Bastien Le Bihan

Josep Masdemont

Gerard Gómez

Stéphanie Lizy-Destrez

DYNAMICS IN THE CENTER MANIFOLD AROUND EQUILIBRIUM POINTS IN PERIODICALLY PERTURBED THREE-BODY PROBLEMS

Keywords: parameterization method, center manifold, four-body model, quasi-bicircular problem, Sun-Earth-Moon system

A new application of the parameterization method is presented to compute invariant manifolds about the equilibrium points of Periodically Perturbed Three-Body Problems (PPTBP). These techniques are applied to obtain high-order semi-numerical approximations of the center manifolds about the points L 1,2 of the Sun-perturbed Earth-Moon Quasi-Bicicular Problem (QBCP), which is a particular case of PPTBP. The quality of these approximations is compared with results obtained using equivalents of previous normal form procedures. Then, the parameterization is used to initialize the computation of Poincaré maps, which allow to get a qualitative description of the periodically-perturbed dynamics near the equilibrium points.

INTRODUCTION

The present paper is part of a project which aims to provide a systematic or near-systematic analysis tool for the motion of a spacecraft about and between the libration points of the Sun-perturbed Earth-Moon system. The dynamics around the Earth-Moon (EM) and Sun-(Earth+Moon) (SEM) collinear libration points L 1 and L 2 are of notable interest. In that context, the literature mainly addresses two four-body models:

A. The Bicircular problem (BCP). Widely used for the study of the EM triangular libration points [START_REF] Simó | The bicircular model near the triangular libration points of the RTBP[END_REF][START_REF] Gómez | Dynamics and Mission Design near Libration Points IV[END_REF] and for transfer optimization [START_REF] Mingotti | Efficient invariant-manifold, low-thrust planar trajectories to the Moon[END_REF][START_REF] Yagasaki | Sun-perturbed earth-to-moon transfers with low energy and moderate flight time[END_REF][START_REF] Topputo | On optimal two-impulse Earth-Moon transfers in a four-body model[END_REF], it considers the Earth and the Moon moving in circular orbits around their barycenter which is moving in circular orbit around the Sun. Such a model is not coherent since the motion of the three primaries is not a solution of the corresponding three-body problem.

B. The Quasi-Bicircular problem (QBCP). This is a bicircular coherent periodic model that has been developed in [START_REF] Andreu | The quasi-bicircular problem[END_REF]. In this framework, the motion of the three primaries is a planar self-consistent solution of the three-body problem along quasi-circular orbits.

Both the BCP and the QBCP provide an all-in-one solution for the study of the dynamics in the Sun-Earth-Moon system. In the present paper, emphasis is made on the collinear libration points L 1 and L 2 , in particular in the Earth-Moon system. Consequently, the QBCP is selected given (i) its coherent nature and (ii) the slightly broader literature available on the subject. Comparisons with results in the BCP are envisaged in the near future.

The description of the phase space around these collinear libration points has been done in the past, either in the Sun-Earth or the Earth-Moon case, and both through pure numerical techniques [START_REF] Gómez | The dynamics around the collinear equilibrium points of the RTBP[END_REF] and semi-analytical procedures [START_REF] Gómez | Dynamics and Mission Design near Libration Points IV[END_REF][START_REF] Andreu | The quasi-bicircular problem[END_REF][START_REF] Jorba | Dynamics in the center manifold of the collinear points of the restricted three body problem[END_REF][START_REF] Farrés | On the High Order Approximation of the Centre Manifold for Odes[END_REF][START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF].

As the reader may know, the span of the semi-numerical approaches are naturally limited since the resulting expansions are not convergent in any open set. However, there usually exists a domain of practical convergence within which the series are seemingly convergent for numerical purposes. In such a domain, they can provide a very compact tool to describe the phase space.

In this paper, the parameterization method is extended to the QBCP framework in order to compute the center manifold of the libration points L 1,2 of the Earth-Moon system.

In section 2, the QBCP is introduced along with the corresponding equations of motion. Then, the extension of the parameterization method to the time-periodic domain requires a suitable form for the linearized vector field, as detailed in section 3.

Section 4 details the building blocks of the parameterization method, with the example of the center manifold of EML 1,2 as a guideline. It is shown that the normal form procedure can still be seen as a subclass (or style) of parameterization methods as in the autonomous case [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF].

In section 5, the case of the center manifold of EML 1,2 is detailed along with numerical results. Two different styles of parameterization are quantitatively compared : the graph and normal form styles. In the EML 2 case, the graph style proves to be useful to partially handle resonances that act as natural obstructions to the semi-analytical normal form. Finally, the approximations of the center manifolds are used to initialize the computation of Poincaré maps. The graph style proves again to be convenient to quickly integrate the equations of motion in the center manifold.

Note that a more complete description of the procedure at hand will be made available in a work under progress [START_REF] Le Bihan | The parameterization method for invariant manifolds in the Quasi-Bicircular Problem[END_REF].

THE QUASI-BICIRCULAR PROBLEM

The Quasi-Bicircular Problem (QBCP) is a restricted fourbody problem introduced by [START_REF] Andreu | The quasi-bicircular problem[END_REF] to describe the motion of a massless spacecraft subjected to the gravitational influence of the Earth, Moon, and Sun, whose own motion is a quasi-bicircular solution of the Three-Body Problem. The resulting system is an Hamiltonian with three degrees of freedom and depending periodically on time. This Hamiltonian takes a simple form when derived in specific synodical systems of reference. As an example, the EM synodical frame is a rotating-pulsating frame centered at the Earth-Moon barycenter B em in such a way that the Earth and the Moon are located at fixed points, as in the usual CRTBP synodical frame (see Figure 1).

Denoting the state by z = (x y z p x p y p z ) T in such a frame, the Hamiltonian of the QBCP takes the form:

H(z, θ) = 1 2 α 1 (p 2 x + p 2 y + p 2 z ) + α 2 (p x x + p y y + p z z) + α 3 (p x y -p y x) + α 4 x + α 5 y -α 6 1 -µ q pe + µ q pm + m s q ps
(1) where:

q 2 pe = (x -µ) 2 + y 2 + z 2 q 2 pm = (x -µ + 1) 2 + y 2 + z 2 q 2 ps = (x -α 7 ) 2 + (y -α 8 ) 2 + z 2
with µ the Earth-Moon mass ratio, and m s the mass of the Sun. Moreover, the α k are trigonometric functions in the variable θ = ω s t where ω s is the pulsation of the Sun. The first coefficients of these functions are available in [START_REF] Andreu | The quasi-bicircular problem[END_REF]. This Hamiltonian is T -periodic with respect to the time t, with T = 2π/ω s .

In the QBCP, due to the effect of the Sun, the libration points are no longer equilibrium points. They are replaced by periodic orbits with the same frequency as the perturbation (see [START_REF] Carlos | On the vertical families of two-dimensional tori near the triangular points of the bicircular problem[END_REF] for the theoretical justification in the L 4,5 BCP case). The so-called dynamical equivalents of EML 1,2 are given on Figure 2, along with the position of their CRTBP counterparts.

x y P q pm q pe q ps B em

B S E M

Fig. 1: The QBCP in the synodical EM synodical reference frame. B is the barycenter of the system (adapted from [START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF]). (b) L2 case Fig. 2: T -periodic orbits that act as dynamical equivalents of the Earth-Moon libration points L 1,2 , in EM coordinates. The central • symbol is the CRTBP geometric position of the libration point. The starting point at t = 0 is plotted on each orbit as well as the direction of motion for t > 0 (red arrow).

A SUITABLE FORM FOR THE LINEARIZED VECTOR FIELD

The so-called parameterization method (PM) has been previously used to compute high-order power series expansions of parameterizations of invariant manifolds of vector fields at fixed points [START_REF] Cabré | The parameterization method for invariant manifolds III: overview and applications[END_REF]. An iconic example is the computation of the center manifold of the Earth-Moon libration point L 1 in the CRTBP [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF]. This paper extends such approach to invariant manifolds of periodically-perturbed vector fields about a periodic orbit with the same frequency.

The first step of such extension is to put the linearized vector field into a form equivalent to the autonomous case. Namely, this requires the origin to be a fixed point and the linearized vector field to be both diagonal and autonomous at this origin. These features can be obtained by applying a suitable change of coordinates, as explained below.

Corresponding change of coordinates

For a system described by a periodically-perturbed Hamiltonian, a change of coordinates that leaves the linearized vector field in proper form can be obtained by implementing the following steps:

1. Autonomize the Hamiltonian by introducing the canonical couple of variables (θ, y θ ) so that the time is managed as a new variable. This is done by simply adding the term ω s y θ to the initial Hamiltonian, forming an extended autonomous Hamiltonian system of degree four.

2. Cancel the terms of order one or, equivalently, translate the origin of coordinates to the periodic orbit which substitutes L 1,2 so that the origin becomes a fixed point.

3. Compute a normal form for the terms of order two, using Floquet theorem. In particular, the differential of the vector field evaluated at the origin will be timeindependent.

4. Use a complexification of the state to get a true diagonal form for the order two.

Such procedure is thoroughly described in [START_REF] Andreu | The quasi-bicircular problem[END_REF] in the QBCP Earth-Moon L 2 case, and in [START_REF] Gómez | Dynamics and Mission Design near Libration Points IV[END_REF] in the BCP L 3,4,5 case. In brief, the composition of these operations defines a change of coordinates, denoted COC, of the form:

     z = P c (θ)ẑ + V (θ) θ = θ y θ = ŷθ + p 0 (θ) + p 1 (ẑ, θ) + p 2 (ẑ, θ) (2)
where p 0 is a 2π-periodic function, p 1,2 are linear functions in ẑ, p is a complex 6 × 6 matrix, and V is a real 6 × 1, all with 2π-periodic coefficients. The final coordinates ẑ = ( xT ŷT ) T = ( x1 x2 x3 ŷ1 ŷ2 ŷ3 ) T are denoted Translated-Floquet-Complexified (TFC). In these coordinates, the Hamiltonian Ĥ takes the form:

Ĥ(ẑ, θ, ŷθ ) = ω s ŷθ + ω 1 ix 1 ŷ1 + ω 2 x2 ŷ2 + ω 3 ix 3 ŷ3 + k≥3 Ĥk (ẑ, θ) (3)
where ω i , i ∈ [ [START_REF] Simó | The bicircular model near the triangular libration points of the RTBP[END_REF][START_REF] Mingotti | Efficient invariant-manifold, low-thrust planar trajectories to the Moon[END_REF]] are real coefficients given in Table 1 for the EML 1,2 case. Moreover, the terms H k (ẑ, θ) for k ≥ 3 are homogeneous Fourier-Taylor polynomials of degree k in the complex variable ẑ that are not explicitly available at this step.

In practice, only the first line of equation ( 2) is incorporated into the numerical implementation.

Corresponding vector field

The change of variables (2) provides an Hamiltonian Ĥ in autonomous diagonal form at order 2. From this form, the Table 1: The coefficients ω i that appear in equation [START_REF] Mingotti | Efficient invariant-manifold, low-thrust planar trajectories to the Moon[END_REF], in the Earth-Moon L 1,2 case. All values are given with the same precision as in [START_REF] Andreu | The quasi-bicircular problem[END_REF].

ω i L 1 L 2 ω 1 -4.
38968496e-01 +1.34709423e-02 ω 2 +2.93720564e+00 +2.16306748e+00 ω 3 +4.22768254e-01 -6.02217885e-02 corresponding vector field can be derived and plugged into the parameterization method:

ż = F (ẑ, θ) = Ω 0 0 -Ω x ŷ + ∞ k=3 J∇ Ĥk (ẑ, θ) (4) with
Ω =   iω 1 0 0 0 ω 2 0 0 0 iω 3   , J = 0 I 3 -I 3 0
The linearized part of (4) has the desired autonomous diagonal form for the initialization of the parameterization method. Moreover, the TFC origin is naturally a fixed point and all the conditions defined at the beginning of this section are satisfied.

Precision of the COC

At this point, one can see that the overall accuracy of the change of coordinates (2) directly depends on the precision with which the periodic coefficients in P c and V are computed. In practice, these coefficients are obtained by a Fourier analysis of 2π-periodic functions integrated along the dynamical equivalent of the libration point (see [START_REF] Gómez | Dynamics and Mission Design near Libration Points IV[END_REF][START_REF] Le Bihan | The parameterization method for invariant manifolds in the Quasi-Bicircular Problem[END_REF]). As in the three-body case, the linear dynamics about the collinear points is a cross product of two centers, and one saddleassociated with a pair of real positive eigenvalues. Naturally, the precision of the integration of the periodic coefficients is limited by the magnitude of the hyperbolic unstable direction associated to the orbit.

To estimate this constraint, let λ u be the eigenvalue associated with the unstable direction. Then an initial error ǫ(0) made along the unstable direction leads, after a time t, to an error:

ǫ(t) ∼ ǫ(0)e t T ln(λu) = ǫ(0)(λ u ) t T ( 5
)
where T is the period of the QBCP. This error is doomed to grow fast: in the case of EML 2 , the initial error is multiplied by a factor roughly equal to 1500 after half a period, and 2 × 10 6 after a full period. The EML 1 case is even worse: the error takes a factor about 21000 (resp. 5 × 10 8 ) after half a period (resp. a full period). On the contrary, in the SEM case, the unstable eigenvalues are quite small: the error is roughly multiplied by 2 after half a period and by 4 after a full period, both in the L 1 and L 2 cases. Such a precision shortfall is of paramount importance for the accuracy of the COC and eventually for the parameterizations of the invariant manifolds about EML 1,2 . In practice, the EML 1 case exhibits an overall poor precision compared to the EML 2 case.

THE PARAMETERIZATION METHOD IN THE QBCP

In this section, the extension of the parameterization method (PM) is introduced, with the example of the center manifold of EML 1,2 as a guideline. Note that, if not stated otherwise, the so-called order of the parameterization method corresponds to the order of the four-variable Taylor expansions that describe the center manifold.

Initialization of the process

In the case of EML 1,2 , it comes from equation ( 4) that

D F (0) = diag(iω 1 , ω 2 , iω 3 , -iω 1 , -ω 2 , -iω 3 ) (6)
Let define the three matrices H, L and N by:

H =         iω 1 0 0 0 0 0 0 0 0 0 ω 2 0 0 iω 3 0 0 0 0 0 0 -iω 1 0 0 0 0 0 0 0 0 -ω 2 0 0 0 -iω 3 0 0         L N
In other words, L spans the 4-dimensional subspace V L ⊂ C 6 tangent to the center manifold W c at the origin. The goal of the PM is here to compute a high order approximation of c , starting with L as its order one approximation. Let denote ẑ = Ŵ (s, θ) the parameterization of W c where s ∈ R 4 are the coordinates of the manifold, and Ŵ (0, θ) = 0, ∀θ.

The invariant manifold is sought in the form of a Fourier-Taylor (FT) expansion, i.e. a power series in the variable s whose coefficients are 2π-periodic Fourier series in the variable θ:

Ŵ (s, θ) = k≥1 Ŵk (s, θ) (7)
with

Ŵk (s, θ) =         Ŵ 1 k (s, θ) . . . Ŵ p k (s, θ) . . . Ŵ 6 k (s, θ)         =             r∈R k w 1 r (θ)s r . . . r∈R k w p r (θ)s r . . . r∈R k w 6 r (θ)s r             (8)
where

R k = r ∈ N 4 , |r| = r 1 + • • • + r 4 = k , s r = s r1 1 . . . s r4 4
, and the coefficients w p r (θ) are trigonometric functions of the form:

w p r (θ) = j w p r,j e ijθ (9)
In practice the sum ( 7) is truncated so that j satisfies |j| ≤ J, with J ∈ N fixed. The dynamics on the manifold is described by a reduced vector field ṡ = f (s, θ) for which f (0) = 0. The vector field f is also sought in the form of a FT series.

The couple ( Ŵ , f ) must satisfy the invariance equation:

F Ŵ (s, θ), θ = D Ŵ s, θ f s, θ + ∂ Ŵ ∂t s, θ (10)
With these ingredients, the order one ( Ŵ1 , f 1 ) is given by:

Ŵ1 (s, θ) = Ls f 1 (s, θ) = Λ L s (11)
where Λ L is the upper left 4 × 4 submatrix of the diagonal matrix Λ = H -1 D F (0)H. From there, the standard procedure is to formally solve the invariance equation [START_REF] Gabern | Effective Computation of the Dynamics Around a Two-Dimensional Torus of a Hamiltonian System[END_REF], starting from [START_REF] Farrés | On the High Order Approximation of the Centre Manifold for Odes[END_REF], by substituting the expansions of Ŵ and f in [START_REF] Gabern | Effective Computation of the Dynamics Around a Two-Dimensional Torus of a Hamiltonian System[END_REF] and find homogeneous terms in increasing order.

The homological equations

Isolating the k-order terms in the invariance equation [START_REF] Gabern | Effective Computation of the Dynamics Around a Two-Dimensional Torus of a Hamiltonian System[END_REF], k > 1, allows to get the k-order homological equation for Ŵk (s, θ) and f k (s, θ):

D F (0) Ŵk -D Ŵk Λ L s -Lf k - ∂ Ŵk ∂t = D Ŵ<k f <k k -F Ŵ<k k ( 12
)
where the dependency in (s, θ) has been omitted for the sake of clarity. The goal is to compute Ŵk and f k in order to satisfy equation [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF]. The terms Ŵ<k , f <k , and F Ŵ<k <k are assumed to have been obtained in previous steps.

First, the right-hand side of ( 12) is computed, which involves the differentiation, sum and product of Fourier-Taylor series, as well as the composition of Fourier-Taylor series with algebraic functions (in particular, the raising to a fractional power). Such operations require a complete Fourier-Taylor algebra that has been implemented from scratch in C++. For more details, see [START_REF] Le Bihan | The parameterization method for invariant manifolds in the Quasi-Bicircular Problem[END_REF]. Then, following the example of [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF], the normal part of ( 12) is separated from its tangent part. The following functions are introduced:

   υ k = H -1
Ŵk

η k = H -1 D Ŵ<k f <k k -f Ŵ<k ) k
Multiplying equation ( 12) by H -1 , the homological equation then take the form:

Λυ k -Dυ k Λ L s -I4 0 f k - ∂υ k ∂t = η k (13)
The diagonal form of Λ allows to solve the homological equations separately on the tangent and normal spaces, spanned by υ L k and υ N k , respectively. In both cases, the adaptation to the non-autonomous case is fairly straightforward as will be detailed in [START_REF] Le Bihan | The parameterization method for invariant manifolds in the Quasi-Bicircular Problem[END_REF]. In brief, the autonomous diagonal form of the linearized vector field ensures that each Fourier coefficient in the Fourier-Taylor series υ p k , p ∈ [ [START_REF] Simó | The bicircular model near the triangular libration points of the RTBP[END_REF][START_REF] Andreu | The quasi-bicircular problem[END_REF]] satisfies a simple linear differential equation in the variable θ, with the following solutions: Normal equations. For p = 5, 6 and for all r ∈ R k , let C r,p = {j ∈ Z, jω s i -(λ pλ L r) = 0} be the set of cross resonances associated to (r, p), where λ L r = λ 1 r 1 + • • • + λ 4 r 4 . It can be shown that C r,p = ∅, and that a solution of the homological equation is:

υ p r (θ) = j∈Z -η p r,j jω s i -(λ p -λ L r) e ijθ (14)
Tangent equations. For p = 1, . . . , 4 and for all r ∈ R k , let I r,p = {j ∈ Z, jω s i -(λ pλ L r) = 0} be the set of internal resonances associated to (r, p). Providing that I r,p = ∅, a solution of this equation is:

υ p r (θ) = j∈Z -η p r,j -f p r,j jω s i -(λ p -λ L r) e ijθ (15)
However, the set of internal resonances is usually not empty. Fortunately, a solution can be built even in the presence of resonances by adjusting the coefficients f p r,j . Several strategies available in the autonomous case (denoted as styles [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF]) are still relevant in the current context. Both styles used in this paper are detailed below: the graph style and the normal form style.

The graph style: It consists in simplifying at most the parameterization of the manifold, by taking υ L k = 0 at each step. That is, for p = 1, . . . , 4 and r ∈ R k : f p r = -η p r , υ p r = 0. Such a parameterization is suitable for all manifolds and is particularly adequate for center manifolds for which there exists an infinite number of internal resonances.

The normal form style: This style consists in simplifying the equations of the dynamics on the manifold, finding a normal form for f . That is, for p = 1, . . . , 4, r ∈ R k , and j ∈ Z:

• if jω s i -(λ p -λ L r) = 0 : f p r,j = 0, υ p r,j = -η p r,j jω s i -(λ p -λ L r) (16a) • if jω s i -(λ p -λ L r) = 0 : f p r,j = -η p r,j , υ p r,j = 0 (16b)
Implementing the normal form style implies to look not only for the zeros but also for the near-zero values of the divisors jω s i -(λ pλ L r). Indeed, numerically speaking, small values for those divisors must be avoided to ensure a certain size for the domain of practical convergence. For this reason, a threshold ε can be implemented in order to select (16a) only when the norm of the corresponding divisor is greater than ε.

With such choices, the normal form style is equivalent to the Hamiltonian normal form approach developed in [START_REF] Andreu | The quasi-bicircular problem[END_REF][START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF], with ε = 0.05.

Final form

Graph and normal form styles

Using the graph style, the center manifold in TFC coordinates takes the form:

ẑ(s, θ) = Ŵ (s, θ) = Ŵ1 (s) + N k≥2         0 Ŵ 2 k (s, θ) 0 0 Ŵ 4 k (s, θ) 0        
One can see that four components of the TFC parameterization are equivalentwithin a scalar factorto the CCM coordinates, and entirely map the center manifold of dimension four. The two remaining components are given as functions of these four central components, therefore defining a graph of the form (x 2 , ŷ2 ) = G(x 1 , x3 , ŷ1 , ŷ3 ). With such a form, it is straightforward to project the current state on the center manifold by using the definition (11) of Ŵ1 (s).

On the contrary, using the normal form style, the equivalent parameterization is a full 6 × 1 Fourier-Taylor vector that does not allow simple projections on the manifold.

Realification of the center manifold

Given the complexification performed in the COC, the vector s is complex, which does not guarantee that W (s, θ) is real. It is then necessary to perform a realification, i.e. a new change of coordinates to ensure that the final result, in physical coordinates, is real. This realification is given by the following equality:

s = C s (17)
with

C = 1 √ 2     1 0 i 0 0 1 0 i i 0 1 0 0 i 0 1    
where s = ( s1 s2 s3 s4 ) T are the complex reduced coordinates of the center manifold (CCM), and s = ( s1 s2 s3 s4 )

T are the real ones (RCM). Injecting (17) in the parameterization, the following real function is obtained:

W : R 4 × R → R 6 (s, θ) → W (s, θ) in EM coordinates
Notations. For the sake of simplicity, and since there is no use of the complex parameterization (CCM coordinates) anymore, the tilde notation is skipped from now on when referring to the RCM coordinates.

Comments on the energy

In the QBCP, the huge potential well of the Sun leads to a severe drop of the order of magnitude of the energy with which the reader might be used to in the CRTBP case. As an example, the energy H Li (t) of the libration orbit around EML 1,2 is always close to -847.5, in normalized EM units. To tackle this issue, the energy of any object is given with respect to the initial energy Li (0) of its associated libration periodic orbit: it is measured by the variable δH t so that the true energy H(t) satisfies H(t) = H Li (0) + δH t . Such a relative estimate is adapted to the use of semi-analytical expansions which are inherently local.

DESCRIPTION OF THE NEIGHBORHOOD OF EML 1,2

Accuracy of the center manifold

The necessary tests of the precision of the parameterization of the center manifold have been focused on the estimation of the orbital error, defined hereafter. Let z(t) and s(t) be the solutions of the Cauchy problem ż = F (z), z(0) = z 0 = W (s 0 , 0) and ṡ = f (s), s(0) = s 0 , respectively. Then, the orbital error e O at time t is defined as:

e O (t, s 0 ) = |W (s(t)) -z(t)| ∞
The orbital error has to be measured on small time spans, since the hyperbolic directions produce exponential errors (see subsection 3.3). In the present paper, the orbital error has been computed up to t = 1, as in [START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF]. Let e 1 O := e O (t = 1), for the sake of brevity. For such a time span, any numerical error on the initial conditions is amplified by a factor of 10 (resp. 20) in the EML 2 (resp. EML 1 ) case.

Since we are working in manifolds of order four or higher, the systematic testing of any kind of error in the complete phase space is a tremendous computational task. Thus, as a first step, one may arbitrarily decrease the number of dimensions along which the error is evaluated. Such "cuts" in the phase space are common for these types of tests (see e.g. [START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF]). After an extensive test campaign involving various cuts, it has been found that the following set of initial conditions acts as a bordeline case for the accuracy of the center manifold, both in the L 1 and L 2 cases:

S -1 = {(s 0 , t 0 = 0) with s 0 = (s 1 0 -s 1 0) T , s 1 ∈ R}
Given that s 2 = s 4 = 0, the motion is restricted to the xyplane in EM coordinates. Moreover, the condition s 1 = -s 3 imposes the initial state z 0 = W (s 0 , 0) to be on the x-axis.

EML 2 case

Figure 3 gives the orbital error e 1 O for the set S -1 of initial conditions, as a function of the x coordinate. The position of EML 2 is given by the point of minimum error, around x = -1.165. The following comments can be made:

• Within the domain of practical convergence, the precision appears to tend to a limit as the order increases.

In particular, there is not much improvement for orders higher than 20 or 22, which is the reason why higher orders have not been displayed.

• More importantly, the radius of practical convergence is undoubtedly better in the graph case. Looking at Figure 3a, one can clearly see the limits of the domain of practical convergence at x 1 = -1.175 and x 2 = -1.13. Those values correspond exactly to the four T /2-periodic resonant orbits highlighted in [START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF], and denoted PO2a-d (see Figure 2b therein). As usual, such low order resonances tend to introduce small divisors in the semi-analytical algorithms and therefore act as a natural obstruction to the existence of a good parameterization. However, the use of the graph style allows to limit to the bare minimum the number of small divisors involved in the solving of the homological equations (see 4.2). Consequently, the domain of practical convergence is increased, as can be seen on Figure 3b.

These remarks remain true for all tested sets of initial conditions.

- (b) Graph style Fig. 3: The orbital error e 1 O for the set S -1 of initial conditions, as a function of the x coordinate, for various orders of the parameterization of the center manifold of EML 2 .

EML 1 case

In the EML 1 , the differences between both styles tend to vanish, mainly because there is no known resonant orbit in the close vicinity of the libration point that would have played a role similar to PO2a-d. Moreover, the overall precision is worse than in the EML 2 case because the EML 1 , as anticipated in subsection 3.3. In particular, numerical simulations in the worst case show that there is not much improvement for orders higher than 16. On the bright side, there is room for improvement since the source of the imprecision has already been identified as the COC instability, which may be partially handled numerically.

Projection method

All the subsequent solutions within the center manifold have been computed using the projection method, described in details in [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF]. In brief: for any initial conditions s 0 in the center manifold, the initial conditions z 0 in EM coordinates are obtained at time t 0 applying z 0 = W (s 0 , ω s t 0 ). Then, the state is integrated using the 6-dimensional physical vector field, until time t = t 1 . Due to numerical instabilities produced by the hyperbolic directions, the overall error tends to grow. Hence, the current state z(t 1 ) can be projected on the parameterized center manifold, which provides a new vector s p . The state z p = W (s p , ω s t 1 ) is then used to start the next integration phase.

In the context of integration by projection, one can get a rough idea of the precision of each solutions by computing the magnitude e P (t 1 ) of the correction performed at each projection, defined as:

e P (t 1 ) = |z(t 1 ) -z p (t 1 )| ∞ (18)
Note that the change of variables involved in the projection process is particularly easy to perform using the graph style, as it has been shown in paragraph 4.3.1.

Variations of the energy

Contrary to the CRTBP case, the energy is not constant along a solution inside the center manifold. In order to get an estimation of the variations of the energy, the xy-planar stroboscopic maps of the center manifold of EML 1,2 have been computed. Namely, for each initial condition of the form s 0 = ( s1 0 s3 0 ) T , t 0 = 0, the EM equations of motion are numerically integrated and a point is stored each time the trajectory crosses the section t = 0 [T ], with T the period of the QBCP. Selecting only the points for which t = 0 [T ] erase all variations ought to the T -periodic behavior of the coefficients that appear in the Hamiltonian (1).

For each individual orbit, the mean energy µ(δH) has been computed along with the associated standard deviation σ(δH). The corresponding results are presented on Figure 4. On this figure, the greater the mean energy, the bigger the orbit. For instance, a mean energy around 0.005 corresponds to a mean distance from the libration point of about 17000 km (resp. 13000 km) in the EML 2 (resp. EML 1 ) case. The stroboscopic maps have been computed in the mean mean energy range [0, 0.011], which encloses the Halo orbit bifurcation for both libration points (see section 5.4).

EML 1 case. The standard deviation is a smooth increasing function of the mean energy value. Moreover, σ(δH) is never greater than 10% of the mean energy, except for very small values of the energy. As an example, σ(0.01) is about 3% of µ(0.01).

EML 2 case. For values of the energy smaller than 0.006, and as the mean energy increases, the standard deviation is more or less converging to 10% of the mean energy.

Around µ(δH) ≃ 0.006, the standard deviation breaks down, which corresponds to the solutions that cross the 2ω s resonance. After this resonance, starting from σ(δH) = 0, the standard deviation regains its smoothness with respect to the mean energy.

Poincaré maps

Poincaré maps provide a qualitative insight into the dynamics inside a center manifold. In this paper, Poincaré maps with a Poincaré section of the form z = 0, p z > 0 have been computed in the EML 1,2 case. Such sections are also usually used in the CRTBP autonomous case [START_REF] Jorba | Dynamics in the center manifold of the collinear points of the restricted three body problem[END_REF][START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF]. In the latter context, an additional condition on the energy of the form δH 0 = h allows to produce two-dimensional Poincaré maps.

In the QBCP case, the energy is no longer constant but its variations remains bounded for any solution in the center manifold, in proportions similar to the results of Figure 4. Hence a relaxed version of the energy condition may be defined: |δH 0 -h| ≤ σ c (h), with σ c (h), a given "energy thickness", representative of the energy variations associated with an initial energy around h. The underlying idea is to define an energy layer such that all solutions that have an initial energy within this layer maintain their energy in it.

In this paper, the "super" constraint σ c (h) = 0 has been imposed in order to limit the number of solutions displayed on the maps. Consequently, looking at a given map, one has to recall that (i) the energy is not constant but bounded along the trajectories (with a maximum deviation around 10%), and (ii) the range of displayed solutions is a representative but non-exhaustive set of solutions inside this energy layer.

In practice, the following process is used to compute the maps: the initial time is set equal to zero. Then, an initial energy value h and a point (s 1 , s 3 ) are selected. Moreover, the condition s 2 = s 4 is imposed, which, with the graph style and the current choice of COC, guarantees that z(t = 0) = 0 in EM coordinates. Imposing δH 0 = h, the corresponding s 2 = s 4 value is computed. Then, this point is used as initial condition for a numerical integration of the equations of motion, plotting a point each time that the trajectory crosses the plane z = 0, p z > 0. For each of these points, the projection method is used to reinitialize the state on the center manifold, and the projection error e P (t) is estimated. The precision e P (t) = 10 -6 is taken as an arbitrary reference accuracy. In practice, it corresponds to a maximum error of about 50 meters in position and 10 -4 meters per second in velocity.

EML 2 case

The results can be seen on Figure 5, for small values of the energy. All solutions comply with the condition e P (t) < 10 -6 . Contrary to the autonomous case, some solutions overlap. In particular, the overlapping seems to increase with the initial energy, which is consistent with the fact that the standard deviation of the energy tends to grow with its mean value.

On each plot, several solutions exhibits specific variation patterns (see for example the innermost green solution of Figure 5c). These patterns are present also when the integration of the equations of motion is performed directly with the reduced vector field. Therefore the discontinuities introduced by the projection method cannot be held responsible for such variations, which are truly characteristic of the periodicallyperturbed system.

In order to test the limit of the description provided by the parameterization method, equivalent Poincaré maps have been computed for higher initial energies. The corresponding results are given on Figure 6. For these two plots, the condition e P (t) < 10 -6 have been relaxed (see caption), but the points that indeed comply with this constraint are displayed in green. As one would expect, it is clear that the precision worsens as the energy increase. Such precision shortage is expected, in particular in the EML 2 neighborhood which corresponds to higher energies than the EML 1 case, in absolute. In particular, it seems difficult to achieve a very good accuracy during the computation of quasi-halo orbits. Even if their influence is reduced by the graph style implementation, it is probable that the orbits resonant with the Sun still participate in the numerical error. A remaining challenge is to dissociate the contribution of the latter to the error from the natural precision decay inherent to the three-body EML 2 case.

EML 1 case

The results for the EML 1 case are given on Figure 7. The same remarks apply. In particular, high energies require the condition e P (t) < 10 -6 to be slightly relaxed. However, the halo orbit bifurcation is more easily obtained. Such a difference may be ought to two factors. First, the Halo bifurcation happens for an energy smaller than in the EML 2 case, both with EML 1 which could act as a natural obstruction to the semianalytical description of this neighborhood. Additional simulations show that, at higher energy, a precision decay very similar to the one displayed on Figure 6 occurs in the EML 1 case, which tends to back up the first hypothesis. (d) δH0 = 0.01 Fig. 7: Low-energy Poincaré maps in the EML 1 case, in EM coordinates. On (a), (b): all solutions comply with the condition e P (t) < 10 -6 . On (c), (d): all solutions comply with the condition e P (t) < 10 -5 . The color scale is just here to visually distinguish the solutions.

CONCLUSION

In this paper, the parameterization method (PM) has been applied to obtain high-order semi-analytical approximations of invariant manifolds about the dynamical equivalents of the collinear libration points in the Sun-perturbed Earth-Moon system, viewed as a Quasi-Bicircular Problem. The overall process can be seen as an extension of previous works in the CRTBP [START_REF] Haro | The parameterization method for invariant manifolds: from rigorous results to effective computations[END_REF][START_REF] Cabré | The parameterization method for invariant manifolds III: overview and applications[END_REF]. Both the method and its applications have been focused on the comparison with normal form computations in the Sun-Earth-Moon QBCP, more specifically in the EML 2 case [START_REF] Andreu | The quasi-bicircular problem[END_REF][START_REF] Andreu | Dynamics in the center manifold around L2 in the quasi-bicircular problem[END_REF]. It has been shown that the advantages of the parameterization methods are particularly relevant in this context. More specifically:

• The different styles available in the PM bring more flexibility to the user, compared to the normal form computations over the Hamiltonian. In particular, the graph style allows to keep to a minimum the number of potential small divisors during the computation of the semianalytical approximations. The visible consequence is that some specific resonances with the Sun are better handled, although they most likely still act as a obstruction for the domain of practical convergence.

• The graph form of the parameterization provides a very simple change of coordinates between the physical variables and the reduced coordinates. This allows to easily project the current state on the center manifold which in turns speeds up the numerical integrations.

• Finally, from a pure numerical perspective, the parameterization method allows to manipulate 4-dimensional Taylor series. On the contrary, the normal form procedure is performed on the 6-dimensional complete state. Working with the parameterization method allows to reduce the dimension of the Fourier-Taylor series, which can be critical at high order, when millions of scalar coefficients are involved.

In spite of these advantages, results on Poincaré maps show that the EML 2 quasi-Halo orbits are still difficult to obtain semi-analytically. On the contrary, they have been more easily computed in the EML 1 case, despite the limited accuracy of the associated change of coordinates (2) that limits the highest available order for the parameterization.

Indeed, the precision of the T -periodic functions that are used throughout the procedure act as a bottleneck for the overall accuracy of the parameterization. Improving these functions would benefit the whole process and help understand how well the resonances are actually handled by this new implementation.

Fig. 4 :

 4 Fig.4: Standard deviation σ of the energy δH(t) as a function of the mean value µ of the energy for some solutions of the planar stroboscopic maps.

Fig. 5 :Fig. 6 :

 56 Fig.5: Low-energy Poincaré maps in the EML 2 case, in EM coordinates. All solutions comply with the condition e P (t) < 10 -6 . The color scale is just here to visually distinguish the solutions.


Eduard Curcȃ
email: curcaeduard@yahoo.ro

On the continuity of Fourier multipliers on Ẇ l,1 R d and Ẇ l,∞ R d

Keywords: Fourier multipliers, Sobolev spaces. MSC 2020 classification: 42B15, 42B05

 concerning homogeneous multipliers of degree zero. We further generalize the result of Kazaniecki and Wojciechowski. We prove that, given an integer l ≥ 1, every multiplier on Ẇ l,1 R d or on Ẇ l,∞ R d is a bounded continuous function on R d . We obtain these results via a substantial simplification of the Riesz products technique used by Kazaniecki and Wojciechowski.

Introduction

In this paper, we study the continuity properties of functions which are Fourier multipliers on the homogeneous Sobolev spaces Ẇ l,1 R d and Ẇ l,∞ R d , where l ≥ 1 is an arbitrary integer.

Given a nonnegative integer l and a parameter 1 ≤ p ≤ ∞, the space Ẇ l,p R d consists of those distributions f on R d for which ∇ l f ∈ L p R d . This space is endowed with the seminorm given by

f Ẇ l,p (R d ) = ∇ l f L p (R d ) = max α∈N d ,|α|=l ∇ α f L p (R d ) .
Given a function m ∈ L 1 loc (R d ), we say that m is a Fourier multiplier on Ẇ l,p R d if, for each Schwartz function f ∈ S(R d ) the distribution m f is tempered and if, in addition, the following estimate holds:

T m f Ẇ l,p ≤ C f Ẇ l,p , ∀ f ∈ S(R d ), (1)
for some constant C < ∞, where T m is defined by the relation

T m f = m f , ∀ f ∈ S(R d ).
The least constant C in the above inequality will be called the norm of m and will be denoted by T m (which is a quantity depending on p and l).

The Fourier transform that we work with is given by the following formula

f (ξ) := R d e -i x,ξ f (x)dx, ∀ f ∈ S(R d ).
Some classical examples of multipliers on Ẇ l,p R d in the case 1 < p < ∞ (for any l), are the functions m j (ξ) := iξ j / |ξ|, defined for ξ ∈ R d \ {0} and any j = 1, 2, ..., d. In this case we have T m j = R j , where R 1 , ..., R d are the Riesz transforms on R d . Let us observe that the functions m j are homogeneous of degree zero, i.e., m j (λξ) = m j (ξ), for all ξ ∈ R d \ {0} and any λ > 0. Also, m j are not continuous at zero. If p = 1, the situation is different. When l = 0, the m j 's fail to be multipliers on Ẇ 0,1 R d = L 1 R d , since the R j 's are not bounded on L 1 R d . In fact, if m is a multiplier on L 1 R d , then it is easy to see that m is the Fourier transform of a finite measure and hence m ∈ C b R d . The case of the multipliers on L ∞ R d is similar. Suppose d ≥ 2. Then there exist Fourier multipliers on Ẇ 1,1 R d which are not Fourier transforms of finite measures (see [START_REF] Poornima | On the Sobolev spaces W k,1 R d . Harmonic analysis[END_REF]Proposition 2.2]). In fact, the proof in [START_REF] Poornima | On the Sobolev spaces W k,1 R d . Harmonic analysis[END_REF] concerning Ẇ l,1 R d applies to all the spaces Ẇ l,p R d , with l ≥ 1 and 1 ≤ p ≤ ∞.

Let us illustrate this when d = 2 and l = 1, via a simple example from [START_REF] Poornima | On the Sobolev spaces W k,1 R d . Harmonic analysis[END_REF]. As a consequence of Ornstein's L 1 non-inequality (see [START_REF] Ornstein | A non-inequality for differential operators in the L1 norm[END_REF]), there exists a distribution u, supported in the unit ball, such that ∂ 2 1 u, ∂ 2 2 u are L 1 functions on R 2 and ∂ 1 ∂ 2 u is not a finite measure. We define m := ∂ 1 ∂ 2 u. Clearly, m is not the Fourier transform of a finite measure, however m is a multiplier on Ẇ 1,p (R 2 ) for all 1 ≤ p ≤ ∞. Indeed,

T m f = ∂ 1 ∂ 2 u * f ,
for any Schwartz function f .

Hence, ∇T m f = ∂ 2 1 u * ∂ 2 f, ∂ 2 2 u * ∂ 1 f and thus, by Young's inequality,

∇T m f L p ≤ ∂ 2 1 u L 1 ∂ 2 f L p + ∂ 2 2 u L 1 ∂ 1 f L p = ∂ 2 1 u L 1 + ∂ 2 2 u L 1 ∇f L p .
Using Ornstein's L 1 non-inequality, Bonami and Poornima proved in 1987 that the only Fourier multipliers on Ẇ l,1 R d which are homogeneous functions of degree zero and continuous outside the origin are the constant functions. More precisely, they proved the following (see [START_REF] Bonami | Nonmultipliers of the Sobolev spaces W k,1 R d[END_REF]Theorem 2]).

Theorem 1 Suppose d ≥ 2, l ≥ 1 are two integers and let Ω be a continuous function on R d \ {0}, homogeneous of degree zero. If Ω is a Fourier multiplier on Ẇ l,1 R d , then Ω is a constant.

When l = 1, this result was generalized by Kazaniecki and Wojciechowski in 2013 as follows (see [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF]Theorem 1.1]).

Theorem 2 Suppose d ≥ 2. If m is a Fourier multiplier on Ẇ 1,1 R d , then m ∈ C b R d .
Since any function homogeneous of degree zero that is continuous on R d has to be constant, we see that Theorem 2 implies Theorem 1 when l = 1. In order to prove Theorem 2, Kazaniecki and Wojciechowski used Theorem 1 and some Riesz product technique reminiscent of [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF]. Also, for technical reasons, some constructions concerning the tranference of multipliers, were involved in the argument. The central role is played by the Riesz products technique, a key tool being a relatively difficult lemma of Wojciechowski (see [9, Lemma 1], [10, Lemma 1]) concerning the L 1 -norm of some trigonometric polynomials.

It seems that, since the above result of Kazaniecki and Wojciechowski, there was no progress in studying the general case of arbitrary multipliers when the regularity of the space is higher than 1.

We follow the ideas in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF] in order to prove a generalisation of Theorem 2 for the case of Ẇ l,1 R d , where l ≥ 1. The proof is also based on the Riesz products technique and the constructions we use are very similar to those in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF]. However, rather than using Wojciechowski's lemma we rely on much easier facts instead (see Lemmas 14 and 16 below). The other ingredient is Theorem 1. We do not use transference theorems and, apart from Bonami and Poornima's result, the proof is quite elementary. Another advantage of our approach is that it also applies to multipliers on Ẇ l,∞ R d . Note that the case of Ẇ 1,∞ R d was not considered in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF].

Our results are the following.

Theorem 3 Suppose d ≥ 2 and l ≥ 1 are integers. If m is a Fourier multiplier on Ẇ l,∞ R d , then m ∈ C b R d . Theorem 4 Suppose d ≥ 2 and l ≥ 1 are integers. If m is a Fourier multiplier on Ẇ l,1 R d , then m ∈ C b R d .

Remark 5

In fact, what we prove in these theorems is that m is a.e. equal to some bounded continuous function.

The proofs go as follows. First, following the idea in the proof of Lemma 3.1 in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF] we show, by simple arguments, that, whenever m is a multiplier on Ẇ l,1 R d or on Ẇ l,∞ R d , we need to have m ∈ C b R d \ {0} . (More specifically, as a preliminary step in our analysis, we define a function which equals m a.e. on R d and is continuous and bounded on R d \ {0}.) Next, using this conclusion, we prove that, if m is a multiplier on Ẇ l,1 R d or on Ẇ l,∞ R d , then m has to be continuous at the origin. For this part of the proof, we use constructions based on Riesz products.

While the proofs of these facts are similar, we start by studying the multipliers on Ẇ l,∞ R d , since the proof is simpler in this setting. For this purpose, we adapt and simplify the ideas in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF] to the case of Ẇ l,∞ R d , with the help of Lemma 14 below. Next, we study multipliers on Ẇ l,1 R d . We show by a duality method that the boundedness of T m implies the existence of bounded solutions for some underdetermined differential equation. We conclude that such solutions do not exist if m is not continuous. Here, the technique is similar to the one in [START_REF] Curcȃ | The divergence equation with L ∞ source[END_REF], which was in turn inspired by the one in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF].

Continuity outside the origin

Suppose that m is a multiplier of Ẇ l,p R d for some 1 ≤ p ≤ ∞ and l ≥ 0. Let us notice that the norm of T m is invariant by dilations and isometries. More precisely, if λ ∈ R\ {0}, respectively R ∈ O(d) and we set m λ (ξ) := m (λξ), respectively m R (ξ) := m(Rξ), ∀ξ ∈ R d , then

T m λ = T m and T m R ∼ l,d T m , (2)
where the T m is the norm of

T m : Ẇ l,p R d → Ẇ l,p R d .
Let us justify this when m ∈ L 1 (R d ); the general case is obtained by approximation. For any d × d real invertible matrix A and any Schwartz function f on R d , we have the following identity:

f A (ξ) = 1 | det A| f A -1 ξ ,
where f A (x) := f (Ax) for any x ∈ R d . Via a change of variables, we find that

T m A f (x) =(2π) -d R d e i x,ξ m (Aξ) f (ξ) dξ = (2π) -d | det A| R d e i x,A -1 ξ m (ξ) f A -1 ξ dξ =(2π) -d R d e i (A -1 ) t x,ξ m (ξ) f A (ξ) dξ = T m f A A -1 t x for a.e. x ∈ R d .
When, A = λI, we get that T m λ f = T m f λ (•/λ). We obtain

∇ l T m λ f L p =λ -l ∇ l T m f λ (•/λ) L p = λ -l λ d/p ∇ l T m f λ L p ≤ T m λ -l λ d/p ∇ l f λ L p = T m ∇ l f L p .
Hence, T m λ ≤ T m for any λ = 0. This gives the first equivalence in (2).

When, A = R, where R is orthogonal, we get

T m R f = T m f R (R•).
Since the absolute value of each entry of R is bounded by 1, we obtain

∇ l T m R f L p l,d ∇ l T m f R (Rx) L p = ∇ l T m f R L p ≤ T m ∇ l f R L p l,d T m ∇ l f L p .
Hence, T m R l,d T m for any orthogonal d × d matrix R. This gives the second equivalence in [START_REF] Curcȃ | The divergence equation with L ∞ source[END_REF].

Let us first observe that the multipliers of Ẇ l,1 R d and the multipliers of Ẇ l,∞ R d are bounded and continuous on R d \ {0}.

Lemma 6 If m is a multiplier on Ẇ l,1 R d , then m ∈ C b R d \ {0} .
Remark 7 We recall that (see Remark 5) the statement means that m has a bounded continuous representative on R d \ {0}.

Proof. We recall that l ≥ 1. We essentially follow the argument in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF]Lemma 3.1]. We first prove that m has a continuous representative on R d \ {0}. Indeed, let ρ be any Schwartz function such that ρ(ξ) = 0, ∀ξ (e.g. a standard Gaussian). Set α j := (δ j k l) 1≤k≤d , where δ j k is the Kronecker delta. Since ∂ α j (T m ρ) ∈ L 1 , we find that m j := (iξ) α j T m ρ is a continuous bounded function, and

(iξ j ) l m(ξ) ρ(ξ) = m j (ξ) for a.e. ξ ∈ R d . (3)
Set, for ξ = 0,

m(ξ) := m j (ξ) (iξ j ) l ρ(ξ) , if ξ j = 0.
It is not clear if the above definition is correct, since the result may depend not only on ξ, but also on the choice of the coordinate ξ j . However, (3) implies first that this definition is correct for a.e. ξ, next, using the continuity of m j , the definition is correct for every ξ = 0 and that, in addition m is continuous on R d \ {0}. Clearly (from (3)), we have that m = m a.e. and

(iξ j ) l m(ξ) ρ(ξ) = m j (ξ) for every ξ ∈ R d \ {0}. ( 4
)
Using the fact that each m j is bounded, we find from (4) that m is bounded on the unit sphere S d-1 . More specifically, we have max

|ξ|=1 | m(ξ)| T m . (5)
Combining ( 5) and ( 2), we find that m is bounded on R d \ {0}.

Lemma 8 If m is a multiplier on Ẇ l,∞ R d , then m ∈ C b R d \ {0} .
Proof. As in the proof of Lemma 6, we first prove that m has a representative which is continuous on R d \ {0}. Set m defined by m (ξ) := m (-ξ). Clearly, m is also a multiplier on Ẇ l,∞ R d , with the same norm as m.

It follows that

∂ l 1 T m ϕ L ∞ (R d ) ≤ T m ∇ l ϕ L ∞ (R d ) , (6)
for any ϕ ∈ C ∞ c (R 2 ). Consider now the normed subspace

V := ∇ l ϕ | ϕ ∈ C ∞ c R d ⊂ C 0 R d β , endowed with the norm induced by C 0 R d β , where β := # α ∈ N d | |α| = l .
Let ρ be as in the proof of Lemma 6. We consider the linear functional L ρ : V → R defined by

L ρ ∇ l ϕ := ρ, ∂ l 1 T m ϕ , ∀ϕ ∈ C ∞ c R d .
Thanks to ( 6), L ρ is well-defined and bounded on V and

L ρ ≤ T m ρ L 1 (R d ) .
Using the Hahn-Banach theorem, we obtain a bounded extension

L ρ of L ρ to C 0 R d β .
Moreover, we can choose

L ρ ∈ C 0 R d β * = M R d β such that its norm equals L ρ . Let (µ α ) |α|=l ∈ M R d β be an element representing L ρ . We have that µ α M (R d ) ≤ T m ρ L 1 (R d ) , (7)
for any multiindex α, with |α| = l. Also, we have

∂ l 1 T m ρ, ϕ = (-1) l ρ, ∂ l 1 T m ϕ = (-1) l L ρ ∇ l ϕ = (-1) l L ρ ∇ l ϕ = (-1) l |α|=l µ α , ∇ α ϕ = |α|=l ∇ α µ α , ϕ , i.e., ∂ l 1 T m ρ = |α|=l ∇ α µ α , (8)
in the sense of tempered distributions on R d .

Taking the Fourier transform in (8), we obtain

(iξ 1 ) l m(ξ) ρ(ξ) = |α|=l (iξ) α µ α (ξ) := m 1 (ξ) a.e. on R d . (9)
Similar identities hold for the partial derivatives ∂ l j T m f , j = 2, . . . , d. Noting that each µ α is a continuous function (since µ α is a finite measure), we continue as in the proof of Lemma 6 and find some m

∈ C(R d \ {0}) such that m = m a.e.
The boundedness of m is obtained exactly as in Lemma 6

Remark 9 In the case where

1 < p < ∞, it is not true that if m is a multiplier on Ẇ l,p R d , then m ∈ C b R d \ {0} . For example if m(ξ) := isgn (ξ 1 )
, then T m is the Hilbert transform on the first coordinate and hence m is a multiplier of any space Ẇ l,p R d , with 1 < p < ∞. However, m is singular on the whole hyperplane {ξ 1 = 0}. Also, if p = 2, any bounded measurable function is a multiplier. Hence, in this case, the multiplier may be even less regular.

It remains to study, in Ẇ l,1 and Ẇ l,∞ , the continuity of the multipliers at the origin.

Almost radial limits

Following [3, Section 2], we will say that a function f : R d \ {0} → C has almost radial limits1 at the origin if the following condition is satisfied.

If (v n ) n≥1 , (w n ) n≥1 ⊂ R d \ {0} are two sequences converging to 0 such that (f (v n )) n≥1 , (f (w n )) n≥1 are convergent sequences and lim n→∞ f (v n ) = lim n→∞ f (w n ), then lim inf n→∞ v n |v n | - w n |w n | > 0. (I)
Note that, if (I) does not hold for f = m, which is bounded, then there exists a sequence

(v n ) n≥1 ⊂ R d \ {0}, converging to 0 and such that v n |v n | → ν ∈ S d-1 , m(v 2n ) → b 1 , m(v 2n+1 ) → b 2 , with b 1 , b 2 ∈ C, b 1 = b 2 .
By considering the possible limits (up to subsequences) of (m(-v 2n )) and (m(-v 2n+1 )), we obtain the following. If m : R d \ {0} → C is a bounded function which does not have almost radial limits at the origin, then there exists a sequence (v n ) n≥1 ⊂ R d \ {0}, converging to 0, and such that (at least) one of the two happens:

v n |v n | → ν ∈ S d-1 , m(v 2n ) → b 1 , m(-v 2n ) → b 1 , m(v 2n+1 ) → b 2 , m(-v 2n+1 ) → b 2 , with b 1 , b 2 ∈ C, b 1 = b 2 , (IIs)
or

v n |v n | → ν ∈ S d-1 , m(v n ) → b 1 , m(-v n ) → b 2 , with b 1 , b 2 ∈ C, b 1 = b 2 . (IIa)
We will refer to the first case as the symmetric case, and to the second as the asymmetric case.

The plan of the proofs of Theorems 3 and 4 consists of establishing the desired results separately in cases (I), (IIs) and (IIa). In case (I), the proof relies on Theorem 1 or on its Ẇ l,∞ variant, Theorem 10 below.

4 Proof of Theorems 3 and 4 in case (I) 4.1 The case of Ẇ l,1 R d First, as in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF], we observe that a bounded function having almost radial limits at 0 also has (genuine) radial limits at 0, and therefore we may define the function

Ω (ξ) := lim n→∞ m (ξ/n) . ( 10
)
Clearly, the function Ω is homogeneous of degree 0. Now, one can easily see that Ω is a multiplier on Ẇ l,1 R d . Indeed, let f be a Schwartz function and let ψ be an arbitrary Schwartz function with ψ L ∞ ≤ 1. Thanks to (2) we have, for any multiindex α with |α| = l and any n ≥ 1,

∇ α T m 1/n f, ψ ≤ T m f Ẇ l,1 .
On the other hand, the dominated convergence theorem gives, with

c = c α,d := ı |α| (2π) -d , ∇ α T m 1/n f, ψ = c R d ξ α m (ξ/n) f (ξ) ψ (ξ) dξ → c R d ξ α Ω (ξ) f (ξ) ψ (ξ) dξ = ∇ α T Ω f, ψ , and hence, | ∇ α T Ω f, ψ | ≤ T m f Ẇ l,1 .
By Lemma 6, we have that Ω ∈ C R d \ {0} . We are now in position to apply Theorem 1 and obtain that Ω is constant. From this and condition (I), we deduce that m is continuous at the origin.

The case of Ẇ l,∞ R d

As above, we conclude that the function Ω defined by ( 10) is a multiplier on Ẇ l,∞ R d . In particular, by Lemma 8, we have that Ω ∈ C R d \ {0} . In order to complete (as above) the proof in this case, it suffices to establish the following analogue of Theorem 1.

Theorem 10 Let d ≥ 2 and l ≥ 1 be some integers and let

Ω ∈ C R d \ {0} ; C be homogeneous of degree zero. If Ω is a multiplier on Ẇ l,∞ R d , then Ω is a constant.
Proof of Theorem 10. We adapt the arguments from [START_REF] De Leeuw | A priori estimates for differential operators in L ∞ norm[END_REF]. As in the proof of Lemma 8, for any Schwartz function ρ with integral 1, one can find some finite measures µ α such that

∂ l 1 T Ω ρ = |α|=l ∇ α µ α , (11)
where Ω (ξ) := Ω (-ξ). Now, if ϕ ∈ S R d and ϕ ε (x) := ϕ(εx) for some ε > 0, then

∂ l 1 T Ω ϕ ε (x) = ε l ∂ l 1 T Ω ϕ (εx), ( 12
)
since Ω is homogeneous of degree zero.

Combining (11) and ( 12), we find that

ε l R d ρ(x) ∂ l 1 T Ω ϕ (εx) dx = ε l |α|=l R d (∇ α ϕ) (εx) dµ α (x). ( 13
)
Since Ω is bounded, ∂ l 1 T Ω ϕ is the inverse Fourier transform of an L 1 function and hence, ∂ l 1 T Ω ϕ is continuous and bounded. Dividing both sides in (13) by ε l and taking ε → 0, we get by the dominated convergence theorem,

∂ l 1 T Ω ϕ (0) = |α|=l µ α (R d ) (∇ α ϕ) (0),
for any ϕ ∈ S R d . This implies that

∂ l 1 T Ω ϕ (x) = |α|=l µ α (R d ) (∇ α ϕ) (x),
for any ϕ ∈ S R d and any x ∈ R d . Hence, by taking the Fourier transform, we get

ξ l 1 Ω (ξ) ϕ (ξ) = |α|=l µ α (R d ) ξ α ϕ (ξ) ,
and we have

ξ l 1 Ω (ξ) = |α|=l µ α (R d ) ξ α =: p 1 (ξ) .
We can write

Ω (ξ) = p 1 (ξ) ξ l 1 , (14)
as an equality of two continuous functions in the domain where ξ 1 = 0. Similarly, there exists a homogeneous polynomial p d of degree l such that

Ω (ξ) = p d (ξ) ξ l d , (15)
as an equality of two continuous functions in the domain where ξ d = 0. From ( 14) and (15), we get

ξ l d p 1 (ξ) = ξ l 1 p d (ξ) everywhere in R d . ( 16
)
By identifying the coefficients in (16), we see that p 1 must be a multiple of ξ l 1 , thus a constant multiple of ξ l 1 (since p 1 is of degree l). Going back to (14), we find that Ω is constant in the region {ξ 1 = 0}. Similarly, Ω is constant in the region {ξ d = 0}, and thus constant (by arguing similarly in other directions).

From now on, we investigate cases (IIs) and (IIa), which are more involved.

Proof of Theorem 3 in case II

We argue by contradiction. We assume that m is not continuous in 0 and we show that (1) does not hold. The following easy lemma will enable us to replace some estimates involving Schwartz functions with similar estimates involving instead functions which are linear combinations of some exponentials. This last type of functions will be used to explicitly construct a sequence of functions violating [START_REF] Bonami | Nonmultipliers of the Sobolev spaces W k,1 R d[END_REF].

Lemma 11 Let m be a multiplier on Ẇ l,∞ R d for some integer l ≥ 0. Consider the set of functions

P m := n j=1 c j e i •,q j n ∈ N * , q 1 , ..., q n ∈ R d \ {0} and c 1 , ..., c n ∈ C .
Let T m : P m → P m be defined by

T m n j=1
c j e i •,q j := n j=1 c j m(q j )e i •,q j , for any n ∈ N * , q 1 , ..., q n ∈ R d \ {0} and c 1 , ..., c n ∈ C.

We have that

T m f Ẇ l,∞ (R d ) ≤ T m f Ẇ l,∞ (R d ) , (17)
for any function f ∈ P m .

Remark 12 Note that, since the exponentials e i •,q j are linearly independent and P m is formed only with (finite) linear combinations of these exponentials, the definition of T m is correct.

Proof. We note that at this point we know that m is continuous and bounded on R d \ {0}. This will be used in the proof below.

Consider a function η ∈ C ∞ c R d whose integral is 1. Fix q ∈ R d \ {0}.
For any small ε > 0, we set ϕ ε q (t) := e i t,q η (εt) on R d . Since ϕ ε q is a Schwartz function, we have

T m ϕ ε q (ξ) = m(ξ)ϕ ε q (ξ) ,
in the sense of tempered distributions on R d . A direct computation gives

ϕ ε q (ξ) = R d e -i t,ξ-q η(εt)dt = 1 ε d R d e -i t, ξ-q ε η(t)dt = 1 ε d η ξ -q ε = (2π) d ε d η q -ξ ε .
Hence,

T m ϕ ε q (ξ) = m(ξ) (2π) d ε d η q -ξ ε , (18)
Note that, since m ∈ L ∞ , the right hand side of (18) is L 1 ; we obtain using the Fourier inversion formula,

T m ϕ ε q (t) = R d e i t,ξ m(ξ) 1 ε d η q -ξ ε dξ = R d e i t,q-εξ m(q -εξ)η(ξ)dξ, (19)
in the sense of tempered distributions. We naturally identify T m ϕ ε q with the right hand side of (19). Now we can prove (17). Let f ∈ P m , with

f (t) = n j=1
c j e i t,q j . Using (19) we have

∇ l n j=1 c j R d
m(q j -εξ)e i t,q j -εξ η(ξ)dξ

L ∞ t = ∇ l T m n j=1 c j ϕ ε q j (t) L ∞ t ≤ T m ∇ l (f (t) η(εt)) L ∞ t .
In other words, for every multiindex α ∈ N d with |α| = l, n j=1 c j R d (q j -εξ) α m(q j -εξ)e i t,q j -εξ η(ξ)dξ

L ∞ t ≤ T m ∇ l f (t) η(εt) L ∞ t + ε T m C f,η ,
where C f,η is a finite constant only depending on f and η. Letting ε → 0 we find that:

∇ α n j=1 c j m(q j )e i t,q j L ∞ t = n j=1 c j q α j m(q j )e i t,q j L ∞ t ≤ T m ∇ l f L ∞ .
Here, we use the fact that η(0) = 1 and the obvious fact that

∇ l f (t) η(0) L ∞ t ≤ lim inf ε→0 ∇ l f (t) η(εt) L ∞ t .
The proof of Lemma 11 is complete.

Remark 13 For simplicity, from now on, we will denote both operators T m and T m by T m . We keep this convention even in the case where p = 1. As we will see this will turn out to be convenient in some computations.

The symmetric case, (IIs)

In what follows we suppose for simplicity that d = 2. Also, we suppose without loss of generality that b 1 = 1, b 2 = 0 and ν = (1, 0). This is possible thanks to the rotation and dilation invariance we have discussed.

We will need the following simple lemma (see the Appendix).

Lemma 14 Fix N ∈ N * . There exists a finite sequence

(σ k ) 1≤k≤N in {0, 1} such that N k=1 σ k k k-1 j=1 1 + i j ≥ 1 π ln N . ( 20
)
Suppose N ∈ N * is fixed and σ 1 , ..., σ N ∈ {0, 1} are some fixed numbers such that inequality (20) holds. We construct, by backward induction on k, a sequence (a k ) 1≤k≤N in R 2 satisfying the following properties: (P1) for each k ∈ {1, ..., N } we have

m ε k a k + 1≤j≤k-1 ε j a j -σ k < 1 4 N , for all ε 1 , ..., ε k ∈ {-1, 0, 1} with ε k = 0; (
ε j a j (1) , a k (2) + 1≤j≤k-1 ε j a j (2) < 1,
for all ε 1 , ..., ε k-1 ∈ {-1, 0, 1};

(P4) for each k ∈ {1, ..., N } we have

a k (2) + 1≤j≤k-1 ε j a j (2)
a k (1) + 1≤j≤k-1 ε j a j (1) < 1 4 N , for all ε 1 , ..., ε k-1 ∈ {-1, 0, 1}.
(A similar construction appears in [3, Subsection 2.2] and in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF].) We proceed by backward induction; we start by constructing a N , then a N -1 , ending eventually with a 1 . At the step N -k we construct the vector a k taking into account the values a N , a N -1 ,....,a k+1 . More precisely, the construction goes as follows. We first modify the sequence (v n ) n≥1 in (IIs) such that v n (2) = 0 for all n ≥ 1. This is possible, since m is continuous on R 2 \ {0}. At each step we choose a k to be a term in the set {v n | n ≡ 0 ( mod 2)} or {v n | n ≡ 1 ( mod 2)} if σ k = 1 or σ k = 0 respectively (recall that b 1 = 1 and b 2 = 0). It remains to see that at each step the term a k can be chosen sufficiently small in order to satisfy the above conditions. Since v n → 0, we can choose a vector a k with both components nonzero and such that |m(ε k a k ) -σ k | < (1/2) 4 -N , for any ε k ∈ {-1, 1}. Since m is continuous outside the origin, there exists r k > 0 such that |m(ξ) -σ k | < 4 -N for any ξ ∈ B(a k , r k ) ∪ B(-a k , r k ). Hence, if a k-1 , ..., a 1 are sufficiently small, then (P1) is satisfied. We have that ν = (1, 0), and hence if a k = v n , for n sufficiently large, and a k-1 , ..., a 1 are sufficiently small, then (P4) is satisfied. It is easy to see that the remaining conditions can be satisfied too.

Consider the set Λ

N := N k=1 ε k a k ε 1 , ..., ε N ∈ {-1, 0, 1} , not all 0 . ( 21
)
Thanks to (P2), for each q ∈ Λ N the representation

q = N k=1 ε k a k , for some ε 1 , ..., ε N ∈ {-1, 0, 1} ,
is unique. Let us also observe that, for each q ∈ Λ N we have (from (P3) and (P4)),

0 < |q(1)| , |q(2)| < 1 (22) and |q(2)| |q(1)| < 1 4 N . ( 23
)
Define the function

R N (t) := -1 + N k=1 1 + i k cos t, a k , t ∈ R 2 . ( 24
)
By (57) (see the Appendix below),

R N (t) = N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k = q∈Λ N c q e i t,q , (25)
for some coefficients c q with |c q | ≤ 1. Thanks to (22) we have that q(1) = 0, for any q ∈ Λ N . This allows us to define the function

h N (t) := q∈Λ N c q (q(1)) l e i t,q , on R 2 . ( 26
)
We claim that

∇ l h N L ∞ (R 2 ) ≤ 4. ( 27
)
Indeed, we have

∂ l 1 h N L ∞ (R 2 ) = R N L ∞ (R 2 ) ≤ 1 + N k=1 1 + 1 k 2 1/2 ≤1 + N k=1 e 1/2k 2 ≤ 1 + e π 2 /12 ≤ 4. ( 28
)
On the other hand, if l 1 ,l 2 are nonnegative integers with l 1 + l 2 = l and l 1 < l, we have (using ( 22), ( 23)),

∂ l 1 1 ∂ l 2 2 h N L ∞ (R 2 ) = q∈Λ N (q(2)) l 2 (q(1)) l-l 1 c q e i t,q L ∞ (R 2 ) ≤ q∈Λ N |q(2)| l 2 |q(1)| l-l 1 = q∈Λ N |q(2)| |q(1)| l 2 ≤ q∈Λ N 4 -N l 2 ≤ |Λ N | 4 -N ≤ 3 N 4 -N ≤ 1.
We are now going to estimate

T m h N Ẇ l,∞ (R 2 )
. Since by (26), h N ∈ P m , with P m as in Lemma 11, we may define T m h N via Lemma 11 (see Remark 13). More specifically, we will prove that

T m h N Ẇ l,∞ (R 2 ) ≥ 1 π ln N -1. ( 29
)
In order to see this, it suffices to prove that

∂ l 1 T m h N L ∞ (R 2 ) ≥ 1 π ln N -1. ( 30
)
We have

∂ l 1 T m h N L ∞ (R 2 ) = T m ∂ l 1 h N L ∞ (R 2 ) = T m R N L ∞ (R 2 ) .
Using (25), we obtain

T m R N (t) = N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 m(ε 1 a 1 + ...ε k a k )   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k . ( 31
)
Introducing the function

Z(t) := N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 σ k   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k , ( 32
)
we have (by the identities ( 57) and ( 59) in the Appendix),

Z(t) = N k=1 iσ k k cos t, a k k-1 j=1 1 + i j cos t, a j .
Lemma 14 yields

Z L ∞ (R 2 ) ≥ |Z(0)| ≥ 1 π ln N . ( 33
)
Also, using property (P1), together with (31) and (32), we get

T m R N -Z L ∞ (R 2 ) ≤ |Λ N | 4 -N ≤ 3 N 4 -N ≤ 1. (34)
Using ( 33), (34) and the triangle inequality, we arrive at

T m R N L ∞ (R 2 ) ≥ Z L ∞ (R 2 ) -T m R N -Z L ∞ (R 2 ) ≥ 1 π ln N -1,
concluding the proof of (30).

By taking N → ∞, (27) and (29) give us that m is not a multiplier on Ẇ l,∞ (R 2 ).

Remark 15 To deal with the case d > 2 we may suppose that ν = (1, 0, ..., 0); we consider constructions like R N ⊗ 1, where R N is defined as above on R 2 and the constant function 1 is defined on R d-2 .

The asymmetric case, (IIa)

This case is very similar to the previous one. We again suppose without loss of generality that b 1 = 1, b 2 = 0 and ν = (1, 0). In a similar way we construct a sequence (a k ) 1≤k≤N satisfying the above properties (P2)--(P4) and (P1') below:

(P1') for each k ∈ {1, ..., N } we have

m ε k a k + 1≤j≤k-1 ε j a j - 1 + ε k 2 σ k < 1 4 N , for all ε 1 , ..., ε k ∈ {-1, 0, 1} with ε k = 0.
With this new sequence (a k ) 1≤k≤N we define Λ N as in (21). We again have ( 22), (23). We also define R N and h N as in ( 24) and ( 26) respectively. The inequality (27) holds in this case too and it remains to show that

∂ l 1 T m h N L ∞ (R 2 ) = T m R N L ∞ (R 2 ) ≥ 1 π ln N -1. (35)
Using (57), we have

T m R N (t) = N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 m(ε 1 a 1 + ...ε k a k )   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k . ( 36
)
Introducing the function

Z(t) := N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 1 + ε k 2 σ k   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k , (37)
we observe that, by ( 57) and (59), we have

Z(t) = N k=1 iσ k k e i t,a k k-1 j=1 1 + i j cos t, a j .
Lemma 14 gives us that

Z L ∞ (R 2 ) ≥ |Z(0)| ≥ 1 π ln N . (38)
Property (P1'), together with (36) and (37), give

T m R N -Z L ∞ (R 2 ) ≤ |Λ N | 4 -N ≤ 3 N 4 -N ≤ 1. (39)
Using (38), (39) and the triangle inequality, we get

T m R N L ∞ (R 2 ) ≥ Z L ∞ (R 2 ) -T m R N -Z L ∞ (R 2 ) ≥ 1 π ln N -1,
concluding the proof of (35). (For the case d > 2, see Remark 15.)

Proof of Theorem 4 in case II

We prove now Theorem 4. Suppose m is not continuous in 0. As in the preceding section, we may assume that m is in one of the cases (IIs) or (IIa). Again we work under the hypothesis

d = 2, b 1 = 1, b 2 = 0 and ν = (1, 0).
Comparing with the strategy we had in the preceding section, there are two main differences. First, since it turns out it is more convenient to work with L ∞ than L 1 , we reformulate the problem by duality. Secondly, we work with genuine trigonometric polynomials on T 2 rather than exponential sums on R 2 . This allows us to use some techniques specific to T 2 that are not available on R 2 .

As in (31) we have

T m R N (t) = N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 m(ε 1 a 1 + ...ε k a k )   ε j =0 i 2j   e i t,ε 1 a 1 +...ε k a k . ( 43
)
For each N we fix a positive integer

M = M (N ) such that M a k ∈ Z 2 for all 1 ≤ k ≤ N . From (43) we get that T m R N is a component-wise 2πM -periodic function. Hence, T m R N (M t) is component-wise 2π-periodic.
We will show that u α in (42) can be chosen to be component-wise 2πM -periodic. In order to prove this we need the following easy lemma.

Lemma 17 Let A > 0 be a real number and suppose u ∈ L ∞ (R 2 ) is given. We consider the sequence of functions

u n (t) := 1 |B n | χ∈Bn u (t + Aχ) , t ∈ R 2 , n ≥ 1,
where

B n := B(0, n) ∩ Z 2 . Then, there exists g ∈ L ∞ (R 2 ), component-wise A-periodic, with g L ∞ (R 2 ) ≤ u L ∞ (R 2 )
and such that u n → g up to a subsequence, in the sense of distributions.

Proof of Lemma 17. Since

u n L ∞ (R 2 ) ≤ u L ∞ (R 2 )
for any n ≥ 1, by the sequential Banach-Alaoglu theorem, there exists

g ∈ L ∞ (R 2 ) with g L ∞ (R 2 ) ≤ u L ∞ (R 2 )
such that u n g in the w * -topology of L ∞ up to a subsequence. (For simplicity we still denote the subsequence by (u n ) n≥1 .) In particular, u n → g in the sense of distributions. Also, we easily get that g is component-wise A-periodic. Indeed, for ϕ ∈ C ∞ c (R 2 ), and any χ 0 ∈ Z 2 , we have

R 2 u n (t + Aχ 0 ) ϕ (t) dt = R 2 u n (t) ϕ (t -Aχ 0 ) dt → R 2 g (t) ϕ (t -Aχ 0 ) dt = R 2 g (t + Aχ 0 ) ϕ (t) dt. (44)
Also,

R 2 u n (t + Aχ 0 ) ϕ (t) dt = 1 |B n | χ∈Bn R 2 u (t + A (χ + χ 0 )) ϕ (t) dt = 1 |B n | χ∈Bn+χ 0 R 2 u (t + Aχ) ϕ (t) dt = 1 |B n | χ∈Bn R 2 u (t + Aχ) ϕ (t) dt + r n = R 2 u n (t) ϕ (t) dt + r n ,
where

r n := 1 |B n | χ∈(Bn+χ 0 )\Bn R 2 u (t + Aχ) ϕ (t) dt - 1 |B n | χ∈Bn\(Bn+χ 0 ) R 2 u (t + Aχ) ϕ (t) dt. Since |(B n + χ 0 ) \B n |, |B n \ (B n + χ 0 )| n and |B n | ∼ n 2 , we have r n → 0. Hence, lim n→∞ R 2 u n (t + Aχ 0 ) ϕ (t) dt = lim n→∞ R 2 u n (t) ϕ (t) dt,
which together with (44) concludes the proof of Lemma 17.

Now, since ∂ l 1 T m R N is component-wise 2πM -periodic, we have ∂ l 1 T m R N = ∂ l 1 T m R N n (with A = 2πM ) for any n ≥ 1. From (42) we get ∂ l 1 T m R N = ∂ l 1 T m R N n = |α|=l ∇ α (u α ) n ,
for any n ≥ 1. Taking n → ∞ and applying Lemma 17 to the functions u α ∈ L ∞ , with A := 2πM , we get

∂ l 1 T m R N = |α|=l ∇ α g α , (45)
for some component-wise 2πM -periodic functions g α ∈ L ∞ (R 2 ) such that

g α L ∞ (R 2 ) ≤ 4 T m (46)
(see ( 41)).

From now on, for each function ψ on R 2 , we write ψ M for the function ψ M (t) := ψ (M t).

Consider the function

G N (t) := -1 + N k=1 (1 + cos t, a k ) , t ∈ R 2 .
Notice that G M N is component-wise 2π-periodic. (We recall here that each M a k belongs to Z 2 .) Also, (T m R N ) M and each g M α are component-wise 2π-periodic. From (45) we get

∂ l 1 (T m R N ) M = |α|=l ∇ α g M α ,
in the sense of distributions on R 2 and hence in the sense of distributions on T 2 . Taking convolution (on the torus T 2 ) with G M N , we get

∂ l 1 (T m R N ) M * G M N = |α|=l ∇ α g M α * G M N . ( 47
)
From the properties of G N it follows that the spectrum of each g M α * G M N and the spectrum of (T m R N ) M * G M N , as functions on the torus T 2 , are included in M Λ N and therefore do not touch the set {0} × Z (see ( 22)). Hence, we can apply the operator ∂ -l 1 in (47) to obtain

(T m R N ) M * G M N = |α|=l ∇ α ∂ -l 1 g M α * G M N .
Hence,

(T m R N ) M * G M N L ∞ (T 2 ) ≤ |α|=l ∇ α ∂ -l 1 g M α * G M N L ∞ (T 2 ) . ( 48
)
We claim that

∇ α ∂ -l 1 g M α * G M N L ∞ (T 2 ) ≤ 8 T m , (49)
for any multiindex α with |α| = l. This estimate is similar to (27).

Indeed, for α = (l, 0) we have, using (46),

∂ l 1 ∂ -l 1 g M α * G M N L ∞ (T 2 ) = g M α * G M N L ∞ (T 2 ) ≤ g M α L ∞ (T 2 ) G M N L 1 (T 2 ) ≤2 g α L ∞ (R 2 ) ≤ 8 T m . (50)
Here, we have used the fact that G M N L 1 (T2 ) ≤ 2. This can be justified as follows. We have N k=1

(1 + cos t, M a k ) ≥ 0 and hence, thanks to (57) and (P2) 2 , we obtain

N k=1 (1 + cos •, M a k ) L 1 (T 2 ) = 1.
We now turn to the proof of (49) for α = (l, 0).

Writing g M α * G M N (t) = q∈Λ N
c q e i t,M q , we get that (note that c q is a Fourier coefficient):

c q ≤ g M α * G M N L ∞ (T 2 ) ≤ 8 T m ,
for all q ∈ Λ N .

Hence, if α = (l 1 , l 2 ), with l 1 + l 2 = l and l 1 < l, we have (using ( 22), (23)),

∂ l 1 1 ∂ l 2 2 ∂ -l 1 g M α * G M N L ∞ (T 2 ) = q∈Λ N (M q(2)) l 2 (M q(1)) l-l 1 c q e i t,M q L ∞ (T 2 ) ≤8 T m q∈Λ N |q(2)| l 2 |q(1)| l-l 1 = 8 T m q∈Λ N |q(2)| |q(1)| l 2 ≤8 T m q∈Λ N 4 -N l 2 ≤ 8 T m |Λ N | 4 -N ≤ 8 T m 3 N 4 -N ≤8 T m . (51)
We see that (50) and (51) imply (49).

We next obtain a contradiction. The starting point is the left-hand side of (48). We claim that

(T m R N ) M * G M N L ∞ (T 2 ) ≥ 1 2π ln N -1. (52)
The method applied to obtain this estimate is similar to the one used to obtain (30). Indeed, by using (57) and (58) (see the Appendix) we have:

(T m R N ) M * G M N (t) = N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 m(ε 1 a 1 + ...ε k a k )   ε j =0 i 4j   e i t,ε 1 M a 1 +...+ε k M a k . (53)
Introducing the function

Z(t) := N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0 σ k   ε j =0 i 4j   e i t,ε 1 M a 1 +...+ε k M a k on T 2 , ( 54
)
we observe that, by ( 57) and (59), we have

Z(t) = N k=1 iσ k 2k cos t, M a k k-1 j=1 1 + i 2j cos t, M a j .
Lemma 16 gives us that

Z L ∞ (T 2 ) ≥ |Z(0)| ≥ 1 2π ln N . (55)
Also, using property (P1), together with (53) and (54), we get

(T m R N ) M * G M N -Z L ∞ (T 2 ) ≤ |Λ N | 4 -N ≤ 3 N 4 -N ≤ 1. (56)
Using (55), (56) and the triangle inequality, we obtain

(T m R N ) M * G M N L ∞ (T 2 ) ≥ Z L ∞ (T 2 ) -(T m R N ) M * G M N -Z L ∞ (T 2 ) ≥ 1 2π ln N -1,
concluding the proof of (52). Now, (48), (49) and (52) allow us to write

1 2π ln N -1 ≤ 8 T m 2 l .
Since N is arbitrary, the last inequality implies that m is not a multiplier on Ẇ l,1 (R 2 ).

The asymmetric case, (IIa)

This case is very similar to the previous one and we skip the proof. We can again suppose by contradiction that m is a multiplier on Ẇ l,1 (R 2 ) and use this result to obtain a representation result similar to the one in (42). The only difference is that now we have to follow the "asymmetric case" as in the proof corresponding to multipliers on Ẇ l,∞ (R 2 ). The functions R N and G N will be constructed as above, starting, as in the case of Ẇ l,∞ (R 2 ), with a sequence (a k ) 1≤k≤N in Q 2 satisfying the conditions (P1'), (P2)-(P4).

7 Appendix

Some useful identities

We quickly recall here some elementary facts and formulas concerning some trigonometric polynomials on the torus.

Fix a finite sequence (a k ) 1≤k≤N in Z d . For each finite sequence α 1 , ..., α N of complex numbers we have the following expansion rule:

N k=1 (1 + α k cos t, a k ) = 1 + N k=1 ε 1 ,...,ε k ∈{-1,0,1} ε k =0   ε j =0 α j 2
  e i t,ε 1 a 1 +...+ε k a k . (57)

A sequence (a k ) k=1,N in Z d will be called dissociated if the only solution to the equation

ε 1 a 1 + ... + ε N a N = ε 1 a 1 + ... + ε N a N ,
with ε 1 , ..., ε N , ε 1 , ..., ε N ∈ {-1, 0, 1} is the trivial solution ε 1 = ε 1 , ..., ε N = ε N . For example any sequence (a k ) 1≤k≤N in Z d which is lacunary on at least one component is dissociated. If (a k ) 1≤k≤N is dissociated and α 1 , ..., α N and β 1 , ..., β N are complex numbers, by using (57) and the relation between convolution and the Fourier transform, we obtain that

N k=1 (1 + α k cos •, a k ) * N k=1 (1 + β k cos •, a k ) = N k=1 1 + α k β k 2 cos •, a k , (58)
as functions on the d-dimensional torus.

The following identity is also useful. We have (1 + c j ) (59)

for any complex numbers c 1 ,..., c N .

A selection lemma

The following interesting fact is taken from [START_REF] Rudin | Real and Complex Analysis[END_REF] (Lemma 6.3, p. 118).

Lemma 18 Suppose z 1 ,..., z N are some complex numbers. Then, there exist σ 1 ,..., σ N ∈ {0, 1} such that

N k=1 σ k z k ≥ 1 π N k=1 |z k | .
The proof is elementary and we skip it.

Let us define two sequences (z 0 k ) 1≤k≤N and (z 1 k ) 1≤k≤N by the expressions

z β k := 1 2 β k k-1 j=1 1 + i 2 β j
for k = 1, ..., N , where β = 0, 1 is an index. Here, the product over an empty set is by convention equal to 1.

It is easy to see that, using Lemma 18 applied to the sequence (z 0 k ) 1≤k≤N we get Lemma 14. Similarly, using Lemma 18 applied to the sequence (z 1 k ) 1≤k≤N we get Lemma 16. (1 + cos t, a j ) , which concludes the proof.

Remark 21 In fact, it is possible to prove Lemma 20 (up to a multiplicative constant) without using Lemma 18. Indeed, the sequence (z k ) 1≤k≤N defined in (61) has a quite simple form: the argument of z k is (k -1) θ N (mod 2π), where θ N := arctan 1/2 √ N . One can choose the sequence (σ k ) 1≤k≤N explicitly: σ k = 1, if -π/4 ≤ (k -1) θ N (mod 2π) ≤ π/4, and σ k = 0, otherwise.

  P2) for each k ∈ {1, ..., N -1} we have 4 |a k (1)| < |a k+1 (1)| and 4 |a k (2)| < |a k+1 (2)| ; (Here, a k (1) and a k (2) are the two coordinates of a k . ) (P3) for each k ∈ {1, ..., N } we have 0 < a k (1) + 1≤j≤k-1

{ε 1 a 1 +

 1 defined on T d . Note that, by (58), we haveg N * G N (t) = ... + ε k a k | ε 1 , ..., ε k ∈ {-1, 0, 1} , ε k = 0}and the projection P A defined byP A f (n) = f (n) if n ∈ A and P A f (n) =0 otherwise, for any trigonometric polynomial f on T d . Observe that (59) and (63) giveP A (g N * G N ) (t) = a j ,and thanks to (62),|P A (g N * G N ) (0G N L 1 (T d ) ≥ g N L ∞ (T d ) P A G N L 1 (T d ) ≥ | g N , P A G N | = |P A (g N * G N ) (0)| ≥ √ N 2π .It remains to observe that,P A G N (t) = N k=1 σ k cos t, a k k-1 j=1

Actually, the definition given in[START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF] Section

2] is slightly different. However, this aspect is unimportant in what follows.

Note that (P2) implies the fact that the sequence (M a k ) 1≤k≤N is dissociated. See the Apendix below.

Acknowledgements

The author thanks Petru Mironescu and Emmanuel Russ for useful discussions and suggestions. He was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

The symmetric case, (IIs)

We need the following analogue of Lemma 14 (see the Appendix):

Lemma 16 Fix N ∈ N * . There exists a finite sequence (σ k ) 1≤k≤N in {0, 1} such that

In what follows (a k ) 1≤k≤N is a sequence in Q 2 satisfying the properties (P1)--(P4) for the sequence (σ k ) 1≤k≤N from Lemma 16 above. It is easy to see that such a sequence exists. Using this sequence we construct the function R N as in (24).

Suppose that m is a multiplier on Ẇ l,1 (R 2 ). Then m defined by m (ξ) := m (-ξ) is also a multiplier on Ẇ l,1 (R 2 ) with the same norm as m. It follows that

for any ϕ ∈ C ∞ c (R 2 ). Consider now the normed subspace

endowed with the norm induced by (L 1 (R 2 ))

. We consider the linear functional L N : V → R defined by

Note that L N is well-defined on V . Thanks to (40), L N is bounded on V . Using the Hahn-Banach theorem, we get that there exists a bounded extension

such that its norm equals L N . Note that, by (28),

be the element representing L N , where α ∈ N 2 are multiindexes. We have that

for any multiindex α, with |α| = l. Also, we have (see Remark 13)

in the sense of distributions on R 2 .

Remarks on Wojciechowski's inequality

We discuss here some inequalities in the spirit of Lemma 14 and Lemma 16. Wojciechowski was the first one to use such inequalities in the proof of non-estimates. In particular, he obtained in [START_REF] Wojciechowski | On the strong type multiplier norms of rational functions in several variables[END_REF] the following relatively difficult estimate (see [

9, Lemma 1], [10, Lemma 1]):

Lemma 19 There exists a constant C > 0 such that, for any integer N ≥ 2 there exist M = M (N ) and a sequence σ 1 ,..., σ N ∈ {0, 1} such that

whenever the sequence

This lemma was already used in conjunction with the Riesz products technique [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF], [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF], [START_REF] Kazaniecki | Ornstein's non-inequalities: Riesz product approach[END_REF]. Lemma 19 was used in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF] to prove that there exists g ∈ L 1 (T 2 ) such that there are no

It was also used in [START_REF] Kazaniecki | Ornstein's non-inequalities: Riesz product approach[END_REF] in order to prove some anisotropic Ornstein-type non-inequalities and in [START_REF] Kazaniecki | On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 1,1 R d[END_REF] to study the continuity of the multipliers on Ẇ 1,1 R d .

Here we want to point out that, in the above-mentioned applications, a weaker form suffices: we only need to know that the lower bound in (60) goes to ∞ when N → ∞. (In the case of the application of Lemma 19 given in [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF], this was observed by Wojciechowski [START_REF] Wojciechowski | On the representation of functions as a sum of derivatives[END_REF]Remark 1].) This weaker version can be achieved by much cheaper arguments than the ones used to obtain Lemma 19. In this direction we mention the following.

Lemma 20 For any integer N ≥ 2 there exists a sequence σ 1 ,..., σ N ∈ {0, 1} such that

Proof. The proof follows the ideas in [START_REF] Curcȃ | The divergence equation with L ∞ source[END_REF]. By applying Lemma 18 to the sequence (z k ) 1≤k≤N , where

we obtain a sequence (σ k ) 1≤k≤N in {0, 1} such that (1 + cos t, a k ) ,