Dietrich H Äfner
email: hafner@math.u-bordeaux1.fr

DECAY AND NON-DECAY OF THE LOCAL ENERGY FOR THE WAVE EQUATION ON THE DE SITTER-SCHWARZSCHILD METRIC

Keywords: 2000 Mathematics Subject Classification. 35B34, 35P25, 35Q75, 83C57 General relativity, De Sitter-Schwarzschild metric, Local energy decay, Resonances

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Introduction

There has been important progress in the question of local energy decay for the solution of the wave equation in black hole type space-times over the last years. The best results are now known in the Schwarzschild space-time. We refer to the papers of Blue-Soffer [START_REF] Blue | Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole[END_REF], Blue-Sterbenz [START_REF] Blue | Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space[END_REF], and Dafermos-Rodnianski [START_REF] Dafermos | A proof of Price's law for the collapse of a self-gravitating scalar field[END_REF] and references therein for an overview. See also the paper of Finster-Kamran-Smoller-Yau for the Kerr space-time [START_REF] Finster | Decay of solutions of the wave equation in the Kerr geometry[END_REF]. Results on the decay of local energy are believed to be a prerequisite for a possible proof of the global nonlinear stability of these space-times. Today global nonlinear stability is only known for the Minkowski space-time (see [START_REF] Christodoulou | The global nonlinear stability of the Minkowski space[END_REF]).

From our point of view one of the most efficient approaches to the question of local energy decay is the theory of resonances. Resonances correspond to the frequencies and rates of dumping of signals emitted by the black hole in the presence of perturbations (see [START_REF] Chandrasekhar | The mathematical theory of black holes[END_REF]Chapter 4.35]). On the one hand these resonances are today an important hope of effectively detecting the presence of a black hole as we are theoretically able to measure the corresponding gravitational waves. On the other hand, the distance of the resonances to the real axis reflects the stability of the system under the perturbation: larger distances correspond to more stability. In particular the knowledge of the localization of resonances gives precise informations about the decay of the local energy and its rate. The aim of the present paper is to show how this method applies to the simplest model of a black hole: the De Sitter-Schwarzschild black hole.

In the euclidean space, such results are already known, especially for non trapping geometries. The first result is due to Lax and Phillips (see their book [START_REF] Lax | Scattering theory[END_REF]Theorem III.5.4]). They have proved that the cut-off propagator associated to the wave equation outside an obstacle in odd dimension ≥ 3 (more precisely the Lax-Phillips semi-group Z(t)) has an expansion in terms of resonances if Z(T ) is compact for a given T . In particular, there is a uniform exponential decay of the local energy. From Melrose-Sjöstrand [START_REF] Melrose | Singularities of boundary value problems. I[END_REF], this assumption is true for non trapping obstacles. Vaȋnberg [START_REF] Vaȋnberg | Asymptotic methods in equations of mathematical physics[END_REF] has obtained such results for general, non trapping, differential operators using different techniques. In the trapping case, we know, by the work of Ralston [START_REF] Ralston | Solutions of the wave equation with localized energy[END_REF], that it is not possible to obtain uniform decay estimates without loss of derivatives. In the exterior domain of two strictly convex obstacles, the local energy decays exponentially with a loss of derivatives, by the work of Ikawa [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of two convex obstacles[END_REF]. This situation is close to the one treated in this paper. We also mention the works Tang-Zworski [START_REF] Tang | Resonance expansions of scattered waves[END_REF] and Burq-Zworski [START_REF] Burq | Resonance expansions in semi-classical propagation[END_REF] concerning the resonances close to the real line and the work of Christiansen-Zworski [START_REF] Christiansen | Resonance wave expansions: two hyperbolic examples[END_REF] for the wave equation on the modular surface and on the hyperbolic cylinder.

Thanks to the work of Sá Barreto and Zworski ([24]) we have a very good knowledge of the localization of resonances for the wave equation on the De Sitter-Schwarzschild metric.

Using their results we can describe an expansion of the solution of the wave equation on the De Sitter-Schwarzschild metric in terms of resonances. The main term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ε derivative loss in the angular directions. For initial data in the complement of a one-dimensional space the local energy is integrable if we permit a (ln -∆ ω ) α derivative loss with α > 1. This estimate is almost optimal in the sense that it becomes false for α < 1 2 . The method presented in this paper does not directly apply to the Schwarzschild case. This is not linked to the difficulty of the photon sphere which we treat in this paper, but to the possible accumulation of resonances at the origin in the Schwarzschild case.

The exterior of the De Sitter-Schwarzschild black hole is given by

(M, g), M = R t × X with X =]r -, r + [ r ×S 2 ω (1.1) g = α 2 dt 2 -α -2 dr 2 -r 2 dω 2 , α = 1 - 2M r - 1 3 Λr 2 1/2 , (1.2)
where M > 0 is the mass of the black holes and Λ > 0 with 9M 2 Λ < 1 is the cosmological constant. r -and r + are the two positive roots of α = 0. We also denoted by dω 2 the standard metric on S 2 .

The corresponding d'Alembertian is

(1.3) g = α -2 D 2 t -α 2 r -2 D r (r 2 α 2 )D r + α 2 r -2 ∆ ω ,
where D • = 1 i ∂ • and -∆ ω is the positive Laplacian on S 2 . We also denote

P = α 2 r -2 D r (r 2 α 2 )D r -α 2 r -2 ∆ ω ,
the operator on X which governs the situation on L 2 (X, r 2 α -2 dr dω). We define

P = r P r -1 ,
on L 2 (X, α -2 dr dω), and, in the coordinates (r, ω), we have

P = α 2 D r (α 2 D r ) -α 2 r -2 ∆ ω + r -1 α 2 (∂ r α 2 ).
We introduce the Regge-Wheeler coordinate given by

(1.4) x ′ (r) = α -2 .
In the coordinates (x, ω), the operator P is given by (1.5)

P = D 2 x -α 2 r -2 ∆ ω + α 2 r -1 (∂ r α 2 ), on L 2 (R × S 2 , dx dω). Let V = α 2 r -2 and W = α 2 r -1 (∂ r α 2
) be the potentials appearing in P . As stated in Proposition 2.1 of [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF], the work of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] implies that for

χ ∈ C ∞ 0 (R) R χ (λ) = χ(P -λ 2 ) -1 χ,
has a meromorphic extension from the upper half plane to C, whose poles λ are called resonances. The set of the resonances is denoted by Res P . We recall the main result of [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF]:

Theorem 1.1 (Sá Barreto-Zworski).
There exist K > 0 and θ > 0 such that for any C > 0 there exists an injective map, b, from the set of pseudo-poles

(1 -9ΛM 2 ) 1 2 3 3 2 M ±N ± 1 2 -i 1 2 N 0 + 1 2 ,
into the set of poles of the meromorphic continuation of (Pλ 2 ) -1 : L 2 comp → L 2 loc such that all the poles in The natural energy space E for the wave equation is given by the completion of

Ω C = {λ; Im λ > -C, |λ| > K, Im λ > -θ| Re λ|}, are in the image of b and for b(µ) ∈ Ω C , b(µ) -µ → 0 as |µ| → ∞. If µ = µ ± ℓ,j = 3 -3 2 M -1 (1-9ΛM 2 ) 1 2 (±ℓ± 1 2 )-i 1 2 (j+ 1 2 ) , ℓ ∈ N, j ∈ N 0 ,
C ∞ 0 (R × S 2 ) × C ∞ 0 (R × S 2 ) in the norm (1.6) (u 0 , u 1 ) 2 E = u 1 2 + P u 0 , u 0 .
It turns out that this is not a space of distributions. The problem is very similar to the problem for the wave equation in dimension 1. We therefore introduce another energy space

E mod a,b (-∞ < a < b < ∞) defined as the completion of C ∞ 0 (R × S 2 ) × C ∞ 0 (R × S 2 ) in the norm (u 0 , u 1 ) 2 E mod a,b = u 1 2 + P u 0 , u 0 + b a S 2 |u 0 (s, ω)| 2 ds dω. Note that for any -∞ < a < b < ∞ the norms E mod a,b
and E mod 0,1 are equivalent. We will therefore only work with the space E mod 0,1 in the future and note it from now on E mod . Let us write the wave equation g u = 0 as a first order system in the following way:

i∂ t v = Lv v(0) = v 0 with L = 0 i -iP 0 .
Let H k be the scale of Sobolev spaces associated to P . We denote H 2 c the completion of H 2 in the norm u 2 2 := P u, u + P u 2 . Then (L, D(L) = H 2 c ⊕ H 1 ) is selfadjoint on E. We denote E k the scale of Sobolev spaces associated to L. Note that because of

(L -λ) -1 = (P -λ 2 ) -1 λ i -iP λ , (1.7)
the meromorphic extension of the cut-off resolvent of P entails a meromorphic extension of the cut-off resolvent of L and the resonances of L coincide with the resonances of P .

Recall that (-∆ ω , H 2 (S 2 )) is a selfadjoint operator with compact resolvent. Its eigenvalues are ℓ(ℓ + 1), ℓ ≥ 0 with multiplicity 2ℓ + 1. We denote (1.8)

P ℓ = r -1 D x r 2 D x r -1 + α 2 r -2 ℓ(ℓ + 1)
the operator restricted to H ℓ = L 2 (R) × Y ℓ where Y ℓ is the eigenspace of the eigenvalue ℓ(ℓ + 1). In the following, P ℓ will be identified with the operator on L 2 (R) given by (1.8).

The operators L ℓ and the spaces E ℓ , E mod ℓ , E k ℓ are defined similarly to the operator L and the spaces E, E mod , E k . Let Π ℓ be the orthogonal projector on E mod ℓ . For ℓ ≥ 1, the space E mod ℓ and E ℓ are the same and the norms are equivalent uniformly with respect to ℓ.

Using Proposition II.2 of Bachelot and Motet-Bachelot [START_REF] Bachelot | Les résonances d'un trou noir de Schwarzschild[END_REF], the group e -itL preserves the space E mod and there exist C, k > 0 such that e -itL u E mod ≤ Ce k|t| u E mod .

From the previous discussion, the same estimate holds for L ℓ with k = 0 uniformly in ℓ ≥ 1. In particular, (L-z) -1 is bounded on E mod for Im z > k, and we denote E mod,-j = (L-z

) j E mod ⊂ D ′ (R × S 2 ) × D ′ (R × S 2 ) for j ∈ N 0 .
We first need a result on P : Proposition 1.2. For ℓ ≥ 1, the operator P ℓ has no resonance and no eigenvalue on the real axis.

For ℓ = 0, P 0 has no eigenvalue in R and no resonance in R \ {0}. But, 0 is a simple resonance of P 0 , and, for z close to 0, we have

(1.9) (P 0 -z 2 ) -1 = iγ z r r| • + H(z),
where γ ∈]0, +∞[ and H(z) is a holomorphic (bounded) operator near 0. Equation (1.9) is an equality between operators from L 2 comp to L 2 loc .

The proof of Proposition 1.2 is given in Section 2.1. For χ ∈ C ∞ 0 (R) we denote henceforth: R χ (λ) = χ(Lλ) -1 χ.

For a resonance λ j we define m(λ j ) by the Laurent expansion of the cut-off resolvent near λ j :

R χ (λ) = ∞ k=-(m(λ j )+1) A k (λ -λ j ) k .
We also define π χ j,k by

π χ j,k = -1 2πi
(-i) k k! R χ (λ)(λ -λ j ) k dλ. (1.10)
The main result of this paper is the following:

Theorem 1.3. Let χ ∈ C ∞ 0 (R). (i) Let 0 < µ / ∈ (1-9ΛM 2 ) 1/2 3 1/2 M 1 2 N 0 + 1 2
such that there is no resonance with Im z = -µ. Then there exists M > 0 with the following property. Let u ∈ E mod such that -∆ ω M u ∈ E mod . Then we have:

(1.11) χe -itL χu = λ j ∈Res P Im λ j >-µ m(λ j ) k=0 e -iλ j t t k π χ j,k u + E 1 (t)u, with (1.12) E 1 (t)u E mod e -µt -∆ ω M u E mod ,
and the sum is absolutely convergent in the sense that

λ j ∈Res P Im λ j >-µ m(λ j ) k=1 π χ j,k -∆ ω -M L(E mod ) 1. (1.

13)

(ii) There exists ε > 0 with the following property. Let g ∈ C([0, +∞[), lim |x|→∞ g(x) = 0, positive, strictly decreasing with x -1 ≤ g(x) for x large. Let u = (u 1 , u 2 ) ∈ E mod be such that g(-∆ ω )

-1 u ∈ E mod . Then we have

(1.14) χe -itL χu = γ rχ r, χu 2 0 + E 2 (t)u, with (1.15) E 2 (t)u E mod g(e εt ) g(-∆ ω ) -1 u E mod .
Remark 1.4. a) By the results of Sá Barreto and Zworski we know that there exists µ > 0 such that 0 is the only resonance in Im z > -µ. Choosing this µ in (i) the sum on the right hand side contains a single element which is

γ rχ r, χu 2 0 .
b) Again by the paper of Sá Barreto and Zworski we know that λ j = b(µ ε ℓ,  ) for all the λ j 's outside a compact set (see Theorem 1.1). For such λ j , we have m j (λ j ) = 0 and π

χ j,k = Π ℓ π χ j,k = π χ j,k Π ℓ is an operator of rank 2ℓ + 1. c) Let E mod,⊥ = {u ∈ E mod ; r, χu 2 = 0}
. By part (ii) of the theorem, for u ∈ E mod,⊥ , the local energy is integrable if (ln -∆ ω ) α u ∈ E mod , for some α > 1, and decays exponentially if -∆ ω ε u ∈ E mod for some ε > 0.

d) In fact, we can replace -∆ ω M by P 2M in the first part of the theorem. And, by an interpolation argument, we can obtain the following estimate: for all ε > 0, there exists δ > 0 such that

(1.16) χe -itL χu = γ rχ r, χu 2 0 + E 3 (t)u, with (1.17) E 3 (t)u E mod e -δt P ε u E mod .
Remark 1.5. In the Schwarzschild case the potential V (x) is only polynomially decreasing at infinity and we cannot apply the result of Mazzeo-Melrose. Therefore we cannot exclude a possible accumulation of resonances at 0. This difficulty has nothing to do with the presence of the photon sphere which is treated by the method presented in this paper.

Remark 1.6. Let u ∈ E mod,⊥ be such that (ln -∆ ω ) α u ∈ E mod for some α > 1. Then we have from part (ii) of the theorem, for λ ∈ R,

∞ 0 χe -it(L-λ) χu dt E mod (ln -∆ ω ) α u E mod . (1.18)
This estimate is almost optimal since it becomes false for α < 1 2 . Indeed we have (λ ∈ R):

R χ (λ)u = i ∞ 0 χe -it(L-λ) χu dt.
Thus from (1.18) we obtain the resolvent estimate

R χ (λ)(ln -∆ ω ) -α L(E mod,⊥ ,E mod ) 1.
It is easy to see that this entails the resolvent estimate

χ(P ℓ -λ 2 ) -1 χ(ln ℓ(ℓ + 1) ) -α 1 |λ| ,
for ℓ ≥ 1. We introduce the semi-classical parameter h 2 = (ℓ(ℓ + 1)) -1 and P = h 2 D 2 x + V (x) + h 2 W (x) as in Section 2.3. Then, for R > 0, the above estimate gives the semi-classical estimate:

χ( P -z) -1 χ | ln h| α h , for 1/R ≤ z ≤ R (see (2.

25) and (2.26))

. Such an estimate is known to be false for α < 1 2 and z = z 0 , the maximum value of the potential

V (x) (see [1, Proposition 2.2]).
Remark 1.7. Let P 1 be the projection on the first variable,

P 1 (u 1 , u 2 ) = u 1 . If u ∈ E mod is such that g(-∆ ω ) -1 (L + i)u ∈ E mod , then P 1 χe -itL χu ∈ C 0 (R × S 2
) and the remainder term in (1.14) satisfies

(1.19) P 1 E 2 (t)u L ∞ (R×S 2 ) g(e εt ) g(-∆ ω ) -1 (L + i)u E mod . Moreover, if u ∈ E mod is such that g(-∆ ω ) -1 (L + i) 2 u ∈ E mod , then χe -itL χu ∈ C 0 ((R × S 2 ) × (R × S 2
)) and the remainder term in (1.14) satisfies

(1.20) E 2 (t)u L ∞ ((R×S 2 )×(R×S 2 )) g(e εt ) g(-∆ ω ) -1 (L + i) 2 u E mod .
The proof of the theorem is based on resolvent estimates. Using (1.7) we see that it is sufficient to prove resolvent estimates for χ(P ℓλ 2 ) -1 χ. This is the purpose of the next section.

Estimate for the cut-off resolvent.

In this section, we obtain estimates for the cut-off resolvent of P ℓ , the operator P restricted to the spherical harmonic ℓ. We will use the description of the resonances given in Sá Barreto-Zworski [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF]. Recall that

(2.1) R χ (λ) = χ(P -λ 2 ) -1 χ,
has a meromorphic extension from the upper half plane to C. The resonances of P are defined as the poles of this extension. We treat only the case Re λ > -1 since we can obtain the same type of estimates for Re λ <

1 using R χ (-λ) * = R χ (λ).
Theorem 2.1. Let C 0 > 0 be fixed. The operators χ(P ℓλ 2 ) -1 χ satisfy the following estimates uniformly in ℓ.

i) For all R > 0, the number of resonances of P is bounded in B(0, R). Moreover, there exists C > 0 such that

(2.2) χ(P ℓ -λ 2 ) -1 χ ≤ χ(P -λ 2 ) -1 χ ≤ C λ j ∈Res P |λ j |<2R 1 |λ -λ j |
for all λ ∈ B(0, R). As usual, the resonances are counted with their multiplicity.

ii) For R large enough,

P ℓ has no resonance in [R, ℓ/R] + i[-C 0 , 0]. Moreover, there exists C > 0 such that (2.3) χ(P ℓ -λ 2 ) -1 χ ≤ C λ 2 , for λ ∈ [R, ℓ/R] + i[-C 0 , C 0 ].
iii) Let R be fixed. For ℓ large enough, the resonances of

P ℓ in [ℓ/R, Rℓ] + i[-C 0 , 0] are the b(µ + ℓ,j
) given in Theorem 1.1 (in particular their number is bounded uniformly in ℓ). Moreover, there exists C > 0 such that

(2.4) χ(P ℓ -λ 2 ) -1 χ ≤ C λ C λ j ∈Res P ℓ |λ-λ j |<1 1 |λ -λ j | , for λ ∈ [ℓ/R, Rℓ] + i[-C 0 , C 0 ].
Furthermore, P ℓ has no resonance in [ℓ/R, Rℓ] + i[-ε, 0], for some ε > 0, and we have

(2.5) χ(P ℓ -λ 2 ) -1 χ ≤ C ln λ λ e C| Im λ| ln λ , for λ ∈ [ℓ/R, Rℓ] + i[-ε, 0].
iv) Let C 1 > 0 be fixed. For R large enough, P ℓ has no resonance in {λ ∈ C; Re λ > Rℓ, and 0 ≥ Im λ ≥ -C 0 -C 1 ln λ }. Moreover, there exists C > 0 such that

(2.6) χ(P ℓ -λ 2 ) -1 χ ≤ C λ e C| Im λ| , for Re λ > Rℓ and C 0 ≥ Im λ ≥ -C 0 -C 1 ln λ .
The results concerning the localization of the resonances in this theorem are proved in [START_REF] Bachelot | Les résonances d'un trou noir de Schwarzschild[END_REF] and [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF], the following figure summarizes the different estimates of the resolvent. In zone I which is compact, the result of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] gives a bound uniform with respect to ℓ (away from the possible resonances). In particular, part i) of Theorem 2.1 is a direct consequence of this work.

Im λ = -C 0 -C 1 ln λ I II III IV λ ∈ C -C 0 R ℓ/R Rℓ ℓ max V
In zone II, the result of Zworski [START_REF] Zworski | Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces[END_REF] gives us a good (uniform with respect to ℓ) estimate of the resolvent. Here, we use the exponential decay of the potential at +∞ and -∞.

By comparison, the corresponding potential for the Schwarzschild metric does not decay exponentially, and our present work cannot be extended to this setting. Note that this problem concerns only zones I and II, but zones III and IV can be treated in the same way.

In zone III, we have to deal with the so called "photon sphere". The estimate (2.4) follows from a general polynomial bound of the resolvent in dimension 1 (see [START_REF] Bony | Microlocalization of resonant states and estimates of the residue of the scattering amplitude[END_REF]).

In zone IV, the potentials ℓ(ℓ + 1)V and W are very small in comparison to λ 2 . So they do not play any role, and we obtain the same estimate as in the free case of -∆ (or as for non trapping geometries).

Estimate close to 0.

This part is devoted to the proof of Proposition 1.2 and of part i) of Theorem 2.1. Since χ(Pλ 2 ) -1 χ has a meromorphic extension to C, the number of resonances in B(0, R) is always bounded and point i) of Theorem 2.1 is clear. It is a classic result (see Theorem XIII.58 in [START_REF] Reed | Methods of modern mathematical physics[END_REF]) that P ℓ has no eigenvalue in R \ {0}. On the other hand, from Proposition II.1 of the work of Bachelot and Motet-Bachelot [START_REF] Bachelot | Les résonances d'un trou noir de Schwarzschild[END_REF], 0 is not an eigenvalue of the operators P ℓ . Moreover, by the limiting absorption principle [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF],

(2.7)

x -α (P ℓ -

(z + i0)) -1 x -α < ∞,
for z ∈ R \ {0} and any α > 1, we know that P ℓ has no resonance in R \ {0}.

We now study the resonance 0 using a technique specific to the one dimensional case. We start by recalling some facts about outgoing Jost solutions. Let

(2.8) Q = -∆ + V (x),
be a Schrödinger operator with V ∈ C ∞ (R) decaying exponentially at infinity. For Im λ > 0, there exists a unique pair of functions e ± (x, λ) such that

   (Q -λ 2 )e ± (x, λ) = 0 lim x→±∞ e ± (x, λ) -e ±iλx = 0
The function e ± is called the outgoing Jost solution at ±∞. Since V ∈ C ∞ (R) decays exponentially at infinity, the functions e ± can be extended, as C ∞ (R) functions of x, analytically in a strip {λ ∈ C; Im λ > -ε}, for some ε > 0. Moreover, in such a strip, they satisfy

|e ± (x, λ) -e ±iλx | =O(e -|x|(Im λ+δ) ) for ± x > 0 (2.9)
|∂ x e ± (x, λ) ∓ iλe ±iλx | =O(e -|x|(Im λ+δ) ) for ± x > 0, (2.10) for some δ > 0. All these properties can be found in Theorem XI.57 of [START_REF] Reed | Methods of modern mathematical physics[END_REF].

Using these Jost solutions, the kernel of (Qλ 2 ) -1 , for Im λ > 0 takes the form

(2.11) R(x, y, λ) = 1 w(λ) e + (x, λ)e -(y, λ)H(x -y) + e -(x, λ)e + (y, λ)H(y -x) ,
where H(x) is the Heaviside function

H(x) = 1 for x > 0 0 for x ≤ 0 ,

and

(2.12)

w(λ) = (∂ x e -)e + -(∂ x e + )e -,
is the wronskian between e -and e + (the right hand side of (2.12) does not depend on x).

In particular, w(λ) is an analytic function on {λ ∈ C; Im λ > -ε}. Since the e ± are always non-zero thanks to (2.9), the resonances are the zeros of w(λ). Such a discussion can be found in the preprint of Tang-Zworski [START_REF] Tang | Potential scattering on the real line[END_REF].

Remark that P ℓ is of the form (2.8). If 0 is a resonance of one of the P ℓ 's with ℓ ≥ 1, the Jost solutions e ± (x, 0) are collinear. In particular, from (2.9) and (2.10), the C ∞ function e + (x, 0) converge to two non zero limits at ±∞ and ∂ x e + (x, 0) goes to 0 as x → ±∞. Since (2.13)

P ℓ = r -1 D x r 2 D x r -1 + α 2 r -2 ℓ(ℓ + 1),
we get, by an integration by parts,

0 = R -R (P ℓ e + )e + dx =ℓ(ℓ + 1) R -R |αr -1 e + | 2 dx + R -R |rD x (r -1 e + )| 2 dx -ir -1 e + D x (r -1 e + ) R -R . (2.14) Since ∂ x (r -1 e + ) = r -1 ∂ x e + -
r -2 α 2 e + , the last term in (2.14) goes to 0 as R goes to +∞. Thus, if ℓ ≥ 1, (2.14) gives e + = 0 and 0 is not a resonance of P ℓ .

We now study the case ℓ = 0. If u ∈ C 2 (R) satisfies P 0 u = 0, we get from (2.13)

r 2 D x r -1 u = -iβ, where β ∈ C a constant. Then u(x) = αr(x) + βr(x) x 0 1 r 2 (t)
where α, β ∈ C constants. Note that r(x) := r(x)

x 0 1 r 2 (t) dt = x r ± + O(1),
as x → ±∞. Since e ± (x, 0) are C ∞ functions bounded at ±∞ from (2.9) which satisfy P 0 u = 0, the two functions e ± (x, 0) are collinear to r and then w(0) = 0 which means that 0 is a resonance of P 0 . The resolvent of P 0 thus has the form

(P 0 -λ 2 ) -1 = Π J λ J + • • • + Π 1 λ + H(λ),
where H(λ) is an analytic family of bounded operators near 0 and Π J = 0.

For all λ = iε with ε > 0, we have

λ 2 (P 0 -λ 2 ) -1 L 2 →L 2 = ε 2 (P 0 + ε 2 ) -1 L 2 →L 2 ≤
1, from the functional calculus. This inequality implies that J ≤ 2 and

Π 2 L 2 →L 2 ≤ 1. If f (x) ∈ L 2
loc is in the range of Π 2 , we have f ∈ L 2 and P 0 f = 0. Then, f ∈ H s for all s and f is an eigenvector of P 0 for the eigenvalue 0. This is impossible because P 0 has no eigenvalue. Thus Π 2 = 0 and J = 1.

So w(λ) has a zero of order 1 at λ = 0. Since e ± (x, 0) = r(x)/r ± , (2.11) implies that the kernel of Π 1 is given by

(2.15) Π 1 (x, y) = 1 w ′ (0)r + r - r(x)r(y) = iγr(x)r(y).
Finally, since iε(P 0 + ε 2 ) -1 → Π 1 as ε → 0 and since P 0 + ε 2 is a strictly positive operator, we get -iΠ 1 u, u ≥ 0 for all u ∈ L 2 comp . In particular, -iiγ > 0 and then γ ∈]0, +∞[.

2.2.

Estimate for λ small in comparison to ℓ.

In this section, we give an upper bound for the cut-off resolvent for λ ∈

[R, ℓ/R]+i[-C 0 , C 0 ]. We assume that λ ∈ [N, 2N ] + i[-C 0 , C 0 ] with N ∈ [R, ℓ/R], and define a new semi-classical parameter h = N -1 , a new spectral parameter z = h 2 λ 2 ∈ [1/4, 4] + i[-4C 0 h, 4C 0 h] and (2.16) P = -h 2 ∆ + h 2 ℓ(ℓ + 1)V (x) + h 2 W (x).
With these notations, have (2.17)

(P ℓ -λ 2 ) -1 = h 2 ( P -z) -1 .
We remark that β 2 := h 2 ℓ(ℓ + 1) ≫ 1 in our window of parameters. The potentials V and W have a holomorphic extension in a sector

(2.18) Σ = {x ∈ C; | Im x| ≤ θ 0 | Re x| and | Re x| ≥ C},
for some C, θ 0 > 0. From the form of α 2 (see (1.2)), there exist κ ± > 0 and functions

f ± ∈ C ∞ (R ± ; [1/C, C]), C > 0, analytic in Σ such that (2.19) V (x) = e ∓κ ± x f ± (x),
for x ∈ Σ and ± Re x > 0. Moreover, f ± have a (non zero) limit for x → ±∞, x ∈ Σ.

Under these hypotheses, and following Proposition 4.4 of [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF], we can use the specific estimate developed by Zworski in [START_REF] Zworski | Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces[END_REF] for operators like (2.16) with V satisfying (2.19). In the beginning of Section 4 of [START_REF] Zworski | Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces[END_REF], Zworski defines a subtle contour Γ θ briefly described in the following figure. Recall that the distorted operator

Σ C 1 C 2 C 1 C 2 -2 κ-ln β 2 κ+ ln β x ∈ C I m x = θ 0 R e x θ C Γ θ
P θ = P | Γ θ is defined by (2.20) P θ u = ( P u) | Γ θ
for all u analytic in Σ and then extended as a differential operator on L 2 (Γ θ ) by means of almost analytic functions. The resonances of P in the sector S θ = {e -2is r; 0 < s < θ and r ∈ ]0, +∞[} = e 2i]-θ,0] ]0, +∞[ are then the eigenvalues of P θ in that set. For the general theory of resonances, see the paper of Sjöstrand [START_REF] Sjöstrand | A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory[END_REF] or his book [START_REF] Sjöstrand | Lectures on resonances[END_REF].

For θ large enough, Proposition 4.1 of [START_REF] Zworski | Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces[END_REF] proves that P has no resonance in [1/4, 4] + i[-4C 0 h, 4C 0 h]. Moreover, for z in that set, this proposition gives the uniform estimate (2.21)

( P θ -z) -1 ≤ C.
Since Γ θ coincides with R for x ∈ supp χ, we have (2.22) χ( Pz) -1 χ = χ( P θz) -1 χ, from Lemma 3.5 of [START_REF] Sjöstrand | Complex scaling and the distribution of scattering poles[END_REF]. Using (2.16), we immediately obtain (2.23)

χ(P ℓ -λ 2 ) -1 χ ≤ C λ 2 ,
which is exactly (2.3).

2.3.

Estimate for λ of order ℓ.

In this part, we study the cut-off resolvent for the energy λ ∈ [ℓ/R, Rℓ] + i[-C 0 , C 0 ]. In this zone, we have to deal with the photon sphere. We define the new semi-classical parameter h = (ℓ(ℓ + 1)) -1/2 and (2.24)

P = -h 2 ∆ + V (x) + h 2 W (x).
As previously, we have

(2.25) (P ℓ -λ 2 ) -1 = h 2 ( P -z) -1 , where (2.26) z = h 2 λ 2 ∈ [1/2R 2 , R 2 ] + i[-3RC 0 h, 0] ⊂ [a, b] + i[-ch, ch],
with 0 < a < b and 0 < c. Note that V is of the form:

z 0 V (x) x 0 x Figure 4. The potential V (x).
In particular, V admits at x 0 a non-degenerate maximum at energy z 0 > 0. On the other hand, for z = z 0 , z > 0, the energy level z is non trapping for p 0 (x, ξ) = ξ 2 + V (x), the principal semi-classical symbol of P . We define P θ by standard distortion (see Sjöstrand [START_REF] Sjöstrand | A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory[END_REF]) and can apply the following general upper bound on the cut-off resolvent in dimension one.

Lemma 2.2 (Lemma 6.5 of [START_REF] Bony | Microlocalization of resonant states and estimates of the residue of the scattering amplitude[END_REF]). We assume that n = 1 and that the critical points of p 0 (x, ξ) on the energy level E 0 are non-degenerate (i.e. the points (x, ξ) ∈ p -1 0 ({E 0 }) such that ∇ p 0 (x, ξ) = 0 satisfy Hess p 0 (x, ξ) is invertible). Then, there exists ε > 0 such that, for

E ∈ [E 0 -ε, E 0 + ε] and θ = N h with N > 0 large enough, (2.27) ( P θ -z) -1 = O(h -M ) z∈Res P |z-z j |<εθ h |z -z j |
for |z -E| < εθ/2 and some M > 0 which depends on N .

Note that there is a slight error in the statement of the lemma in [START_REF] Bony | Microlocalization of resonant states and estimates of the residue of the scattering amplitude[END_REF]. Indeed, M depends on N , and in the proof of this lemma, the right hand side of (6.18), O(ln(1/θ)), must be replaced by O(θh -1 ln(1/θ)).

Recall that, from Proposition 4.3 [START_REF] Barreto | Distribution of resonances for spherical black holes[END_REF], which is close to the work of Sjöstrand [START_REF] Sjöstrand | Semiclassical resonances generated by nondegenerate critical points[END_REF] on the the resonances associated to a critical point, there exists an injective map b(h) from

(2.28) Γ 0 (h) = µ j = z 0 -ih |V ′′ (x 0 )|/2(j + 1/2); j ∈ N 0 ,
the set of resonances of P such that

(2.29) b(h)(µ) -µ = o(h), µ ∈ Γ 0 (h),
and such that all the resonances in [a/2, 2b] + i[-ch, ch] are in the range of b(h). In particular, the number of resonances of P is bounded in [a/2, 2b] + i[-ch, ch]. Furthermore, the operator

P has no resonance in Ω(h) = [a/2, 2b] + i[-εh, ch],
for any ε > 0 and h small enough.

Using a compactness argument, we get (2.27) for all z ∈ [a, b] + i[-ch, ch]. Thus, from (2.25), (2.26), χ( Pz) -1 χ = χ( P θz) -1 χ, the estimate λ h -1 = ℓ(ℓ + 1) λ for λ ∈ [ℓ/R, Rℓ] + i[-C 0 , 0], Lemma 2.2 and the previous discussion, we get (2.30)

χ(P ℓ -λ 2 ) -1 χ ≤ C λ C z j ∈Res P |λ-λ j |<1 1 |λ -λ j | , for λ ∈ [ℓ/R, Rℓ] + i[-C 0 , C 0 ]
and (2.4) follows.

On the other hand, P has no resonance in Ω(h) and in this set (2.31)

χ( P -z) -1 χ    h -M on Ω(h), 1 | Im z| on Ω(h) ∩ {Im z > 0}.
We can now apply the following version, due to Burq [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF], of the so-called "semi-classical maximum principle" introduced by Tang-Zworski [START_REF] Tang | From quasimodes to resonances[END_REF].

Lemma 2.3 (Burq). Suppose that f (z, h) is a family of holomorphic functions defined for 0 < h < 1 in a neighborhood of

Ω(h) = [a/2, 2b] + i[-ch, ch],
with 0 < a < b and 0 < c, such that

|f (z, h)|    h -M on Ω(h), 1 | Im z| on Ω(h) ∩ {Im z > 0}.
Then, there exists h 0 , C > 0 such that, for any 0 < h < h 0 ,

(2.32) |f (z, h)| ≤ C | ln h| h e C| Im z|| ln h|/h , for z ∈ [a, b] + i[-ch, 0].
This lemma is strictly analogous to Lemma 4.7 of [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF]. Combining (2.25), (2.26), λ h -1 λ with this lemma, we obtain (2.33)

χ(P ℓ -λ 2 ) -1 χ ≤ C ln λ λ e C| Im λ| ln λ , for λ ∈ [ℓ/R, Rℓ] + i[-ε, 0],
for some ε > 0.

Estimate for the very large values of λ.

Here, we study the resolvent for |λ| ≫ ℓ. More precisely, we assume that

λ ∈ [N, 2N ] + i[-C ln N, C 0 ],
for some C > 0 fixed and N ≫ ℓ. We define the new semi-classical parameter h = N -1 and

z = h 2 λ 2 ∈ h 2 [N 2 /2, 4N 2 ] + ih 2 [-4CN ln N, 4C 0 N -1 ] ⊂ [a, b] + i[-ch| ln h|, ch],
for some 0 < a < b and 0 < c. Then, P ℓ can be written

P ℓ -λ 2 = h -2 ( P -z),
where P = -h 2 ∆ + µV (x) + νW (x), with µ = ℓ(ℓ + 1)h 2 , ν = h 2 . For N ≫ ℓ, the coefficients µ, ν are small, and the operator P is uniformly non trapping for z ∈ [a, b]. We can expect a uniform bound of the cut-off resolvent in [a, b] + i[-ch| ln h|, ch]. Such a result is proved in the following lemma.

Lemma 2.4. For all χ ∈ C ∞ 0 (R), there exist µ 0 , ν 0 , h 0 , C > 0 such that, for all µ < µ 0 , ν < ν 0 and h < h 0 , P has no resonance in

[a, b] + i[-ch| ln h|, ch]. Moreover (2.34) χ( P -z) -1 χ ≤ C h e C| Im z|/h , for all z ∈ [a, b] + i[-ch| ln h|, ch]. Assume first Lemma 2.4. For λ ∈ [N, 2N ] + i[-C ln N, C 0 ], we have χ(P ℓ -λ 2 ) -1 χ = h 2 χ( P -z) -1 χ ≤Che C| Im z|/h ≤ C |λ| e 4C| Im λ| ,
and the estimate (2.6) follows.

Proof of Lemma 2.4. For µ and ν small and fixed, the estimate (2.34) is already known. The proof can be found in the book of Vaȋnberg [START_REF] Vaȋnberg | Asymptotic methods in equations of mathematical physics[END_REF] in the classical case and in the paper of Nakamura-Stefanov-Zworski [START_REF] Nakamura | Resonance expansions of propagators in the presence of potential barriers[END_REF] in our semi-classical setting. To obtain Lemma 2.4, we only have to check the uniformity (with respect to µ and ν) in the proof of [20, Proposition 3.1].

• Limiting absorption principle.

The point is to note that (2.35)

A = xhD x + hD x x,
is a conjugate operator for all µ, ν ≪ 1. Let g ∈ C ∞ 0 ([a/3, 3b]; [0, 1]) be equal to 1 near [a/2, 2b]. The operator g( P )Ag( P ) is well defined on D(A), and its closure, A, is self-adjoint. The operator P is of class C 2 (A). Recall that P is of class C r (A) if there exists z ∈ C \ σ( P ) such that R ∋ t → e itA ( Pz) -1 e -itA , is C r for the strong topology of L 2 (see [2, Section 6.2] for more details).

We have

(2.36) ih -1 [ P , A] = 4 P -4µV -4νW -2µxV ′ -2νxW ′ .
In particular, for µ and ν small enough, we easily obtain

(2.37) 1 [a/2,2b] ( P )i[ P , A]1 [a/2,2b] ( P ) ≥ ah1 [a/2,2b] ( P ).
Note that this Mourre estimate is uniform with respect to µ, ν.

It is also easy to check that (2.38)

x -1 A ≤ C ( P + i) -1 [ P , A] ≤ Ch ( P + i) -1 [[ P , A], A] ≤ Ch 2 ( P + i) -1 [ P , [ P , A]] ≤ Ch 2 ( P + i) -1 A[ P , [ P , A]] ≤ Ch 2 ,
uniformly in µ, ν.

The regularity P ∈ C 2 (A), the Mourre estimate (2.37) and the upper bound (2.38) are the key assumptions for the limiting absorption principle. In particular, from, for instance, the proof of Proposition 3.2 in [START_REF] Alexandrova | Semiclassical scattering amplitude at the maximum point of the potential[END_REF] which is an adaptation of the theorem of Mourre [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF], we obtain the following estimate: For α > 1/2, there exist µ 0 , ν 0 , h 0 , C > 0, such that (2.39)

x -α ( P -z) -1 x -α ≤ Ch -1 , for all µ < µ 0 , ν < ν 0 , h < h 0 and z ∈ [a/2, 2b] + i]0, ch]. In particular, (2.40) χ( P -z) -1 χ ≤ Ch -1 . for z ∈ [a/2, 2b] + i[0, ch].
• Polynomial estimate in the complex.

The second point of the proof is to obtain a polynomial bound of the distorted resolvent. To obtain such bounds, we use the paper of Martinez [START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF]. In this article, the author studies the resonances of Q = -h∆ + V (x) where V is a C ∞ (R n ) function which can be extended analytically in a domain like Σ (see (2.18)) and decays in this domain. If the energy level z 0 is non trapped for the symbol q(x, ξ) = ξ 2 + V (x), the operator Q has no resonance in [z 0δ, z 0 + δ] + i[Ah ln h, 0] for a δ small enough and any A > 0. Moreover,

(2.41) (Q θ -z) -1 ≤ Ch -C for z ∈ [z 0 -δ, z 0 + δ] + i[Ah ln h, 0].
Here Q θ denotes the distorted operator outside of a large ball of angle θ = Bh| ln h|, with B ≫ A.

Of course, P satisfies the previous assumption on Q, for µ and ν fixed small enough. But, following line by line the proof of (2.41) in [START_REF] Martinez | Resonance free domains for non globally analytic potentials[END_REF]Section 4], one can prove that (2.41) is uniformly true for µ, ν ≪ 1. This means that there exist µ 0 , ν , h 0 C > 0 such that (2.42)

χ( P -z) -1 χ = χ( P θ -z) -1 χ ≤ Ch -C , for all µ < µ 0 , ν < ν 0 , h < h 0 and z ∈ [a/2, 2b] + i[ch ln h, 0].
• Semi-classical maximum principle.

To finish the proof, we use a version of the semi-classical maximum principle. This argument can be found in [20, Proposition 3.1], but we give it here for the convenience of the reader.

We can construct a holomorphic function f (z, h) with the following properties:

|f | ≤ C for z ∈ [a/2, 2b] + i[ch ln h, 0], |f | ≥ 1 for z ∈ [a, b] + i[ch ln h, 0], |f | ≤ h M for z ∈ [a/2, 2b] \ [2a/3, 3b/2] + i[ch ln h, 0],
where M is the constant C given in (2.42). We can then apply the maximum principle in The cut-off resolvent estimates for P ℓ give immediately cut-off resolvent estimates for L ℓ .

Proposition 3.1. Let χ ∈ C ∞ 0 (R). Then the operator χ(L ℓλ) -1 χ sends E mod ℓ into itself we have uniformly in ℓ:

(3.1) χ(L ℓ -z) -1 χ L(E mod ℓ ) z χ(P ℓ -z 2 ) -1 χ
Proof. Using Theorem 2.1, (1.7), the equivalence of the norms E mod a,b as well as the fact that we can always replace u by χu, χ ∈ C ∞ 0 (R), χχ = χ we see that it is sufficient to show:

χ(P ℓ -z 2 ) -1 χu H 1 χ(P ℓ -z 2 ) -1 χ u H 1 , (3.2) χ(P ℓ -z 2 ) -1 χu H 1 z χ(P ℓ -z 2 ) -1 χ u L 2 , (3.3) χ(P ℓ -z 2 ) -1 P ℓ χu L 2 z χ(P ℓ -z 2 ) -1 χ u H 1 . (3.4)
Using complex interpolation we see that it is sufficient to show:

χ(P ℓ -z 2 ) -1 χu H 2 χ(P ℓ -z 2 ) -1 χ u H 2 , (3.5) χ(P ℓ -z 2 ) -1 χu H 2 z 2 χ(P ℓ -z 2 ) -1 χ u L 2 , (3.6) χ(P ℓ -z 2 ) -1 P ℓ χu L 2 χ(P ℓ -z 2 ) -1 χ u H 2 . (3.7)
We start with (3.7) which follows from

χ(P ℓ -z 2 ) -1 P ℓ χ = χ(P ℓ -z 2 ) -1 χP ℓ + χ(P ℓ -z 2 ) -1 [P ℓ , χ]u.
Let us now observe that

P ℓ χ(P ℓ -z 2 ) -1 χu =[P ℓ , χ](P ℓ -z 2 ) -1 χ + χ(P ℓ -z 2 ) -1 P ℓ χu = χ(P ℓ + i) -1 [P ℓ , [P , χ]](P ℓ -z 2 ) -1 χu + χ(P ℓ + i) -1 [P ℓ , χ](P ℓ -z 2 ) -1 (P ℓ + i)χu + χ(P ℓ -z 2 ) -1 P ℓ χu.
From this identity we obtain (3.5) and (3.6) using (3.7) (for (3.5)) and the uniform boundedness of (P ℓ + i) -1 [P ℓ , [P ℓ , χ]].

Resonance expansion for the wave equation.

For the proof of the main theorem we follow closely the ideas of Vaȋnberg [START_REF] Vaȋnberg | Asymptotic methods in equations of mathematical physics[END_REF]Chapter X.3]. If N is a Hilbert space we will denote by L 2 ν (R; N ) the space of all functions v(t) with values in N such that e -νt v(t)

∈ L 2 (R; N ). Let u ∈ E mod ℓ and v(t) = e -itL ℓ u t ≥ 0, 0 t < 0.
Then v ∈ L 2 ν (R; E ℓ ) for all ν > 0. We can define ṽ

(k) = ∞ 0 v(t)e ikt dt
as an element of E for all k with Im k > 0. The function ṽ depends analytically on k when Im k > 0. Also, on the line Im k = ν the function belongs to L 2 (R; E ℓ ). We have the inversion formula:

v(t) = 1 2π ∞+iν -∞+iν
e -ikt ṽ(k) dk and the integral converges in L 2 ν (R; E ℓ ) for all ν > 0. From the functional calculus we know that ṽ(k) = -i(L ℓk) -1 u for all k with Im k > 0. We therefore obtain for all t ≥ 0:

e -itL ℓ u = 1 2πi ∞+iν -∞+iν (L ℓ -k) -1 e -ikt u dk, (3.8)
where the integral is convergent in L 2 ν (R; E ℓ ). In the following, we denote by

R ℓ χ (k) the meromorphic extension of χ(L ℓ -k) -1 χ. Lemma 3.2. Let χ ∈ C ∞ 0 (R), N ≥ 0.
Then, there exist bounded operators B j ∈ L(E mod,-q ℓ ; E mod,-j-q ℓ ), j = 0, . . . , N , q ∈ N 0 and B ∈ L(E mod,-q ℓ ; E mod,-N -1-q ℓ ), q ∈ N 0 such that

(3.9) R ℓ χ (k) = N j=0 1 (k -i(ν + 1)) j+1 B j + 1 (k -i(ν + 1)) N +1 B R ℓ χ (k)χ, for some χ ∈ C ∞ 0 (R) with χ χ = χ.
Proof. We by induction over N . For N = 0, we write

(L ℓ -k) -1 + 1 k -i(ν + 1) = 1 k -i(ν + 1) (L ℓ -i(ν + 1))(L ℓ -k) -1 . and choose B 0 = -χ 2 . Then (3.10) R ℓ χ (k) - 1 k -i(ν + 1) B 0 = 1 k -i(ν + 1) B χ, χ R ℓ χ (k)χ, where B χ, χ = χ(L ℓ -i(ν + 1)) χ, with χ = χ χ, is in the space L(E m,-q ℓ ; E m,-1-q ℓ ).
Let us suppose that the lemma is proved for N ≥ 0. We put

(3.11) B N +1 = 1 (k -i(ν + 1)) N +1 B χ 2 χ Using (3.10), we get R ℓ χ (k) = N j=0 1 (k -i(ν + 1)) j+1 B j + 1 (k -i(ν + 1)) N +1 B R ℓ χ χ = N +1 j=0 1 (k -i(ν + 1)) j+1 B j + 1 (k -i(ν + 1)) N +2 B B χ, χ R ℓ χ χ, (3.12)
with χ ∈ C ∞ 0 (R) with χ χ = χ. This proves the lemma.

Let us define

R ℓ χ (k) = R ℓ χ (k) - 1 j=0 1 (k -i(ν + 1)) j+1 B j . Then, Lemma 3.2 implies, for Im k ≤ ν, R ℓ χ (k) L(E mod ℓ ;E mod,-2 ℓ ) 1 k 2 R ℓ χ (k) L(E mod ℓ ;E mod ℓ ) . (3.13) Now observe that (3.14) ∞+iν -∞+iν B j (k -i(ν + 1)) -j-1 e -ikt dk = 0. Therefore (3.8) becomes: χe -itL ℓ χu = 1 2πi ∞+iν -∞+iν R ℓ χ (k)e -ikt u dk,
where the previous integral is absolutely convergent in L(E mod ℓ ; E mod,-2 ℓ ).

We first show part (i) of the theorem. Integrating along the path indicated in Figure 5 we obtain by the Cauchy theorem:

Im k = -ln | Re k| -Rℓ -µ ν -µ Γ 3 Γ 5 Γ 1 k ∈ C Γ Γ 4 Γ 2 X -X -Rℓ Rℓ Figure 5. The paths Γ j . (3.15) 1 2πi X+iν -X+iν e -ikt R ℓ χ (k)u dk = λ j ∈Res P ℓ Im λ j >-µ m(λ j ) k=0 e -iλ j t t k π χ j,k u+ 5 j=1 1 2πi Γ j e -itλ R ℓ χ (λ)u dλ. Let I j = 1 2πi Γ j e -itλ R ℓ χ (λ)u dλ. Using (2.6), (3.1
) and (3.13), we have, for t large enough,

I 3 E mod,-2 ℓ X+iν X-i ln X e -ist R ℓ χ (s)u E mod,-2 ℓ ds ν -ln X 1 X 2 e ts+C|s| ds u E mod ℓ e tν t X -2 u E mod ℓ . (3.16)
We now take the limit X goes to +∞ in the L(E mod ℓ ; E mod,-2 ℓ ) sense in (3.15). The integrals I 3 and I 5 go to 0 thanks to (3.16) and, in the integrals I 2 and I 4 , the paths Γ • are replaced by paths which extend Γ • in a natural way and which go to ∞. We denote them again by Γ • . We remark that (3.17)

Γ 4 ∪Γ 1 ∪Γ 2 B j (k -i(ν + 1)) -j-1 e -ikt dk = 0,
where the integral is absolutely convergent in L(E mod ℓ ; E mod,-2 ℓ ). On the other hand, we have the estimate, for t large enough, and a similar estimate holds for I 4 . Since all these estimates hold in L(E mod ℓ ), (3.18) and (3.19) give the estimate of the rest (1.12) with M = (C + 1)/2. The estimate (1.13) follows from (1.10), Theorem 2.1 iii) and Proposition 3.1.

Let us now show part (ii) of the theorem. We choose 0 > -µ > sup{Im λ; λ ∈ (Res P )\{0}} and the integration path as in part (i) of the theorem. We first suppose e ε ′ t > ℓ for some ε ′ > 0 to be chosen later. Then the estimate for I 1 can be replaced by

I 1 E mod ℓ e ((C+1)ε ′ -µ)t u E mod ℓ .
Let us now suppose ℓ ≥ e ε ′ t . On the one hand we have the inequality:

χe -itL ℓ χ L(E mod ℓ ) 1,
since the norms on E mod ℓ and on E ℓ are uniformly equivalent for ℓ ≥ 1. On the other hand by the hypotheses on g we have 1 ≤

g(e 2ε ′ t ) g(ℓ(ℓ + 1))

.

It follows:

χe -itL χ L(E mod ℓ )
g(e 2ε ′ t ) g(ℓ(ℓ + 1)) .

This concludes the proof of the theorem if we choose ε ′ sufficiently small and put ε := min{2ε ′ , µ -(C + 1)ε ′ }.

Proof of Remark 1.4 d). We note that for u ℓ ∈ D(P ℓ ), we have Estimate (1.12) can be written

P ℓ u ℓ , u ℓ = r -1 D x r 2 D x r -1 + V ℓ(ℓ +
E 1 (t) E mod e -µt -∆ ω M χ 0 u E mod ,
with χ 0 ∈ C ∞ 0 (R) and χ 0 χ = χ. Let χ j ∈ C ∞ 0 (R), j = 1, . . . , 2M with χ j+1 χ j = χ j for j = 0, . . . , 2M -1. Remark that there exists C > 0 such that √ V > 1/C on the support of χ 2M . Using the radial decomposition u = ℓ u ℓ , we get Proof of Remark 1.7. We only prove the first part since the proof of the second part is analogous. As we are localized in space, we can use the Sobolev embedding H 2 (R 3 ) ֒→ C 0 (R 3 ).

-∆ ω M χ 0 u E mod sup ℓ ℓ 2M χ 0 u ℓ E mod
Using P u 1 ≤ L u 1 u 2 E mod , it is sufficient to consider (L + i)χe -itL χ. For this purpose, we write Lχe -itL χ =(L + i) -1 [L, [L, χ]]e -itL χ + (L + i) -1 [L, χ]e -itL [L, χ] + (L + i) -1 [L, χ]e -itL χ(L + i) + χe -itL [L, χ] + χe -itL χL. (3.24) It is easy to check that the operators (L + i) -1 [L, [L, χ]] and (L + i) -1 [L, χ] can be extended to bounded operators on E mod . Note also that all commutators can be multiplied on the left and on the right by a cut-off function χ ∈ C ∞ 0 (R) with χχ = χ without changing them. By Theorem 1.3, the left hand side of (3.24)

Figure 1 .

 1 Figure 1. The resonances of P near the real axis.

Figure 2 .

 2 Figure 2. The different zones in Theorem 2.1.

Figure 3 .

 3 Figure 3. The set Σ and the contour Γ θ .

[a/ 2 , 3 . 1 .

 231 2b] + i[ch ln h, 0] to the subharmonic function ln χ( Pz) -1 χ + ln |f (z, h)| + M c Im z h , the lemma with (2.40) and (2.42). Proof of the main theorem 3.Resolvent estimates for L ℓ .

I 1 ee

 1 -i(Rℓ+s-i(µ+ln s ))t R ℓ χ (Rℓ + si(µ + ln s ))u -µt e -ln s t e C(ln s +µ) ds u E mod ℓ e -µt u E mod ℓ ,(3.19)

1 )

 1 u ℓ , u ℓ ≥ V ℓ(ℓ + 1) u ℓ , u ℓ ,

sup ℓ ℓ 2M - 1 (

 1 P + 1)χ 0 u ℓ E mod = sup ℓ ℓ 2M -1 χ 1 (P + 1)χ 0 u ℓ E mod sup ℓ χ 2M (P + 1)χ 2M -1 (P + 1) • • • χ 1 (P + 1)χ 0 u ℓ E mod (P + 1) 2M u E mod . (3.22)Finally, for the interpolation argument, we use the fact that (ℓ L(E ℓ ;E ℓ ) = 1, for ℓ ≥ 1.

  is equal to Lγ rχ r, χu 2 0 + LE 2 (t), in E mod,-1 . By the same theorem, the right hand side of (3.24) is equal to α + E 2 (t) where α is a constant in time andE 2 (t) E mod g(e εt ) g(-∆ ω ) -1 (L + i)u E mod .It follows α = Lγ rχ r, χu 2 0 and thus the first part of the remark.

Acknowledgments: We would like to thank A. Bachelot for fruitful discussions during the preparation of this article. This work was partially supported by the ANR project JC0546063 "Equations hyperboliques dans les espaces temps de la relativité générale : diffusion et résonances".


Marine Spiteri

Marion Henini

Raffaella Goglia

Pierre Combris

Louis Georges Soler

Relationship between voluntary nutrition labeling, food composition and prices: the example of biscuits and pastries

mean prices are significantly different in the two sub-samples (α=5%) Sub-sample A: products with extended nutritional

Main Results

First, the results show that voluntary nutrition labeling is rather well developed (figure 1). Indeed, only 16% of biscuits/pastries of the sample have no nutrition labeling at all. 51% of the products present basic nutritional information and 33% of the products have extended nutritional information.

Second, we observe that products with extended nutritional information are not of a better nutritional quality (table 2). Indeed, we compared nutrients contents (protein, fat, saturated fat, carbohydrates, sugars, fiber and sodium) of products with extended nutritional information versus other products (without nutritional label or with a basic one) for 8 categories of products. Statistical comparisons show that in most of the cases, nutrient contents of the two sub-populations are not significantly different (for 47 tests out of 56 performed). Nevertheless, we note that nutritional compositions differ significantly according to the level of detail of the nutritional label for cookies and chocolate filled wafers:

-cookies with extended nutritional information have protein, fat and fiber contents significantly higher and carbohydrates contents significantly lower; -chocolate filled wafers with extended nutritional information have carbohydrates and sugars contents significantly lower and fiber contents significantly higher.

Methods

Food quality database

In France, Ministries in charge of Food and Agriculture, Health, and Consumption decided to create an Observatory of Food Quality (Oqali) in 2008. The goal was to set up an independent system of observation in order to assess the nutritional composition of food items marketed by all the brands existing in the French market. In 2010, the Oqali database contains about 15,000 items, representing around 30% of the processed food marketed in France. Each product is described by several parameters such as the nutrient content (energy, protein, fat, saturated fat, carbohydrates, sugars, fiber and sodium), the nutrition labeling, the ingredient list, nutrition and health claims, the serving size, the nutrition guidelines included on the packaging and the brand.

In this study, we analyze nutrition labeling, nutritional composition and prices of biscuits and pastries items, keeping out those with nutrition claims for which label is mandatory. To fill out data, physico-chemical analyses of nutritional composition have been performed for some items with incomplete nutrition labeling on the packaging. Finally, our sample of biscuits and pastries is composed of 1427 items, 136 of which were chemically analyzed.

Socio-economic database

The mean price of each food item is assessed by matching Oqali database with data provided by TNS Kantar/Worldpanel which records in-store purchases made by representative households in France during the year in progress. Analyses First, a detailed analysis of nutrition labeling was performed on the data. This analysis allows the assessment of the presence/absence of various types of labels (with basic or extended information) according to the category of products (table 1). Second, we compared the nutrient contents of products with and without extended nutrition labeling, using either the information available on the packaging, or analytical results when this information was missing. Third, we compared the prices of food items according to the presence/absence of extended nutritional information on their packaging. Two-sample t statistic tests were used when conditions for parametric test were fulfilled, Mann-Whitney tests otherwise. Third, products with extended nutritional label appear to be more expensive for 8 categories out of 10. Nevertheless, statistical tests show that those differences in prices are significant only for 2 categories of products: coated sponge cakes filled with jam and wafers filled with chocolate/hazelnut (table 3). Further investigation will account for products promoted as traditional recipes which are generally more expensive and do not have systematically a nutritional label. To prevent some food related illnesses, health policies may influence consumers' behaviors through appropriate food labeling. In some countries, like the US, nutrition labeling is mandatory, whereas it is still optional in the EU, except when a nutrition claim is made in the labeling or the advertising of a foodstuff. In fact, the choice between voluntary versus mandatory labeling depends on several factors related, on the demand side, to the potential impacts of nutrition labeling on consumers' health, and on the supply side, to the possible extension of labeling if it only relies on firms' initiatives. Focusing on the supply side, the purpose of this study is to assess the development of voluntary nutrition labeling in France, and to determine the relationship between the presence of nutritional label, the food composition and prices. Analyses are performed on biscuits/pastries data, using detailed data on nutritional characteristics and prices of food items.

Figure 1 :

 1 Figure 1: Nutrition labeling in biscuits/pastries (no nutrition labeling: n=222; group 1: n=721; group 1 + minerals or vitamins: n=15; group 2: n=372; group 2 + minerals or vitamins: n=97)

Table 2 :

 2 Food composition of 8 categories of biscuits and pastries according to the type of nutrition labeling on the packaging

* mean nutrient contents are significantly different in the two sub-samples (adjusted α=0.7%) Sub-sample A: products with extended nutritional information Sub-sample B: products with basic or no nutrition information n: number of observations ; sd : standard deviation

Table 1 : types of labels Basic nutritional information Group 1 : energy, protein, fat, carbohydrates Group 1+: group 1 + minerals or vitamins Extended nutritional information Group 2 : energy, protein, fat, saturated fat, carbohydrates, sugars, fiber, sodium Group 2+: group 2 + minerals or vitamins

 1

Table 3 :

 3 Mean prices of 10 categories of biscuits and pastries according to the type of nutrition labeling on the packaging

	Category	Sub-sample	Number of observations	Mean price (€/kg)	Standard deviation

ConClusions and peRspeCtives

The presence of an extended nutritional label on the packages is not correlated with a better nutritional quality or higher prices for biscuits and pastries. This work is a preliminary one and should be extended to other food groups. Moreover, nutritional label is more and more developed on packages, including those of low-price products. So it would be interesting to repeat this survey over time. The final aim is to provide useful insights to discuss the opportunity of voluntary/mandatory nutrition labeling.


Annabelle Le Parc

Study of casein aggregation into micelle in mammary epithelial cell

est destinée au dépôt

Caseins, a family of acidic phosphoproteins which represent 78% of bovine milk proteins, interact with colloidal calcium phosphate (CCP) and aggregate to organize into a supramolecular structure: the casein micelle. The structure of the micelle determines the functional characteristics of the milk protein fraction and has an impact on food processing. Despite the importance of the nutritional and functional value of casein micelle, the intrinsic organisation and the mechanisms of formation of these structures in the mammary epithelial cell (MEC) has not been fully established.

The aim of this work is to obtain information about the specific arrangement of the caseins in the structure and to characterize the interactions between the various components of the micelle. To understand the structural complexity of the casein micelle, we believe it is essential to gain new knowledge about the formation of this structure within the MEC in order to elucidate the role of the individual casein and of the CCP in the aggregation process. Since the aggregation of the caseins appears to be initiated in the endoplasmic reticulum (ER), and proceeds during their transport to the apical surface, the dynamic of elaboration of the micelle will be studied in vivo. One project is to characterize the initial micellar states formed in the ER of rat MEC. We will then follow the evolution of this basic structure in the secretory pathway. The possible role of disulfide bridges in the formation and the stabilization of the micellar states will be investigated. In addition, we will try to precise the role of the calcium in the aggregation process; we will analyse the repartition of calcium along the secretory pathway of MEC by energy-filtering transmission electron microscopy.

With this aim, we have purified microsomes using differential centrifugation techniques. Morphological observations by electron microscopy indicate that the rough ER fraction was almost pure, very few other structures such as mitochondria and casein micelles being observed. Permeabilisation of rough microsomes with saponine, at pH 6.8, induces the release of soluble resident proteins (PDI, BIP and calreticuline) of the ER but not that of the membranous chaperone calnexine. The conditions allowing the conservation of the aggregated state of the caseins will be determined. This approach is currently applied to characterize the native micelles in the ER. The micelles obtained from rough microsomes will be isolated by density sucrose gradient and their structure compared to that of the casein micelle found in rat milk. The size, homogeneity, stoechiometry of the caseins and type of interactions within the structure, as well as the occurrence of disulfide bounds will be investigated.

INRA, UR 1196, Génomique et Physiologie de la Lactation, F-78352 Jouy-en-Josas, France.

INRA, URM 1253, Science et Technologie du Lait et de l'oeuf, F-35042 Rennes, France.