Quasi invariant modified Sobolev norms for semi linear reversible PDEs
aa r X i v : . [ m a t h - ph ] A ug Quasi invariant modified Sobolev norms for semilinear reversible PDEs.
Erwan Faou and Benoˆıt Gr´ebertNovember 19, 2018
Abstract
We consider a general class of infinite dimensional reversible differential sys-tems. Assuming a non resonance condition on the linear frequencies, we constructfor such systems almost invariant pseudo norms that are closed to Sobolev-likenorms. This allows us to prove that if the Sobolev norm of index s of the initialdata z is sufficiently small (of order ǫ ) then the Sobolev norm of the solution isbounded by 2 ǫ during very long time (of order ǫ − r with r arbitrary). It turns outthat this theorem applies to a large class of reversible semi linear PDEs includingthe non linear Schr¨odinger equation on the d-dimensional torus. We also applyour method to a system of coupled NLS equations which is reversible but notHamiltonian.We also notice that for the same class of reversible systems we can provea Birkhoff normal form theorem that in turn implies the same bounds on theSobolev norms. Nevertheless the technics that we use to prove the existence ofquasi invariant pseudo norms is much more simple and direct. Contents .2.1 Nonlinear Schr¨odinger equation on the torus . . . . . . . 83.2.2 Coupled NLS on the torus . . . . . . . . . . . . . . . . . . 9 The control of the high index Sobolev norms of the solution of nonlinear partialdifferential equations during long time is a difficult and interesting problem, inparticular on compact manifolds where there is no dispersion effects and thus notime decay of the solutions of the linear part. Recently a series of works gave asolution to this problem by using the Birkhoff normal form theory applied to someHamiltonian nonlinear PDEs including in particular the nonlinear Schr¨odingerequation on a d-dimensional torus and the nonlinear wave equation on the circle(see [7],[2],[6],[9] and [4]), the Klein Gordon equation on Zoll manifolds (see[5]) or the nonlinear quantum oscillator on R d (see [10]). The method consistsin obtaining a normal form for the corresponding Hamiltonian function H inconvenient Sobolev type phase spaces P s in such a way that in the new variables H decomposes into the sum of a Hamiltonian N (the normal form), whose flowpreserves the Sobolev norms, and a remainder Hamiltonian R whose vector field, X R satisfies (here r is an arbitrary integer and ||·|| s denotes the standart Sobolevnorm) k X R ( z ) k s ≤ C k z k r +1 s , for z ∈ P s small enough.Then a standard bootstrap procedure shows that if the initial data, z , is suffi-ciently small, say k z k s ≤ ǫ = ǫ ( r, s ), then the solution remains under control, k z ( t ) k s ≤ ǫ during very long time | t | ≤ ǫ r . The aim of this paper is to obtain the same dynamical result for reversible PDEsthat are not necessarily Hamiltonian by a more direct and simple method. Actu-ally we generalize to the infinite dimension the classical algorithm of constructionof approximate integrals of motion (see for instance [3] section 4 and referencesquoted therein). In [3] this generalization is done for reversible Hamiltoniansystems, constructing almost invariant actions but not almost invariant pseudonorms. In this short article we want to stress out that the construction actuallyworks for reversible systems that are not Hamiltonian and that this construc-tion leads directly to bounds on Sobolev norms for a large class of semi linearreversible PDEs. We also mention that our approach is totally self contained. he reversibility property allows to solve exactly the so called homologicalequation. Here exactly means that we do not solve it modulo terms in normalform (i.e. corresponding to resonant monomials which are actually absent in thereversible context, see Lemma 4.4).At the same time the simplification in the resolution of the homological equationhas a cost: we can consider only non resonant cases (see Definition 2.3), whereasthe Birkhoff normal form technics (see [6] [5] [10]) allows to deal with resonantcases. Notice that a similar approach, mimicking for instance [9], would providea Birkhoff normal form result for infinite dimensional reversible system and thuswould allow to consider resonant reversible system. We denote N = Z d or N d (depending on the concrete application) for some d ≥ a = ( a , . . . , a d ) ∈ N , we set | a | = max (cid:0) , a + · · · + a d (cid:1) . We define the set Z = N × {± } . For j = ( a, δ ) ∈ Z , we define | j | = | a | and wedenote by j the index ( a, − δ ). By a slight abuse of notation we will also denoteby N the set { j = ( a, +1) , a ∈ N } . For z = ( z j ) j ∈Z ∈ C Z we define the involution ρ via the formula ρ ( z ) j = z ¯ j We will say that z is real if ρ ( z ) = ¯ z where ¯ z = (¯ z j ) j ∈Z and where for any ζ ∈ C ,¯ ζ denotes the complex conjugate of ζ .For a given real number s ≥
0, we consider the Hilbert space P s = ℓ s ( Z , C )made of elements z ∈ C Z such that k z k s := X j ∈Z | j | s | z j | < ∞ . Let U be a an open set of P s , let ω = ( ω j ) j ∈Z ∈ R Z and let F be a continuousvector field from U to P F ( z ) = ( F j ( z )) j ∈Z . We consider the following differential system on P i ˙ z j = ω j z j + F j ( z ) , j ∈ Z . (2.1) eq1 .2 Hypothesis We first describe the hypothesis needed on the vector field F (H1) Regularity condition : for all s > d/ P s → P s z = ( F j ( z )) j ∈Z is continuous and has a zero of order at least 2 at the origin, in such a waythat k F ( z ) k s ≤ C k z k s for z sufficiently small (here C is a constant depending on s ).(H2) Reality condition : F ¯ j ( z ) = − ¯ F j ( z ) for all real z, i.e. when ρ ( z ) = ¯ z in such a way that equation (2.1) ¯ j is the complex conjugate of equationequation (2.1) j (provided the vector ω j satisfies a similar condition, see(2.6) below) and that the flow Φ t associated to the differential system (2.1)preserves the reality of the initial datum: Φ t ( z ) is real for all t when z isreal.(H3) Reversibility condition : for all z , ρ ( F j ( z )) = − F j ( ρ ( z )) , ∀ j ∈ Z and for all real z in such a way that the flow Φ t associated with the differential system (2.1)satisfies ρ (Φ t ( z )) = Φ − t ( ρ ( z ))for all real z .We now translate these hypothesis on the coefficients of the Taylor polyno-mials of F . We first need some more notations:Let ℓ ≥ j = ( j , . . . , j r ) ∈ Z r , we define µ ( j ) asthe third largest integer between | j | , . . . , | j r | . Then we set S ( j ) = | j i r | − | j i r − | where | j i r | and | j i r − | denote the largest and the second largest integer between | j | , . . . , | j r | .In the following, for ℓ = ( ℓ , . . . , ℓ m ), we use the notation z ℓ = z ℓ . . . z ℓ m . efinition 2.1 Let k ≥ , M > and ν ∈ [0 , + ∞ ) , and let F j ( z ) = k X m =2 X ℓ ∈Z m a j ℓ z ℓ , j ∈ Z , (2.2) TM where a jℓ are complex numbers.We say that F ∈ T M,νk if there exist a constant C depending on M such that ∀ m = 2 , . . . , k, ∀ ℓ ∈ Z m , ∀ j ∈ Z , | a j ℓ | ≤ C µ ( j, ℓ ) M + ν µ ( j, ℓ ) + S ( j, ℓ ) M . (2.3) Ereg
Notice that this definition is the analog of the polynomial spaces used in [9, 4].We learn from [9] that if F ∈ T M,νk then F satisfies the regularity hypothesis (H1)for s ≥ ν + d/ k z k s ≤
1. The best constant in the inequality (2.3) definesa norm | Q | T M,νk for which T M,νk is a Banach space. We set T ∞ ,νk = \ M ∈ N T M,νk which is a Frechet space.One easily verifies that a polynomial vector field F of the form (2.2) satisfiesthe reality condition (H2) if and only if ∀ m = 2 , . . . , k, ∀ ℓ ∈ Z m , ∀ j ∈ Z , a ¯ j ¯ ℓ = − ¯ a j ℓ (2.4) rever and that F satisfies the reversibility condition (H3) if and only if ∀ m = 2 , . . . , k, ∀ ℓ ∈ Z m , ∀ j ∈ Z , a ¯ j ¯ ℓ = − a j ℓ . (2.5) real Note that (H2) and (H3) imply that a j ℓ ∈ R . Definition 2.2
A vector field F is in the class T if • There exists s ≥ such that for any s ≥ s , F ∈ C ( U , P s ) for someneighborhood U of the origin in P s . • F exhibits a zero of order at least 2 at the origin. • For all k ≥ , there exists ν ≥ such that the Taylor expansion of degree k of F around the origin belongs to T ∞ ,νk . • The coefficients of the Taylor expansion of F satisfy (2.4) and (2.5) . We now describe the hypothesis on the frequencies vector.First we assume the symmetry ω ¯ j = − ω j , j ∈ Z (2.6) omsym hich ensures the reversibility of the linear part of (2.1). We also assume anupper bound of the frequencies of the form ∀ a ∈ N , | ω a | ≤ C | a | m (2.7) Eboundomega for some constants
C > m > j = ( j , . . . , j r ) ∈ Z r , and denote by j i = ( a i , δ i ) ∈ N × {± } for i = 1 , . . . , r . We set Ω( j ) = δ ω a + · · · + δ r ω a r . Definition 2.3
A frequencies vector ω ∈ R Z is non resonant if for any integer r ≥ , there exists two constants γ > and α > such that for any j ∈ Z i with ≤ i ≤ r , one has | Ω( j ) | ≥ γµ ( j ) α (2.8) A.2 except if j = ¯ j . Note that the condition j = ¯ j is equivalent to the fact that z j only depends onthe actions, I ℓ = z ℓ z ¯ ℓ , ℓ ∈ N . Our setting is very close to the Hamiltonian case. Actually we can endow thephase space P s with the canonical symplectic structure i P j ∈N dz j ∧ dz ¯ j . Thenthe linear part of (2.1) corresponds to the Hamilton equations associated withthe harmonic oscillator H = X j ∈N ω j z j z ¯ j . The nonlinear part of (2.1) is also Hamiltonian if and only if there exists a regularfunction P such that F j = ∂P∂z ¯ j j ∈ N ,F ¯ j = − ∂P∂z j j ∈ N . (2.9) Ham
In this case the total Hamiltonian function reads H ( z ) = H ( z ) + P ( z ) , (2.10) Edecomp nd the system (2.1) can hence be written ˙ z j = − iω j z j − i ∂P∂z ¯ j ( z ) j ∈ N ˙ z ¯ j = iω j z ¯ j + i ∂P∂z j ( z ) j ∈ N . (2.11) Eham3
Writing down the Taylor polynomial of order k ≥ P as P ( z ) = k X m =3 X ℓ ∈Z m a ℓ z ℓ , the reversibility condition (H2) is equivalent to P ( ρ ( z )) = P ( z ), which is actuallytrue for H , while the reality condition equivalent to P ( z ) ∈ R for real z (i.e. ρ ( z ) = ¯ z ), which again is true for H . Theorem 3.1
For any r ≥ , there exists s ( r ) > and for each s > s thereexist ǫ s > , C s > and a continuous function N ( r ) s : B (0 , ǫ s ) → R + where B (0 , ǫ s ) denotes the ball of radius ǫ s centered at the origin in P s , such that(i) (cid:12)(cid:12) N ( r ) s ( z ) − k z k s (cid:12)(cid:12) ≤ C s k z k s for all z ∈ B (0 , ǫ s ) (ii) if t z ( t ) is a solution of the reversible system (2.1) then (cid:12)(cid:12)(cid:12)(cid:12) dd t N ( r ) s ( z ( t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C s k z ( t ) k r +1 s for all time t such that z ( t ) ∈ B (0 , ǫ s ) . The proof is postponed to section 4. The dynamical consequences are given inthe following
Corollary 3.2
For any r ≥ there exists s ( r ) > and for each s > s ( r ) there exist ǫ s > , C s > such that if z ∈ P s satisties k z k s = ǫ < ǫ s then thesolution z ( t ) of (2.1) with initial datum z is a function in C ([ − T ǫ , T ǫ ] , P s ) with T ǫ ≥ ǫ r . Furthermore k z ( t ) k s ≤ ǫ, ∀ t ∈ [ − T ǫ , T ǫ ] . roof. We follow the standard bootstrap argument. Let t z ( t ) be the localsolution to (2.1) with initial datum z . This solution is defined and of class C in an interval ( − T, T ) for some
T > T ≥ ǫ r . Take ǫ s given by Theorem 3.1 but corresponding to r + 2 instead of r . Let T be thesupremum of the times 0 < t < T such that k z ( t ′ ) k s ≤ ǫ for all t ′ ∈ [ − t, t ] . As2 ǫ < ǫ s we can apply assertion (ii) of Theorem 3.1 to get for t ∈ ( − T , T ), | N ( r ) s ( z ( t )) − N ( r ) s ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t dd t N ( r ) s ( z ( t ′ )) dt ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C s | t | (2 ǫ ) r +1 . Then using assertion (i) of the same theorem we deduce that for t ∈ ( − T , T ), k z ( t ) k s ≤ k z k s + C s (2 ǫ ) + C s | t | (2 ǫ ) r +3 . Therefore, reducing eventually ǫ s , we obtain that for t ∈ ( − T , T ) and | t | ≤ ǫ r k z ( t ) k s ≤ / ǫ. Hence by definition of T and continuity of t
7→ k z ( t ) k s we conclude that T ≥ T ≥ ǫ r . We first consider Hamiltonian non linear Schr¨odinger equations of the form i∂ t ψ = − ∆ ψ + V ⋆ ψ + ∂ g ( x, ψ, ¯ ψ ) , x ∈ T d (3.1) E1 where V ∈ C ∞ ( T d , C ) has real Fourier coefficients, and g ∈ C ∞ ( T d × U , C )where U is a neighborhood of the origin in C . We assume that for all z ∈ C ,we have g ( x, z, ¯ z ) = g ( x, ¯ z, z ), and that g ( x, z, ¯ z ) = O ( | zi | ). Notice that forsuch a semi linear Schr¨odinger equation (i.e. with a nonlinear term that dependsonly on x and on ψ ( x ) but not on the derivative of ψ ), the reality condition, g ( x, z, ¯ z ) ∈ R , yields naturally to Hamiltonian equations (i.e. with a nonlinearterm that can be written ∂ g ( x, ψ, ¯ ψ )). In other words, the reversible setting ishere more restrictive than the Hamiltonian setting. The Hamiltonian functionalis given by H ( ψ, ¯ ψ ) = Z T d |∇ ψ | + ¯ ψ ( V ⋆ ψ ) + g ( x, ψ, ¯ ψ ) d x. Let φ a ( x ) = (cid:0) π (cid:1) d/ e ia · x , a ∈ Z d be the Fourier basis on L ( T d ). With thenotation N = Z d and φ j ( x ) = φ a ( ± x ) for j = ( a, ± ∈ Z we write ψ = X j ∈N z j φ j ( x ) and ¯ ψ = X j ∈N z ¯ j φ ¯ j ( x ) . urther we set F j = ∂P∂z ¯ j j ∈ N ,F ¯ j = − ∂P∂z j j ∈ N , where P ( z ) = Z T d g ( x, X j ∈N z j φ j ( x ) , X j ∈N z ¯ j φ ¯ j ( x ))d x. Then equation (3.1) can (formally) be written i ˙ z j = ω j z j + F j ( z ) , j ∈ Z (3.2)where the frequency vector ω j defined by ω a = | a | + ˆ V a for a ∈ N and therelation ω j = − ω ¯ j for all j ∈ Z , satisfies (2.7) with m = 2. Now the hypothesis g ( x, ψ, ¯ ψ ) = g ( x, ¯ ψ, ψ ) , ensures that P ( ρ ( z )) = P ( z ) and thus implies the reversibility condition (H2).The reality condition is also satisfied since g ( x, z, ¯ z ) is real. The fact that thenonlinearity F belongs to T can be verified using the regularity of g and theproperties of the basis functions φ a , see [9, 6]. In this situation, it can be shownthat the non resonance condition is fulfilled for a large set of potential V (see [6]or [9]). To generate a reversible PDE that is not Hamiltonian, we have to consider sys-tems of coupled PDEs. As example of a system of coupled partial differentialequations we consider a pair of NLS equations coupled via the nonlinear terms.This kind of system is used in nonlinear optics (see for instance [1, 11] and refer-ences quoted therein). From the mathematical point of view the interest of thisexample is that the reversible context is much more rich than the Hamiltonianone. We consider the system for ( ψ, φ ) given by i ˙ ψ = − ψ xx + V ⋆ ψ + ∂ ¯ ψ g ( x, ψ, ¯ ψ, φ, ¯ φ ) , (3.3) i ˙ φ = − φ xx + V ⋆ φ + ∂ ¯ φ g ( x, ψ, ¯ ψ, φ, ¯ φ ) . (3.4)Assume as in the previous section that V , V ∈ C ∞ ( T d , C ) have real Fouriercoefficients, and that g , g ∈ C ∞ ( T d × U , C ) where U is a neighborhood ofthe origin in C . We assume that g i ( x, z, ¯ z, ζ, ¯ ζ ) = g i ( x, ¯ z, z, ¯ ζ, ζ ), and that This hypothesis is for instance satisfied when g only depends on the modulus of | ψ | like in theGross-Pitaevskii equation. Notice that this condition is not necessary in the Hamiltonian case. i ( x, z, ¯ z, ζ, ¯ ζ ) = O ( | z | + | ζ | ) for i = 1 ,
2. Thus the system fulfills the threeconditions : reality, reversibility and regularity. Nevertheless, in general thissystem is Hamiltonian only if g = g . For instance take g ( ψ, ¯ ψ, φ, ¯ φ ) = | ψ | | φ | and g ( ψ, ¯ ψ, φ, ¯ φ ) = | ψ | | φ | to obtain a reversible but non Hamiltonian system.The frequencies are given by ω a := | a | + ˆ V ( a ) , ω a := | a | + ˆ V ( a ) , a ∈ Z d . We can adapt results of [9] to prove that the non resonances condition is fulfilledfor a large set of potential ( V , V ) (see also [6] section 3.4). We adapt the classical algorithm of construction of the approximate integrals ofmotion (see for instance [3, 8]). We Taylor expand the vector field F as F = r X k =2 F ( k ) + ˜ F ( r +1) (4.1) F where F ( k ) = ( F ( k ) j ) j ∈Z ∈ T ∞ ,νk . (4.2) tildeF Here, each F ( k ) j is a homogeneous polynomial of degree k and ˜ F ( r +1) is a remain-der term satisfying k ˜ F ( r +1) ( z ) k s ≤ C s k z k r +1 s . for some constant C s depending on s . We search the almost invariant pseudonorm N ( r ) s under the same form: N ( r ) s ( z ) = r X k =2 N s,k ( z ) (4.3) N where for all k ≥ N s,k ( z ) is an even (i.e. satisfying N s,k ( ρ ( z )) = N s,k ( z ) for all z ) continuous homogeneous polynomial of degree k with in particular N s, ( z ) = k z k s . More precisely we will search the polynomials N s,k ( z ) in the class Γ γk that wenow define: Definition 4.1
Let γ > and k ∈ N . A formal homogeneous polynomial ofdegree k on P s Q ( z ) = X j ∈Z k b j z j s in the class Γ γk if there exists a constant C > such that | b j | ≤ Cµ ( j ) γ β ( j ) s (1 + S ( j )) for all j ∈ Z k where µ ( j ) and S ( j ) are defined in section 2 and β ( j ) is theproduct of the two largest index between | j | , . . . , | j k | .Furthermore we say that Q is even (resp. odd) when b j = b ¯ j , ( resp. b j = − b ¯ j ) j ∈ Z k . Notice that z
7→ k z k s is in Γ (in that case we have always j = j and byconvention µ ( j ) = 1). Remark also that even polynomials are real valued forreal z , provided the coefficients b j are all real. The proof of the following lemmais postponed to the Appendix Lemma 4.2 (i) If ≤ γ < s − / , then for all k ≥ the space Γ γk is includedin the space of continuous polynomials from P s to R and in particular if Q ∈ Γ γk ,there exists a constant C > (depending on s and k ) such that | Q ( z ) | ≤ C k z k ks . (ii) Let Q ∈ Γ γk with ≤ γ < s − / and k ≥ , then the map z
7→ ∇ Q iscontinuous from P s into P − s and in particular, if Q ∈ Γ γk , there exists a constant C > such that k∇ Q ( z ) k − s ≤ C k z k k − s . The second technical lemma whose proof is again postponed to the appendixlinkgs the two classes of polynomials defined above. For a vector field F andfunctional G defined on P s let us define (formally) the Lie derivative of G along F by L F G := X ℓ ∈Z F ℓ ∂G∂z ℓ . Then the following result holds true:
Lemma 4.3
Let γ, ν ≥ be given real numbers and m, n ≥ two integers.For a given s , let G ∈ Γ γn and F ∈ T ∞ ,νm , and assume s ≥ max( γ, ν + 4) then L F G ∈ Γ γ + ν +4 n + m − . Moreover if F satisfies the Hypothesis (H2) and (H3), then if G is even L F G is odd. For Q ∈ Γ γk for some γ > k ≥ ω -derivative ∂ ω by theformula ∂ ω Q ( z ) = X ℓ ∈Z ω ℓ z ℓ ∂Q∂z ℓ . he key to prove Theorem 3.1 relies on the construction of iterative solutions ofhomological equations. The next Lemma shows how it is possible to solve them. Lemma 4.4
Let k be an integer, let G ∈ Γ γk be an odd homogeneous polynomialof degree k and let ω be a non resonant vector of frequencies satisfying (2.8) .The homological equation ∂ ω N = G has a unique solution N ∈ Γ γ + αk which is furthermore an even polynomial. Proof.
Write G ( z ) = X j ∈Z k a j z j and search N ( z ) = X j ∈Z k b j z j satisfying the homological equation ∂ ω N = G . With these notations, the lastequation is equivalent to Ω( j ) b j = a j , j ∈ Z k . (4.4) triv Notice that, since G is odd, a ℓ = − a ¯ ℓ and thus a ℓ = 0 when ℓ = ¯ ℓ . But, as ω isnon resonant, Ω( j ) = 0 ⇐⇒ j = ¯ j . Therefore equation (4.4) is always solvable by setting b j = Ω( j ) − a j , j ∈ Z k . Then, the fact that N belongs to Γ γ + αk is a consequence of (2.8). Furthermore,since Ω(¯ j ) = − Ω( j ) and G is odd, we deduce that N is even. Remark 4.5
In the Hamiltonian case, this miracle does not occur: the homo-logical equation cannot be solve exactly and we have to add so called normalterms which correspond to the resonant monomials z j with j = ¯ j . (see for in-stance [9]) Remark 4.6
In the previous Lemma, if the coefficients of G are real, then asthe frequencies ω j are real, the coefficients of N remain real. ow we have the tools to prove Theorem 3.1. Proof of Theorem 3.1.
We have dd t N ( r ) s = − i X j ∈Z ( ω j z j + F j ( z )) ∂N ( r ) s ∂z j Inserting the Taylor expansions (4.1) and (4.3) and equating the terms of thesame degree we get the recursive homological equations, k = 2 , . . . , r − ∂ ω N s,k +1 = G k +1 where G k is determined by G k +1 = − k X m =2 L F ( k +2 − m ) N s,m , k = 2 , . . . , r − . (4.5) eq:G Now by Lemma 4.3 and Lemma 4.4 these formal equations can be solved verifyingat each step that G k ∈ Γ ( k − αk is odd and that N s,k ∈ Γ ( k − αk is even. Moreoveras the coefficients of the vector fields F ( m ) , m = 2 , . . . , r are real, we see that thecoefficients of the polynomials N s,k remain real at each step.We then verify estimate (i) by using Lemma 4.2. To verify (ii) we remark thatby construction dd t N ( r ) s = Q r +1 − i X j ∈Z ˜ F j ∂N ( r ) s ∂z j where Q r +1 is a polynomial of degree r + 1 in Γ ( r − αr +1 . Thus using again Lemma4.3 we have k Q r +1 ( z ) k s ≤ C k z k r +1 s . On the other hand (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈Z ˜ F j ( z ) ∂N ( r ) s ∂z j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˜ F ( z ) k s k∇ N ( r ) s ( z ) k − s and we conclude using (4.2) and Lemma 4.2 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈Z ˜ F j ( z ) ∂N ( r ) s ∂z j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k z k r +1 s . Proof of Lemma 4.2. i) Let Q ( z ) = P j ∈Z k b j z j be in Γ γk with γ < s − /
2. We have | Q ( z ) | ≤ C X j ∈Z k µ ( j ) γ β ( j ) s (1 + S ( j )) | z j |≤ C X j ∈Z k µ ( j ) γ β ( j ) s (1 + S ( j )) Π km =1 | j m | s Π km =1 | j m | s | z j m | , where we used the notation j = ( j , . . . , j k ) for a generic multi-index in Z k .By symmetry of the right hand side of the last inequality, we can reduce the sumto the indices j that are ordered in the sense that | j | ≥ | j | ≥ . . . ≥ | j k | so that β ( j ) = | j || j | .First remark that by Cauchy-Schwarz inequality, for any s > /
2, one has X ℓ | ℓ | s − s | z ℓ | ≤ C k z k s (5.1) peou Then we obtain using Cauchy-Schwarz inequality for each index j , . . . , j k | Q ( z ) | ≤ C X j ∈Z k S ( j )) | j | s − γ Π km =4 | j m | s Π km =1 | j m | s | z j m |≤ C k z k ( k − s X | j |≥| j |≥| j | | j | − | j | ) | j | s − γ Π m =1 | j m | s | z j m | . Now as s − γ > /
2, and using again the Cauchy-Schwarz inequality and (5.1),we have that X j | j | s | z j || j | s − γ ≤ C k z k s . Eventually, as j (1 + | j | ) − is a ℓ sequence and as the convolution product ofa ℓ sequence with a ℓ sequence gives rise to a ℓ sequence, we obtain X | j |≥| j | | j | − | j | ) | j | s | z j || j | s | z j | ≤ C k z k s which concludes the proof of assertion (i). ii) Let Q ( z ) = P j ∈Z k +1 b j z j be in Γ γk with γ < s − /
2. We have k∇ Q ( z ) k − s = X ℓ ∈Z | ℓ | − s (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂z ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C X ℓ ∈Z | ℓ | − s X j ∈Z k | b ℓ, j || z j | . . . | z j k | ≤ C ( k !) X ℓ ∈Z | ℓ | − s X j ∈Z k> | b ℓ, j || z j | . . . | z j k | where Z k> denotes the set of ordered k -uples ( j , . . . , j k ) such that | j | ≥ . . . ≥ | j k | .Then we use again (5.1) to obtain k∇ Q ( z ) k − s ≤ C k z k k − s × X ℓ ∈Z | ℓ | − s X | j |≥| j |≥| j | µ ( ℓ, j , j , j ) γ β ( ℓ, j , j , j ) s (1 + S ( ℓ, j , j , j )) | z j || z j || z j | . We have to decompose the last sum depending on whether | ℓ | ≤ | j | or not. First case | ℓ | ≤ | j | In that case we can write X ℓ ∈Z | ℓ | − s X | j |≥| j |≥| j | , | ℓ | µ ( ℓ, j , j , j ) γ β ( ℓ, j , j , j ) s (1 + S ( ℓ, j , j , j )) | z j || z j || z j | . (5.2) pouetpouet For a fixed ℓ , the sum in j can be bounded by X | j |≥ ℓ | j | γ | z | + X | j |≤ ℓ ℓ γ | z | ≤ Cℓ γ k z k s using (5.1), provided s − γ > /
2. Hence the expression (5.2) is bounded by C k z k s X ℓ ∈Z | ℓ | − s − γ ) X | j |≥| j | | j | s | z j || j | s | z j | (1 + | j | − | j | ) . As the sequence b = ( | j | s | z j | ) j ∈Z belongs to ℓ ( Z ) and the sequence a = ((1 + | j | ) − ) j ∈Z belongs to ℓ ( Z ), the convolution a ⋆ b belongs to ℓ ( Z ) and k a ⋆ b k ≤ C k z k s . Therefore | X j b j ( a ⋆ b ) j | ≤ C k z k s hich leads to X ℓ ∈Z | ℓ | − s X | j |≥| j |≥| j | , | ℓ | µ ( ℓ, j , j , j ) γ β ( ℓ, j , j , j ) s (1 + S ( ℓ, j , j , j )) | z j || z j || z j | ≤ C k z k s as expected. Second case | ℓ | ≥ | j | In this case we can write X ℓ ∈Z | ℓ | − s X | j | , | ℓ |≥| j |≥| j | µ ( ℓ, j , j , j ) γ β ( ℓ, j , j , j ) s (1 + S ( ℓ, j , j , j )) | z j || z j || z j | ≤ C k z k s X ℓ ∈Z | ℓ | − s X | j | | j | s ℓ s (1 + || j | − | ℓ || ) | z j | ≤ C k z k s X ℓ ∈Z | ( a ⋆ b ) ℓ | ≤ C k z k s where we used (5.1) and the notations introduced in the previous case. Proof of Lemma 4.3.
Let G ( z ) = P j ∈Z n b j z j be in Γ γn and F = ( F j ) j ∈Z a homogeneous vector fieldin T ∞ ,νm with F j ( z ) = X ℓ ∈Z m a j ℓ z j ℓ , j ∈ Z . One has L F G ( z ) = X k F k ( z ) ∂G∂z k ( z )= X k ∈Z X ℓ ∈Z m X j ∈Z n a k ℓ z ℓ n − X i =1 b j ...j i kj i +1 ...j n z j . So in view of the symmetry in the estimates of the coefficients a or b , one has toprove that, there exist an integer N and a constant C > ℓ ∈ Z m and j ∈ Z n , one has X k ∈Z µ ( k, ℓ ) N + ν ( µ ( k, ℓ ) + S ( k, ℓ )) N µ ( k, j ) γ β ( k, j ) s (1 + S ( k, j )) ≤ Cµ ( ℓ , j ) α β ( ℓ , j ) s (1 + S ( ℓ , j )) . (5.3) groumph To prove this relation, we will show that there exist an integer M and a constant C > k ∈ Z , ℓ ∈ Z m and j ∈ Z n , the ollowing relation holds: (cid:20) µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) (cid:21) M µ ( k, ℓ ) ν +4 µ ( k, j ) γ β ( k, j ) s ≤ Cµ ( ℓ , j ) α β ( ℓ , j ) s . (5.4) siduele Indeed, if this relation is satisfied, then taking N = M + 4, the relation (5.3)reduces to X k ∈Z µ ( k, ℓ ) + S ( k, ℓ )) (1 + S ( k, j )) ≤ C (1 + S ( ℓ , j )) . Now as µ ( k, ℓ ) ≥ ℓ ∈ Z m and j ∈ Z n , one has(1 + S ( k, ℓ ))(1 + S ( k, j )) ≥ (1 + S ( ℓ , j ))we conclude using the fact that X k ∈Z (1 + S ( k, ℓ )) − ≤ C where the constant is independent of ℓ ∈ Z m .The rest of the proof consists in showing (5.4).We assume without lost of generality that ℓ and j are ordered (i.e. | ℓ | ≥ | ℓ | ≥ . . . ≥ | ℓ m | and | j | ≥ | j | ≥ . . . ≥ | j n | and we consider three different cases: First case : µ ( k, ℓ ) ≤ µ ( ℓ , j ) and µ ( k, j ) ≤ µ ( ℓ , j ).In this case it remains to prove (choosing α = ν + γ + 4) that there exist M and C such that uniformly with respect to ℓ ∈ Z m and j ∈ Z n (cid:20) µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) (cid:21) M β ( k, j ) s ≤ Cβ ( ℓ , j ) s . (5.5) siduelemenos This is trivially true (with M = 0 and C = 1) if | j | ≥ | k | since then β ( k, j ) ≤ β ( ℓ , j ). Now, if | j | ≤ | k | , then β ( k, j ) = | k || j | and • either S ( k, ℓ ) ≤ | k | / | ℓ | ≥ | k | / β ( ℓ , j ) ≥| ℓ || j | ≥ β ( k, j ) and (5.5) is satisfied with M = 0 and C = 2 s . • or S ( k, ℓ ) ≥ | k | / µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) β ( k, j ) ≤ µ ( k, ℓ )1 + | k | / | k || j |≤ | ℓ || j | | k | | k | / ≤ β ( ℓ , j )and (5.5) is satisfied with M = s and C = 2 s . Second case : µ ( k, ℓ ) > µ ( ℓ , j ).In this case, µ ( k, ℓ ) = min( | ℓ | , | k | ) and µ ( ℓ , j ) ≥ min( | ℓ | , | j | ) and therefore in( | k | , | ℓ | ) ≥ | j | . This in turn implies µ ( k, ℓ ) ≤ | ℓ | , β ( k, j ) = | j || k | and µ ( k, j ) ≤ µ ( ℓ , j )and in the other hand µ ( ℓ , j ) ≥ | j | and β ( ℓ , j ) = | ℓ || ℓ | . Thus µ ( ℓ , j ) β ( ℓ , j ) ≥ | j || ℓ || ℓ | and µ ( k, ℓ ) β ( k, j ) ≤ | ℓ || j || k | and • either S ( k, ℓ ) ≤ | k | / | ℓ | ≥ | k | / µ ( ℓ , j ) β ( ℓ , j ) ≥ µ ( k, ℓ ) β ( k, j ) and β ( ℓ , j ) ≥ β ( k, j )and (5.4) is satisfied with M = 0, α = ν + γ + 2 and C = 2 s (here we use s ≥ ν + 4). • or S ( k, ℓ ) ≥ | k | / µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) β ( k, j ) ≤ β ( ℓ , j )but furthermore µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) µ ( k, ℓ ) β ( k, j ) ≤ | ℓ | | k || j | | k | / ≤ | ℓ || ℓ || j | | k | | k | / ≤ β ( ℓ , j ) µ ( ℓ , j )and (5.4) is satisfied with M = s , α = ν + γ + 2 and C = 2 s (here we useagain that s ≥ ν + 4) . Third case : µ ( k, j ) > µ ( ℓ , j ).As in the second case, µ ( k, j ) > µ ( ℓ , j ) implies min( | k | , | j | ) ≥ | ℓ | . This in turnimplies µ ( k, j ) = min( | k | , | j | ) , β ( k, j ) = | j | max( | k | , | j | ) and µ ( k, ℓ ) ≤ µ ( ℓ , j )and in the other hand µ ( ℓ , j ) ≥ | ℓ | and β ( ℓ , j ) = | j || j | . Thus µ ( ℓ , j ) β ( ℓ , j ) ≥ | ℓ || j || j | and µ ( k, j ) β ( k, j ) ≤ | j || j || k | and either S ( k, ℓ ) ≤ | k | / | ℓ | ≥ | k | / µ ( ℓ , j ) β ( ℓ , j ) ≥ µ ( k, j ) β ( k, j ) and β ( ℓ , j ) ≥ β ( k, j )and (5.4) is satisfied with M = 0, α = ν + γ + 2 and C = 2 s (here we use s ≥ γ ). • or S ( k, ℓ ) ≥ k/ µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) β ( k, j ) ≤ β ( ℓ , j )but furthermore µ ( k, ℓ ) µ ( k, ℓ ) + S ( k, ℓ ) µ ( k, ℓ ) β ( k, j ) ≤ | ℓ || j || k || j | | k | / ≤ | ℓ || j || j | | k | | k | / ≤ β ( ℓ , j ) µ ( ℓ , j )and (5.4) is satisfied with M = s , α = ν + γ + 2 and C = 2 s (here we useagain that s ≥ γ ) . References [1] G.P. Agrawal and R.W. Boyd (Eds),
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