | . By (7.5), it holds | ω w θ j | ≥ j − , | ω w θ j | ≤ j + and ω w θ j ≤ N + 1, ( | j | ≤ N ). If | j | > rN , the following inequalityholds |h ω w θ , I θ ( l − k ) i| ≥ | ω w θ j − ω w θ j | − | X j = j ,j ω w θ j ( l j − k j ) |≥ ( | j | − ( | j | + 1)) − ( N + 1)( r − ≥ ( | j | + | j | )( | j | − | j | ) − ( N + 1)( r − − ≥ γ | j | N α = γM l,k N α . M l,k > rN , the set Θ θr,lk is empty. It only needs to calculate the sumof the set Θ θr,lk with ( l, k ) being in the following set ^ O w θ ,rr ∗ +3 ,N, b := (cid:8) ( l, k ) ∈ O w θ ,rr ∗ +3 ,N, b (cid:12)(cid:12) M l,k ≤ rN (cid:9) which is bounded by ♯ ^ O w θ ,rr ∗ +3 ,N, b ≤ r (4 N ) r +2 (7.11)There exists j with 4 rN ≥ | j | > N such that l j + l − j − k j − k − j = 0 and (cid:12)(cid:12) ∂g w θ ( v ) ∂v w θ j (cid:12)(cid:12) > | j | m . (7.12)Denote a set b Θ θr,lk := (cid:26) V ( x ) ∈ Θ θm (cid:12)(cid:12)(cid:12)(cid:12) |h ω w θ ( V ) , I θ ( l − k ) i| ≤ rγN α − (cid:27) . When ( l, k ) ∈ ^ O w θ ,rr ∗ +3 ,N, b , using the fact Θ θr,lk ⊂ b Θ θr,lk and (7.12), it implies thatmeas Θ θr,lk ≤ meas b Θ θr,lk ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∂g w θ ( v ) ∂v j (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12) rγN α − ≤ r ) m +1 γN α − m − . (7.13)From (7.11) and (7.13), it holds thatmeas (cid:0) [ ( l,k ) ∈ O wθ,rr ∗ +3 ,N, b Θ θr,lk (cid:1) = meas (cid:0) [ ( l,k ) ∈ ^ O wθ,rr ∗ +3 ,N, b Θ θr,lk (cid:1) ≤ r + m +4 r m +3 γN α − m − r − . (7.14)Using the same method, the following inequality holds truemeas( [ ( l,k ) ∈ O wθ,rr ∗ +3 ,N, ∪ O wθ,rr ∗ +3 ,N, Θ θr,lk ) ≤ m +2 r +3 √ r m +2 γN α − m − r − . (7.15)In view of (7.15) and (7.14), one has meas(Θ θm \ e Θ θm,N ) ≤ r ∗ +3 X r =3 X ( l,k ) ∈ ( O wθ,rr ∗ +3 ,N, ∪ O wθ,rr ∗ +3 ,N, ∪ O wθ,rr ∗ +3 ,N, ) meas Θ θr,lk < r ∗ +4+ m r m +3 ∗ γN α − m − r ∗ − . Now Theorem 1 is obtained by Theorem 3 and Lemma 7.1. The transformation ψ ( x, t ) = P j ∈ Z u j ( t ) φ j ( x ) is from ℓ p to H p ([0 , π ] , C ) and satisfies k u ( t ) k p ≤ k ψ ( x, t ) k H p ([0 , π ] , C ) = sup ≤| n |≤ p k D nx ψ ( x, t ) k L ≤ (2 π ) p k u ( t ) k p . Take e ε ≪ V ( x ) ∈ e Θ m , if the initial value of ψ ( x, t ) to (2.2)fulfilling k ψ ( x, k H p ([0 , π ] , C ) ≤ ε/ < e ε/ , then it holds that k ψ ( x, t ) k H p ([0 , π ] , C ) ≤ ε, forany | t | ≺ ε − ( r ∗ − . .2 Proof of Theorem 2 The following statements deal with the solution to equation (2.11). It is obvious that j ( j ∈ Z ∗ ) is the eigenvalue of ( − ∂ xx ) under periodic boundary condition and φ j ( x ) := √ π e i jx is the corresponding eigenfunction. Precisely,( − ∂ xx ) φ j ( x ) = j φ j ( x ) ∀ j ∈ Z ∗ . For any ψ ∈ H p +1 / ([0 , π ] , C ), ψ ( x, t ) = X j ∈ Z ∗ b ψ j ( t ) φ j ( x ) , where b ψ j ( t ) := R π ψ ( x, t ) φ − j ( x )d x . In order to transform equation (2.11) into an infinitedimensional Hamiltonian system under a standard symplectic form, I will use a tool givenin [30] u ( t ) = ( u j ( t )) j ∈ Z ∗ , u j ( t ) := b ψ j ( t ) | j | , for any j ∈ Z ∗ . (7.16)It is easy to get that if ψ ∈ H p +1 / ([0 , π ] , C ) then the corresponding Fourier coeffi-cients vector ( b ψ j ) j ∈ Z ∗ ∈ ℓ p +1 / ( Z ∗ , C ) and u ∈ ℓ p ( Z ∗ , C ). Moreover, there exist constants˜ C , ˜ C > C k u k p ≤ k ψ k H p +1 / ([0 , π ] , C ) ≤ ˜ C k u k p . (7.17)Under transformation (7.16), equation (2.11) therefore can be written into the followingHamiltonian system with respect to symplectic from w defined in (3.5), for any j ∈ Z ∗ , ˙ u j = − i sgn( j ) ∂H w ( u, ¯ u ) ∂ ¯ u j ˙¯ u j = i sgn( j ) ∂H w ( u, ¯ u ) ∂u j (7.18)with the Hamiltonian H w ( u, ¯ u ) := H w + P w ( u, ¯ u ) , (7.19)where H w := X j ∈ Z ∗ ω j | u j | , ω j := sgn( j )( − j + b V j ) = sgn( j )( − j + v w j | j | m ) ∈ R . (7.20)By assumptions B - B in Theorem 2, P w ( u, ¯ u ) has a zero at origin at last order 3 withthe following form P w ( u, ¯ u ) := + ∞ X r =3 X | l + k | = r M ( l,k )= i ∈ MPw r ⊆ Z ( ˜ P w ) ir,lk Y t ∈ Z ∗ | t | lt + kt u l ¯ u k , where M P w r ⊂ Z is a symmetric set( ˜ P w ) ir,lk := 1(2 π ) r − l ! k ! ˆ ∂ l + k F∂ψ l ∂ψ k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ( − i ) , l ! = Y j ∈ Z ∗ l j ! . C > r ≥ i ∈ M P w r ⊆ Z , | ( ˜ P w ) ir,lk | ≤ C r − h i i β and ( ˜ P w ) ir,lk = 1(2 π ) r − l ! k ! ˆ ∂ l + k F∂ψ l ∂ψ k (cid:12)(cid:12)(cid:12)(cid:12) (0 , ( − i ) = 1(2 π ) r − l ! k ! ˆ ∂ l + k F∂ψ k ∂ψ l (cid:12)(cid:12)(cid:12)(cid:12) (0 , ( i ) = ( ˜ P w ) − ir,kl which means that the power series P w ( u, ¯ u ) has ( β, C .From (7.20), the origin is the elliptic equilibrium point of the equation (7.18). UsingTheorem 3, Lemma 7.1 and (7.17), for any V ∈ ˜Θ m , there exists ˜ ε ≪ < ε < ˜ ε , if k ψ (0 , x ) k H p +1 / ([0 , π ] , C ) < ε < ˜ ε, then it satisfies k ψ ( t, x ) k H p +1 / ([0 , π ] , C ) < ε, for any | t | ≺ ε − r ∗ +1 . Now the proof of Theorem 4 is given in this section.
Proof.
For any θ ∈ { , } , denote g ( − ,w θ ) ( u, ¯ u ) := r ∗ +3 X r =3 P w θ r ( u, ¯ u ) , R N ( − ,w θ ) ( u, ¯ u ) := 0 , R T ( − ,w θ ) ( u, ¯ u ) := ∞ X r = r ∗ +4 P w θ r ( u, ¯ u ) , where P w θ r ( u, ¯ u ) is an r -degree homogeneous polynomial of P w θ ( u, ¯ u ). Thus (3.9) can berewritten as H ( − ,w θ ) = H w θ + g ( − ,w θ ) + R N ( − ,w θ ) + R T ( − ,w θ ) , defined in B p ( R ∗ ) . (8.1)To start with, the results hold at rank r = 0. For any R < R ∗ and any N satisfying(5.28), I will look for a bounded Lie-transformation T w θ to eliminate the non-normalizedmonomials of Γ N ≤ g ( − ,w θ )3 . The Lie-transformation T w θ is constructed from 1-time flowΦ tS (0) wθ of the following equations, ˙ u j = − i sgn θ ( j ) ∇ ¯ u j S (0) w θ ( u, ¯ u ) , ˙¯ u j = i sgn θ ( j ) ∇ u j S (0) w θ ( u, ¯ u ) , j ∈ Z ∗ , θ ∈ { , } , where S (0) w θ is undetermined. Under transformation T w θ the new Hamiltonian H (0 ,w θ ) hasthe following form, H (0 ,w θ ) = H ( − ,w θ ) ◦ T w θ = ( H w θ + g ( − ,w θ ) + R T ( − ,w θ ) ) ◦ Φ S (0) wθ = H w θ + { H w θ , S (0) w θ } w θ + g ( − ,w θ )3 (8.2) r ∗ +3 X t =4 g ( − ,w θ ) t + X ν ≥ ( H w θ ) ( ν,S (0) wθ ) + X ν ≥ r ∗ +3 X t =3 ( g ( − ,w θ ) t ) ( ν,S (0) wθ ) + X ν ≥ ∞ X t = r ∗ +4 ( R T ( − ,w θ ) t ) ( ν,S (0) wθ ) , (8.3) where ( . ) ( ν,S (0) wθ ) is defined in (5.30). The auxiliary Hamiltonian function S (0) w θ are obtainedby solving the following homological equation { H w θ , S (0) w θ } w θ + Γ N ≤ P w θ = Z w θ . (8.4)Using Remark 4.2, Γ N> P w θ and Γ N ≤ P w θ are still having ( β, θ )-type symmetric coefficientssemi-bounded by C ( θ, −
1) = C θ > . From Lemma 5.2, Z w θ is ( θ, γ, α, N )-normal formof Γ N ≤ P w θ and the Hamiltonian vector field of S (0) w θ satisfies k X w θ S (0) wθ k p ≤ c C θ p +1 N α γ k u k p k u k for any ( u, ¯ u ) ∈ B p (2 R ) . (8.5)From (8.4), the following holds true(8 .
2) = Z w θ ( u, ¯ u ) + Γ N> P w θ . The Lie-transformation T w θ satisfiessup ( u, ¯ u ) ∈ B p ( R ) kT w θ ( u, ¯ u ) − ( u, ¯ u ) k p = sup ( u, ¯ u ) ∈ B p ( R ) k Φ S (0) wθ ( u, ¯ u ) − ( u, ¯ u ) k p = sup ( u, ¯ u ) ∈ B p ( R ) (cid:13)(cid:13) Z t =0 X w θ S (0) wθ ◦ Φ τS (0) wθ ( u, ¯ u )( τ ) dτ (cid:13)(cid:13) p . (8.6)Use the bootstrap method to estimate T w θ . First, assume thatΦ tS (0) wθ : B p ( R ) → B p (2 R ) , for any t ∈ [0 , . (8.7)By (8.5)-(8.7), the following inequality holds truesup ( u, ¯ u ) ∈ B p ( R ) kT w θ ( u, ¯ u ) − ( u, ¯ u ) k p ≤ sup ( u, ¯ u ) ∈ B p (2 R ) (cid:13)(cid:13) Z t =0 X w θ S (0) wθ ( τ ) dτ (cid:13)(cid:13) p ≤ C θ c N α γ p +1 (2 R ) . (8.8)Since R is small enough, from (5.28) and (8.8), the transformation T w θ satisfiessup ( u, ¯ u ) ∈ B p ( R ) kT w θ ( u, ¯ u ) − ( u, ¯ u ) k p ≤ R, which means T w θ : B p ( R ) → B p (2 R ) . (8.9)Denote T w θ (0) := T w θ . By (5.28), (8.6) and (8.8), it is verified that (5.29) holds for rank r = 0: sup ( u, ¯ u ) ∈ B p ( R ) kT w θ (0) ( u, ¯ u ) − ( u, ¯ u ) k p ≤ C ( θ, p, r ∗ ) R − r ∗ +1 . Set Z (0 ,w θ ) := Z ( − ,w θ ) + Z w θ , R N (0 ,w θ ) := R N ( − ,w θ ) + Γ N> g ( − ,w θ )3 . (8.10)52ince Z w θ and g ( − ,w θ )3 having ( β, θ )-type symmetric coefficients, then Z (0 ,w θ ) and R N (0 ,w θ ) are still having ( β, θ )-type symmetric coefficients. Denote the r ∗ + 3-degree polynomial ofpower series (8.3) as g (0 ,w θ ) and the remainder as R T (0 ,w θ ) , i.e., g (0 ,w θ ) := r ∗ +3 X t =4 g (0 ,w θ ) t , R T (0 ,w θ ) := X t>r ∗ +3 R T (0 ,w θ ) t , where for any 4 ≤ t ≤ r ∗ + 3, g (0 ,w θ ) t := g ( − ,w θ ) t + ( H w θ ) ( t − ,S (0) wθ ) + t − X n ′ =1 ( g ( − ,w θ ) t − n ′ ) ( n ′ ,S (0) wθ ) and for any t > r ∗ + 3 R T (0 ,w θ ) t := ( H w θ ) ( t − ,S (0) wθ ) + t − X n ′ =1 ( g ( − ,w θ ) t − n ′ ) ( n ′ ,S (0) wθ ) + t − X n ′ =1 ( R T ( − ,w θ ) t − n ′ ) ( n ′ ,S (0) wθ ) . From Remark 5.4 and Lemma 4.1, g (0 ,w θ ) and R T (0 ,w θ ) have ( β, θ )-type symmetric coef-ficients. In order to estimate them, one needs to estimate the coefficients of functions( H w θ ) ( t − ,S (0) wθ ) , P t − n ′ =1 ( g ( − ,w θ ) t − n ′ ) ( n ′ ,S (0) wθ ) and P t − n ′ =1 ( R T ( − ,w θ ) t − n ′ ) ( n ′ ,S (0) wθ ) . By Remark 5.4, when θ = 0, for any | l + k | = t ≥
4, any 1 ≤ n ′ ≤ t − i ∈ M ( g ( − ,w t − n ′ ) ( n ′ ,S (0) w , it holds X ( l ,k ,i ) ∈A (cid:0) ( g ( − ,w t − n ′ )( n ′ ,S (0) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( g ( − ,w ) t − n ′ ) ( n ′ ,S (0) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ C t − h i i β ( 144 N α +2 c β γ ) n ′ n ′ ! n ′ − Y n =0 ( t − n ′ + n + 1) (8.11)and when θ = 1, it holds (cid:12)(cid:12)(cid:0) (˜ g ( − ,w ) t − n ′ ) ( n ′ ,S (0) w ) (cid:1) it,lk (cid:12)(cid:12) ≤ C t − h i i β ( 72 cN α +1 β γ ) n ′ n ′ ! n ′ − Y n =0 ( t − n ′ + n + 1) . (8.12)By equation (8.4), when θ = 0 it follows X ( l ,k ,i ) ∈A (cid:0) ( Zw
03 )( t − ,S (0) w (cid:1) it,lk | (cid:0) ( Z w ) ( t − ,S (0) w ) (cid:1) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i}≤ h i i β (cid:0) N α +2 c β γ (cid:1) t − C t − ( t − t − Y n =0 (4 + n ); (8.13)when θ = 1 it follows | (cid:0) ( ˜ Z w ) ( t − ,S (0) w ) (cid:1) it,lk | ≤ C t − h i i β (cid:0) c N α β γ (cid:1) t − t − t − Y n =0 (4 + n ) . (8.14)53hen N satisfies (5.28), using (8.12)-(8.14), in the case θ = 0 it holds that X ( l ,k ,i ) ∈A ( g (0 ,w it,lk (cid:12)(cid:12) ( g (0 ,w ) ) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ X ( l ,k ,i ) ∈A ( g ( − ,w it,lk | ( g ( − ,w ) ) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i} + X ( l ,k ,i ) ∈A (cid:0) ( Zw
03 )( t − ,S (0) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( Z w ) ( t − ,S (0) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i} + X ( l ,k ,i ) ∈A (cid:0) ( g ( − ,w t − n ′ ) i ( n ′ ,S (0) w (cid:1) it,lk t − X n ′ =1 (cid:12)(cid:12)(cid:0) ( g ( − ,w ) t − n ′ ) ( n ′ ,S (0) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ h i i β ( C c β N α γ ) t − =: 1 h i i β (cid:0) C (0 , (cid:1) t − . When θ = 1 it follows (cid:12)(cid:12) ( e g (0 ,w ) ) it,lk (cid:12)(cid:12) ≤ | ( e g ( − ,w ) ) it,lk | + | (cid:0) ( g Z w ) ( t − ,S (0) w ) (cid:1) it,lk | + t − X n ′ =1 (cid:12)(cid:12)(cid:0) ( e g ( − ,w ) t − n ′ ) ( n ′ ,S (0) w (cid:1) it,lk (cid:12)(cid:12) ≤ (cid:0) C (1 , (cid:1) t − h i i β . Similarly, R T (0 ,w θ ) ( u, ¯ u ) has still ( β, θ )-type symmetric coefficients semi-bounded by C ( θ, r < r ∗ . By these assumptions, there exist areal number ˜ R < R ∗ and a Lie-transformation which changes Hamiltonian (8.1) into thefollowing form H ( r,w θ ) = H w θ + Z ( r,w θ ) + R N ( r,w θ ) + g ( r,w θ ) + R T ( r,w θ ) , which is defined in B p ( R r ) ( R < ˜ R < R ∗ ), where R r := r ∗ − r r ∗ R . One should construct abounded Lie-transformation T w θ r to eliminate the non-normalized monomials of Γ N ≤ g ( r,w θ ) r +4 .Because g ( r,w θ ) r +4 have ( β, θ )-type symmetric coefficients, by Remark 4.2, the coefficients ofΓ N ≤ g ( r,w θ ) r +4 and Γ N> g ( r,w θ ) r +4 are ( β, θ )-type symmetric coefficients semi-bounded by C ( θ, r ).Make use of the 1-time flow of the following equation, for any j ∈ Z ∗ ˙ u j = i sgn θ ( j ) ∂ ¯ u j S ( r ) w θ ( u, ¯ u ) , ˙¯ u j = − i sgn θ ( j ) ∂ u j S ( r ) w θ ( u, ¯ u ) , to define a Lie-transformation T w θ r , under which the new Hamiltonian has the followingform formally, H ( r +1 ,w θ ) := H ( r,w θ ) ◦ T w θ r = H w θ + Z ( r,w θ ) + R N ( r,w θ ) + { H w θ , S ( r ) w θ } w θ + g ( r,w θ ) r +4 (8.15)+ r ∗ +3 X t = r +5 g ( r,w θ ) t + X ν ≥ ( H w θ ) ( ν,S ( r ) wθ ) + X ν ≥ ( Z ( r,w θ ) + g ( r,w θ ) + R N ( r,w θ ) ) ( ν,S ( r ) wθ ) + X ν ≥ ( R T ( r,w θ ) ) ( ν,S ( r ) wθ ) . (8.16) S ( r ) w θ can be obtained by solving the following homologicalequation { H w θ , S ( r ) w θ } w θ + Γ N ≤ g ( r,w θ ) r +4 = Z r +4 . (8.17)From Lemma 5.2, Z r +4 is ( θ, γ, α, N )-normal form of Γ N ≤ g ( r,w θ ) r +4 and(8 .
15) = Z r +4 + Γ N> g ( r,w θ ) r +4 . The Hamiltonian vector field X w θ S ( r ) wθ satisfiessup ( u, ¯ u ) ∈ B p ( R r ) k X w θ S ( r ) wθ ( u, ¯ u ) k p ≤ C ( θ, r )) r +2 ( r + 4) p +1 c r +3 N α γ R r +3 . (8.18)Using (8.18) and bootstrap method, suppose thatΦ tS ( r ) wθ : B p ( R r +1 ) → B p ( R r ) , (8.19)for any t ∈ [0 , ( u, ¯ u ) ∈ B p ( R r ) kT w θ r ( u, ¯ u ) − ( u, ¯ u ) k p = sup ( u, ¯ u ) ∈ B p ( R r ) k Φ S ( r ) wθ ( u, ¯ u ) − ( u, ¯ u ) k p = sup ( u, ¯ u ) ∈ B p ( R r ) (cid:13)(cid:13) Z t =0 X w θ S ( r ) wθ ◦ Φ τS ( r ) wθ ( u, ¯ u )( τ ) dτ (cid:13)(cid:13) p ≤ C ( θ, r )) r +2 ( r + 4) p +1 c r +3 N α γ R r +3 . (8.20)By (5.28) and (8.20), the transformation T r satisfiessup ( u, ¯ u ) ∈ B p ( R ) kT w θ r ( u, ¯ u ) − ( u, ¯ u ) k p ≤ δ/ R r − R r +1 ) / , which verifies (8.19). Denote T wθ ( r +1) := T wθ ( r ) ◦ T w θ r . By (8.20) and (8.19), noting that R < ˜ R < R ∗ <
1, it holdssup ( u, ¯ u ) ∈ B p (cid:0) ( R r + R r +1 ) / (cid:1) kT wθ ( r +1) ( u, ¯ u ) − ( u, ¯ u ) k p ≤ sup ( u, ¯ u ) ∈ B p (cid:0) Rr + Rr +12 (cid:1) (cid:0) kT wθ ( r ) ◦ T w θ r ( u, ¯ u ) − T w θ r ( u, ¯ u ) k p + kT w θ r ( u, ¯ u ) − ( u, ¯ u ) k p (cid:1) ≤ sup ( u, ¯ u ) ∈ B p ( R r ) kT ( r ) ( u, ¯ u ) − ( u, ¯ u ) k p + 16 (cid:0) C ( θ, r ) (cid:1) r +2 ( r + 4) p +1 c r +3 N α γ R r +3 . (8.21)Because C ( θ, t ) ≤ C ( θ, t + 1) for any positive integer t , from (8.21), one has that sup ( u, ¯ u ) ∈ B p (cid:0) ( R r + R r +1 ) / (cid:1) kT wθ ( r +1) ( u, ¯ u ) − ( u, ¯ u ) k p ≤ r +3 X t =3 N α γ (cid:0) C ( θ, t − (cid:1) t − t p +1 c t − R t − + 16 N α γ (cid:0) C ( θ, r + 1) (cid:1) r +2 ( r + 4) p +1 c r +3 R r +3 r ≤ N α γ r +4 X t =3 (cid:0) C ( θ, t − (cid:1) t − t p +1 c t − R t − ≤ R − r ∗ +1 . Z ( r +1 ,w θ ) := Z ( r,w θ ) + Z r +4 , R N ( r +1 ,w θ ) := R N ( r,w θ ) + Γ N> g ( r,w θ ) r +4 . (8.22)By Remark 5.4 and Lemma 4.1, Z ( r +1 ,w θ ) and R N ( r +1 ,w θ ) have ( β, θ )-type symmetriccoefficients. Denote g ( r +1 ,w θ ) = r ∗ +3 X t = r +5 g ( r +1 ,w θ ) t , R T ( r +1 ,w θ ) = X t>r ∗ +3 R T ( r +1 ,w θ ) t , where g ( r +1 ,w θ ) t := g ( r,w θ ) t + ( Z r +4 − Γ N ≤ g ( r,w θ ) r +4 ) ( t − r − r +2 ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =0 ( R N ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) + P [ t − ( r +4) r +2 ] n ′ =0 ( g ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =2 ( Z ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) , when ( r + 2) | ( t − g ( r,w θ ) t + P [ t − r +2 ] n ′ =0 ( R N ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) + P [ t − ( r +4) r +2 ] n ′ =0 ( g ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =2 ( Z ( r,w θ ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) wθ ) , when ( r + 2) ∤ ( t − R T ( r +1 ,w θ ) t := ( Z r +4 − Γ N ≤ g ( r,w θ ) r +4 ) ( t − r − r +2 ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =1 (cid:0) R N ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − ( r +4) r +2 ] n ′ =1 (cid:0) g ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =1 (cid:0) Z ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − r ∗− r +2 ] n ′ =0 (cid:0) R T ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) , when ( r + 2) | ( t − P [ t − r +2 ] n ′ =1 (cid:0) R N ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − ( r +4) r +2 ] n ′ =1 (cid:0) g ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − r +2 ] n ′ =1 (cid:0) Z ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) + P [ t − r ∗− r +2 ] n ′ =0 (cid:0) R T ( r,w θ ) t − n ′ ( r +2) (cid:1) ( n ′ ,S ( r ) wθ ) , when ( r + 2) ∤ ( t − a ] denotes the integer part of the real number a . Using Lemma 4.1 and Remark5.4, from the fact that g ( r,w θ ) , R T ( r,w θ ) , R N ( r,w θ ) and Z ( r,w θ ) have ( β, θ )-type symmetriccoefficients semi-bounded by C ( θ, r ), then g ( r +1 ,w θ ) and R T ( r +1 ,w θ ) also have ( β, θ )-typesymmetric coefficients.When θ = 0, using Remark 5.4, the followings estimates hold: for any | l + k | = t with M ( l, k ) = i ∈ M ( g ( r,w r +4 ) ( t − r − r +2 ,S ( r ) w , X ( l ,k ,i ) ∈A (cid:0) ( Zr +4 − Γ N ≤ g ( r,w r +4 )( t − r − r +2 ,S ( r ) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( Z r +4 − Γ N ≤ g ( r,w ) r +4 ) ( t − r − r +2 ,S ( r ) w ) (cid:1) i ( l ,k ,i ) r,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ (cid:0) C ( θ, r ) (cid:1) r +2 h i i β (cid:0) β +2 ( r + 4) (cid:0) C ( θ, r ) (cid:1) r +2 c N α +1 γ (cid:1) t − r − r +2 (2 N ) ( t − r − r +2) ( t − r − r +2 )! t − r − r +2 Y n =1 (cid:0) t + 1 − n ( r + 2) (cid:1) ;(8.23) | l + k | = t with M ( l, k ) = i ∈ M ( g ( r,w t − n ′ ( r +2) ) ( n ′ ,S ( r ) w , X ( l ,k ,i ) ∈A (cid:0) ( g ( r,w t − n ′ ( r +2))( n ′ ,S ( r ) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( g ( r,w ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ (cid:0) C ( θ, r ) (cid:1) t − n ′ ( r +2) − h i i β (cid:0) β +2 ( r + 4) (cid:0) C ( θ, r ) (cid:1) r +2 c N α +1 γ (cid:1) n ′ (2 N ) ( r +2) n ′ n ′ ! n ′ Y n =1 (cid:0) t +1 − n ( r + 2) (cid:1) ;(8.24)for any | l + k | = t with M ( l, k ) = i ∈ M ( Z ( r,w t − n ′ ( r +2) ) ( n ′ ,S ( r ) w , X ( l ,k ,i ) ∈A (cid:0) ( Z ( r,w t − n ′ ( r +2))( n ′ ,S ( r ) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( Z ( r,w ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ (cid:0) C ( θ, r ) (cid:1) t − n ′ ( r +2) − h i i β (cid:0) β +2 ( r + 4) (cid:0) C ( θ, r ) (cid:1) r +2 c N α +1 γ (cid:1) n ′ (2 N ) ( r +2) n ′ n ′ ! n ′ Y n =1 (cid:0) t +1 − n ( r + 2) (cid:1) , (8.25) for any | l + k | = t with M ( l, k ) = i ∈ M ( R N ( r,w t − n ′ ( r +2) ) ( n ′ ,S ( r ) w X ( l ,k ,i ) ∈A (cid:0) ( R N ( r,w t − n ′ ( r +2))( n ′ ,S ( r ) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( R N ( r,w ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ (cid:0) C ( θ, r ) (cid:1) t − n ′ ( r +2) − h i i β (cid:0) β +2 ( r + 4) (cid:0) C ( θ, r ) (cid:1) r +2 c N α +1 γ (cid:1) n ′ (2 N ) ( r +2) n ′ n ′ ! n ′ Y n =1 (cid:0) t + 1 − n ( r + 2) (cid:1) . (8.26) and for any | l + k | = t with M ( l, k ) = i ∈ M ( R T ( r,w t − n ′ ( r +2) ) ( n ′ ,S ( r ) w X ( l ,k ,i ) ∈A (cid:0) ( R T ( r,w t − n ′ ( r +2))( n ′ ,S ( r ) w (cid:1) it,lk (cid:12)(cid:12)(cid:0) ( R T ( r,w ) t − n ′ ( r +2) ) ( n ′ ,S ( r ) w ) (cid:1) i ( l ,k ,i ) t,lk (cid:12)(cid:12) · max {h i i , h i − i i}≤ (cid:0) C ( θ, r ) (cid:1) t − n ′ ( r +2) − h i i β (cid:0) β +2 ( r + 4) (cid:0) C ( θ, r ) (cid:1) r +2 c N α +1 γ (cid:1) n ′ (2 N ) ( r +2) n ′ n ′ ! n ′ Y n =1 (cid:0) t + 1 − n ( r + 2) (cid:1) . (8.27) By (8.23)-(8.27) and assumption (5.28), for any r + 5 ≤ t ≤ r ∗ + 3, | l + k | = t and i ∈ M g ( r +1 ,wθ ) t , the following estimate holds X ( l ,k ,i ) ∈A ( g ( r +1 ,wθ )) it,lk | ( g ( r +1 ,w θ ) ) i ( l ,k ,i ) t,lk | · max {h i i , h i − i i} ≤ (cid:0) C ( θ, r + 1) (cid:1) t − h i i β , (8.28) which means that g ( r +1 ,w ) ( u, ¯ u ) has ( β, C ( θ, r + 1) > R T ( r +1 ,w θ ) ( u, ¯ u ) and R N ( r +1 ,w ) ( u, ¯ u ) are also of ( β, θ )-type symmetric coef-ficients semi-bounded by C ( θ, r + 1) >
0. 57 cknowledgement
This paper is supported in part by Science and Technology Commission of ShanghaiMunicipality (No. 18dz2271000).
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