G Divyang
email: divyang@tifrbng.res.in

R Émi Bhimani

Carles

NORM INFLATION FOR NONLINEAR SCHR ÖDINGER EQUATIONS IN FOURIER-LEBESGUE AND MODULATION SPACES OF NEGATIVE REGULARITY

Keywords: 2010 Mathematics Subject Classification. 35Q55, 42B35 (primary), 35A01 (secondary) Nonlinear Schrödinger equations, Ill-posedness, Fourier-Lebesgue spaces, modulation spaces

We consider nonlinear Schrödinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schrödinger equations in a general abstract functional setting.

where ψ = ψ(t, x) ∈ C, σ ∈ N, µ ∈ {1, -1}. We prove some ill-posedness results in Fourier-Lebesgue and modulation spaces, involving negative regularity in space.

We recall the notion of well-posedness in the sense of Hadamard.

Definition 1.1. Let X, Y ֒→ S ′ (R d ) be a Banach spaces. The Cauchy problem for (1.1) is well posed from X to Y if, for all bounded subsets B ⊂ X, there exist T > 0 and a Banach space X T ֒→ C([0, T ], Y ) such that:

(i) For all φ ∈ X, (1.1) has a unique solution ψ ∈ X T with ψ |t=0 = φ.

(ii) The mapping φ → ψ is continuous from (B, • X ) to C([0, T ], Y ).

The negation of the above definition is called a lack of well-posedness or instability. In connection with the study of ill-posedness of (1.1) and nonlinear wave equations Christ, Colliander, and Tao introduced in [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] the notion of norm inflation with respect to a given (Sobolev) norm, saying that there exist a sequence of smooth initial data (ψ n (0)) n≥1 and a sequence of times (t n ) n≥1 , both converging to 0, so that the corresponding smooth solution ψ n , evaluated at t n , is unbounded (in the same space).

The solutions to (1.1) is invariant under the scaling transformation (1.2) ψ(t, x) → λ 1/σ ψ λ 2 t, λx , λ > 0.

The homogeneous Sobolev space Ḣs (R d ) is invariant exactly for s = s c , where

s c = d 2 - 1 σ .
Another important invariance of (1.1) is the Galilean invariance: if ψ(t, x) solves (1.1), then so does

(1.
3) e iv•x-i|v| 2 t/2 ψ(t, x -vt)

for any v ∈ R d . This transform does not alter the L 2 (R d ) norm of the function.

From these two invariances, well-posedness is not expected to hold in H s (R d ) as soon as s < max(0, s c ). In this paper, we consider the case of negative regularity, s < 0, in Fourier-Lebesgue and modulation spaces, instead of Sobolev spaces. Kenig, Ponce and Vega [START_REF] Kenig | On the ill-posedness of some canonical dispersive equations[END_REF] established instability for the cubic NLS in H s (R) for s < 0. Christ, Colliander and Tao [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] generalized this result in H s (R d ) for s < 0 and d ≥ 1. In the periodic case x ∈ T d , instability in H s (T d ) for s < 0 was established in [START_REF]Instability of the periodic nonlinear Schrödinger equation[END_REF] (d = 1) and [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF] (d ≥ 1). Stronger results for the cubic NLS on the circle were proven by Molinet [START_REF] Molinet | On ill-posedness for the one-dimensional periodic cubic Schrödinger equation[END_REF]. In [START_REF] Oh | A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces[END_REF]Theorem 1.1], Oh established norminflation for (1.1) in the cubic case σ = 1, in H s (R) for s ≤ -1/2 and in H s (R d ) for s < 0 if d ≥ 2. He actually proved that the flow map fails to be continuous at any function in H s , for s as above. Norm inflation in the case of mixed geometries, x ∈ R d × T n , for sharp negative Sobolev regularity in (1.1), is due to Kishimoto [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF], who also considers nonlinearities which are not gauge invariant.

The general picture to prove ill-posedness results is typically as following, as explained in e.g. [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF]: at negative regularity, one relies on a transfer from high frequencies to low frequencies, while to prove ill-posedness at positive regularity, one uses a transfer from low frequencies to high frequencies. In particular, the proofs are different whether a negative or a positive regularity is considered.

Stronger phenomena than norm inflation have also been proved, showing that the flow map fails to be continuous at the origin from H s to H k even for (some) k < s, and so a loss of regularity is present. This was proven initially for 0 < s < s c by Lebeau [START_REF] Lebeau | Perte de régularité pour les équations d'ondes sur-critiques[END_REF] in the case of the wave equation, then in [START_REF] Carles | Geometric optics and instability for semi-classical Schrödinger equations[END_REF] (cubic nonlinearity) and [START_REF] Alazard | Loss of regularity for super-critical nonlinear Schrödinger equations[END_REF][START_REF] Thomann | Instabilities for supercritical Schrödinger equations in analytic manifolds[END_REF] for NLS. In the case of negative regularity, an infinite loss of regularity was established in [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF] for (1.1) in H s (R d ) (d ≥ 2 and s < -1/(2σ + 1)), and in the periodic case x ∈ T d in [START_REF] Carles | Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces[END_REF], in Fourier-Lebesgue spaces. Typically, the NLS flow map fails to be continuous at the origin from H s (R d ) to H k (R d ), for any k ∈ R.

1.2. Fourier-Lebesgue spaces. The Fourier-Lebesgue space F L p s (R d ) is defined by

F L p s (R d ) = f ∈ S ′ (R d ) : f F L p s := f • s L p < ∞ ,
where the Fourier transform is defined as

f (ξ) = F f (ξ) = 1 (2π) d R d e -ix•ξ f (x)dx, f ∈ S(R d ),
and where 1 ≤ p ≤ ∞, s ∈ R, and ξ s = (1 + |ξ| 2 ) s/2 (ξ ∈ R d ). For p = 2, F L 2 s = H s the usual Sobolev space. For s = 0, we write F L p 0 (R d ) = F L p (R d ).

The scaling (1.2) leaves the homogeneous F Lp s (R d )-norm (replace the Japanese bracket

• s with the length |•| s in the definition of F L p s (R d )) invariant for s = s c (p)
, where

s c (p) := d 1 - 1 p - 1 σ .
Of course when p = 2, we recover the previous value s c . On the other hand, the Galilean transform (1.3) does not alter the F L p (R d ) norm of ψ, and so wellposedness is not expected to hold in F L p s for s < max(0, s c (p)). Note however that the recent results from [START_REF] Harrop-Griffiths | Sharp well-posedness for the cubic NLS and mKdV in H s (R)[END_REF] show that this heuristical argument is not always correct: in the case p = 2, d = 1 = σ, well-posedness may hold for min(0,

s c (2)) = -1 2 < s < max(0, s c (2)) = 0. Therefore, s < min(0, s c (p)
) is a safer assumption to obtain ill-posedness results. In this paper, we consider cases where s < 0.

In [24, Theorem 1], Hyakuna-Tsutsumi established local well-posedness for the cubic NLS in

F L p (R d ) for p ∈ (4/3, 4) \ {2}. Later this result is generalized in [23, Theorem 1.1] for p ∈ [1, 2].
Our first results concern norm inflation of the type discussed above:

Theorem 1.2. Assume that 1 ≤ p ≤ ∞, d, σ ∈ N and s < min (0, s c (p)).
For any δ > 0, there exists ψ 0 ∈ F L p s (R d ) and T > 0 satisfying ψ 0 F L p s < δ and 0 < T < δ, such that the corresponding solution ψ to (1.1) exists on [0, T ] and

ψ(T ) F L p s > δ -1 .
As discussed above, in the case s c (p) > 0, norm inflation is expected in F L p s (R d ) for 0 < s < s c (p), but with different arguments. The proof of Theorem 1.2 is inspired by the two-scale analysis of Kishimoto [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF]. We also prove norm inflation with an infinite loss of regularity: the initial regularity must be sufficiently small, and we leave out the cubic one-dimensional nonlinearity.

Theorem 1.3. Let σ ∈ N, s < -1
2σ+1 and assume dσ ≥ 2. There exist a sequence of initial data (ψ n (0)) n≥1 in S(R d ) such that

ψ n (0) F L p s -→ n→∞ 0, ∀p ∈ [1, ∞],
and a sequence of times t n → 0 such that the corresponding solutions

ψ n to (1.1) satisfies ψ n (t n ) F L p k -→ n→∞ ∞, ∀k ∈ R, ∀p ∈ [1, ∞].
Remark 1.4. There is no general comparison between the assumptions on s in Theorems 1.2 and 1.3: for p = 1, min(0, s c (1)) = -1/σ < -1/(2σ + 1), while if s c (p) ≥ 0, we obviously have min(0, s c (p)) = 0 > -1/(2σ + 1).

1.3. Modulation spaces. We now turn our attention to the theory of modulation spaces. The idea of modulation spaces is to consider the decaying properties of space variable and its Fourier transform simultaneously. Specifically, we consider the short-time Fourier transform (STFT) (sliding-window transform/wave packet transform) of f with respect to Schwartz class function g ∈ S(R d ):

V g f (x, ξ) = R d f (t)g(t -x)e -iξ•t dt, (x, ξ) ∈ R 2d ,
whenever the integral exists. Then the modulation spaces

M p,q s (R d ) (1 ≤ p, q ≤ ∞, s ∈ R) is defined as the collection of tempered distributions f ∈ S ′ (R d ) such that f M p,q s = V g f L p x (1 + |ξ| 2 ) s/2 L q ξ < ∞,
with natural modification if a Lebesgue index is infinite. For s = 0, we write M p,q 0 (R d ) = M p,q (R d ). When p = q = 2, modulation spaces coincide with usual Sobolev spaces H s (R d ). For the last two decades, these spaces have made their own place in PDEs and there is a tremendous ongoing interest to use these spaces as a low regularity Cauchy data class for nonlinear dispersive equations; see e.g. [START_REF] Baoxiang | Isometric decomposition operators, function spaces E λ p,q and applications to nonlinear evolution equations[END_REF][START_REF] Bényi | Local well-posedness of nonlinear dispersive equations on modulation spaces[END_REF][START_REF] Ruzhansky | Modulation spaces and nonlinear evolution equations[END_REF][START_REF] Bhimani | The Hartree-Fock equations in modulation spaces[END_REF][START_REF] Wang | The global Cauchy problem for the NLS and NLKG with small rough data[END_REF][START_REF] Wang | Harmonic Analysis Method for Nonlinear Evolution Equations[END_REF][START_REF] Oh | Global well-posedness of the one-dimensional cubic nonlinear Schródinger equation in almost critical spaces[END_REF]. Using the algebra property and boundedness of Schrödinger propagator on M p,q s (R d ), (1.1) is proved to be locally well-posed in M p,1 s (R d ) for 1 ≤ p ≤ ∞, s ≥ 0, and in M p,q s (R d ) for 1 ≤ p, q ≤ ∞ and s > d(1 -1/q), via fixed point argument; see [START_REF] Baoxiang | Isometric decomposition operators, function spaces E λ p,q and applications to nonlinear evolution equations[END_REF][START_REF] Bényi | Local well-posedness of nonlinear dispersive equations on modulation spaces[END_REF][START_REF] Bhimani | Functions operating on modulation spaces and nonlinear dispersive equations[END_REF]. Using uniform-decomposition techniques, Wang and Hudzik [START_REF] Wang | The global Cauchy problem for the NLS and NLKG with small rough data[END_REF] established global well-posedness for (1.1) with small initial data in M 2,1 (R d ). Guo [START_REF] Guo | On the 1D cubic nonlinear Schrödinger equation in an almost critical space[END_REF] proved local well-posed for the cubic NLS in M 2,q (R) (2 ≤ q ≤ ∞), and later Oh and Wang [START_REF] Oh | Global well-posedness of the one-dimensional cubic nonlinear Schródinger equation in almost critical spaces[END_REF], established global existence for this result. In [START_REF] Chaichenets | On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space Mp,q(R)[END_REF], Chaichenets et al. established global well-posedness for the cubic NLS in M p,p ′ (R) for p sufficiently close to 2. The well-posedness problems for some other PDEs in M p,q s (R d ) are widely studied by many authors, see for instance the excellent survey [START_REF] Ruzhansky | Modulation spaces and nonlinear evolution equations[END_REF] and references therein. We complement the existing literature on well-posedness theory for (1.1) with Cauchy data in modulation spaces. First, observe that, in view of Proposition 2.6 below,

ψ (λ •)) M 2,q s λ -d 2 max (1, λ s ) ψ M 2,q s , if 1 ≤ q ≤ 2, λ -d(1-1 q ) max (1, λ s ) ψ M 2,q s , if 2 ≤ q ≤ ∞
, for all λ ≤ 1 and s ∈ R. Invoking the general belief that ill-posedness at positive regularity is due to the transfer from low frequencies (0 < λ ≪ 1) to high frequencies, the scaling (1.2) suggests that ill-posedness occurs in M 2,q s (R d ) if

s < s c = d 2 -1 σ if 1 ≤ q ≤ 2, d 1 -1 q if 2 ≤ q ≤ ∞.
The following analogue of Theorem 1.2 then appears rather natural.

Theorem 1.5. Let d, σ ∈ N and assume that

• s < min d 2 -1 σ , 0 when 1 ≤ q ≤ 2, and • s < min d 1 -1 q -1 σ , 0 when 2 ≤ q ≤ ∞.
For any δ > 0, there exists ψ 0 ∈ M 2,q s (R d ) and T > 0 satisfying ψ 0 M 2,q s < δ and 0 < T < δ such that the corresponding solution ψ to (1.1) exists on [0, T ] and

ψ(T ) M 2,q s > δ -1 .
We also have some infinite loss of regularity of the flow map (1.1) at the level of modulation spaces with negative regularity. We no longer assume p = 2, and show a stronger result, provided that the negative regularity s is sufficiently small, and (again) that we discard the one-dimensional cubic case.

Theorem 1.6. Let σ ∈ N, s < -1 2σ+1 and assume dσ ≥ 2. There exists a sequence of initial data (ψ n (0)) n≥1 in S(R d ) such that

ψ n (0) M p,q s -→ n→∞ 0, ∀p, q ∈ [1, ∞],
and a sequence of times t n → 0 such that the corresponding solutions ψ n to (1.1)

satisfies ψ n (t n ) M p,q k -→ n→∞ ∞, ∀k ∈ R, ∀p, q ∈ [1, ∞].
Remark 1.7. Contrary to the Fourier-Lebesgue case, the assumption regarding s is always weaker in Theorem 1.5 than in Theorem 1.6 (recall that the cubic onedimensional case is ruled out in Theorem 1.6).

1.4. Comments and outline of the paper. As pointed out before, the numerology regarding the norm inflation phenomenon (Theorems 1.2 and 1.5) is probably sharp, up to the fact that the minimum should be replaced by a maximum in the assumption on s, and that at positive regularity, different arguments are required.

On the other hand, we believe that the restriction s < -1 2σ+1 in Theorems 1.3 and 1.6 is due to our approach, and we expect that the result is true under the mere assumption s < 0 if dσ ≥ 2, and for s

< -1/2 if d = σ = 1.
The analogue of our results remains true if we replace ∆ by the generalized dispersion of the form ∆ η = d j=1 η j ∂ 2 xj , η j = ±1. The (1.1) associated ∆ η (with the non uniform signs of η j ) arises in the description of surface gravity waves on deep water, see e.g. [START_REF] Sulem | The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse[END_REF].

In [START_REF] Sugimoto | Local well-posedness for the Davey-Stewartson equation in a generalized Feichtinger algebra[END_REF], Sugimoto-Wang-Zhang established some local well-posedness results for Davey-Stewartson equation in some weighted modulation spaces. We note that our method of proof can be applied to get norm-inflation results for Davey-Stewartson equation, and infinite loss of regularity in the spirit of [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF], in some negative modulation and Fourier-Lebesgue spaces.

Theorems 1.3 and 1.6 cover any smooth power nonlinearity in multidimension, and power nonlinearities which are at least quintic in the one-dimensional case. Our method our proof seems too limited to prove loss of regularity in the case of the cubic nonlinearity on the line. It turns out that the method followed to treat the cubic nonlinearity on the circle in [START_REF] Carles | Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces[END_REF] seems helpless in the case of the line. On the other hand, Theorems 1.2 and 1.5 include the cubic one-dimensional Schrödinger equation.

The rest of this paper is organized as follows, In Section 2, we recall various properties associated to modulation spaces. In Section 3, we prove Theorem 1.2, and we adapt the argument in Section 4 to prove Theorem 1.5. In Section 5, we show how the theory of weakly nonlinear geometric optics makes it possible to prove loss of regularity at negative regularity for (1.1). A general framework where multiphase weakly nonlinear geometric optics is justified is presented in Section 6, and it is applied in Section 7 to prove Theorems 1.3 and 1.6.

Notations. The notation A

B means A ≤ cB for a some constant c > 0, Let (Λ ε ) 0<ε≤1 and (Υ ε ) 0<ε≤1 be two families of positive real numbers.

• We write Λ ε ≪ Υ ε if lim sup ε→0 Λ ε /Υ ε = 0. • We write Λ ε Υ ε if lim sup ε→0 Λ ε /Υ ε < ∞. • We write Λ ε ≈ Υ ε if Λ ε Υ ε and Υ ε Λ ε .

Preliminary: modulation spaces

Feichtinger [START_REF] Feichtinger | Modulation Spaces on Locally Compact Abelian Groups[END_REF] introduced a class of Banach spaces, the so-called modulation spaces, which allow a measurement of space variable and Fourier transform variable of a function, or distribution, on R d simultaneously, using the short-time Fourier transform (STFT). The STFT of a function f with respect to a window function g ∈ S(R d ) is defined by

(2.1) V g f (x, y) = R d f (t)g(t -x)e -iy•t dt, (x, y) ∈ R 2d ,
whenever the integral exists. For x, y ∈ R d , the translation operator T x , and the modulation operator M y , are defined by

T x f (t) = f (t -x) and M y f (t) = e iy•t f (t).
In terms of these operators the STFT may be expressed as

(2.2) V g f (x, y) = f, M y T x g = e -ix•w (f * M w g * ) (x),
where f, g denotes the inner product for L 2 functions, or the action of the tempered distribution f on the Schwartz class function g, and g * (y) = g(-y). Thus

V : (f, g) → V g (f ) extends to a bilinear form on S ′ (R d ) × S(R d ), and V g (f ) defines a uniformly continuous function on R d × R d whenever f ∈ S ′ (R d ) and g ∈ S(R d ). Definition 2.1 (Modulation spaces). Let 1 ≤ p, q ≤ ∞, s ∈ R and 0 = g ∈ S(R d ).
The weighted modulation space M p,q s (R d ) is defined to be the space of all tempered distributions f for which the following norm is finite:

f M p,q s = R d R d |V g f (x, y)| p dx q/p (1 + |y| 2 ) sq/2 dy 1/q , for 1 ≤ p, q < ∞. If p or q is infinite, f M p,q
s is defined by replacing the corresponding integral by the essential supremum.

Remark 2.2. The definition of the modulation space given above, is independent of the choice of the particular window function. See [START_REF] Gröchenig | Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis[END_REF]Proposition 11.3.2(c)].

We recall an alternative definition of modulation spaces via the frequency-uniform localization techniques, providing another characterization which will be useful to prove Theorem 1.5. Let Q n be the unit cube with the center at n, so (

Q n ) n∈Z d constitutes a decomposition of R d , that is, R d = ∪ n∈Z d Q n . Let ρ ∈ S(R d ), ρ : R d → [0, 1] be a smooth function satisfying ρ(ξ) = 1 if |ξ| ∞ ≤ 1 2 and ρ(ξ) = 0 if |ξ| ∞ ≥ 1. Let ρ n be a translation of ρ, that is, ρ n (ξ) = ρ(ξ -n), n ∈ Z d . Denote σ n (ξ) = ρ n (ξ) ℓ∈Z d ρ ℓ (ξ) , n ∈ Z d .
Then (σ n (ξ)) n∈Z d satisfies the following properties:

(2.3)

               |σ n (ξ)| ≥ c, ∀ξ ∈ Q n , supp σ n ⊂ {ξ : |ξ -n| ∞ ≤ 1}, n∈Z d σ n (ξ) ≡ 1, ∀ξ ∈ R d , |D α σ n (ξ)| ≤ C |α| , ∀ξ ∈ R d , α ∈ (N ∪ {0}) d .
The frequency-uniform decomposition operators can be exactly defined by

n = F -1 σ n F . For 1 ≤ p, q ≤ ∞, s ∈ R, it is known [18] that f M p,q s ≍   n∈Z d n (f ) q L p (1 + |n|) sq   1/q
, with natural modifications for p, q = ∞. [START_REF] Wang | Harmonic Analysis Method for Nonlinear Evolution Equations[END_REF][START_REF] Gröchenig | Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis[END_REF][START_REF] Ruzhansky | Modulation spaces and nonlinear evolution equations[END_REF]). Let p, q, p j ,

Lemma 2.3 ([
q j ∈ [1, ∞], s, s j ∈ R (j = 1, 2). Then (1) M p1,q1 s1 (R d ) ֒→ M p2,q2 s2 (R d ) whenever p 1 ≤ p 2 and q 1 ≤ q 2 and s 2 ≤ s 1 . In particular, H s (R d ) ֒→ M p,q s (R d ) for 2 ≤ p, q ≤ ∞ and s ∈ R. (2) M p1,q1 s1 (R d ) ֒→ M p2,q2 s2 (R d ) for q 1 > q 2 , s 1 > s 2 and s 1 -s 2 > d/q 2 -d/q 1 . (3) M p,q1 (R d ) ֒→ L p (R d ) ֒→ M p,q2 (R d ) holds for q 1 ≤ min{p, p ′ } and q 2 ≥ max{p, p ′ } with 1 p + 1 p ′ = 1. (4) M min{p ′ ,2},p (R d ) ֒→ F L p (R d ) ֒→ M max{p ′ ,2},p (R d ), 1 p + 1 p ′ = 1. (5) S(R d ) is dense in M p,q (R d ) if p and q are finite. (6) M p,p (R d ) ֒→ L p (R d ) ֒→ M p,p ′ (R d ) for 1 ≤ p ≤ 2 and M p,p ′ (R d ) ֒→ L p (R d ) ֒→ M p,p (R d ) for 2 ≤ p ≤ ∞. (7) The Fourier transform F : M p,p s (R d ) → M p,p s (R d ) is an isomorphism. (8) The space M p,q s (R d ) is a Banach space. (9) The space M p,q s (R d ) is invariant under complex conjugation. Theorem 2.4 (Algebra property). Let p, q, p i , q i ∈ [1, ∞] (i = 0, 1, 2). If 1 p1 + 1 p2 = 1 p0 and 1 q1 + 1 q2 = 1 + 1 q0 , then (2.4) M p1,q1 (R d ) • M p2,q2 (R d ) ֒→ M p0,q0 (R d );
with norm inequality f g M p 0 ,q 0 f M p 1 ,q 1 g M p 2 ,q 2 . In particular, the space

M p,q (R d ) is a pointwise F L 1 (R d )-module, that is, we have f g M p,q f F L 1 g M p,q .
Proof. The product relation (2.4) between modulation spaces is well known and we refer the interested reader to [START_REF] Bényi | Local well-posedness of nonlinear dispersive equations on modulation spaces[END_REF] and since

F L 1 (R d ) ֒→ M ∞,1 (R d ), the desired inequality (2.4) follows.
For f ∈ S(R d ), the Schrödinger propagator e i t 2 ∆ is given by

e i t 2 ∆ f (x) = 1 (2π) d R d e ix•ξ e -i t 2 |ξ| 2 f (ξ)dξ.
The first point in the following statement was established in [START_REF] Bényi | Unimodular Fourier multipliers for modulation spaces[END_REF], and the second, in [37, Proposition 4.1].

Proposition 2.5 ([4, 37]).

(1

) Let t ∈ R, p, q ∈ [1, ∞]. Then e i t 2 ∆ f M p,q ≤ C(t 2 + 1) d/4 f M p,q where C is some constant depending on d. (2) Let 2 ≤ p ≤ ∞, 1 ≤ q ≤ ∞. Then e i t 2 ∆ f M p,q ≤ (1 + |t|) -d( 1 p -1 2 ) f M p ′ ,q .
For (1/p, 1/q) ∈ [0, 1] × [0, 1], we define the subsets

I 1 = {(p, q); max(1/p, 1/p ′ ) ≤ 1/q}, I * 1 = {(p, q); min(1/p, 1/p ′ ) ≥ 1/q}, I 2 = {(p, q); max(1/q, 1/2) ≤ 1/p ′ }, I * 2 = {(p, q); min(1/q, 1/2) ≥ 1/p ′ }, I 3 = {(p, q); max(1/q, 1/2) ≤ 1/p}, I * 3 = {(p, q)
; min(1/q, 1/2) ≥ 1/p}. We now define the indices:

µ 1 (p, q) =      -1/p if (1/p, 1/q) ∈ I * 1 , 1/q -1 if (1/p, 1/q) ∈ I * 2 , -2/p + 1/q if (1/p, 1/q) ∈ I * 3 , and
µ 2 (p, q) =      -1/p if (1/p, 1/q) ∈ I 1 , 1/q -1 if (1/p, 1/q) ∈ I 2 , -2/p + 1/q if (1/p, 1/q) ∈ I 3 .
The dilation operator f λ is given by

f λ (x) = f (λx), λ > 0. Proposition 2.6 (See Theorem 3.2 in [17]). Let 1 ≤ p, q ≤ ∞, s ∈ R. There exists a constant C > 0 such that for all f ∈ M p,q s (R d ), 0 < λ ≤ 1, we have C -1 λ dµ1(p,q) min{1, λ s } f M p,q s ≤ f λ M p,q s ≤ Cλ dµ2(p,q) max{1, λ s } f M p,q s .

Norm inflation in Fourier-Lebesgue spaces

Define

µ σ (z 1 , . . . , z 2σ+1 ) = σ+1 ℓ=1 z ℓ 2σ+1 m=σ+2 zm . Definition 3.1. For ψ 0 ∈ L 2 (R d ), define U 1 [ψ 0 ](t) = e i t 2 ∆ ψ 0 , U k [ψ 0 ](t) = -i k 1 ,...,k 2σ+1 ≥1 k 1 +•••+k 2σ+1 =k t 0 e i (t-τ ) 2 ∆ µ σ U k1 [ψ 0 ], ..., U k2σ+1 [ψ 0 ] (τ )dτ, k ≥ 2.
It is known that the solution ψ of (1.1) can be written as a power series expansion [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation[END_REF], and [START_REF] Iwabuchi | Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions[END_REF][START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF] for later refinements of the method. Definition 3.2. Let A > 0 be a dyadic number. Define the space M A as the completion of C ∞ 0 (R d ) with respect to the norm [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation[END_REF][START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF]). Let A > 0 be a dyadic number.

ψ = ∞ k=1 U k [ψ 0 ], see
f MA = ξ∈AZ d f L 2 (ξ+QA) , Q A = [-A/2, A/2) d . Lemma 3.3 ([
(1) M A ∼ A M 1 , and for all ǫ > 0, H

d 2 +ǫ ֒→ M 1 ֒→ L 2 .
(2) M A is a Banach algebra under pointwise multiplication, and

f g MA ≤ C(d)A d/2 f MA g MA ∀f, g ∈ M A .
(3) Let A ≥ 1 be a dyadic number and φ ∈ M A with ψ 0 MA ≤ M. Then, there exists C > 0 independent of A and M such that

U k [ψ 0 ](t) MA ≤ t k-1 2σ (CA d/2 M ) k-1 M, for any t ≥ 0 and k ≥ 1. (4) Let (b k ) ∞ k=1 be a sequence of nonnegative real numbers such that b k ≤ C k 1 ,...,k 2σ+1 ≥1 k 1 +•••+k 2σ+1 =k b k1 • • • b k2σ+1 ∀k ≥ 2.
Then we have

b k ≤ b 1 C k-1 0 , ∀k ≥ 1, where C 0 = π 2 6 C(2σ + 1) 2 1/(2σ) b 1 .
Corollary 3.4 (See Corollary 1 in [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF]). Let A ≥ 1 be dyadic and

M > 0. If 0 < T ≪ (A d/2 M ) -2σ
, then for any ψ 0 ∈ M A with ψ 0 MA ≤ M : (i) A unique solution ψ to the integral equation associated with (1.1),

ψ(t) = e i t 2 ∆ ψ 0 -i t 0 e i (t-τ ) 2 ∆ µ σ (ψ(τ ))dτ exists in C([0, T ], M A ). (ii)
The solution ψ given in (i) has the expression

(3.1) ψ = ∞ k=1 U k [ψ 0 ] = ∞ ℓ=0 U 2σℓ+1 [ψ 0 ]
which converges absolutely in C([0, T ], M A ).

k j ≥ 1 such that k 1 + • • • + k 2σ+1 = 2σ. Now since U 2σ [ψ 0 ] ≡ 0, it follows that U 4σ [ψ 0 ] ≡ 0 and so on. Thus, U k [ψ 0 ](t) ≡ 0 for all k ∈ 2σN.
The general idea from [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation[END_REF] to prove instability is to show that one term in the sum (3.1) dominates the sum of the other terms, and rules out the continuity of the flow map. Usually, the first Picard iterate accounting for nonlinear effects, that is, U 2σ+1 [ψ 0 ] in our case, does the job. The proof of Theorems 1.2 and 1.5 indeed relies on this idea, for a suitable ψ 0 as in [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF].

Let N, A be dyadic numbers to be specified so that N ≫ 1 and 0 < A ≪ N . We choose initial data of the following form

(3.2) ψ 0 = RA -d/p N -s χ Ω ,
for a positive constant R and a set Ω satisfying

Ω = η∈ (η + Q A ),
for some

⊂ {ξ ∈ R d : |ξ| ∼ N } such that # ≤ 3. Then we have ψ 0 F L p s ∼ R, ψ 0 MA ∼ RA d( 1 2 -1 p ) N -s .
In fact, we have

ψ 0 p F L p s = R p A -d N -sp Ω (1 + |ξ| 2 ) ps/2 dξ = R p A -d N -sp η η+QA
(1 + |ξ| 2 ) ps/2 dξ.

Since A < N and |η| ∼ N, we have [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF]). There exists C > 0 such that for any ψ 0 satisfying (3.2) and k ≥ 1, we have

N 2 (1 + |ξ| 2 ) N 2 for ξ ∈ η + Q A and so N ps (1 + |ξ| 2 ) ps/2 N ps . As # ≤ 3 and |η + Q A | ∼ A d , we infer that ψ 0 p F L p s ∼ R p . Lemma 3.6 (See Lemma 3.6 in
supp U k [ψ 0 ](t) ≤ C k A d , ∀t ≥ 0.
The next result is the analogue of [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF]Lemma 3.7].

Lemma 3.7. Let ψ 0 given by (3.2), s < 0 and 1 ≤ p ≤ ∞. Then there exists C > 0 depending only on d, σ and s such that following holds.

U 1 [ψ 0 ](T ) F L p s ≤ CR, ∀T ≥ 0, (3.3) U k [ψ 0 ](T ) F L p s ρ k-1 1 C k A -d/p RN -s • s L p (QA) , (3.4)
where

ρ 1 = RN -s A d(1-1 p ) T 1 2σ .
Proof. The Schrödinger group is a Fourier multiplier,

U 1 [ψ 0 ](T ) F L p s = (e i t 2 |•| 2 ψ 0 ) • s L p = ψ 0 F L p s ≤ CR, hence (3.
3). We note that

I := U k [ψ 0 ](T ) F L p s ≤ • s L p (supp U k [ψ0](t)) sup ξ∈R d U k [ψ 0 ](t, ξ) ≤ • s L p (supp U k [ψ0](t)) k1+•••+k2σ+1=k t 0 |v k1 (τ )| * • • • * |v k2σ+1 (τ )| L ∞ dτ,
where

v k ℓ is either U k ℓ [ψ 0 ] or U k ℓ [ψ 0 ]
. By Young and Cauchy-Schwarz inequalities,

v k1 * • • • * v k2σ+1 L ∞ ≤ v k1 * v k2 L ∞ v k3 * • • • * v k2σ+1 L 1 ≤ v ∨ k1 v ∨ k2 L 1 2σ+1 ℓ=3 v k ℓ L 1 ≤ v k1 L 2 v k2 L 2 2σ+1 ℓ=3 v k ℓ L 1 ≤ 2σ+1 ℓ=3 supp U k ℓ [ψ 0 ] 1/2 2σ+1 ℓ=1 U k ℓ [ψ 0 ] L 2 .
Thus, we have

I ≤ • s L p (supp U k [ψ0](t)) I 1 ,
where

I 1 := k1+•••+k2σ+1=k t 0 2σ+1 ℓ=3 supp U k ℓ [ψ 0 ](τ ) 1/2 2σ+1 ℓ=1 U k ℓ [ψ 0 ](τ ) L 2 dτ. By Lemma 3.3 (3) (with M = CRN -s A d 2 -d p ), we have, for all k ≥ 1, U k [ψ 0 ](t) L 2 ≤ U k [ψ 0 ](t) MA ≤ Ct k-1 2σ C 2 RA d/2 N -s A d 2 -d p k-1 RN -s A d 2 -d p .
Note that, by Lemma 3.6,

I 1 k1+•••+k2σ+1=k t 0 2σ+1 ℓ=3 A d/2 2σ+1 ℓ=1 τ k ℓ -1 2σ RA d(1-1 p ) N -s k ℓ -1 RN -s A d 2 -d p dτ (RN -s ) k A d(2σ-1) 2 A d(1-1 p )(k-2σ-1) A ( d 2 -d p )(2σ+1) t 0 τ k-2σ-1 2σ dτ A d(1-1 p )(k-1) A -d/p (RN -s ) k t k-1 2σ .
Since s < 0, for any bounded set D ⊂ R d , we have

|{ ξ s > λ} ∪ D| ≤ |{ ξ s > λ} ∪ B D | , ∀λ > 0,
where

B D ⊂ R d is the ball centered at origin with |D| = |B D |. This implies that ξ s L p (D) ≤ ξ s L p (BD )
. In view of this and performing simple change of variables (ξ = C k/d ξ ′ ), we obtain

• s L p (supp U k [ψ0](t)) ≤ • s L p ({|ξ|≤C k/d A}) C k • s L p ({|ξ|≤A}) ,
and the lemma follows.

In the next lemma we establish a crucial lower bound on

U 2σ+1 [ψ 0 ]. Lemma 3.8. Let 1 ≤ p ≤ ∞, 1 ≤ A ≪ N and = {N e d , -N e d , 2N e d } where e d = (0, . . . , 0, 1) ∈ R d . If 0 < T ≪ N -2 , then we have U 2σ+1 [ψ 0 ](T ) F L p s RA -d p N -s ρ 2σ 1 • s L p (QA) ,
where

ρ 1 = RN -s A d-d p T 1 2σ .
Proof. Note that

U 2σ+1 [ψ 0 ](T, ξ) = ce -i T 2 |ξ| 2 Γ σ+1 ℓ=1 ψ 0 (ξ ℓ ) 2σ+1 m=σ+2 ψ 0 (ξ m ) T 0 e i t 2 Φ dtdξ 1 ...dξ 2σ+1 , where Γ = (ξ 1 , . . . , ξ 2σ+1 ) ∈ R (2σ+1)d : σ+1 ℓ=1 ξ ℓ - 2σ+1 m=σ+2 ξ m = ξ , Φ = |ξ| 2 - σ+1 ℓ=1 |ξ ℓ | 2 + 2σ+1 m=σ+2 |ξ m | 2 .
By the choice of initial data (3.2), we have

Γ σ+1 ℓ=1 ψ 0 (ξ ℓ ) 2σ+1 m=σ+2 ψ 0 (ξ m ) = Γ σ+1 ℓ=1 RA -d/p N -s χ Ω (ξ ℓ ) 2σ+1 m=σ+2 RA -d/p N -s χ Ω (ξ m ) = RA -d/p N -s 2σ+1 Γ 2σ+1 ℓ=1 χ Ω (ξ ℓ )dξ 1 . . . dξ 2σ+1 = RA -d/p N -s 2σ+1 C Γ 2σ+1 ℓ=1 χ η ℓ +QA (ξ ℓ )dξ 1 . . . dξ 2σ+1 ,
where the sum is taken over the non-empty set

C = (η 1 , . . . , η 2σ+1 ) ∈ {±N e d , 2N e d } 2σ+1 : σ+1 ℓ=1 η ℓ - 2σ+1 m=σ+2 η m = 0 . For ξ ∈ Q A , we have |ξ i | 2 ≤ |ξ| 2 ≤ A 2 ≪ N 2 and so |Φ| N 2 . Then | t 2 Φ(ξ)| ≪ 1 for 0 < T ≪ N -2 .
In view of this, together with the fact that the cosine function decreasing on [0, π/4], we obtain

T 0 e i t 2 Φ(ξ) dt ≥ Re T 0 e i t 2 Φ(ξ) dt ≥ 1 2 T.
Taking the above inequalities into account, we infer

(3.5) U 2σ+1 [ψ 0 ](T, ξ) RA -d/p N -s 2σ+1 (A d ) 2σ T χ (2σ+1) -1 QA (ξ).
Hence, we have

U 2σ+1 [ψ 0 ](T ) F L p s RA -d/p N -s 2σ+1 (A d ) 2σ T • s L p ((2σ+1) -1 QA) RA -d p N -s ρ 2σ 1 • s L p (QA) ,
where

ρ 1 = RN -s A d-d p T 1 2σ
.

For the convenience of reader, we compute the L p -norm of weight

• s on the cube Q A . Lemma 3.9. Let A ≫ 1, d ≥ 1, s < 0 and 1 ≤ p < ∞. We define f p s (A) =      1 if s < -d p , (log A) 1/p if s = -d p , A d/p+s if s > -d p . Then we have f p s (A) • s L p (QA) f p s (A) and f ∞ s (A) = • s L ∞ (QA) ∼ 1.
In particular, f p s (A) A d p +s for any s < 0.

Proof. We first compute the • L p -norm on ball of radius R 1 in R d , say B R1 (0). Since • s is radial, we have

I(R 1 ) := BR 1 (0) 1 (1 + |ξ| 2 ) -sp/2 dξ = 2π d/2 Γ(d/2) R1 0 r d-1 (1 + r 2 ) -sp/2 dr.
Notice that (1 + r 2 ) -sp/2 ≥ max{1, r -sp }, and assuming that R 1 ≫ 1, we obtain:

I(R 1 ) 1 0 r d-1 max{1, r -sp } dt + R1 1 r d-1 max{1, r -sp } dr = 1 0 r d-1 dr + R1 1 1 r -sp-d+1 dr.
Using conditions on s, we have

I(R 1 ) (f p s (R 1 )) p . Notice that Q A ⊂ B √ dA/2 (0), we have • s L p (QA) ≤ I( √ dA/2)
1/p f p s (A). On the other hand, we notice that 1 + r 2 ≤ 2 if 0 < r < 1 and 1 + r 2 ≤ 2r 2 if 1 < r < R 2 for some appropriate R 2 . Using this together with the above ideas, we obtain f p s (A)

• s L p (QA)
. This completes the proof.

Proof of Theorem 1.2. By Corollary 3.4, we have the existence of a unique solution to (1.1) in M A up to time T whenever

ρ 1 = RN -s A d(1-1 p ) T 1/(2σ) ≪ 1. In view of Lemma 3.7 and since ρ 1 < 1, ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) F L p
s can be dominated by the sum of the geometric series. Specifically, we have

∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) F L p s A -d/2 RN -s f p s (A) ∞ ℓ=2 ρ 2σℓ 1 A -d/2 RN -s f p s (A)ρ 4σ 1 . (3.6)
By Corollary 3.4 and the triangle inequality, we obtain

ψ(T ) F L p s = ∞ ℓ=0 U 2σℓ+1 [ψ 0 ] F L p s ≥ U 2σ+1 [ψ 0 ](T ) F L p s -U 1 [ψ 0 ](T ) F L p s - ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) F L p s .
In order to ensure

ψ(T ) F L p s U 2σ+1 [ψ 0 ](T ) F L p s , we rely on the conditions U 2σ+1 [ψ 0 ](T ) F L p s ≫ U 1 [ψ 0 ](T ) F L p s , (3.7) U 2σ+1 [ψ 0 ](T ) F L p s ≫ ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) F L p s . (3.8)
To use Lemma 3.8, we require

(i) T ≪ N -2 .
In view of Lemma 3.7, to prove (3.7) it is sufficient to prove

(ii) Rρ 2σ 1 A -d p N -s f p s (A) ≫ R, with ρ 1 = RN -s A d-d p T 1 2σ
. Finally, in view of Lemmas 3.7, 3.8 and 3.9, and (3.6), to prove (3.8) it is sufficient to prove:

(iii) ρ 1 ≪ 1, (iv) Rρ 2σ 1 A -d p N -s f p s (A) ≫ Rρ 4σ 1 A -d p N -s f p s (A).
We now choose A, R and T so that conditions (i)-(iv) are satisfied. To this end, we set

R = (log N ) -1 , A ∼ (log N ) -2σ+2 |s| N, T = (A d( 1 p -1) N s ) 2σ .
Then we have

ρ 1 = RN -s A d-d p T 1 2σ = (log N ) -1 ≪ 1.
Hence, condition (iii) is satisfied and so condition (iv). Note that

T = (log N ) -2σ+2 |s| d( 1 p -1)2σ N d( 1 p -1)2σ+2σs .
Since s < d 1 -1 p -1 σ and log N = O(N ǫ ) for any ǫ > 0, we have

T ≪ N -2 ,
and hence (i) is satisfied. By Lemma 3.9, we have f p s (A) A d p +s for any s < 0 and A ≥ 1 and so

Rρ 2σ 1 A -d p N -s f p s (A) log N ≫ (log N ) -1 = R and hence (ii) is satisfied. Thus, we have ψ(T ) F L p s U 2σ+1 [ψ 0 ](T ) F L p s log N. Since ψ 0 F L p s ∼ R = (log N ) -1 and T ≪ N -2
, we get norm inflation by letting N → ∞.

Norm inflation in modulation spaces

The proof of Theorem 1.5 follows the same general lines as the proof of Theorem 1.2 from the previous section. Let N, A be dyadic numbers to be specified so that N ≫ 1 and 0 < A ≪ N . We choose initial data of the following form (4.1)

ψ 0 = RA -d/2 N -s χ Ω , if 1 ≤ q ≤ 2, RA -d/q N -s χ Ω , if 2 ≤ q ≤ ∞, where Ω = η∈ (η + Q A ), Q A = [-A/2, A/2), for some ⊂ {ξ ∈ R d : |ξ| ∼ N } such that # ≤ 3.

4.1.

A priori estimates: 1 ≤ q ≤ 2. Then we have, for any s ∈ R,

ψ 0 H s ∼ R, ψ 0 MA ∼ RN -s . Lemma 4.1. Let q ∈ [1, 2]
, ψ 0 given by (4.1), s < 0. Then there exists C > 0 depending only on d, σ and s such that following holds.

U 1 [ψ 0 ](T ) M 2,q s ≤ CR, ∀T ≥ 0, (4.2) U k [ψ 0 ](T ) M 2,q s ρ k-1 C k A -d/2 RN -s (1 + |n|) s ℓ q (0≤|n|≤A) , (4.3) where ρ = RN -s A d/2 T 1 2σ .
Proof. By Lemma 2.3 and Proposition 2.5, we have

U 1 [ψ 0 ](T ) M 2,q s ψ 0 (T ) M 2,q s ψ 0 (T ) M 2,1 s R,
hence (4.2). By Plancherel theorem and (2.3), for s < 0, we have

U k [ψ 0 ](T ) M 2,q s = (1 + |n|) s σ n U k [ψ 0 ](T ) L 2 ℓ q ≤ sup ξ∈R d U k [ψ 0 ](t, ξ) (1 + |n|) s σ n L 2 Qn∩ supp U k [ψ0](t) ℓ q ≤ sup ξ∈R d U k [ψ 0 ](t, ξ) (1 + |n|) s ℓ q (0≤|n|≤CA) .
This yields the desired inequality in (4.3).

Lemma 4.2. Let s < 0, q ∈ [1, 2], 2 ≤ A ≪ N and = {N e d , -N e d , 2N e d } where e d = (0, . . . , 0, 1) ∈ R d . If 0 < T ≪ N -2 , then we have U 2σ+1 [ψ 0 ](T ) M 2,q s RA -d 2 N -s ρ 2σ (1 + |n|) s ℓ q (0≤|n|≤A)
,

where ρ = RN -s A d/2 T 1 2σ .
Proof. By (2.3), we note that

U 2σ+1 [ψ 0 ](T ) q M 2,q s = n∈Z d n (U 2σ+1 [ψ 0 ](T )) q L 2 (1 + |n|) sq = n∈Z d σ n U 2σ+1 [ψ 0 ](T ) q L 2 (1 + |n|) sq n∈Z d 1 (1 + |n|) -sq Qn | U 2σ+1 [ψ 0 ](ξ, T )| 2 dξ q/2
, where Q n is a unit cube centered at n ∈ Z d . Arguing as before in the proof of Lemma 3.8 (specifically, by (3.5)), for ξ

∈ Q A = [-A/2, A/2) d , we have U 2σ+1 [ψ 0 ](T, ξ) RA -d/2 N -s 2σ+1 (A d ) 2σ T χ (2σ+1) -1 QA (ξ).
It follows that

U 2σ+1 [ψ 0 ](T ) M 2,q s RA -d/2 N -s 2σ+1 (A d ) 2σ T   ⌊A/2⌋ |n|=⌊-A/2⌋ 1 (1 + |n|) -sq   1/q RA -d/2 N -s ρ 2σ   ⌊A/2⌋ |n|=⌊-A/2⌋ 1 (1 + |n|) -sq   1/q , where the floor function is ⌊x⌋ = max (m ∈ Z | m ≤ x) and ρ = RN -s A d/2 T 1 2σ .

4.2.

A priori estimates: 2 ≤ q ≤ ∞. Then we have, for any s ∈ R,

ψ 0 F L q s ∼ R, ψ 0 MA ∼ RA d( 1 2 -1 q ) N -s .
Lemma 4.3. Let s < 0 and 2 ≤ q ≤ ∞. Then there exists C > 0 depending only on d, σ and s such that following holds.

U 1 [ψ 0 ](T ) M 2,q s ≤ CR, ∀T ≥ 0, (4.4) U k [ψ 0 ](T ) M 2,q s ρ k-1 2 C k A -d/q RN -s (1 + |n|) s ℓ q (0≤|n|≤A) , (4.5)
where

ρ 2 = RN -s A d(1-1 q ) T 1 2σ .
Proof. By Lemma 2.3, we have

U 1 [ψ 0 ](T ) M 2,q s U 1 [ψ 0 ](T ) F L q s ≤ CR.
The proof of (4.5) is similar to Lemmas 4.1, (4.3) and 3.7, (3.4), we omit the details.

The next lemma is the analogue of Lemmas 3.8 and 4.2, so we leave out its proof.

Lemma 4.4. Let s < 0, 2 ≤ q ≤ ∞, 1 ≤ A ≪ N and = {N e d , -N e d , 2N e d } where e d = (0, . . . , 0, 1) ∈ R d . If 0 < T ≪ N -2 , then we have U 2σ+1 [ψ 0 ](T ) M 2,q s RA -d q N -s ρ 2σ 2 (1 + |n|) s ℓ q (0≤|n|≤A) ,
where

ρ 2 = RN -s A d-d q T 1 2σ .
4.3. Proof of Theorem 1.5.

Lemma 4.5. Let s < 0. In the limit A → ∞, we have, for 1 ≤ q < ∞,

(1 + |n|) s ℓ q (0≤|n|≤A) ∼ A→∞ g q s (A) :=      1 if -sq > d, (log A) 1/q if -sq = d, A 1/q+s if -sq < d,
and

(1 + |n|) s ℓ ∞ (0≤|n|≤A) ∼ A→∞ 1.
Proof. Since (1 + |ξ|) sq is a decreasing function in |ξ|, in view of the integral test and Lemma 3.9, we have, for 1 ≤ q < ∞,

(1 + |n|) s ℓ q (0≤|n|≤A) =   0≤|n|≤A 1 (1 + |n|) -sq   1/q ∼ A→∞ |ξ|≤A dξ (1 + |ξ|) -sq 1/q ∼ A→∞ A 0 r d-1 dr (1 + r) -sq 1/q
, hence the result for q finite. The case q = ∞ is straightforward.

To prove Theorem 1.5, we distinguish two cases. First case: 1 ≤ q ≤ 2. By Corollary 3.4, we have the existence of solution to (1.1) in M A up to time T whenever ρ = RN -s A d/2 T 1/(2σ) ≪ 1. In view of Lemma 3.7 and since

ρ < 1, ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) M 2,q
s can be dominated by the sum of a geometric series. Specifically, we have

∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) M 2,q s A -d/2 RN -s g q s (A) ∞ ℓ=2 ρ 2σℓ
A -d/2 RN -s g q s (A)ρ 4σ . (4.6) By Corollary 3.4 and the triangle inequality, we obtain

ψ(T ) M 2,q s = ∞ ℓ=0 U 2σℓ+1 [ψ 0 ] M 2,q s ≥ U 2σ+1 [ψ 0 ](T ) M 2,q s -U 1 [ψ 0 ](T ) M 2,q s - ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) M 2,q s .
In order to ensure ψ(T ) M 2,q s U 2σ+1 [ψ 0 ](T ) M 2,q s , we rely on the conditions

U 2σ+1 [ψ 0 ](T ) M 2,q s ≫ U 1 [ψ 0 ](T ) M 2,q s , (4.7) U 2σ+1 [ψ 0 ](T ) M 2,q s ≫ ∞ ℓ=2 U 2σℓ+1 [ψ 0 ](T ) M 2,q s . (4.8)
In view of Lemmas 4.1 and 4.2, (4.7) amount to the condition (4.9)

Rρ 2σ A -d 2 N -s g q s (A) ≫ R
T ≪ N -2 , ρ ≪ 1, Rρ 2σ A -d 2 N -s g q s (A) ≫ Rρ 4σ A -d/2
N -s g q s (A). We now choose A, R and T so that conditions (4.9) and (4.10) are satisfied. To this end, we set

R = (log N ) -1 , A ∼ (log N ) -2σ+2 |s| N, T = (A -d/2 N s ) 2σ .
Then we have

ρ = RN -s A d/2 T 1 2σ = (log N ) -1 ≪ 1. Note that T = (log N ) -2σ+2 |s| d( 1 2 -1)2σ N -dσ+2σs . Since s < d 2 -1 σ and log N = O(N ǫ ) for any ǫ > 0, we have T ≪ N -2 .
We have g q s (A) A d q +s for any s < 0 and A ≥ 1. Thus, for 1 ≤ q ≤ 2, we have

Rρ 2σ A -d 2 N -s g q s (A) (log N ) -(2σ+1) (log N ) 2σ+2 A d( 1 q -1 2 ) log N ≫ (log N ) -1 = R,
and (4.9) is satisfied. Thus, we have ψ(T

) M 2,q s U 2σ+1 [ψ 0 ](T ) M 2,q s log N. Since ψ 0 M 2,q s R = (log N ) -1 and T ≪ N -2
, we get norm inflation by letting N → ∞. This completes the proof for q ∈ [1, 2]. Second case: 2 ≤ q ≤ ∞. By Corollary 3.4, we have the existence of solution to (1.1) in M A up to time T whenever ρ 2 = RN -s A d(1-1 q ) T 1/(2σ) ≪ 1. By Lemmas 4.3 and 4.4, the conditions

(4.11) T ≪ N -2 , ρ 2 ≪ 1 and Rρ 2σ 2 A -d q N -s g q s (A) ≫ R ensures that ψ(T ) M 2,q s U 2σ+1 [ψ 0 ](T ) M 2,q s ∼ Rρ 2σ
1 A -d q N -s g q s (A). We now choose A, R and T so that conditions (4.11) satisfied. To this end, we set

R = (log N ) -1 , A ∼ (log N ) -2σ+2 |s| N, T = (A d( 1 q -1) N s ) 2σ .
Then we have

ρ 2 = RN -s A d-d q T 1 2σ = (log N ) -1 ≪ 1. Note that T = (log N ) -2σ+2 |s| d( 1 q -1)2σ N d( 1 q -1)2σ+2σs .
Since s < d 1 -1 q -1 σ and log N = O(N ǫ ) for any ǫ > 0, we have

T ≪ N -2 .
Note that g q s (A) A d q +s for any s < 0 and A ≥ 1 and so

Rρ 2σ 2 A -d q N -s g q s (A) log N ≫ (log N ) -1 = R. Thus, we have ψ(T ) M 2,q s U 2σ+1 [ψ 0 ](T ) M 2,q s log N. Since ψ 0 M 2,q s R = (log N ) -1 and T ≪ N -2
, we get norm inflation by letting N → ∞. This completes the proof of Theorem 1.5.

Norm inflation as a by-product of geometric optics

The proof of Theorems 1.3 and 1.6 follows the same strategy as in [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF]: through a suitable rescaling, we turn the ill-posedness result into an asymptotic result, which can be expressed in the framework of weakly nonlinear geometric optics. More precisely, we change the unknown function ψ to u, via (5.1)

u ε (t, x) = ε 2-J 2σ ψ(εt, x),
where the parameter ε will tend to zero. For ψ solution to (1.1), u ε solves (5.2)

iε∂ t u ε + ε 2 2 ∆u ε = µε J |u ε | 2s u ε .
The case J = 1 corresponds to weakly nonlinear geometric optics (WNLGO), as defined in [START_REF]Semi-Classical Analysis for Nonlinear Schrödinger Equations[END_REF]. As noticed in [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF] in the framework of Sobolev spaces, a phenomenon of infinite loss of regularity can be proved via this WNLGO setting, under the assumption s < -1/(2σ) in the analogue of Theorems 1.3 and 1.6. Like in that paper, in order to weaken the assumption on s to s < -1/(2σ + 1), we will have to consider some value J > 1, and perform some "asymptotic sin", in the sense that we change the hierarchy in an asymptotic expansion involving the limit ε → 0.

The heuristic idea is the same as in [START_REF]Instability of the periodic nonlinear Schrödinger equation[END_REF]: when negative regularity is involved, the zero Fourier mode plays a stronger role than (large) non-zero modes, which come with a small factor. With the scaling (5.1) in mind, our goal is to show that we may consider initial data (for u) of the form

u(0, x) = j =0 e ij•x/ε α j (x),
that is containing only rapidly oscillatory terms, and such that the evolution under (5.2) creates a non-trivial non-oscillatory term.

To be more specific, recall the strategy of multiphase nonlinear geometric optics (see [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF] for more details): we plug an ansatz of the form u(t, x) = j e iφj (t,x)/ε a j (t, x) into (5.2), and order the powers of ε. The most singular term is of order ε 0 , it is the eikonal equation:

∂ t φ j + 1 2 |∇φ j | 2 = 0.
In the case of an initial phase φ j (0, x) = j • x, no caustic appears, and the global solution is given by

(5.3) φ j (t, x) = j • x - |j| 2 2 t.
In the sequel, we consider such phases, for j ∈ Z d . The next term in the hierarchy is of order ε 1 , but as evoked above, we "cheat", and incorporate some nonlinear effects even if J > 1 (and J < 2),

(5.4)

∂ t a j + j • ∇ x a j = -iµε J-1 φ k 1 -φ k 2 +•••+φ k 2σ+1 =φj a k1 āk2 . . . a k2σ+1 ,
where we have used ∇φ j (t, x) = j. Again in view of the specific form of the phase (5.3), the condition on the sum involves a resonant condition, (k 1 , k 2 , . . . , k 2σ+1 ) ∈ R j , where

R j = (k ℓ ) 1≤ℓ≤2σ+1 , 2σ+1 ℓ=1 (-1) ℓ+1 k ℓ = j, 2σ+1 ℓ=1 (-1) ℓ+1 |k ℓ | 2 = |j| 2 .
In the cubic case σ = 1, those sets are described exactly:

Lemma 5.1 (See [START_REF] Colliander | Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation[END_REF][START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF]). Suppose σ = 1.

• If d = 1, then R j = {(j, ℓ, ℓ), (ℓ, ℓ, j) ; ℓ ∈ Z}. • If d ≥ 2, then (k 1 , k 2 , k 3 ) ∈ R j
precisely when the endpoints of the vectors k 1 , k 2 , k 3 , j for four corners of a non-degenerate rectangle with k 2 and j opposing each other, or when this quadruplet corresponds to one of the following two degenerate cases:

(k 1 = j, k 2 = k 3 ) or (k 1 = k 2 , k 3 = j).
The above lemma explains why our approach distinguishes the one-dimensional case and the multi-dimensional case, and in particular why the cubic one-dimensional case is left out in Theorems 1.3 and 1.6.

5.1.

Multi-dimensional case. The leading idea in [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF] is to start from three nontrivial modes only, in the case d ≥ 2, and create at least one new mode (possibly more if σ ≥ 2), corresponding to j = 0.

Lemma 5.2. Let d ≥ 2 and σ ∈ N * . Define k 1 , k 2 , k 3 ∈ Z d as k 1 = (1, 0, . . . , 0), k 2 = (1, 1, 0, . . . , 0), k 3 = (0, 1, 0, . . . , 0).
For initial data of the form u(0, x) = α(x)

3 j=1 e ikj •x/ε , α ∈ S(R d ) \ {0},
we have a 0|t=0 = 0 and ∂ t a 0|t=0 = ε J-1 c 0 α(x), with c 0 = ♯R 0 ≥ 1. This lemma is straightforward, in view of (5.4), and since (k

1 , k 2 , k 3 ) ∈ R 0 if σ = 1, (k 1 , k 2 , k 3 , k 1 , k 1 , . . . , k 1 ) ∈ R 0 if σ ≥ 2.
5.2. One-dimensional case. In the one-dimensional case, we have a similar result, provided that the nonlinearity is at least quintic, in view of [START_REF] Carles | Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces[END_REF]Lemma 4.2] (and Example 4.3 there):

Lemma 5.3. Let d = 1 and σ ≥ 2. Define k 1 , k 2 , k 3 , k 4 , k 5 ∈ Z as (k 1 , k 2 , k 3 , k 4 , k 5 ) = (2, -1, -2, 4, 3).
For initial data of the form u(0, x) = α(x)

5 j=1 e ikj x/ε , α ∈ S(R) \ {0},
we have a 0|t=0 = 0 and ∂ t a 0|t=0 = ε J-1 c 0 α(x), with c 0 = ♯R 0 ≥ 1. 5.3. How to conclude. Supposing that we can prove that the geometric expansion recalled above provides an approximation u app for the solution u to (5.2), suitable in the sense that the error is measured in a sufficiently strong norm, the idea is that both u |t=0 and u -u app are small in spaces involving negative regularity in x, while Lemma 5.2 or 5.3 implies that u app is large in many spaces.

6. (Very) weakly nonlinear geometric optics 6.1. A convenient functional framework. Throughout this section, we denote by X a Banach algebra in the space variable, that is

(6.1) ∃C > 0, f g X ≤ C f X g X , ∀f, g ∈ X.
We suppose that X is translation invariant and, denoting by

τ k f (x) = f (x -k), (6.2) τ k f X = f X , ∀k ∈ R d , ∀f ∈ X.
We assume in addition that the multiplication by plane wave oscillations leaves the X-norm invariant, (6.3)

f e k X = f X , ∀k ∈ R d
, where e k (x) = e ik•x .

Note that this assumption rules out Sobolev spaces H s (R d ), unless s = 0. Finally, we assume that the Schrödinger group acts on X, at least locally in time: Assumption 6.1. There exists T 0 such that e i t 2 ∆ maps X to X for t ∈ [0, T 0 ], and

∃C > 0, e i t 2 ∆ L(X,X) ≤ C, ∀t ∈ [0, T 0 ].
In [START_REF] Carles | Multiphase weakly nonlinear geometric optics for Schrödinger equations[END_REF][START_REF] Giannoulis | High-frequency averaging in semi-classical Hartree-type equations[END_REF], the case X = F L 1 (M ) was considered, with M = T d (a choice resumed in [START_REF] Carles | Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces[END_REF]) or R d . In [START_REF]Geometric optics and instability for NLS and Davey-Stewartson models[END_REF][START_REF] Mouzaoui | High-frequency averaging in the semi-classical singular Hartree equation[END_REF], the choice X = F L 1 ∩ L 2 (R d ) was motivated by the presence of more singular nonlocal nonlinearities. We shall consider later two sorts of X spaces:

F L 1 ∩ F L p or F L 1 ∩ M 1,1 .
We denote by

Y =    (a j ) j∈Z d , j∈Z d a j X < ∞    = ℓ 1 (X),
and

Y 2 =    (a j ) j∈Z d ∈ Y, j∈Z d j 2 a j X + j ∇a j X + ∆a j X < ∞    .
We suppose that u solves (5.2) with initial data

u(0, x) = j∈Z d α j (x)e ij•x/ε .
6.2. Construction of the approximate solution. The approximate solution is given by

u app (t, x) = j∈Z d a j (t, x)e iφj (t,x)/ε ,
where φ j is given by (5.3) and the a j 's solve (5.4), with initial data α j .

Lemma 6.2. Let d ≥ 1, σ ∈ N * and J ≥ 1.

• If (α j ) j∈Z d ∈ Y , then there exists T > 0 independent of ε ∈ [0, 1] and a unique solution (a j ) j∈Z d ∈ C([0, T ]; Y ) to the system (5.4), such that a j|t=0 = α j for all j ∈ Z d .

• If in addition (α j ) j∈Z d ∈ Y 2 , then (a j ) j∈Z d ∈ C([0, T ]; Y 2 ).
Proof. In view of Duhamel's formula for (5.4), a j (t, x) = a j (0, x-jt)-iλ (k1,k2,...,k2σ+1)∈Rj t 0 a k1 āk2 . . . a k2σ+1 (s, x-j(t-s))ds, the first point of the lemma is straightforward, as an easy consequence of (6.2) and (6.1), and a fixed point argument. The second point requires a little bit more care: it was proven in [10, Proposition 5.12] in the case X = F L 1 (Wiener algebra), and the proof relies only on the properties of X required at the beginning of this section.

From now on, we assume (α j ) j∈Z d ∈ Y 2 .

6.3. Error estimate. First, in view of the assumptions made in this section, a standard fixed point argument yields, in view of (6.1) and Assumption 6.1:

Lemma 6.3. Let d ≥ 1, σ ∈ N * and J ≥ 1. If u 0 ∈ X, then there exists T ε > 0 and a unique solution u ∈ C([0, T ε ]; X) to (5.2) such that u |t=0 = u 0 .
To construct the approximate solution u app , we have discarded two families of terms:

• Non-resonant terms, involving the source term

r 1 := µε J j (k1,k2,...,k2σ+1) ∈Rj a k1 āk2 . . . a k2σ+1 e i(φ k 1 -φ k 2 +•••+φ2σ+1)/ε .
• Higher order terms, involving

r 2 := ε 2 2 j ∆a j e iφj /ε .
Indeed, u app solves (6.4)

iε∂ t u app + ε 2 2 ∆u app = µε J |u app | 2σ u app + r 1 + r 2 .
Duhamel's formula for u -u app =: w reads

w(t) = -iµε J-1 t 0 e iε t-τ 2 ∆ |u| 2σ u -|u app | 2σ u app (τ )dτ -iµε J-1 j=1,2 t 0 e iε t-τ 2 ∆ r j (τ )dτ.
In view of our assumptions on X, we readily have, thanks to Minkowski inequality,

w(t) X ≤ C t 0 u app (τ ) 2σ X + w(τ ) 2σ X w(τ ) X dτ + Cε -1 j=1,2 t 0 e iε t-τ 2 ∆ r j (τ )dτ X ,
for some C independent of ε ∈ [0, 1] and t ∈ [0, T 0 ]. In view of the second point of Lemma 6.2, we readily have

t 0 e iε t-τ 2 ∆ r 2 (τ )dτ X ε 2 .
By construction, r 1 is the sum of terms of the form g(t, x)e ik•x/ε-ωt/(2ε) , with k ∈ Z d , ω ∈ Z, and the non-resonance property reads exactly |k| 2 = ω.

Lemma 6.4. Let k ∈ R d , ω ∈ R, with |k| 2 = ω. Define D ε (t, x) = t 0 e iε t-τ 2 ∆ g(τ,
x)e ik•x/ε-iωτ /(2ε) dτ.

Then we have

D ε (t, x) = -2iε |k| 2 -ω e iε t-τ 2 ∆ g(τ, x)e ik•x/ε-iωτ /(2ε) t 0 + 2iε |k| 2 -ω t 0 e iε t-τ 2 ∆ e ik•x/ε-iωτ /(2ε) i 2 (ε∆g + 2k • ∇g) + ∂ t g (τ, x) dτ.
In particular, for t ∈ [0, T 0 ],

D ε (t) X ε ||k| 2 -ω| g L ∞ ([0,t];X) + ∆g L ∞ ([0,t];X) + |k| ∇g L ∞ ([0,t];X) + ∂ t g L ∞ ([0,t];X) .
Proof. The last estimate follows directly from the identity of the lemma, (6.3) and Assumption 6.1, so we only address the identity, which is essentially established in [10, Lemma 5.7] (up to the typos there). Setting η = ξ -k/ε, the (spatial) Fourier transform of D is given by where we have denoted

D ε (t, ξ) = e -iεt|η+k/
θ = ε η + k ε 2 - ω ε = ε|η| 2 + 2k • η θ1 + |k| 2 -ω ε θ2 .
Integrate by parts, by first integrating e iτ θ2/2 :

e iε t 2 |ξ| 2 D ε (t, ξ) = - 2i θ 2 e iτ θ/2 b (τ, η) t 0 + 2i θ 2 t 0 e iτ θ/2 i θ 1 2 b (τ, η) + ∂ t b (τ, η) dτ.
The identity follows by inverting the Fourier transform.

We infer:

Proposition 6.5. Let d ≥ 1, σ ∈ N * , J ≥ 1, and (α j ) j∈Z d ∈ Y 2 . Then for T as in Lemma 6.2, u -u app L ∞ ([0,T ];X) ε.
Proof. First, Lemma 6.2 and (5.4) imply that we also have

(∂ t a j ) j∈Z d ∈ C([0, T ]; Y ).
Then, in view of these properties and Lemma 6.4, we have

t 0 e iε t-τ 2 ∆ r 1 (τ )dτ X ε J+1 ,
where we have used the fact that in the application of Lemma 6.4, |k| 2 -ω ≥ 1, since now k ∈ Z d and ω ∈ Z. We infer

w(t) X ≤ C t 0 u app (τ ) 2σ X + w(τ ) 2σ X w(τ ) X dτ + Cε J + Cε,
where C is independent of ε ∈ [0, 1] and t ∈ [0, T ]. Lemmas 6.2 and 6.3 yield w ∈ C([0, min(T, T ε )]; X). Since w |t=0 = 0, the above inequality and a standard continuity argument imply that u ∈ C([0, T ]; X) provided that ε > 0 is sufficiently small, along with the announced error estimate.

7. Norm inflation with infinite loss of regularity 7.1. Proof of Theorem 1.3. For 1 < J < 2 to be fixed later, let u ε defined by (5.1), and consider the initial data given by Lemma 5.2 (if d ≥ 2) or Lemma 5.3 (if d = 1 and σ ≥ 2). We apply the analysis from Section 6 with X = F L 1 ∩ F L ∞ . This is obviously a Banach algebra, (6.1) holds, thanks to Young inequality, the Xnorm is invariant by translation, and by multiplication by plane wave oscillations as in (6.3). Assumption 6.1 is satisfied with C = 1 for any T 0 > 0, since the Schrödinger group is a Fourier multiplier of modulus one. We can therefore invoke the conclusions of Lemma 5.2 (if d ≥ 2), Lemma 5.3 (if d = 1 and σ ≥ 2), and Proposition 6.5 (in all cases). In order to translate these properties involving u solving (5.2) in terms of ψ solving (1.1), we use the following lemma: Proof. We have obviously

I ε (f, j)(ξ) = f ξ - j ε .
The first point is thus trivial. For the second one, if p is finite,

I ε (f, j) p F L p s = ξ ps f ξ - j ε p dξ.
Note that, for s ≤ 0, e -it |j| 2 2ε âj t, ξ -j ε , so we infer, at least for ε sufficiently small ((a j (τ )) j∈Z d ∈ Y from Lemma 6.2),

I ε (f, j) p
u app (τ ) F L p k ε J-1 , ∀p ∈ [1, ∞], ∀k ∈ R.
On the other hand, since X = F L 1 ∩ F L ∞ ⊂ F L p , Proposition 6.5 yields

u(τ ) -u app (τ ) F L p k u(τ ) -u app (τ ) X ε, ∀k ≤ 0, hence, if J < 2, u(τ ) F L p k ε J-1 , ∀p ∈ [1, ∞], ∀k ≤ 0.
Therefore, ψ(ετ

) F L p k ε J-2 2σ × ε J-1 , ∀p ∈ [1, ∞]
, ∀k ∈ R. The right hand side is unbounded as ε → 0 provided that J -2 2σ + J -1 < 0, that is, J < 2σ + 2 2σ + 1 .

Then given s < -1/(2σ + 1), we can always find a J ∈]1, 2[ satisfying (7.1) and the above constraint. Theorem 1.3 follows in the case k ≤ 0, by taking for instance ε n = 1/n and t n = ε n τ . In the case k > 0, we just recall the obvious estimate ψ(ετ ) F L p k ≥ ψ(ετ ) F L p , and the proof of Theorem 1.3 is complete. 7.2. Proof of Theorem 1.6. In the case of modulation spaces, the proof goes along the same lines as above, up to adapting the space X and Lemma 7.1.

We choose X = F L 1 ∩ M 1,1 . Theorem 2.4 shows that the Banach algebra property (6.1) is satisfied. F L 1 is translation invariant, and for M 1,1 ,

V g (τ k f ) (x, y) = R d f (t -k)g(t -x)e -iy•t dt = e -iy•k R d f (t)g(t + k -x)e -iy•t dt = e -iy•k V g (f )(x -k, y),
and thus

τ k f M 1,1 = V g (τ k f ) L 1
x,y = f M 1,1 . We have used already the fact that (6.3) is satisfied on F L 1 . On M 1,1 , this is the case too, since for k ∈ R d , V g (f e k ) (x, y) = R d f (t)e ik•t g(t -x)e -iy•t dt = V g (f ) (x, y -k), and so f e k M 1,1 = V g (f e k ) L 1

x,y = V g (f ) L 1

x,y = f M 1,1 . Finally, Assumption 6.1 is satisfied thanks to Proposition 2.5, and we can again invoke Lemma 5.2, Lemma 5.3, and Proposition 6.5.

Like before, Lemma 5.2 and Lemma 5.3 show that there exists τ > 0 independent of ε such that a 0 (τ ) M p,q k ε J-1 , ∀p, q ∈ [1, ∞], ∀k ∈ R.

By the same asymptotic decoupling phenomenon as in the case of F L p spaces, we infer u app (τ ) M p,q k ε J-1 , ∀p, q ∈ [1, ∞], ∀k ∈ R.

In view of Lemma 2.3, X ֒→ M p,q k for all p, q ≥ 1 and all k ≤ 0, and so u(τ ) -u app (τ ) M p,q k u(τ ) -u app (τ ) X ε, ∀p, q ∈ [1, ∞], ∀k ≤ 0.

The analogue of Lemma 7.1 is the following: • For all s ∈ R, p, q ∈ [1, ∞], and f ∈ M p,q s (R d ), I ε (f, 0) M p,q s = f M p,q s . • Let j ∈ R d \ {0}. For all s ≤ 0, there exists C = C(j) independent of p, q ∈ [1, ∞] such that for all f ∈ S(R d ),

I ε (f, j) M p,q s ≤ Cε |s| f M p,q s .
Proof. The first point is proven like (6.3) above. For the second point, write I ε (f, j) M p,q s = V g f x, y -j ε L p At this stage, we can repeat the same arguments as in the previous subsection, and Theorem 1.6 follows.

1 . Introduction 1 . 1 .

 111 General setting. We consider the nonlinear Schrödinger (NLS) equations of the form (1.1)i∂ t ψ + 1 2 ∆ψ = µ|ψ| 2σ ψ, x ∈ R d ; ψ(0, x) = ψ 0 (x),

Remark 3 . 5 .

 35 By Definition 3.1, we obtain U k [ψ 0 ](t) = 0 unless k ≡ 1 mod 2σ. For instance, U k [ψ 0 ](t) ≡ 0 for all k ∈ 2σN. To see this, fix σ ∈ N. Then clearly U 2σ [ψ 0 ] ≡ 0 because there does not exist

ε| 2 / 2 t 0 e 2 t 0 e 0 e

 20200 iετ |η+k/ε| 2 /2 b (τ, η) e -iωτ /(2ε) dτ = e -iεt|η+k/ε| 2 /iτ θ/2 b (τ, η) dτ = e -iεt|η+k/ε| 2 /2 t iτ θ2/2 e iτ θ1/2 b (τ, η) dτ,

Lemma 7 . 1 .

 71 Let d ≥ 1. For f ∈ S ′ (R d ) and j ∈ R d , denote I ε (f, j)(x) = f (x)e ij•x/ε . • For all s ∈ R, p ∈ [1, ∞], and f ∈ F L p s (R d ), I ε (f, 0) F L p s = f F L p s . • Let j ∈ R d \ {0}. For all s ≤ 0, there exists C = C(j) independent of p ∈ [1, ∞] such that for all f ∈ S(R d ), I ε (f, j) F L p s ≤ Cε |s| f F L p |s| .

Lemma 5 . 2 and

 52 point of the lemma in the case p finite. The case p = ∞ follows from the same estimate, controlling the supremum of a product by the product of the suprema.With u |t=0 as in Lemma 5.2 or Lemma 5.3, the above lemma yields, in view of (5.1), and for s < 0,ψ(0) F L p s ε J-22σ +|s| . This sequence of initial data is small (in F L p s for all p) Lemma 5.3 show that there exists τ > 0 independent of ε such thata 0 (τ ) F L p k ε J-1 , ∀p ∈ [1, ∞], ∀k ∈ R.By construction, ûapp (t, ξ) = j∈Z d

Lemma 7 . 2 .

 72 Let d ≥ 1. For f ∈ S ′ (R d ) and j ∈ R d , denote I ε (f, j)(x) = f (x)e ij•x/ε .

  used Peetre inequality a + b s ≤ a s b |s| . The lemma follows.

The first author benefits from a fellowship by the Henri Lebesgue Center, which he wishes to thank. The first author also wishes to thank DST-INSPIRE and TIFR-CAM for the academic leave.


Nasser

L Wendling
email: laurent.wendling@parisdescartes.fr

I Debled-Rennesson

H Nasser

Multilevel Polygonal Descriptor Matching Defined By Combining Discrete Lines And Force Histogram Concepts

Keywords: Discrete lines, force histogram, polygonal representation, multilevel structure

  L'archive ouverte pluridisciplinaire

INTRODUCTION

Shape representation methods are usually split into main categories: region-based approaches or contour-based approaches [START_REF] Loncaric | A survey of shape analysis techniques[END_REF][START_REF] Jain | Statistical pattern recognition: a review[END_REF][START_REF] Zhang | Shape-based image retrieval using generic fourier descriptor[END_REF]. The choice of one representation instead of another one generally relies on the application under consideration. We focus on contour and more specifically on its polygonal description, consisting in a finite chain of straight line segments closing in a loop. Such a representation is widely used in several research fields such as graphic symbol recognition [START_REF] Cordella | Symbol recognition in documents: a collection of techniques?[END_REF][START_REF] Santosh | Document Image Analysis -Current Trends and Challenges in Graphics Recognition[END_REF][START_REF] Baird | The evolution of document image analysis[END_REF][START_REF] Santosh | Graphical symbol recognition[END_REF] or Geographic Information System [START_REF] Duckham | Efficient generation of simple polygons for characterizing the shape of a set of points in the plane[END_REF].

Segments often serves a common basis for both syntactic and structural approaches. Already in the 90s a string-matching technique to the problem of recognizing and classifying polygons has been defined by Maes [START_REF] Maes | Polygonal shape recognition using string matching techniques[END_REF] but this method is limited when the polygonal approximation of the object is inconsistent. Another 1 Université Paris Descartes, LIPADE, 45, rue des Saints-Pères, 75270 Paris, France: 2 Université de Lorraine, LORIA, UMR 7503, Vandoeuvre-lès-Nancy, F-54506, France.

works focus for instance on the definition of a complete system for architectural drawing analysis [START_REF] Dosch | A complete system for the analysis of architectural drawings[END_REF] or symbol spotting [START_REF] Rusiñol | Symbol spotting in vectorized technical drawings through a lookup table of region strings[END_REF].

Main well-known drawbacks of polygonal approximation rely on sensitivity to the noise or to over-segmentation effect considering the raster contour. Highly concave curves tend also to provide a bad discontinuity behavior on the achieved polygonal representation.

Since the last decades, numerous works have been carried out to assess the polygonal contour best fitting a raster contour. For line extraction, the Hough transform [START_REF] Dana | Generalizing the hough transform to detect arbitrary shapes[END_REF] has been widely used. But, applications are reasonably limited in case of degraded images due to its high computational cost. However, robust vectorization algorithms exist in line drawing images as in engineering drawings [START_REF] Janssen | Adaptive vectorization of line drawing images[END_REF][START_REF] Hilaire | Robust and accurate vectorization of line drawings[END_REF][START_REF] Song | An object-oriented progressive-simplification based vectorization system for engineering drawings: Model, algorithm, and performance[END_REF]. The use of such low-level primitives varies widely in accordance with the complexity of the considered object. Efficient methods were also defined to take into account noise and/or over-segmentation by refining the process at different scales [START_REF] Debled-Rennesson | Multiorder polygonal approximation of digital curves[END_REF][START_REF] Dilip | Polygonal representation of digital curves[END_REF][START_REF] Kerautret | Meaningful scales detection along digital contours for unsupervised local noise estimation[END_REF][START_REF] Vacavant | A combined multiscale/irregular algorithm for the vectorization of noisy digital contours[END_REF], etc. Despite existing relevant approximation works finding the "best" polygon is always a challenge which highly depends on the application under consideration.

In other words, since polygonal contour descriptors are based on the boundary of a shape, they cannot capture the internal structure of a shape. Furthermore, these methods are generally not suited to handle disjoint shapes or shapes with holes because the boundary information is not available. Moreover, a common drawback is error-prone raster to vector conversion inducing loss of information, which may result in lower recognition rates. On the other hand, region-based methods are more suited to general applications. However, they are generally more computationally intense and most approaches need to normalize (centroid position, re-sampling) the image to process with affine properties. Such a normalization introduce errors, sensitivity to noise, and thus inaccuracy in the recognition process.

In this paper we consider a powerful aim consisting in describing the shape at several scales from width of discrete lines [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF]. Considering several widths allows to limit noise effect and limit artifact while limiting to some extent the influence of over-segmentation according to the processed widths. It allows to go progressively from an accurate representation to a more rough one while highlighting the main directions of the parts of the shape. Then we combine such description with a feature descriptor able to efficiently handle with holes or split parts of an object while integrating in its structure both relational parts and spatial variations [START_REF] Matsakis | A new way to represent the relative position between areal objects[END_REF]. This method can be assumed as hybrid; it combines a structural description of shapes with a statistical shape descriptor.

The outline of the paper is as follows. First the main frameworks required to define this new method are recalled in Section 2, that is discrete lines, blurred segments, and force histogram. The multi-polygonal structure is described in Section 3 as well as three basic matching strategies allowing to handle these representations. An experimental study performed on two common databases is given in Section 4 to show the interest of the proposed approach. Finally perspectives are provided in the conclusion.

BACKGROUND

Discrete Lines

The arithmetical definition of discrete lines [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF] is used in our method to decompose a discrete shape contour into discrete line segments.

A discrete line D(a, b, µ, ω), whose main vector is (b, a), lower bound µ and thickness ω (with a, b, µ and ω being integer such that gcd(a, b) = 1) is the set of integer points (x, y) verifying µ ≤ ax -by < µ + ω.

In this work, we study sequences of points, corresponding to shape contours, and we compute polygonal approximations containing all the points of the sequences.

More precisely, let us consider a sequence of points S b , it is named a blurred segment of width ν if there exists a discrete line D(a, b, µ, ω) containing all the points of S f and such that ω-1 max(|a|,|b|) ≤ ν. This discrete line is called the optimal bounding line of S b .

A linear algorithm was proposed in [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear time[END_REF] to incrementally obtain the characteristics of the optimal bounding line of a sequence of points. It relies on the linear and incremental computation of the convex hull of the scanned point sequence as well as on the arithmetical and geometrical properties of discrete lines.

Based on this algorithm, an interesting geometrical structure can be computed on a shape contour: the width-ν tangential cover. It consists in the sequence of all maximal blurred segments of width ν (M BSν ) located on the shape contour (see Fig. 1). The common zone of several M BSν can contain a point with high curvature, called dominant point. A method to detect dominant points from a width-ν tangential cover of a contour is proposed in [START_REF] Nguyen | A discrete geometry approach for dominant point detection[END_REF][START_REF] Nasser | Dominant point detection based on discrete curve structure and applications[END_REF], a polygonal representation of the contour is then deduced. Different values of width then induce several polygonal simplifications of a same contour (see Fig. 4).

Fig. 1 On the left, the width-1 tangential cover of a piece of a shape contour. On the center, the width-2 tangential cover of the same piece. On the right, the width-1 tangential cover of a noisy discrete circle.

Force Histogram

Force Histogram of a shape

The histogram of forces was initially defined to assess the spatial relation between two binary objects but it can be easily used to provide a discriminate signature (and descriptor) characterizing one object. We recall here basic notions aiming at describing such a kind of signature which efficiently integrates both whole shape and spatial description following a set of directions. A full description of the underlying theoretical developments can be found in [START_REF] Matsakis | Relations spatiales structurelles et interprtation dimages[END_REF][START_REF] Matsakis | A new way to represent the relative position between areal objects[END_REF][START_REF] Tabbone | Matching of graphical symbols in line-drawing images using angular signature information[END_REF][START_REF] Matsakis | A general approach to the fuzzy modeling of spatial relationships[END_REF]. The attraction force between two points at a distance d is given by:

∀d ∈ R * + , ϕr(d) = 1/d r
with r, the kind of force processed, e.g. r = 0 for constant forces, which are suitable considering one object and r = 2 for gravitational ones. The handling of segments is considered to decrease the computation time instead of directly studying any pair of points. Let I 1 and I 2 be two segments beared by a line of angle θ k from the frame, D θ k I1I2 the distance between them and |.| the length of a segment. The calculation of the attraction force fr of a segment with regard to another is given by:

fr(|I 1 |, D θ k I1I2 , |I 2 |) = |I1|+D θ k I 1 I 2 +|I2| D θ k I 1 I 2 +|I2| |I2| 0 ϕr(u -v)dvdu
Considering one raster object A, following a direction θ k it can be entirely described by the set of segments beared by a pencil of parallel lines of angle θ k from the orthogonal frame. Let us take one line, denoted D θ k η . The set of S segments corresponds to: A θ k (η) = ∪{I i } i=1,S and the mutual attraction of these segments is given by:

F (θ, A θ k (η), A θ k (η)) = i∈1..S j∈1..S fr(|I i |, D θ k IiIj , |I j |)
Due to superimposed segments, constant forces are suitable in our study. An approximation of gravitational force impact might be done using points while avoiding superimposed points and linked points at a null distance. The last constraint can be eventually removed by using a sub-pixel modeling. Another way can be to introduce progressive forces (as in [START_REF] Matsakis | Relations spatiales structurelles et interprtation dimages[END_REF]) but this generally leads to minor the impact of gravitational ones considering compact shapes.

Force histogram of a polygon

Using raster data all the pencils of lines D η θ which entirely describe A are then processed and the associated mutual attraction summed. Considering a polygonal description of A, noted P A , the calculation is based on a set of regular polygons having 3 or 4 vertices (see Fig. 2). All the lines D η θ bearing a polygonal point are projected on a perpendicular straight line to set the intervals describing A [START_REF] Matsakis | Relations spatiales structurelles et interprtation dimages[END_REF][START_REF] Matsakis | A new way to represent the relative position between areal objects[END_REF][START_REF] Matsakis | A general approach to the fuzzy modeling of spatial relationships[END_REF]. Then A consists in a set of trapezium (or triangle) whose height is given by q Object A Fig. 2 Set of triangles or trapezium describing an object A following a direction θ.

the width of intervals, combined with bases to calculate the associated attraction and summed (Riemann sum).

The calculation of F AA (θ), respectively F P A (θ), remains to an assessment of the forces exerted by an object with itself in the direction θ.

Let us consider a set of p directions Θ = {θ k } k=1,p∧θ k ∈[-π,+π) with a constant step. Finally the calculation of F AA onto Θ defines a spatial relational descriptor, denoted F A for raster data and F P A for polygonal one. The calculation of the complexity depends on the sorting of the projected vertices. That is in O(pv ln(v)) with v the number of vertices of the polygon.

MULTI-POLYGONAL MATCHING

Multi-Polygonal Structure

Let A and B be two shapes and let P A = ∪ ωi∈Ω (P Aω i ) and P B = ∪ ωi∈Ω (P Bω i ) be their multi-polygonal representation obtained using n increasing width Ω = {ω 1 , ω 2 . . . ωn} with ω i < ω i+1 and P Aω i the polygon calculated from the width w i . The force histogram (or distribution) is calculated on each representation from the set of points defining each polygon following p directions Θ (see previous section). Then the multi-polygonal descriptor associated to a shape A is denoted by:

F P A = ωi∈Ω F P Aω i

Multi-Polygonal Descriptor Properties

By axiomatic definitions of the function F , the following properties can be checked and easily introduced in the matching process following the specificity of the application under consideration:

-Translation as polygon are processed independently of their location in an image. Only points and the whole shape of the polygon are considered.

-Symmetry according to a mirror representation of the polygon. In this case processing the directions in the opposite direction allows to directly take into account the symmetry property.

-Homothety according to a normalized distribution. Considering real points, the global shape of the polygon is the same at different scales. Obviously considering discrete data, high disparities might occur at very low scales. However the whole representation of the force histogram remains stable in most of the cases due to the low sensitivity to noise of the approach [START_REF] Matsakis | A new way to represent the relative position between areal objects[END_REF][START_REF] Tabbone | Matching of graphical symbols in line-drawing images using angular signature information[END_REF].

-Rotation because the approach is isotropic. A rotation of the polygon relies on circular shifts on the achieved force histogram. An assessment of the rotation (not invariant) can be carried out by minimizing the distance between two superimposed force histograms at different shifts.

Such properties rely on each polygonal representation P Aω i but they can be easily extended to the whole description F P A as they are calculated from the same located shape and so apply along a common principle at each associated level F P A ω i . It is important to notice that if a shape contains several holes having or not parts of shapes implying a set of external and internal polygons, it is always possible to calculate the associated distribution due to the bi-linearity property. That is trapezium describing the holes are not integrated in the calculation of F . In other words, it is not useful to process holes during the matching step as they are directly taken into account inside of the force histogram.

Matching Strategies

Three schemes able to match two multi polygonal distributions (MPD) F P A and

F P B
are defined in this section. Let M be a metric, or a similarity ratio, calculating a score between two distributions.

Weighted histogram matching

The first scheme, denoted by M 1 relies on the calculation of a global distribution

G(F P A

) related to a shape A by aggregating the distributions achieved at each level. Let θ k be a direction, which corresponds to the k th bin of the distribution. A weighted scheme similar to [START_REF] Dubois | A general approach to parameter evaluation in fuzzy digital pictures[END_REF] is used here as follows (assuming that ω 0 = 0).

G F P A (θ k ) = i=1,n (ω i -ω i-1 )F P Aω i (θ k )
And so on, considering all the p directions (see section 2.

2) to define the average distribution G(F P A

). The distance ∆ M1 between two MPD A and B is directly given by:

∆ M1 (A, B) = M G F P A , G F P B
Considering raster data, a similar scheme was successively applied to classify chest radiograph image view but considering F-signatures computed on a set of image binarisation [START_REF] Santosh | Angular relational signature-based chest radiograph image view classification[END_REF]. Here the modeling is based on polygon handling and the weight is set from the difference between successive width values.

Level polygonal feature set matching

The second scheme, noted M 2 , relies on an assessment level per level of both distributions which are weighted as in M 1 by the difference of two successive levels, that is:

∆ M2 (A, B) = 1 n i=1,n (ω i -ω i-1 )M F P Aω i , F P Bω i
The interest of this scheme is to study the similarity of pairwise distributions at each level ω i . The aggregation is quite similar as the one of Krishnapuram et al. [START_REF] Krishnapuram | Quantitative analysis of properties and spatial relations of fuzzy image regions[END_REF] who modeled standard relationships between fuzzy sets described by level cuts. However we do not ensure a convex hull representation and we put no hypothesis about polygonal level membership.

Key direction matching

At last, a third scheme M 3 is based on the matching of a set of meaningful direction values, denoted by Φ, consisting in local minimum and maximum points extracted from the derivative of each distribution. Let us consider a distribution A at width ω i , Φ A ωi corresponds to the set of directions corresponding to the loci of both extrema and minima of F P Aω i , that is:

Φ A ωi = {θ k |θ k ⊂ Θ\{θ 1 , θp}∧ (F P Aω i (θ k-1 ) < F P Aω i (θ k ) > F P Aω i (θ k+1 ) ∨F P Aω i (θ k-1 ) > F P Aω i (θ k ) < F P Aω i (θ k+1 ))}
Due to the circular representation (modulo the size p), θ 1 and θp are separately processed. Considering two objects A and B, the matching is similar to M 2 but limited to the set of suitable directions Φ A ωi and Φ B ωi extracted at each level ω i :

∆ M3 (A, B) = 1 n i=1,n (ω i -ω i-1 )M F P Aω i Φ A ω i ∪Φ B ω i , F P Bω i Φ A ω i ∪Φ B ω i
In some extent, the underlying idea of the well-known curvature scale space method [START_REF] Mokhtarian | Silhouette-based isolated object recognition through curvature scale space[END_REF] is followed. However the derivative does not require Gaussian filters to smooth the distribution in order to focus on key points. The computation of force histogram integrates the whole shape according to progressive directions and the studied distributions are most of the time regular with no high disparities.

EXPERIMENTAL STUDY

Databases and metrics

Two well-known shape databases provided by Sharvit et al. [START_REF] Sharvit | Symmetry-based indexing of image databases[END_REF] are used to test our methods. The first database B1 consists in nine categories with 11 shapes in each cluster and the second one B2 consists in 18 categories of 12 shapes. Considering B1 a few of the shapes are occluded (airplanes and hands) and some shapes are partially represented (rabbits, men, and hands). There are also distorted objects (tools) and heterogeneous shapes in the same cluster (animals) (see Fig. 3). The database B2 contained also shapes having similar disturbance.

trix. That is, for and the value q i f h i from 0 to p. Shannon theory ng. Here, we set pled values of q i This algorithm ze N • N and M reduce the come possible values s of q i are also increment Dx is refore, when we by cos h/2. Simisin h/2. represented by a m is defined by: at the properties over to the disly, due to errors n Section 6 that are small. tion of the shape odulo p. Thereier transform on hieved by ignoreeping the mag-, respectively, be model shape for l transform. After ansform F, the efined as follows:

; FR l q ðpÞ FR l q ð0Þ
! . . ; FR l m ðpÞ FR l m ð0Þ
.

-transform, the between the two a shape recogniiency of the prosted on several et al. [START_REF] Mokhtarian | Silhouette-based isolated object recognition through curvature scale space[END_REF] kindly ttp://www.lems. ve been used. tegories with 11 hape is matched against all the other shapes of the database (at all 9801 shape comparisons are made) and we count in the nth (n from 1 to 11) nearest neighbors the number of times the test image was correctly classified. A few of the shapes are occluded (clusters airplanes and hands) and some shapes are partially represented (clusters rabbits, men, and hands). There are also distorted objects (cluster 6) and very heterogeneous shapes in the same cluster (animals). Furthermore, to demonstrate scale invariance and robustness to noise, the experiments were tested again with the database of Fig. 7 increased in scale by 400%, scaled down by 50%, and degraded with salt-and-pepper noise (common in binary images) with different degrees. We can remark that the results of the matching are quite similar to those obtained with the original database (compare Figs. 9-11 with Fig. 8). More significantly, we have run the experiments on a certain shape which has been scaled and rotated (see Fig. 12). When two shapes are similar the distance is close to zero and we can see from Fig. 12 that with only one exception, the distance measures are small. The exception is due to pixelation errors resulting from the scaling of the smallest image which has been scaled down by a factor of 32 from the largest. Also a shape from a different class is included in the Fig. 12 to have a reference of how small the metric is for the scaled/rotated dogs. On average the distance is 9 times greater between the dogs and the rabbit images than between the dogs theirself. We also compare our method with two recent approaches which give rise interesting results for several test databases (see [START_REF] Baird | The evolution of document image analysis[END_REF] and [37] for comparative studies):

• The approach of Bernier and Landry [START_REF] Baird | The evolution of document image analysis[END_REF] based on a polar representation of the contour points with respect to the centroid of the shape. • The approach of Zhang and Lu [37] defined on the generic Fourier descriptor transform from a polar-raster sampled shape image. Such databases are widely used to compare feature descriptors which are more adapted to classify raster pattern due to noise, artifacts and distorted parts. For instance, the generic Fourier descriptor [START_REF] Zhang | Shape-based image retrieval using generic fourier descriptor[END_REF] provides high recognition rates on these databases. Here we consider only polygonal representations of shapes which are obviously more sensitive to noise effect and pattern distortions. The aim is to show the interest of such multi-polygonal structure through the matching strategies M 1 , M 2 and M 3 . Several metrics and similarity ratio ∆ will be tested as Jaccard, Tanimoto, Chi 2 and Hellinger. Due to the weak number of samples the evaluation process is similar to a leave one out considering the distance of each sample with the remaining database (as k = 1) to avoid a bias.

Evaluation

Multi-scale structure

We provide in Fig. 4 an example of polygonal decomposition following four widths to better visualize the structure used in this paper. Two samples belong to the same cluster (fish) and another belong to the cluster hand.

It may be noted that the small elements (polygonal contour in green in Fig. 4) of the structure are better preserved at low width (for instance the fins of the Shape ⍵=1 ⍵=2 ⍵=4 ⍵=3 fishes) and a larger width concentrates rather meaningful directions of the shape (the palm of the hand is smoothed as well as fish fins).

Few shapes and the force histogram calculated on associated polygon defined from the width ω 1 are given in Fig. 5. The difference between the shapes can be directly seen in the distributions. For instance the arm raised induced a directional perturbation visible on the second force histogram. The last two force histogram samples indicate the main direction of the shape (plane and tool) while integrating the width of the objects (see the pick of the tool force histogram).

Number of directions

Tests are done on Sharvit B1 to study the impact of the underlying parameters. Five increasing widths are considered for the multi-polygonal representation of all the shapes. The influence of the number of directions p is studied (see Table 1 The achieved results (using the Jaccard index) show a good behavior of the three methods except for a large number of directions. Many spatial directions introduce more irrelevant information during matching and thus reduce the impact of significant directions.

Distance and Similarity

Following the previous study, p is set to 32 directions (16 is also a good compromise). Five usual similarity ratio and distance are studied (see Table 2). We also add a contour based approach Sa which consider circular directions (here 360) defined from the centroid of the shape as in [START_REF] Bernier | A new method for representing and matching shapes of natural objects[END_REF]. The three methods are few sensitive to usual metrics (or similarity). This good behavior, as well as the good recognition rates, shows that the information stored in the force histogram distributions, which integrates both the organization of the spatial relationships and the disparities of the shape in the same distribution, are discriminating and robust at the metric.

Methods

Increasing number of cuts

The impact of increasing number of widths is considered (see Fig. 6). A step set at 0.5 between each width is considered. The number of directions p is set to 16 and Tanimoto index is applied.

An increase in recognition rates is obtained for all three methods. The method M 1 is sensitive to the number of processed polygons. So, the weighted sum of force histogram becomes less discriminating when polygons of increasing width are added. The method M 2 remains stable most of the time, showing the interest of a matching level per level, that is following the same width. The behavior of the method M 3 is interesting as the score increases gradually, showing the influence of main directions at high level.

Test on Sharvit B2

Another test is carried out on the database B2. We consider values of parameters similar to those used to make tests on B 1 , that is p = 32, five widths and the Jaccard index. We also consider a polygonal representation method, denoted by KM , which provide a polygon close to the raster contour of a shape [START_REF] Dilip | Polygonal representation of digital curves[END_REF]. The force histogram is calculated on it to make a comparison. The results are close to those ones obtained using B1 despite an increasing number of samples showing the good behavior of our approaches. Handling with a multi-level approach seems to better take into account the main directional regions of a shape than a relevant method KM where the polygonal contour fits well the pattern but only at one scale.

Methods

Discussion

Both methods M 1 and M 2 provide interesting results while having a robust behavior independently of the metric used.

Most of the recognition rates obtained with M 2 are better than those ones of M 1 showing the interest of handling the polygons, level per level, instead of averaging the amount of associated force histograms.

Fig. 7 .Fig. 3

 73 Fig. 7. A database of 99 shapes made available by Sharvit on his website [30].

Fig. 4

 4 Fig. 4 Multi-Polgonal representation following width.

Fig. 5

 5 Fig. 5 Force histogram calculated on Polygons (width ω 1 ).

Fig. 6

 6 Fig. 6 Recognition rates. Increasing number of cuts.

Table 1

 1 ). Recognition rates obtained using the Jaccard index.

	Directions	128	64	32	16
	M 1	68.9 84.4 83.3	83.3
	M 2	72.2 85.6	85.6	87.8
	M 3	67.8 76.7 81.1	80.0

Table 2

 2 Recognition rates. Metrics and Similarity

		M 1	M 2	M 3	S A
	Jaccard	83.3	85.6	81.1	74.4
	Tanimoto	83.3	85.6	81.1	74.4
	L 1	82.2	85.6	82.2 74.4
	Chi 2	84.4 85.6	82.2	73.3
	Hellinger	84.4 85.6	82.2	66.7

Table 3

 3 Recognition rates (32 bins -Jaccard Index)

		M 1	M 2	M 3	KM	S A
	Rate	84.3 86.7	81.5	83.4	63.0

Considering M 3 the results are a bit disappointing even few directions are processed (lower than 13% in average). This emphasizes that the relational curve between the key directions remains a discriminant information. Furthermore this method has an interesting behavior when the number of processed polygons increases.

CONCLUSION

Original methods aiming at representing a shape from a set of distributions associated to polygons have been proposed in this paper. These approaches combined two powerful concepts: discrete lines and force histogram. Achieved results on two databases are very promising. Furthermore the methods are fast and ensure a robust matching.

New developments integrating the interval between key directions or enabling to process with a different number of polygons per representation are under consideration. As the larger the value of the width, the rougher the aspect of the polygon is, further investigations are also dedicated to the search of the optimal description of each sample of a database.