N Burq

B Ducomet

DECAY OF LOCAL ENERGY FOR SOLUTIONS OF THE FREE SCHRÖDINGER EQUATION IN EXTERIOR DOMAINS by

In this article, we study the decay of the solutions of Schrödinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no non-trapping assumptions) or some weakly trapping situations Résumé. -On s'intéresse dans cet article à la décroissance temporelle de la solution du problème extérieur pour l'équation de Schrödinger, dans le cas d'obstacles. Les situations auquelles on s'intéresse sont le cas général (pas d'hypothèse géométrique) ou alors des situations faiblement captantes.

Introduction

Since the early works of Lax and Phillips [START_REF] Lax | Scattering theory[END_REF][START_REF] Melrose | Geometric scattering theory[END_REF] on exterior Dirichlet problem for the wave operator ∂ 2

t -∆, it is well known that geometry of the boundary plays a crucial role in the time decay properties of the solution whether it is trapping or not. If the boundary is not trapping, it is known [START_REF] Morawetz | Decay of solutions of the exterior problem for the wave equation[END_REF] that local energy decays at exponential rate. For the same problem, under very general trapping assumptions, N. Burq proved [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] that there exists an exponentially small neighbourhood of the energy real axis, free of resonances, which implies a logarithmic decay rate of local energy.

For the Schrödinger operator ı∂ t +∆ which enjoys different spectral properties the only available result to our knowledge, concerning the exterior non-trapping case, is due to Tsutsumi [START_REF] Tsutsumi | Local energy decay of solutions to the free Schrödinger equation in exterior domains[END_REF](see also Hayashi [11] for star shaped geometries and Rauch [START_REF] Rauch | Local decay of scattering solutions to Schrödinger's equation[END_REF] for the potential case), and reads as follows.

Consider a smooth compact obstacle Θ ⊂ R d such that Ω = R d \ Θ is connected.

(1.1)

           ı∂ t + ∆ u = 0 in R + × Ω, u(x, t) = 0 on R + × ∂Ω, u | t=0 = u 0 ∈ L 2 (Ω) in Ω,
Denoting L 2 R (Ω) := {u ∈ L 2 (Ω), Supp u ⊂ Ω ∩ B(0, R)}, one get [START_REF] Tsutsumi | Local energy decay of solutions to the free Schrödinger equation in exterior domains[END_REF] N. Burq is partially supported by ANR projects ANR-13-BS01-0010-03, (ANAÉ) and ANR-16-CE40-0013 (ISDEEC).

Theorem 1 (Tsutsumi). -Let d 3 and Ω a non-trapping exterior domain (i.e. such that any singularity of the Green function for the Dirichlet problem for the wave equation in Ω goes to infinity as t → ∞). Let U (t) be the evolution operator associated to problem (1.1), and R 1 , R 2 > 0. There exists C > 0 such that for all t > 1

(1.2) U (t) L 2 R 1 (Ω),L 2 R 2 (Ω) C t d 2
.

In this article our purpose is to generalize the result of Tsutsumi in the general trapping case by adapting the strategy of [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] to a generalized Schrödinger equation, with a Laplace-Beltrami operator

1 (x) divA(x)∇,
replacing the Laplacian ∆, and the results of Ikawa [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of two convex obstacles[END_REF][START_REF] Ikawa | On the poles of the scattering matrix for two strictly convex obstacles[END_REF] in the case where O is the union of several disjoint convex obstacles. An extra motivation for this work is the relevance of this kind of information for global existence of solutions to related non-linear Schrödinger's equations (see [START_REF] Tsutsumi | Global solutions of the nonlinear Schrödinger equation in exterior domains[END_REF] [28] [START_REF] Burq | Smoothing effect for Schrödinger boundary value problems[END_REF]).

Acknowledgements. We would like to thank C. Zuily for various discussions on the content of this article and a referee for signaling several imprecisions in a first version of the paper and raising comments which lead to significant improvements (see (2.14) in Theorem 5).

Hypotheses and results

Let us consider a compact set Θ ⊂ R d with C ∞ boundary such that Θ ⊂ B(0, a) for a a > 0. We denote by P the operator defined by ∂ i a ij (x)∂ j ,

in Ω := Θ c associated to the weight (x) ∈ C ∞ (Ω) such that (x) 0 > 0 and the symmetric and positive definite matrix A with entries A := (a ij ) ∈ C ∞ (Ω) satisfying the ellipticity condition (2.2) n i,j=1 a i,j (x)ξ i ξ j C|ξ| 2 , ∀(x, ξ) ∈ T * Ω, C > 0.

We suppose that ("asymptotic flatness") for any multi-index α such that |α| 1, there exist β > 0 and M α > 0 such that ∀x ∈ R d \B(0, a)

(2.3) ∂ α x (x) -1 + n i,j=1 ∂ α x a i,j (x) -δ i,j M α x β+|α| ,
where we denote by • the usual "japanese bracket" (2.4) x = 1 + |x| 2 1/2 .

In the following we will also denote by P the self-adjoint extension of (2.1) in H = L 2 (Ω, (x) dx) with domain D(P ) = {u ∈ H 2 (Ω), u| ∂Ω = 0}. We shall denote by H k = D(P k/2 ) endowed with the norm

u 2 H k = P k/2 u L 2 + u 2 L 2 ,
and by H -k the image of L 2 by the map u → (P + Id)u, endowed with the norm

u H -k = (P + Id) -1 u L 2 .
If we denote by R(z) the outgoing resolvent (z 2 -P ) -1 of P in Ω, z -P -1 u 0 is the solution of the problem (2.5)

         z 2 -P v = u 0 in Ω, v | ∂Ω = 0, x |x| • ∇ x -i √ z v = o |x|→+∞ (|x| (1-n)/2 ), ( v is outgoing)
and we have

R(z)f = i ∞ 0 e -it √ z u(t) dt,
where u is the solution de (1.1) and the square root is defined on C \ R + by

ρe iθ = √ ρe iθ/2 , ρ > 0, θ ∈ (-2π, 0). It is known [21] that z → R(z) extends from Im z < 0 (the physical half plane) to a region O (if d is odd, O is the complex plane C, and if d is even, O is the simply connected covering of C * ) as a meromorphic operator from L 2 comp (Ω) into H 2 loc (Ω). Then the resolvent R(z) = R( √ z)
extends meromorphically from C * to its universal cover (even dimensions) and to the two leaves cover (odd dimensions).

Let us recall the general result [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF][START_REF] Burq | Lower bounds for shape resonances widths of long range Schrödinger operators[END_REF] Theorem 2 (Burq). -Assume that the operator P satisfies the assumption (AE) in Remark 2.5. There exist a, D > 0 such that, for any χ ∈ C ∞ 0 (R d ), in the conditions of Theorem 3, the operator χR(z)χ has an analytic continuation, as a continuous operator in H, in the region

(2.6) Ω = z ∈ C; Im z e -a √ | Re z| : |z| D .
Moreover, in this region, there exist A > 0 and C > 0 such that

(2.7) χR(z)χ L(H) Ce A √ | Re z| .
From these general estimates, following Vodev, we deduce resolvent estimates with space weights (see [START_REF] Vodev | Exponential bounds of the resolvent for a class of noncompactly supported perturbations of the Laplacian[END_REF]Corollary 1.2]) Theorem 3 (Vodev). -There exists positive constants C 1 , C 2 and 0 such that R(z), which is holomorphic in Im ξ < 0, does not have any pole in the region

(2.8) V := z ∈ C : | Im z| < 0 e -C2 √ | Re z| ∩ {|z| > C 1 },
where (1 + i0) 1/2 > 0. Moreover for any s > 1/2 and for any z ∈ V the resolvent R satisfies the bound

(2.9) e -x R(z)e -x L(H,H) C e C3 √ | Re z| .
This result will imply as in [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF], a logarithmic decay for the solution of the problem (1.1). More precisely our first result reads Theorem 4. -For all R 1 , R 2 > 0 and any k > 0, there exists C > 0 such that for any initial condition u 0 ∈ D(P k ) supported in B(0, R 1 ) ∩ Ω, one has for any t > 1

(2.10)

B(0,R1)∩Ω |u(x, t)| 2 dx 1/2 C (log t) 2k u 0 H k .
Our second result concerns some weakly trapping situations where resolvent estimates stronger than (2.9) are available. For the wave equation, such situations were investigated by Ikawa in a series of articles (see [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of two convex obstacles[END_REF], [START_REF] Ikawa | On the poles of the scattering matrix for two strictly convex obstacles[END_REF], [START_REF] Ikawa | Precise informations on the poles of the scattering matrix for two strictly convex obstacles[END_REF], [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF]), Gérard [START_REF] Gérard | Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes[END_REF] and more recently by Petkov-Stoyanov [START_REF] Petkov | Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles[END_REF].

We now assume that the operator P satisfies

-(E)
The metric in Section 2 is exactly euclidean outside of a fixed ball.

-(H N ) The weighted resolvent χ(x)R(z)χ(x) which is holomorphic in Im z < 0 admits a holomorphic extension in a region

(2.11) {z ∈ C; | Re z| C, Im z c|z| -N < C},
and satisfies for some s > 1 2 and some compactly supported function with large enough support, χ,

(2.12) χ(x)R(z)χ(x) L(H,H) C|z| N .
Remark 2.1. -In the litterature, motivated by wave equation expansion, authors usually study the resolvent

R(z) = R(z 2 ) = (-∆ -z 2 ) -1 .
A straightforward calculation shows that stating Assumption H N in this context amounts simply to replace the domain (2.11) by

{z ∈ C; | Re z| C, Im z c|z| -N -1 2 },
and the estimate (2.12) by

χ(x)R(z)χ(x) L(H,H) C|z| 2N .
As a consequence, H -1 2 is actually equivalent to a non trapping assumption while for larger N , it corresponds to a weakly trapping one. Theorem 5. -Assume that (E) and (H N ) are satisfied.

Then -If 0 < N , for any real k > 0, there exists C k > 0 such that for any t 1,we have

(2.13) x -d/2- e -itP (1 -iP ) k x -d/2- L(H) C k max 1 t d 2 , log(t) t k N -If -1 2 N < 0, for any real 0 k < d/2 -(1 + 2N
), and any > 0, there exists C k, > 0 such that for any t 1,we have

(2.14) x -d/2 e -itP (1 -iP ) k x -d/2 L(H) C k 1 t d 2 - . Remark 2.2.
-Notice that as noticed by a referee, since the operator [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of two convex obstacles[END_REF][START_REF] Ikawa | On the poles of the scattering matrix for two strictly convex obstacles[END_REF][START_REF] Ikawa | Precise informations on the poles of the scattering matrix for two strictly convex obstacles[END_REF][START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF], Burq [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] showed that the assumptions (H N ) above are satisfied for any N > -1 2 (actually the loss with respect to the non trapping estimates, N = -1 2 is only logarithmic). See also [START_REF] Christianson | Local smoothing for the Schrödinger equation with a prescribed loss[END_REF][START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF]. In the case of degenerate obstacles the work of Ikawa [START_REF] Ikawa | Poles of scattering matrices for two degenerate convex bodies[END_REF][START_REF] Ikawa | Spectral and scattering theory and applications[END_REF] suggests that Assumption (H N ) should be in this case true for some N > 0 (depending on the order of vanishing of the curvature). It is satisfied for some semi-classical models (see [START_REF] Burq | Zworski Geometric control in the presence of a black box[END_REF]Section 6.4]).

x d/2-((1 -iP ) -k x -d/
Remark 2.4. -The self adjointness of P could probably (see [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF]) be replaced by the assumption that iP is maximal dissipative. For simplicity, we do not pursue in this direction here.

Remark 2.5. -Another natural assumption would be Assumption (AE). -The operator P satisfies (AE) if and only if it satisfies (2.1) (2.2) and there exists R 0 > 0, θ 0 > 0, 0 > 0 such that its coefficients are analytic in the domain

Λ θ0, 0,R0 = {z = rω ∈ C d ; ω ∈ C d dist(ω; S d-1 ) < 0 , r ∈ C, |r| R 0 , arg(r) ∈ [-θ 0 , θ 0 ]},
(where S d-1 is the real sphere) and satisfies (2.3) in Λ θ0, 0,R0 .

We are not aware of precise low frequency resolvent estimates under assumption (AE) for boundary value problems, though it is likely that they should hold. Provided such estimates were true, we could generalize our results to this more general setting 3. Gluying estimates: from space-local results to space-weighted estimates In this section we prove that it is enough to prove the results in Theorem 5 with the weight x -1 replaced by a cut off χ(x), χ ∈ C ∞ 0 (R d ). In the remaining of this section, we assume that Theorem 5 is true with x -1 replaced by a cut-off χ ∈ C ∞ 0 (R d ). Namely -If 0 < N , for any real k > 0, there exists C k > 0 such that for any t 1,we have

(3.1) χ(x) e -itP (1 -iP ) k/2 χ(x) L(H) C k max 1 t d 2 - , log(t) t k N -
if (E) and H N are satisfied.

-If -1 2 N < 0, for any real 0 k < d/2 -(1 + 2N ), and any > 0, there exists C k, > 0 such that for any t 1,we have

(3.2) χ(x) e -itP (1 -iP ) k/2 χ(x) L(H) C k t -d 2 + .
The idea is to use, as a black box, some results by Bony and Häffner that we recall below (see [2, Theorems 1-iii), 5-i)]

Theorem 6 (Bony-Häfner). -Assume that there is no obstacle and that the operator P 0 satisfies assumptions (2.1), (2.2) and furthermore is non trapping.

Then we have

(3.3) ∃C > 0; ∀|t| 1, x -d/2 e itP0 x -d/2 L(H -d/2 ;L 2 ) C t -d/2+ .
Furthermore, for all φ ∈ C ∞ 0 (]0, +∞[) and µ 0 there exists h 0 > 0 and C > 0 such that for all 0 < h < h 0 , and all t = 0, (3.4)

x -µ e itP0 φ(h

2 P 0 ) x -µ L(L 2 )
th -1 -µ .

Corollary 3.1. -Assume that there is no obstacle and that the operator P 0 satisfies assumptions (2.1), (2.2) and furthermore is non trapping. Then for any 0 < ν < µ, ν < 1, k ∈ R, there exists C > 0,

(3.5) x -µ e itP0 x -µ L(H k (R d );L 1 (-1,1);H k+ν (R d )))
C.

Proof.

-For simplicity we just prove the result for k = 0, the general case beeing similar. For fixed h > 0,

x -µ e itP0 φ(h 2 P 0 ) x -µ H ν (R d ) ∼ h -ν x -µ e itP0 φ(h 2 P 0 ) x -µ L 2 (R d )
, and consequently according to (3.4)

(3.6) x -µ e itP0 φ(h 2 P 0 ) x -µ L(L 2 (R d );L 1 (-1,1);H ν (R d )) Ch -ν 1 0 th -1 -µ dt Ch 1-ν max(1, h µ-1 ) Ch δ ,
where δ > 0 and the estimate (3.7) follows from the dyadic decomposition

u = Ψ(P 0 )u + +∞ j=1 φ(2 -2j P 0 )u 0 ,
and summation.

We now turn to proving the same type of estimate under assumption H N in the presence of an obstacle Proposition 3.2. -Assume that the operator satisfies (E) and (H N ). Then for any 0 < ν < µ, ν < -2N , there exists C > 0,

(3.7) x -µ e itP0 φ(h 2 P 0 ) x -µ L(H k (R d );L 1 (-1,1);H k+ν (R d )))
C.

Proof. -Again, for simplicity we just prove the result for k = 0. Let us first prove this estimate with the weight x -µ replaced by a cut off χ(x). We start from the formula

χ(x)e itP φ(h 2 P )χ(x) = z=0 - e itz φ(h 2 z)χ(x)(P + z) -1 χ(x)dz,
and perform the change of variables ζ = h -2N z, which gives

χ(x)e itP φ(h 2 P )χ(x) = h 2N ζ=0 - e ih 2N tζ φ(h 2+2N ζ))χ(x)(P + h 2N ζ)) -1 χ(x)dz,
and according to (2.11) the r.h.s. is holomorphic in the region ζ < c. We use the quasi-holomorphic extension of the function

φ h (ζ) = φ(h 2+2N ζ), φ k h defined by φ k h (x + iy) = k j=0 d k dx k φ h (x) (iy) j j! ρ(y), with ρ ∈ C ∞ 0 (0, c) equal to 1 near 0, which satisfies |∂ z φ k h (x + iy))| C|ρ (y)| + Ch 2(1+N )(k+1) |y| k |ρ(y)|.
As a consequence, using Green's formula and (2.12), we obtain (using that the support of

φ h is included in the set |ζ| h -2-2N ) (3.8) χ(x)e itP φ(h 2 P )χ(x) L(L 2 ) = h 2N ζ=x+iy,y>0 e ih 2N tζ ∂ ζ φ k h (ζ)χ(x)(P + h 2N ζ) -1 χ(x)|dζ| , Ch 2N h -2-2N h -2N c y=0 e -th 2N y (|ρ (y)| + Ch 2(1+N )(k+1) |y| k )dy Ch -2-2N e -th 2N + C k ( h 2 t ) k+1 .
And consequently, using this estimate for |t| h -2N -and the trivial bound (by 1) otherwise, we get

χ(x)e itP φ(h 2 P )χ(x)u 0 L 1 (-1,1);H ν h -ν (h -2N -+ O(h ∞ )),
and the result follows if > 0 is small enough. Now we extend the result by replacing the χ(x) weight on the left by x -µ . Let

u(t) = (1 -χ(x))e itP χ(x)u 0 , solution of (3.9) (i∂ t + P )u = -[P, χ]e itP χ(x)u 0 .
Since P is euclidean or asymptotically flat, if χ = 1 on a sufficiently large domain, and since v and [P, χ]u are supported on the region where χ = 1, we can replace P in (3.9) by an operator P 0 on L 2 (R d ) (no obstacle Θ), whose coefficients coincide with the coefficients of P on the support of 1 -χ and which is non trapping (because, P 0 can be taken arbitrarily close to the euclidean Laplacian in C ∞ -norm). It consequently satisfies (3.3) and (3.4), and we now have

(i∂ t + P 0 )u = -[P 0 , χ]e itP χ(x)u 0 ⇒ (1 -χ)u = e itP0 (1 -χ)u 0 + i t 0
e i(t-s) P 0 [P 0 , χ]e isP χu 0 ds.

The contribution of the first term is easily dealt with. To estimate the contribution of the second term, we choose ν 1 < -2N and ν 2 < 1 such that ν = ν 1 + ν 2 -1 and χ equal to 1 near the support of ∇χ, and write (3.10)

x -µ t 0 e i(t-s)P0 [P 0 , χ]e isP χu 0 ds

L 1 ((-1,1)t;H ν 1 +ν 2 -1 ) (t,s)∈(-1,1) 2 x -µ e i(t-s)P0 χ[P 0 , χ] χe isP χu 0 H ν dsdt (t,s)∈(-1,1) 2 x -µ e i(t-s)P0 χ H ν 1 -1 →H ν 1 +ν 2 -1 χe isP χ L 2 →H ν 1 dsdt u 0 L 2 C u 0 L 2 .
This shows that we can replace the χ(x) weight on the left by x -µ . To replace the χ(x) weight on the right by x -µ , we argue by duality and proceed in a similar way: here we have to deal with operators norms of the adjoint operator

x -µ 1 -1 e -isP0 x -µ f (s)ds,
and bound its norm from L ∞ H -1 to L 2 , knowing that the similar bound holds when the weight on the left is replaced by a cut-off. We express (1 -χ)e -isP0 in terms of e -isP and the proof follows by similar considerations as previously.

Let us now come back to the proof of Theorem 5 knowing that (3.1) and (3.2) hold. As a first step, we show that we can replace the cut off χ on the left in (3.2) by the weight x -d/2 . We write again

x -d/2 (1 -χ)(1 -iP ) k e -itP χ = (1 -iP ) k (1 -iP ) -k x -d/2 (1 -χ)(1 -iP ) k x d/2 x -d/2 e -itP χ,
and by standard pseudodifferential calculus (away from the boundary) we have that the operator

A = (1 -iP ) -k x -d/2 (1 -χ)(1 -iP ) k x d/2 ,
is, modulo smoothing and decaying terms, equal to

A(1 -χ 1 ) (if χ is equal to 1 on the support of χ 1 )
and is bounded on

H k . As a consequence, it is enough to estimate x -d/2 (1 -χ 1 )e -itP χ from L 2 to H k . Let u = e itP χu 0 . The function v = (1 -χ 1 )χu satisfies (3.11) (i∂ t -P )v = -[P, χ 1 ]u.
We can, as before, replace here P by P 0 which satisfies (3.3), (3.4) and according to (3.11) and Duhamel's formula, we have

v = e -itP0 χu 0 + i t s=0 e -i(t-s)P0 [P, χ 1 ]u(s)ds.
The first term is estimated by applying (3.3). The idea to estimate the non homogeneous term is to use for e -i(t-s)P0 the estimate (3.3) and for [P, χ]u(s) the estimate (3.2). An issue here is the loss of one derivative due to the fact that the operator P is of order 2 (and hence the bracket is of order 1). To gain back this loss of one derivative, we have to estimate carefully the contributions of the regions |t -s| 1/2 and |s| 1/2. Let us first estimate the (easy) part when both |s|, and |t -s| are larger than 1/2. In this case the loss of derivatives can be recovered by using the slack of d 2 1 derivatives in (3.3), while the k derivatives from the H k norm are estimated using (3.1), (3.2) (according whether k > 0 or k < 0), giving (3.12)

x

-d 2 v H k C t s=0 1 1 + |t -s| d 2 - 1 1 + |s| d 2 - u 0 L 2 C 1 + |t| d 2 -2 u 0 L 2 .
Let us now focus on the contribution of |t -s| 1/2. In this case we have |s| ∼ |t| and we bound the non homogeneous part by

(3.13) |t-s|<1/2 x -d 2 e -i(t-s)P0 χ L(H k-1+ ;H k ) [P, χ 1 ] L(H k+ ;H k-1+ ) χe -isP χ L(L 2 ;H k+ ) ds C x -d 2 e -itP0 χ L(H k-1+ ;L1(0,1);H k ) sup |s-t|<1/2 χe -isP χ L(L 2 ;H k+ )
and we use Corollary 3.1 and (3.1) or (3.2) to conclude.

Let us now deal with the contribution of the region |s| 1/2. We proceed similarly and get a bound (with k + < d/2)

(3.14) |s|<1/2 x -d 2 e -i(t-s)P0 χ L(H -;H k ) [P, χ 1 ] L(H 1-;H -) χe -isP χ L(L 2 ;H 1-) ds C sup |σ-t|<1/2 x -d 2 e -iσP0 χ L(H -;H k ) χe -isP χ L(L 2 ;L 1 (0,1/2);H 1-) ,
and we use (3.3) to estimate the first term in the r.h.s and Proposition 3.2 for the second term.

As a consequence, we just proved that in (3.1), (3.2) we can put in the left x -d 2 instead of χ(x). By duality we hence also have a similar estimate with x -d 2 instead of χ(x) in the right (and now χ(x) in the left). We now re-perform the same procedure which allows to replace again χ(x) in the left by x -d 2 . This ends the proof of the insertion of the proper weight in (3.1), (3.2). ). Theorem 4 amounts to proving that for any k > 0 there exists C > 0 such that for any u 0 satisfying u 0 = χu 0 , one has

χe itP u 0 C log(t) 2k (P + i) k/2 u 0 L 2 .
Since the L 2 norm is conserved by the evolution e itP , it is by interpolation clearly enough to prove the result for even integers k. In that case, if χ is equal to 1 on the support of χ, we have

v 0 = (P + i) k/2 u 0 = (P + i) k/2 χu 0 = χ(P + i) k/2 χu 0 = χv 0
and Theorem 4 amounts to proving Theorem 7. -For any even integer k > 1 there exists a constant C k > 0 such that for any t > 1

(4.1) χ(x) e -itP (1 -iP ) k χ(x) L(H) C k (log t) 2k .
4.2. Proof of Theorem 7. -Let us notice first that an application of [3, Théorème 3], using only the weaker estimate

(4.2) χ(x)R(z)χ(x) L(H,H) C e C3| Re z| , in the domain (4.3) V := z ∈ C : | Im z| < 0 e -C2| Re z| ∩ {|z| > C 1 },
implies directly (at least in the high frequency regime), see (4.1)

(4.4) χ(x) e -itP (1 -iP ) k χ(x) L(H) C k (log t) k .
To get the proper decay rate ( C k (log t) 2k ), we shall use as in [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] a time cut-off, together with a complex deformation of the contour in the ξ = z 2 -variable which is adapted to Schrödinger operators.

In the rest of this section we shall prove Theorem 7. We first deal with the low frequencies. According to [3, Section B.2] we have Theorem 8. -For any θ ∈ [0, π 4 [ there exist δ > 0 and r, C > 0 such that the resolvent R(z) admits an analytic expansion in the sector

{z = ρe iφ ∈ C; 0 < ρ < r, φ ∈ [-π -θ, θ]},
and is uniformly bounded in this sector. Now we start with the classical formula

(4.5) e -itP χ(P ) (1 -iP ) k x = 1 2iπ +∞-i0 ξ=-∞-i0 e itξ χ(ξ) 1 -iξ k 1 ξ + P dξ x.
Let χ be a quasi-analytic extension of χ i.e. a function satisfying

|∂ z χ(z)| C N | z| N , ∀N, such that Ω = supp(∂ z χ) ⊂ x + iy; |x| ∈ [ r 4 , r 2 ], 0 y C 8 . r 2 supp (∂ z χ) Figure 1. A first contour Γ.
By contour deformation on the contour Γ of Figure 1, denoting by Γ -the region situated between Γ and the real axis, we get

(4.6) e -itP χ(P ) (1 -iP ) k x = 1 2iπ Γ e -itξ 1 2iπ χ(ξ) 1 -iξ k 1 ξ + P dξ x - 1 π Ω∩Γ- e -itξ ∂ z χ(η) 1 -iξ k 1 ξ + P |dη| x,
which is estimated in norm by

C r 0 e -tδη dη + C N Ω∩Γ- e -t (ξ) (ξ) N dξ C t .
It remains to estimate the contribution of (1 -χ(P )).

For U 0 ∈ H put V = e -itP χ(x)U 0 . We observe that for any s 0

(4.7) V s U 0 .
Using Fourier-Laplace transform, for any t > 0, x ∈ H we have

(4.8) e -itP 1 -χ(P ) (1 -iP ) k x = 1 2iπ +∞-i0 ξ=-∞-i0 e itξ 1 -χ(ξ) 1 -iξ k 1 ξ + P dξ x.
The idea of the proof consists in decomposing in (4.8) the integral into two parts, the first one corresponding to "low" frequencies, |ξ| log 2 (t), for which we perform contour deformations (after taking a suitable quasi-analytic extension of χ), using the assumptions on the resolvent, while the other part corresponds to "high" frequencies for which we simply use that

|(1 -iξ) -k | log(t) -2k .
In order to be able to perform this decomposition and still use contour deformation arguments, we take a convolution with a gaussian function and perform the decomposition on the convolution side.

Furthermore, instead of decomposing the Cauchy data (which would introduce a δ t=0 / ∈ L 1 loc (R t ) singularity at 0), it is better to transform the equation into a non-homogeneous one and perform the decomposition on the right hand side.

Let ψ ∈ C ∞ R t , equal to 0 for t < 1/3 and to 1 for t > 2/3, and let U = 1 (1-iP ) k ψV be solution of

∂ t + iP U = ψ t 1 -χ(P ) (1 -iP ) k V t . (4.9) y = εe -a √ x 1 2 1 Figure 2. A second contour Γ.
We have

U (t) = t 0 e -i(t-s)P ψ (s) 1 -χ(P ) (1 -iP ) k V s ds. (4.10)
Let c 0 > 0 and c 1 > 0 to be chosen later. We have

U (t) = t s=0 ds Im ξ=0 - dξψ (s)e i t-s ξ 1 -χ(ξ) (1 -iξ) k 1 ξ + P V s +∞ -∞ c 0 π e -c0 λ-ξ log t 2 dλ = s ξ |λ|<c1log t + s ξ |λ| c1log t = I 1 + I 2 .
(4.11)

4.2.1. Estimating I 1 .
-To estimate the contribution of the term I 1 the idea is to deform the integration contour in the ξ variable into the contour Γ of Figure 2 defined by

Γ = - 1 2 , 1 2 ∪ - 1 2 , - 1 2 + iεe -a √ 1/2 ∪ - 1 2 , - 1 2 + iεe -a √ 1/2 ∪ Γ, Γ = x + εe -a √ |x| , |x| 1 2 .
To be able to perform such a deformation, we first replace the function χ by a quasi-analytic extension, i.e. a function

χ : z ∈ C → χ(z), which coincides with χ on R, is supported in {Re z ∈ (-1, 1)}, equal to 1 in Re z ∈ (-1 2 , 1 2 ) and such that ∀N ∈ N, ∃C N ; ∀z ∈ C, |∂ z χ(z)| C N | z| N .
Then we have to check that for any s ∈ [0, 1] the operator χ(x)R(ξ)e -isP χ(x) is holomorphic with respect to the ξ variable in the domain below Γ and satisfies (uniformly with respect to s) an estimate of the type (2.9).

For Im ξ < 0 the two families of operators (4.12) e isξ χ(x)R(ξ)χ(x) -i s 0 χ(x)e -iσ(P +ξ) χ(x)dσ and χ(x)R(ξ)e -isP χ(x), coincide for s = 0, and satisfy the same differential equation

(4.13) ∂ s w = iξw -iχ(x)e -isP χ(x),
hence, these two families are equal for Im ξ < 0, and the family on the l.h.s. gives the proper analytic continuation. As a consequence, it is possible in the integral defining I 1 to deform the contour into the contour Γ (notice that the deformation near infinity is justified, for fixed λ, by the fast decay of the integrand ).

According to Green's formula (notice that according to the support condition on χ, the only contour contribution is given by Γ), we get that

I 1 = s ξ∈ Γ |λ|<c1log t +i t s=0 ds z∈Ω ψ (s)e i t-s ξ ∂ z χ(z) (1 -iz) k 1 z + P V s +∞ -∞ c 0 π e -c0 λ-z log t 2 |z|dλ,
where

Ω = supp(∂ z χ) ⊂ x + iy; |x| ∈ 1 2 , 1 , 0 y εe -a √ |x| .
From (4.7), using the fact that the operator χ(x)R(ξ)e -isP χ(x) is uniformly bounded on H with respect to z ∈ Ω and s ∈ [0, 1] for any t 2, we get that the contribution of the second term is, for any K > 0, bounded by (4.14) χ(x)

s z∈Ω |λ|<c1 log(t) C N U 0 z∈Ω e -t-1 Re z (Re z) K dz C K U 0 (t -1) K+1 .
From (2.9), we also have for any t > 1

(4.15) χ(x) s ξ∈ Γ |λ|<c1 log(t) C √ c 0 +∞ η=-∞ |λ|<c1 log(t) e -(t-1)ε0e -a|η| 1/2 +A|η| 1/2 -c0(λ-η/log t) 2
dη dλ U 0 .

Choose now c 2 > 0 such that √ c 2 a < 1, and

ϕ = -(t -1)ε 0 e -a|η| 1/2 + A|η| 1/2 -c 0 (λ -η/log t) 2 .
Then if |η| c 2 log 2 t, we also have

(4.16) ϕ √ c 2 A log t -(t -1)ε 0 t -√ c2a . Choose c 1 ∈]0, c 2 [.
J(u) = 1 s=0 Im ξ=0 - |λ| c 1 log(t) ψ s e i u-s ξ 1 (1 -iξ) k 1 ξ + P V s c 0 π e -c0 λ-ξ log(t) 2 .
For t 1, we have J(t) = I 2 (t) and for any u ∈ R,

(4.22) (∂ u + iP )J(u) = 1 s=0 Im ξ=0 - |λ| c 1 log(t) ψ s ie i u-s ξ 1 -iξ k V s c 0 π e -c0 λ-ξ log(t) 2 := K(u),
which implies (4.23) J(t) = e -itP J(0) + t 0 e -i(t-s)P K(s)ds.

To estimate the norm of J t in H, we shall use the fact that e -isP is a contraction on H for s 0, and we bound separately K u for u 1, J 0 and 1 0 K u du. Since the r.h.s. in (4.10) (the analog of K(u)) vanishes for u > 1 as well as the Cauchy data (U (0)), we expect the main contribution to come from the third term.

For u ∈ [1, t], we deform in (4.22) the contour in the ξ variable into the contour defined by Im ξ = log(t), which gives according to (4.7) using k > 1 and supp(ψ

) ⊂ [1/3, 2/3] K(u) +∞ -∞ e -(u-2/3) log(t) 1 (1 + |ξ|) k dξ U 0 C k e -log(t)/3 U 0 . (4.24)
Now we bound J(0). Let us study for example the contribution to (4.21) of the region λ > c 1 log(t). We deform the contour into the contour Γ = Γ + ∪ Γ -, with

Γ + = {z = 1 + η -i log(t); η > 0}, Γ -= {z = 1 + η -i/2; η 0} ∪ [1 -i/2, 1 -i log(t)]. (4.25)
For ξ ∈ Γ -, we have for any s ∈ [0, 1] and any λ ∈ c 1 √ log t, +∞

(4.26) e -isξ V s 1 -iξ k 1 ξ + P c 0 2π e -c0 λ-ξ log(t) 2 C (1 + |ξ|) k e -δ(λ 2 +ξ 2 / log 2 t) U 0 (δ > 0).
As a consequence we deduce that the contribution of Γ -to J(0) is bounded in norm by (4.27) C log(t)

λ c1 log(t) e -δλ 2 U 0 = O e -ε log 2 t U 0 . For ξ ∈ Γ + and s ∈ [1/3, 2/3] we have (4.28) e -isξ V s 1 -iξ k 1 ξ + P c 0 π e -c0 λ-ξ √ log t 2 Ce -log(t)/3 1 (1 + |η|) k e -c0 λ-η log(t) 2 U 0 , hence, using k > 1, the contribution of Γ + to J(0) is bounded in norm by (4.29)
Ce -log(t)/3 U 0 .

The contributions to J 0 of the region λ < -c 1 log(t) are bounded similarly.

Finally, it remains to bound (4.30)

1 0 K(u) du 1 0 K(u) 2 du 1/2 . From Plancherel formula, +∞ -∞ K(u) 2 du = C +∞ -∞ i 1 -iξ k V ψ (ξ) |λ| c1 log(t) e -c0 λ-ξ log t 2 dλ 2 dξ = C +∞ -∞ H(ξ) 2 dξ, (4.31)
where, for any |ξ|

> c1 2 log 2 t H(ξ) = |λ| c1 log(t) 1 1 -iξ k e -c0 λ-ξ log t 2 dλ V ψ ξ C (log t) 2k V ψ (ξ) , (4.32)
and for |ξ| c1 2 log 2 t

(4.33) H(ξ) |λ|>c1 log(t) e -δ(λ 2 +ξ 2 / log 2 t) dλ V ψ (ξ) Ce -log 2 t V ψ (ξ) .
Then from (4.30), (4.31) (4.32) and (4.33), using

(4.34) +∞ -∞ V ψ ξ 2 dξ = +∞ -∞ ψ V s 2 ds C 1 0 |ψ s | 2 ds U 0 2 , (let us recall that V (s) = e isB χ(x)U 0 ⇒ V (s) χ(x)U 0 C U 0 ), we deduce that (4.35) 1 0 K(u) du C 1 log t 2k + e -ε log t U 0 .
From (4.23), (4.24) (4.27), (4.29) and (4.35), we obtain finally

I 2 C (log t) 2 U 0 ,
which ends the proof of Theorem 7.

The case of a weakly trapping obstacle. Proof of Theorem 5

Theorem 5 exhibits two quite different regimes:

1. the first one (-1 2 N < 0) corresponds to non trapping (N = -1/2) or weakly trapping (-1 2 < N < 0) situations for which the (space truncated) Schrödinger evolution is smoothing and consequently we gain derivatives, 2. the second one (0 < N ) corresponds to stronger trapping, for which this smoothing effect is no more true.

We shall first focus on this second regime N > 0 and give the proof of (2.13) under the assumption (E).

Applying now formally the Neumann expansion

(I + S(k)) -1 = j 0 (-1) k (I + S(0)) -1 (S(z) -S(0)) k (I + S(0)) -1 ,
near z = 0, we conclude that R(z) satisfies (5.1) and (5.2), if and only if G(0) is one to one.

In order to show that suppose that w ∈ L 2 R (Ω) satisfying G(0)w = 0. From the previous definitions of α 1,2 and β 1,2 we have

R a (z 0 )(α 1 w) = 0 for |x| < R, R 0 (0)(α 2 w) = 0 for |x| > R + 1, so w(x) = α 1 (x)w(x) = (P -z 0 )R a (z 0 )(α 1 w) = 0 for |x| < R, w(x) = α 2 (x)w(x) = P 0 R 0 (0)(α 2 w) = 0 for |x| > R + 1. Using the fact that (5.5) R a (z 0 )(α 1 w) + R 0 (0)(α 2 w) = 0 for R + 1/3 < |x| < R + 2/3,
we get

(5.6)

(P 0 + ζ 1 (x))R a (z 0 )(α 1 w) = 0 for R + 1/3 < |x| < a, (5.7
) R a (z 0 )(α 1 w) = 0 for |x| = a, and (5.8)
(P 0 + ζ 2 (x))R 0 (0)(α 2 w) = 0 for |x| < R + 2/3,
where (5.9)

ζ 1 (x) = -z 0 if R + 1/2 < |x| a, 0 if R + 1/3 < |x| < R + 1/2,

and

(5.10)

ζ 2 (x) = 0 if |x| < R + 1/2, -z 0 if R + 1/2 < |x| < R + 2/3.
From (5.6) and (5.8) we see that R a (z 0 )(α 1 w) is Hölder continuous for R + 1/3 < |x| a, and that R 0 (0)(α 2 w) is Hölder continuous for |x| R + 2/3. Therefore after (5.5) R a (z 0 )(α 1 w) and R 0 (0)(α 2 w) are also Hölder continuous for |x| a.

Applying the maximum principle [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], we get

max{|R a (z 0 )(α 1 w)|; |x| = R + 1/3} max{|R a (z 0 )(α 1 w)|; |x| = R + 2/3},
and max{|R 0 (0)(α 2 w)|; |x| = R + 2/3} max{|R 0 (0)(α 2 w)|; |x| = R + 1/3}. But from (5.5) it follows that in fact max{|R a (z 0 )(α 1 w)|; |x| = R + 1/3} = max{|R a (z 0 )(α 1 w)|; |x| = R + 2/3},
then applying the maximum principle to R 0 (0)(α 2 w) in {R + 1/3 < |x| a} and using (5.7) we obtain

R a (z 0 )(α 1 w) = 0 if R + 1/3 < |x| < a.
Finally using equation G(0)w = 0 together with (5.10), we get

R 0 (0)(α 2 w) = 0 in R d ,
and finally that R a (z 0 )(α 1 w) = 0 in Ω R , so we conclude that w vanishes identically and so G(0) is actually one to one, which ends the proof. 5.2. Proof of Theorem 5 (case N > 0). -In order to prove the decay, we shall use, as previously, a temporal cut-off with a contour deformation in the complex plane.

For any U 0 ∈ H, let

V = 1 (1 -iP ) k e -itP χ(x)U 0 .
As previously (4.7) and (4.8) are still valid.

We start from (4.10), and write

χ(x)U (t) = χ(x) t s=0 ds Im ξ=0 - dξψ (s)e i t-s ξ 1 (1 -iξ) k 1 ξ + P V s +∞ -∞ c 0 π e -c0 λ-ξ log(t) t 1 2N 2 dλ = χ(x) s ξ |λ|<c1 t log(t) 1 2N +χ(x) s ξ |λ| c1 t log(t) 1 2N = I 1 (U 0 ) + I 2 (U 0 ).
(5.11) 5.2.1. Estimate of I 1 . -Following Section 4.2.1, it is possible in the integral defining I 1 , to deform the integration contour on the new contour Γ defined as (5.12)

Γ = 0, D + ıcD -N ∪ 0, -D + ıcD -N ∪ z = ρ + ıcρ -N : |ρ| D = Γ 1 ∪ Γ 2 ∪ Γ 3 .
Let us first estimate the contributions of

Γ 1 ∪ Γ 2 . D y = cx -N cD -N Figure 3. A third contour Γ.
We have to be more careful here than in Section 4.2 because this analysis will give most of the time the leading term (and an O(t -) estimate is no more sufficient).

We distinguish two cases according whether d is odd or even.

d odd.

From (5.1), the main contribution of the operator χ(x)R(ξ)e isP χ(x)

for ξ ∈ Γ 1 ∪ Γ 2 and s ∈ [0, 1] for any t 2, is B d-2 ξ d-2 2 with B d-2 bounded. If Γ 1 = {re iθ1 , r ∈ [0, a]}, we have (5.13) χ(x) s ξ∈Γ1 C U 0 a 0 e -tr sin θ1 r d-2 2 dr C U 0 t d 2 ,
and the same estimate is true on Γ 2 . d even.

From (5.2), we can check that the main contribution from

χ(x)R(ξ)e isP χ(x) forξ ∈ Γ 1 ∪ Γ 2 , s ∈ [0, 1] and any t 2, is B 1,0 ξ d-2 2
log ξ where B 1,0 is a bounded operator. Let ξ = z 2 , so that A becomes

A := 2 Γ1 e itz 2 z d-1 log z dz + 2 Γ2 e itz 2 z d-1 log z dz,
where Γ j , j = 1, 2 are (after deformation) segments in the half plane {Im z > 0}.

Putting ζ = t 1/2 z, we get

A := 2 t d/2 Γ1,t∪ Γ2,t e iζ 2 ζ d-1 log ζ - 1 2 log t dζ =: A 1 + A 2 .
As the integrand in A 2 which corresponds to the term 1 2 log t is no more singular at 0, it is possible to deform the contour Γ 1,t ∪ Γ 2,t into a small horizontal segment {s + iβ, c 1 < s < c 2 }, with β > 0 which gives (5.15)

|A 2 | = log t t d/2 Γ1,t∪ Γ2,t e iζ 2 ζ d-1 dζ C log t t d/2 e -2βt .
Similarly the contribution of Γ .

It just remains to estimate the contribution of Γ 3 . From (2.12), we also have for any t > 1

(5.18) χ(x)

s ξ∈Γ3 |λ|<c1 t log(t) 1 2N C √ c 0 +∞ η=-∞ |λ|<c1 t log(t) 1 2N e -(t-1)c|η| -N +N log(|η|)-c0 λ-η log(t) t 1 2N 2 dη dλ U 0 .
Let now c 2 > 0 to be fixed later and let

ϕ = -(t -1)c|η| -N + N log(|η|) -c 0 λ -η log(t) t 1 2N 2 .
Then if |η| c 2 t log(t) 1 N , we also have (5.19) ϕ -(t -1)c

-1 N 2 log(t) t + N (log(t) -log(c 2 )) -(c -1 N 2 -(N + 1)) log(t).
Let us choose c 1 ∈]0, c 2 [.

There exists δ > 0 such that for any |λ| < c 1

t log(t) 1 2N and if |η| c 2 t log(t) 1 N , then λ -η log(t) t 1 2N 2 δ   λ 2 + η log(t) t 1 2N 2   , (5.20) which implies ϕ N log(η) -c 0 δ λ 2 + η 2 log(t) t 1 N . (5.21)
We deduce that there exists ε > 0 such that (5.22)

|η|>c2 t log(t) 1 N e A|η| 1/2 -c0δ η log(t) t 1 2N 2 = O e - t log(t) 1 2N
. Taking c 2 > 0 small enough, from (5.13), (5.18), (5.19), (5.22), (5.15), (5.16), (5.17), and (5.22) we get (5.23) χ 1 I 1 Ct -d U 0 .

Estimate of

I 2 for k > 1. -Let (5.24) J(u) = 1 s=0 Im ξ=0 - |λ|>c 1 t log(t) 1/2N ψ s e i u-s ξ 1 (1 -iξ) k 1 ξ + P V s c 0 π e -c0 λ-ξ log(t) t 1 2N 2 .
For t 1, we have J(t) = I 2 (t) and for any u ∈ R

(5.25) (∂ u + iP )J(u) = 1 s=0 Im ξ=0 - |λ|>c 1 t log(t) 1/2N ψ s ie i u-s ξ 1 -iξ k V s c 0 π e -c0 λ-ξ log(t) t 1 2N 2 := K(u), which implies (5.26) J(t) = e -itP J(0) + t 0
e -i(t-s)P K(s)ds.

To estimate the norm of J t in H, we use the fact that e -isP is a contraction on H for s 0, and we bound separately K u for u 1, J 0 and 1 0 K u du. Since the r.h.s. in (4.10) (the analog of K(u)) vanishes for u > 1 as well as the Cauchy data U (0), we expect the main contribution to be coming from the third term.

For u ∈ [1, t], we have

K(u) = 2 3 s= 1 3 ψ s V s Q(u, s)ds, and
(5.27) Q(u, s) = Im ξ=0 - |λ|>c 1 t log(t) 1/2N ie i u-s ξ 1 -iξ k c 0 π e -c0 λ-ξ log(t) t 1 2N 2
.

We deform in (5.27) the contour in the ξ variable into the contour

Γ = ξ + i t log(t) 1 N ; ξ ∈ R .
Using that k > 1, we bound the integral by

e -(u-s) t log(t) 1 N +c0 2 t log(t) 1 N
, which for > 0 small enough decays faster than any polynomial in t (because for u We shall now deform the contour in the ξ variable in (5.29) into the contour

1, s ∈ 1 3 , 2 3 , u -s > 1 3 ). So we get (5.28) K(u) C m t -m . Now we bound (5.29) J(0) = 1 s=0 Im ξ=0 - |λ|>c 1 t log(t) 1/2N ψ s e i u-s ξ 1 (1 -iξ) k 1 ξ + P V s c 0 π e -c0 λ-ξ log(t) t 1
Γ = {z = 1 + η -i/2; η 0} ∪ 1 -i/2, 1 -i t log(t) 1 N ∪ z = 1 + η -i t log(t) 1 N ; η > 0 = Γ -∪ Γ 0 ∪ Γ + . -1 2 1 -( t log(t) ) 1/N Figure 4. A third contour Γ.
For ξ ∈ Γ -∪ Γ 0 , we have for any s ∈ 1 3 , 2 3 and any λ ∈ c 1

t log(t) 1/2N , +∞ (5.30) e -isξ V s 1 -iξ k 1 ξ + P c 0 2π e -c0 λ-ξ log(t) t 1 2N 2 C (1 + |ξ|) k e -δ(λ 2 + (ξ) 2 log(t) t 1 N e c0 2 t log(t) 1 N U 0 (δ > 0).
As a consequence, using again that k > 1, we deduce that (for > 0 small enough) the contribution of Γ - to J(0) is bounded in norm by (5.31) e c0 2 t

log(t) 1 N λ c1 log(t) e -δλ 2 U 0 = O e -γ t log(t) 1/N U 0 , γ > 0.
For ξ ∈ Γ + and s ∈ [1/3, 2/3] we have

(5.32) e -isξ V s 1 -iξ k 1 ξ + P c 0 π e -c0 λ-ξ log(t) t 1 2N 2 Ce (c0 2 -3 ) t log(t) 1/N 1 (1 + |η|) k e -c0 λ-(1+η) log(t) t 1/N 2 U 0 .
Hence, using k > 1, we see that the contribution of Γ + to J(0) is again bounded in norm by (5.33) O e -γ( t log(t) )

1/N U 0 .
The contributions to J 0 of the region λ < - -In this section, we are going to recover the case 0 < k 1 from the estimates for k = 0 and k = 2 and an interpolation argument. This interpolation argument cannot be performed on the final result, but we have to apply it on the operators I 2 .

We follow the k dependence of the operators I 1 and I 2 explicitely and write I k 1 , I k 2 . When k < 1, I k 2 is only defined by the relation

I k 2 (U 0 ) = χ(x)(1 + iP )
-k e -itP χ(x)U 0 -I 1 (U 0 ), where we observe that the integral defining I k 1 still makes sense. According to the estimates in Section 5.2.1, we have

I q 1 L(L 2 )
C t d/2 , q = 0, k, which implies that

I 0 2 L(L 2 )
I 0 1 L(L 2 ) + χ(x)(1 + iP ) -k e -itP χ(x) L(L 2 ) C.

On the other hand, for 0 < k 1, I p 1 is clearly the p 2 -interpolate between I 0 1 and I 2 1 and χ(x)(1 + iP ) -p e -itP χ(x) is the p 2 -interpolate between χ(x)e -itP χ(x) and χ(x)(1 + iP ) -2 e -itP χ(x). As a consequence, we get We now deform the integral above on the new contour Γ, noticing that the deformation is licit due to the factor (1+iξ) k (1-i ξ) k+2 and the boundedness of the resolvent in the domain between the real axis and the contour Γ. Our purpose is to get estimates uniform with respect to > 0 and pass to the limit → 0. The contributions of Γ 1 and Γ 2 are now estimated exactly as in the previous section, and we only have to estimate the contribution of Γ 3 uniformly with respect to > 0.

D y = cx -N cD -N
From (2.12), we have for any t > 0 (with a constant C independent on > 0)

4. Proofs of theorem 4 4. 1 .

 41 Proof of theorem 4. -Let χ ∈ C ∞ 0 equal to 1 on B(0, R 1

By taking the parametrization Γ 1 =

 1 {re iθ1 , r ∈ [0, a]} et Γ 2 = {re iθ2 , r ∈ [0, b]},

2N 2 dξ 1 t log(t) 1 2N

 211 dλ ds. Let us study for example the contribution to (5.29) of the region c < λ (the other contribution being similar).

5 From ( 5 .

 55 e -c0 λ-ξ log(t) 37) H(ξ)|λ|>c1 log(t) e -δ(λ 2 +ξ 2 log(t) t 1/N ) dλ V ψ (ξ) Ce -δ t log(t) 1/N V ψ (ξ) .Hence, from (5.34), (5.35) (5.36) and (5.37)recall that V (s) = e isB χ(x)U 0 ⇒ V (s) χ(x)U 0 C U 0 ), we deduce that (26), (5.28) (5.31), (5.33) and (5.39), we get finally

3 .

 3 Estimate of I 2 for 0 < k < 1.

2 . 5 . 3 . 2 N

 2532 Collecting estimates(5.23) and (5.40) gives (2.13) ends the proof of Theorem 5. Proof of Theorem 5 (case -1 < 0). -As previously we first focus on the case where assumption (E) holds. We start from a representation formula, for > 0 fixed. IfU (t) = e itP (1 + iP ) k (1 -i P ) k+2 χ(x)U 0 ,we have (5.41)x -d/2 U (t) = χ(x) Im ξ=0 - dξe itξ (1 + iξ) k (1 -i ξ) k+2 1 ξ + P χ(x)U 0

( 5 .

 5 42) Γ = 0, D + ıcD -N ∪ 0, -D + ıcD -N ∪ z = ρ + ıcρ -N : |ρ| D = Γ 1 ∪ Γ 2 ∪ Γ 3 .

Figure 5 .

 5 Figure 5. A fourth contour Γ (recall that -1 2 < N < 0).

e

  -tc|η| -N |η| k 1 + |η| dη U 0 Ct -d/2 U 0 H .Letting → 0 gives (2.14) under assumption (E).

5.1. Analytic properties of the resolvent. -Adapting a result of Tsutsumi (see Lemma 2.3 in [START_REF] Tsutsumi | Local energy decay of solutions to the free Schrödinger equation in exterior domains[END_REF]) to the euclidean case (E) with (x) = 1 and a ij (x) = δ ij for |x| > R, we have

in the region W = {z ∈ O ; |z| K}, where the operators B j , j = 0, 1, 2, ... are bounded from

and the expansions converge uniformly in operator norm.

, where the operators B mj , m, j = 0, 1, 2, ...

and the expansions also converge uniformly in operator norm.

Proof: After [START_REF] Vainberg | On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behaviour for large time of solutions on nonstationary problems[END_REF] we define the operator

where α 1,2 and β 1,2 are two C ∞ (R d ) cut off such that

where I is the identity operator.

After [START_REF] Vainberg | On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behaviour for large time of solutions on nonstationary problems[END_REF] we know that S(z) is compact from

, where z → (I + S(z)) -1 is meromorphic. Now as we know [START_REF] Vainberg | On exterior elliptic problems polynomially depending on a spectral parameter, and the asymptotic behaviour for large time of solutions on nonstationary problems[END_REF] that expansions (5.1) and (5.2) are valid for the free operator R 0 (z)(α 2 •) ≡ (∆ + z 2 ) -1 , it follows that they also hold for G(z) and S(z).

As I + S(0) = ∆G(0) in Ω R we see that G(0)w = R 0 (0)w → 0 for |x| → ∞. Moreover as the solution of    P 0 u = 0 in Ω, u| ∂Ω = 0, u → 0 as |x| → ∞. is u ≡ 0, we get from Fredholm theory that a necessary and sufficient condition for I + S(0) to have a bounded inverse is that G(0) is one to one.


Making the art of fun freely accessible": the politics of leisure in the community art movement in Britain in the 1970s and 1980s.

In the summer of 1969, residents of Kentish Town and Chalk Farm (North London) were invited to come on board a converted double-decker, the "Fun Art Bus", to enjoy an out-of-the-ordinary ride. Depending on the day, they would be entertained to a theatre performance, a poetry reading or a puppet show. The initiative was one among many other activities proposed by Inter-Action, a theatre and arts collective founded in London by American Ed Berman in April 1968. Literally and symbolically, the bus was a vehicle which allowed the opening up of spaces for creative expression, in boroughs where cultural venues were scarce or non-existent. The group's intention, suggested in its name, was to facilitate local residents' engagement with the arts, as spectators and/or as participants in inclusive projects. Yet by developing people's access to artistic expression, community arts groups such as Inter-Action hoped to encourage people to become more involved in the life of the community, therefore using the arts as a catalyst:

One of [Inter-Action's] main aims is to make the arts, especially drama and the media, useful and relevant to local community life, young people and the educational process. Another primary aim is to experiment with the arts and the media to develop new socially-rooted or educational applications for the purpose of an improved and more responsible community life. (Inter-Action Trust, 1) Across the UK, in inner cities, on council estates, in new towns or, more rarely, in rural villages, similar initiatives appeared, whether inspired by art collectives or generated by residents themselves

Community arts practices therefore raised significant questions on the concepts of leisure, culture and the arts, and on how these concepts relate. What is art and who can be an artist? Does artistic production need to be sanctioned by an authority to be recognised as art? Why not consider leisure activities as conducive to potentially empowering forms of artistic expression and invest them as 1 Some projects specialised in one artistic practice while others developed a multi-disciplinary approach and offered activities in theatre, music, video, photography, printing, pottery or mural painting... such? Couldn't leisure time -defined as freedom from economically-rewarding work -be considered as a legitimate context for the production of art works, where such art work was not meant for a market but for the cultural development of the community? This article contends that the notion of leisure is pivotal to the way the community arts movement conceptualised its practices. Access to leisure, with its rich associations with play, "fun" and recreation, with its dimension of pleasure and emphasis on imagination, was given pride of place in the movement's practical and political framework. "Making the art of fun freely accessible", to quote Ed Berman, can be identified as a provocative political aim: that of opening alternative spaces and of harnessing the dynamic and subversive dimension of creative expression, particularly in areas lacking the specific resources and amenities. In a crucial way, community arts projects fought against the marginalisation of popular expressions, asserting instead the equal validity of all cultural forms and signifying practices. The community arts movement helped redefine art around the concept of expression, away from the figure of the individual artist and towards the possibility of co-authorship in the production of collective meanings.

This research is based on the study of archives from several organisations: Inter-Action in London, (founded in 1968), Westminster Endeavour for Liaison and Development (WELD), in Handsworth, (Birmingham, founded in 1968), Trinity Arts in Small Heath (Birmingham, founded in 1972), Jubilee Arts in Sandwell (West Midlands, founded in 1974), Tower Hamlets Arts Project (London, founded in 1975). The documents used include correspondence, grant applications, annual reports, press cuttings, photographs, posters, leaflets. Interviews were conducted with photographers Brendan Jackson, from Jubilee Arts, and David Hoffman, who worked with Tower Hamlets Arts Project, as well as Graham Peete, a printer who was a member of Telford Arts. In 2015, Brendan Jackson launched a website which makes hundreds of photographs produced by members of Jubilee Arts accessible in digital format, as well as videos made from old film footage. This material was also used in this article.

"Games for the imagination" (Jubilee Arts): expanding creative expression through leisure.

When going through the archives of community arts groups, studying documents such as annual reports for instance, one is struck by the sheer variety of activities offered by the different organisations in theatre, film-making and video, photography, dance, mural painting, handicrafts, printing and publishing, poster-making, community bookshops, poetry, music, etc. The multidisciplinary approach was aimed at eliciting forms of creative expression from people whose views and opinions were seldom, if ever, given the chance to be heard. Leisure and artistic activities were (pleasurable) means to an end, that of enabling people to authorise themselves to take part and express themselves through the means they chose2 . In the words of Jubilee Arts member Kate Organ:

Jubilee is a community arts project, what that means is that we aren't community workers in Smethwick nor are we trying to teach the people here how to be Laurence Oliviers in their own back rooms. What we're trying to do is to get the people to express something about the way they live, about their own area, in fact to make changes in their area and take control of some aspects within their own area through the arts. We're artists so we do it through the arts but anyone can do it in any area. (Jubilee Arts Archive, 1977) The organisation of summer activities for children and teenagers, known as "play schemes" in the jargon of community artists, was a crucial moment in the calendar, and an important part of the work of community arts projects. Carnivals were often the highlight of the summer weeks, culminating in coloured processions, games and performances involving local residents and impressive numbers of children. Community arts organisations conceived their role in terms of facilitating the participation of as many members of the community as possible in collective activities: the best way to "engage" people of all ages was to give them a sense of being part of something which related to and belonged to them. Being providers of a structure for the organisation of creative activities at a local level, community arts projects acted as catalysts for the expression of people's aspirations and visions, enabling communities to re-imagine themselves.

The provision of spaces for leisure activities with an emphasis on artistic expression was therefore useful as a tool, at the level of practical strategies used by community artists, to generate a sense of group identity, to ignite a desire to act together, and foster a recognition of common needs and interests. Through participation community arts practices, people could find a sense of purpose and the desire to act for the development of local community life. Members of Jubilee Arts, based in the borough of Sandwell, recognised this:

A community arts group must make available a structure for collective action which the community has not had hitherto the opportunity to explore -a structure which gives access not primarily to the products, but to the processes of a whole range of creative activities. In time, the control and organisation of such a project must become the responsibility of the local community as much as that of the artists involved. Only then will it be accepted as a legitimate organ of self-expression by the people with whom it works" (Jubilee Arts, 1977, 2) The intention was close to the contemporary efforts of French cultural "animateurs". "Animation culturelle" sought to elicit forms of agency among socially disadvantaged social groups, by making participation in leisure activities one of the levers of social action and popular education (Meister; Augustin; Moser et al). In a similar way, community artists worked primarily with children, teenagers, school drop-outs, unemployed people, pensioners, women, members of ethnic minorities, in an effort to create the conditions for expression, through the learning and sharing of creative skills outside of formal contexts. 3 Within that framework, leisure, encompassing a broad field of practices, was conceived as a fundamental imaginative activity, which should be made available to all. To Steve Trow, founding member of Jubilee Arts, community arts were based on the "conviction that the creation of original work, rooted in local cultures, local experience and aspirations, has a potency and a resonance that can re-shape our perceptions of what is valuable and what may be possible." (Trow,1). Making leisure accessible thus became a political aim of the community arts movement.

Access to leisure as a political aim.

A photograph from the Jubilee Arts Archive taken around 1974 shows Steve Trow, dressed up as "Mr. No-All" (sic), wearing a large white coat and a top hat, acting as a kind of clownish Pied Piper and followed by a group of children. They form a miniature street demonstration, and one of the banners reads "Wanted: a place to play". The event was one of the first activities taken on by the young members of Jubilee Arts in the summer, and the aim was to gather as many children as possible in Sandwell to show them where to find their local play centre. The picture illustrates the concerns of early community arts projects regarding the dearth of cultural facilities in the concrete environment of new council estates. They chose to intervene precisely in these places:

Sandwell is a large metropolitan borough in the West Midlands. 300 000 people live in it. It's a very heavily industrialised area and used to be a lot of small towns which have now been formed into a large metropolitan borough. It doesn't have a city centre, the borough has no professional arts group and as far as provision for play for children goes, it has only twelve play centres to service the entire borough. (Jubilee Arts Archive) 4 The community arts organisations which appeared in the late 1960s and early 1970s all started from the same identification of a severe lack of leisure activities in the urban areas where they established themselves. Kentish Town, Small Heath, Handsworth, Tower Hamlets, Telford, Smethwick... these councils, among many others, were all urban working-class areas suffering from economic deprivation. The provision of cultural resources in these areas seemed to have seldom been a consideration of planners or local authorities.

The lack of access to leisure activities in deprived urban areas, because of poor provision, therefore highlighted the broader issue of the failures of post-war town planning. In the context of reconstruction, town planning was enrolled in the endeavour to raise living standards. This chimed in with the emphasis on increased state controls over the economy and coordinated efforts to extend state provision. However, the construction of new council estates and the demolition of derelict innercity housing as part of "slum clearance" was preferred to the alternative option of renovating existing housing in working-class neighbourhoods. (Greed,115,280) Close-knit working-class communities were dispersed on new estates, in which high-rise buildings replaced the more traditional terraced houses. 5 Working-class forms of sociability were considerably undermined as a result. Furthermore, the housing estates built in the 1950s and 1960s were designed and planned in ways which did not take into account the needs of the populations which were to be housed in them:

Planning was based upon a top-down rather than a bottom-up approach, with very little involvement of, and hardly a word of protest from, the people, who were meant to be the beneficiaries of the planning system. (Greed,153) A critique of the shortcomings of post-war urban planning policies emerged from the late 1960s. Community activists were at the forefront of campaigns meant to hold local authorities and planning agencies to account. 6 They lent their support to tenants' associations, in their denunciation of housing problems (insalubrious buildings, disrepair, lack of facilities, etc.) 7 . In many instances, they played a 4 Sylvia King makes the same observation: "Sandwell has a population of nearly 300,000. That's the same size as Bradford or Belfast. It has a very small Victorian municipal art gallery/museum in Wednesbury, a 100-seater tiny amateur theatre, and no bookshop. There's one two-screen cinema (about to close) and an old single screen showing exclusively Asian films. It's hard to imagine that that's all there might be in say, Belfast!" in "Making Mansions", Remaking Birmingham: the Visual Culture of Urban Regeneration. Liam Kennedy, Ed. Routledge: 1984. p. 56.

5

A process observed from the late 1950s in the East London borough of Bethnal Green by sociologists Michael Young and Peter Willmott, Family and Kinship in East London, Harmondsworth: Penguin, first ed. 1957. 6 The Community Development Projects (a national programme initiated by the Home Office in 1969 and ended in 1976) provides an interesting example of the contradictions between a conception held at national government level that poverty could be tackled through specific piecemeal policies at local level, and the conclusions of social scientists and community workers involved in the projects arguing that poverty was structural. CDP workers advocated for a radical change in the way community development was addressed, away from paternalistic social pathology models. See Craig, 2011. 7 Photographer Bill Dolce, a member of Bootle Art and Action (Merseyside), and photographer Paul Carter, active in Blackfriars Settlement (South London), both underline one of the roles of photography in a community arts and community action context: photographs could be used as evidence to denounce bad housing and insalubriousness, role in advocating greater consultation and involvement of the population in planning decisions affecting their very conditions of existence. A report written in 1987 by members of Free Form, a North London community arts organisation created in 1969, emphasised the role played by community artists and community activists in this shift:

Once the scale of the building failures began to be realised, people started to challenge both the power of the architectural profession, which had produced the vision, and that of the planning profession, which had helped put it into practice. […] There were increasing demands by the public for a greater say in the decisions which affected them, and they were being supported by some professionals who were beginning to redefine their role, as well as by the voluntary sector. Since its beginnings in the early 1970s, community arts and community architecture have been recognised, established and are flourishing nationally and internationally. (Free Form Arts Trust Ltd.,2) Community arts organisations, through their encouragement of people's expression on issues affecting their lives, were ferments of agitation and resistance at a community level. In several respects, the community arts movement shared similar concerns with those of the Contemporary Centre for Cultural Studies in Birmingham 8 . While Cultural Studies developed fundamental theoretical insights into the processes of transformation of working-class cultures in post-war Britain, community arts acted on a concrete level and sought to give a voice to the people whose lives were affected, whose concerns were disregarded and whose cultural expressions were dismissed if ever acknowledged. Community arts organisations were perfectly aware of the undermining if not wilful destruction of working-class culture:

Small Heath is a generally run-down inner-ring area in the midst of redevelopment, renewal and general improvement. The population [...] is predominantly working class, the culture of which has been almost totally suppressed in that opportunities for cultural expression rarely exist. (Trinity Arts) 9 Community artists denounced the combination of social deprivation and cultural marginalisation in the neighbourhoods they worked in. Developing access to facilities and resources was clearly identified as a priority. The strategy used was to raise awareness about the state of cultural provision locally so as to elicit demands from within the community itself for more cultural activities and for a commitment from local authorities in the long term, in a bottom-up approach.

When Jubilee goes into a community, we're aren't trying to drop goodies on the people from heaven, what we're trying to do is work with local people for them to identify needs in their own area, not necessarily artistic needs, but for example are there play facilities for children, suitable facilities for old folks, anything like that, youth clubs... And for them to identify that need, and for us to help them find the right channels of resources, of communications, with the bodies that can help like the local authority, what we will do is that we will instigate the project, by doing a piece of street theatre, doing a pub show, doing a playscheme, but from there on end, we will respond to how that community takes up what we've set up. (Jubilee Arts Archive, 1977) In terms how of they worked, therefore, community arts organisations adopted very direct and pragmatic modes of action.

9

The story could be repeated in many of the environments where community arts projects developed. Inter-Action, based in Kentish Town, North London, for example, was an area notorious for its crime rate and youth delinquency at the turn of the 1960s.

local authorities, the Arts Council or specific programmes such as Urban Aid 10 , they favoured contacts between members of the community, and lent their support to other community associations. They intervened on a very local scale, through what could be considered as microscopic forms of intervention but also as minor acts of resistance. One recurring mode of action offers a telling illustration of this. In the 1970s, several community arts organisations purchased converted doubledeckers: Inter-Action, Trinity Arts and Jubilee Arts all bought their own bus in the 1970s, while the Islington Bus Company based its activity around one (it was used as a lending library). On a practical level, these vehicles made excellent mobile resources, which could easily circulate to places where no play provisions existed. They were easily identified in the neighbourhoods on their different routes, and provided points of focus for leisure-based activities in areas which lacked them. They were equipped with video and photo equipment, materials for puppet shows or theatre performances, could be used as libraries or stage sets, and were transformed into bases for temporary playschemes deployed on patches of waste-ground 11 . Women's groups, tenants' associations, as well as pensioners' groups also used them as meeting spaces. Thankful users from the Bermuda Mansions Tenants Association in Walsall wrote a cover letter emphasising the important part [Jubilee Art's Bus played] in highlighting a number of tenants' grievances, redundancies at local firms and helping the unemployed to find a use for their enforced free time, [and] the importance of the bus to the youth of Sandwell and the roll (sic) it plays in taking art to the people who would otherwise not bother to seek art. (Jubilee Arts Archive, c. 1984) On a more symbolic level, therefore, these friendly and brightly-coloured buses did provide a muchneeded resource, a safe space, as well as a point of convergence and they were to some extent capable of generating collective initiatives. Opening channels for people's expression took different forms in diverse community arts practices, and of course the buses were only one among many other ways of achieving this. As an aim, promoting self-expression from within communities always remained firmly connected to principles of community development and self-help.

Leisure and empowerment: enabling communities to reclaim control

In its efforts to make access to leisure and creativity a reality, the community arts movement stressed the collective dimension of these imaginative activities, and endeavoured to channel energies in order to trigger collective action on issues affecting people's lives. The community arts movement shared similar analyses with the radicalism of 1970s "community action" politics, characterised by the emergence of local, collective initiatives on issues identified by the members of a community. Peter Hain defined community action as "a style of political action through which people gain the confidence to agitate for their rights and the ability to control their own destinies" (Hain,21). These objectives were embedded in a larger ambition to generate processes of empowerment at the level of the whole community.

The concept of empowerment can be defined as a social process whereby individuals as well as social groups develop self-confidence and skills, as well as a critical consciousness of their material 10 Urban Aid was a policy implemented from 1975 that was meant to tackle social inequalities in inner cities, particularly by allocating money to community centres or law centres. (Greed,119) 11 "We were very much celebrities, because our bus was bigger and more impressive and more sophisticated and it had a darkroom in it. We didn't just do Play-doh or stencilling. We were quite the sort of radical people of the play bus world. Because we weren't just a play bus, we were an arts bus." Jubilee Arts Archive, "Interview with Kate Organ", 2015.

conditions, and begin to take action on problems identified by them and affecting their lives. Through collective action, people reclaim control over decision-making processes and become agents who gain power over their own destiny. From the 1960s and through the 1970s, the concept belonged to a radical rhetoric of social and political change from below (Bacqué, Biewener). Community arts organisations were driven by such objectives.

One example of a local initiative which federated the energies of a whole community towards a common purpose was the Tower Hamlets Arts Project (London). The collective emerged from an initial resistance to an arts project proposed in 1974 by the Greater London Arts Council and funded by Thames TV, for which there had been no consultation with the local population. Ten thousand pounds were to be allocated to professional artists for the temporary use of commercial billboards, the idea being to bring the work of fine artists to the people: "Eyesites" was quickly nicknamed "Eyesores" in the neighbourhood. People active in the borough's rich voluntary sector objected to what they considered to be money inappropriately spent as well as a paternalistic gesture (Braden,23). They organised public meetings in order to debate over and propose an alternative project, defended their case to Thames TV and eventually managed to reverse the initial decision and win the allocated sum 12 . The rationale of Tower Hamlets Arts Project was that money should be used to support existing arts and media groups, to help them sustain themselves in the long term and expand their activities in the neighbourhood. Instead of bringing art to the people, artistic expression produced within the community should be encouraged. Tower Hamlets Arts Project coordinated the activities of the different community arts groups, with activities in video, photography, music, murals, theatre, creative writing... The Tower Hamlets Art Project Community Bookshop was created then, and still exists to this day as the Bricklane Bookshop 13 . The budget made provision for the "Big Show", a large collaborative event consisting in taking over the exhibition space of the renowned Whitechapel Art Gallery for an entire month. The whole project had a definite impact both in terms of the development of provision for the arts locally and in the vision the local population gained of itself through its achievements. Symbolically, members of the community had managed to challenge an initial project been imposed to them without consultation and which had no relevance to their desires, needs and aspirations. Instead, their project had from the start been defined as active and inclusive, long-term and open to discussions instead of top-down, and limited in terms both of duration and participation of the public.

The Greater London Arts Association proposal had the unforeseen and beneficial effect of making a public issue of the arts in the borough. It also raised public consciousness over the high degree of creative activity that had been achieved by residents and professionals locally. (Braden,24) Maggie Pinhorn, a film-maker involved in the Basement Project (a community video group active in the Tower Hamlets Arts Project) described her role as being an enabler of other people's expressions, as someone who accompanied processes of personal empowerment: I am in the business of building up people's confidence in order that they can express themselves creatively and use their imaginations. That is possibly the most political act that you can be doing. [...] What I can see as a result of people having been involved in that kind of work is that they are not going to do it in terms of film or video or anything else, but in their personal lives. So you might build up somebody's confidence enough for them to complain about their housing conditions. To go along and demand a decent flat to live in. To get a job, to think and to write more, to do whatever else for themselves to improve their own quality of life. (Heinz & 12 People felt that "the scheme was irrelevant to the borough because it did not meet any outstanding needs nor encourage enough involvement and the participation necessary to be of long term benefit to the community." THAP News, vol. 1, N°1, January 76, p. 101.

13

See http://bricklanebookshop.org/history/index.html Accessed on April 22 nd , 2016.

Wade, 182)

The "Fun Farm" is another example of a collective endeavour involving a local community. It was one of the many activities developed under the umbrella of Inter-Action: a derelict building was refurbished on a patch of land in Kentish Town and transformed into an "urban" farm complete with animals and activities such as gardening, horse-riding, pottery, a repair shop, women's groups... "Land that was dead and cut off from the housing estates […] has been reclaimed and through a voluntary body turned into a real leisure-producing enterprise 14 ." The organisation flourished because of the active participation of the local community, across age groups, in the project. The skills and the confidence developed by participants had direct positive effects on people's own lives.

The notion of "self-help" is a leitmotiv in the discourse of community artists and how they conceived their action in the community. Away from the Victorian acceptation of the term, with its emphasis on virtuous individual reform [START_REF] Smiles | Self-Help, With Illustrations of Conduct and Perseverance[END_REF], community artists tended to use the notion of self-help as a synonym for empowerment, to describe the process whereby people would seek to reclaim control over their community's destiny and improve conditions for themselves though their own agency.

These examples offer illustrations of how economically deprived, working-class communities found the strength and resource, with the help and experience of community arts organisations, to develop their own initiatives and act locally on issues affecting their own lives. Leisure, used as a point of entry for participants, became a lever for processes of empowerment both on an individual and collective level.

Democratisation of culture vs cultural democracy.

As can be seen through the example of Tower Hamlets Arts Project, the activities enabled by community arts organisations challenged the narrow association of art with the taste of an elite (equated with "high art"). Instead, the community arts movement defended the notion that artistic production should not be reserved to an exclusive class of people, mainly middle-class, but be enjoyed, practised and experienced by all sections of society. In its annual report for the year 1987-88, Telford Community Arts, made this idea very clear:

[This is] not the Royal Opera, but for the people of the Wrekin 15 ... To make their own art that is imaginative, exciting and effective, and expresses the interests of the working-class. We challenge the notion that the arts are something for "other people" to do, for the "well-off", for the "well-educated", or other privileged sections of society. (Telford Community Arts, 2)

Community arts embraced an anthropological definition of culture and artistic expression: they understood culture as the framework for the communication of meanings which are produced, contested, transformed and shared within social groups. Subverting elitist conceptions of "high" and "low" culture, the ambition was to fight for communities' capacity to define their own cultural meanings, and for the recognition of these meanings as valid and worthy. Such a proposition topples hierarchical categories between what makes "good" or "bad" art. Instead, it places emphasis on the processes that artistic expression involves, on the social context in which it takes place, on the meaningful interactions it fosters between people, on the uses to which it is put. Paul Carter, a photographer involved in the Blackfriars Settlement (South London) and initiator of the "Photography

Project" wrote about the role played by photographs produced by participants in the project:

Many of the photographs used in the project are not what many people would consider good photography. They are not great images with a universal message able to transcend time and culture. They are very humble images. The important thing is that they work in the context. They are made by people of the community for the community.

[…] I think the photographs produced are art. They are not elitist art. They are the people's art. They are people's expression and search for themselves and the power to create the kind of life they want for themselves. (Carter,93) Throughout the 1960s and 1970s, the Centre for Contemporary Cultural Studies in Birmingham broke new ground in the study of ideology, power and culture: from a strong Marxist perspective initially, the Centre both recognised culture as the sphere of production of conflicting meanings and signifying practices, and described the mechanisms of reproduction of ideological domination 16 . The perpetuation of the unequal distribution of the means of cultural production in society was one such mechanism. Community arts organisations embodied a struggle to intervene on this problematic: it addressed this imbalance by tackling the problem of access to the means of cultural production and circulation, and by fighting for the recognition of cultural pluralism and of people's agency as cultural producers.

As well as redefining "art" to include very diverse creative practices (theatre, video, photography, creative writing, silk-screening, murals, etc.), community arts recognised that processes were just as important as products, and underlined the importance of the context and uses of artistic production. The notions of access, participation and collaboration, central to community arts, blurred the traditional distinction between artist and spectator, professional and amateur. These perspectives undermined the classic concept of the artist as an individual endowed with genius, and encouraged instead processes of collective authorship 17 . Such conceptual shifts allowed to move away from the logic of the commodification of art, offering instead the idea that the means of expression and cultural production should be collectively shared and control, and remain outside of the commercial sphere.

In this fight, radical community arts practitioners defined their commitment to cultural democracy as opposed to the aim of cultural democratisation, a policy defended by the Arts Council. Through the 1970s, community arts organisations and the Arts Council were locked in a tug-of-war over competing definitions of art and culture and over which art forms and practices should be funded. The Charter of the Arts Council, redefined in 1967, determined two objectives: "to develop and improve the knowledge, understanding and practice of the arts" and "to increase the accessibility of the arts to the public throughout Great Britain." From the early 1970s, the Arts Council seemed to consider community arts as a minor practice, a worthwhile experimental venture, but certainly not as part of the artistic traditions it usually supported. It did consider and respond to grant applications from community arts organisations through its "new activities committee" (1969-70) and "experimental arts committee" Council allocated a fraction of its funding to community arts projects throughout the 1970s 18 . The Community Arts Committee was maintained until 1979.

Yet the two entities supported contradicting projects: the Arts Council construed its aims around the notion of the democratisation of high culture, interpreting its Charter to mean encouraging accessibility to the great works of art throughout the country, and privileging appreciation over practice 19 . In contrast to this, from the late 1970s onwards, the Association of Community Artists defended the notions of cultural democracy and cultural pluralism, which reflected a much more bottom-up approach to cultural production. To Owen Kelly, author of one of the few reference books published at the time on the community arts movement, cultural democracy meant "producing the right conditions within which communities can have their own creative voices recognised and given sufficient space to develop and flourish" (6):

[Cultural democracy] revolves around the notion of plurality, and around equality of access to the means of cultural production and distribution. It assumes that cultural production happens within the context of wider social discourses, and that [cultural production] will produce not only pleasure but knowledge. ( 101) Understood as a radical social and political project, cultural democracy confronted the reproduction of elite culture, and worked to de-construct the ideological domination of bourgeois taste. The recognition of the need to defend cultural pluralism contained definite political overtones. Artist Su Braden wrote in 1978:

The truth is that people make culture. They make it in towns and cities, in villages and hamlets, on housing estates and in suburbs, in Hampstead and in Hull. It is to do with self-expression and social needs. It is active, not passive. It is neither a sub-culture nor an alternative. It is active and to be lived rather than passive and to be appreciated. ( 174)

Community arts in the era of Thatcherism

By the beginning of the 1980s, the community arts movement appeared more structured. Even though the Arts Council stepped back from directly funding organisations in 1979, it encouraged a devolutionary process involving an increased role for Regional Arts Associations in supporting community arts 20 . Regional Associations became the main source of public funding for community arts organisations 21 , alongside local authorities, while private sources such as the Calouste Gulbenkian Foundation remained supportive of the movement (Hewison & Holden).

However, the cultural policy of the Conservative government led by Margaret Thatcher was a source of worry for the community arts sector. As the annual funding to the Arts Council was reduced and some of its committees axed in 1979 and 1980 (the Community Arts and the Photography Committees 18

In 1972, the Arts Council financed fifty-seven organisations and artists described as belonging to the category « Community Arts » for a total sum of £176,000. The following year, £350,000 were attributed to seventy-five projects (Nigg & Wade,30;Kelly,15). A million pounds was reserved to community arts in 1978 (Sinclair,224).

19

In particular Roy Shaw, General Secretary of the Arts Council, took position for the democratisation of culture in his essay « Arts for All », c. 1985. 20 The Arts Council of Great Britain and the Regional Arts Associations: Towards a new relationship, published in May 1980. Arts Council Archives, ACGB/21/58, London, Blythe House.

21

Regional Arts Associations began to recognise the positive social impact of community arts and started funding projects in the 1970s. There remained regional differences in terms of financial commitment. The Greater London Arts Association, Northern Arts and West Midlands Arts were particularly supportive. respectively), the Prime Minister made no mystery about her traditional conception of art: art was high art, heritage, the national institutions and the national canon 22 . The State should remain aloof from intervention in the domain of artistic production for fear of generating a form of official State art. On the other hand, the government developed incentives for private businesses to increase their patronage of the arts, and encouraged the view that the arts were to be considered as a productive, wealth-generating economic sector 23 . These neoliberal values could not have been more at odds with those held by the more radical community artists. Already a minor voice in the artistic field through the 1970s, the community arts movement was now forced to adopt strategies of resistance against the new Conservative rhetoric and policies. As the Association of Community Artists was compelled to adopt charity status, under new directives in 1980 requiring that organisations applying for public money should be charities (therefore blunting their political edge), its campaigning activity was taken over by the Shelton Trust, which expressed its radical commitment to "an egalitarian and plural society by the extension of democratic practice to all social relationships" (Shelton Trust quoted by Higgins, 34).

In the transformed ideological context, the necessity of access to leisure and creativity in collective, emancipatory practices was reasserted and included in a critique of cultural industries and the mechanisms of cultural hegemonies. In 1986, the Shelton Trust reaffirmed the core political project of community arts practices around the concept of cultural democracy: Culture and Democracy: The Manifesto read as an unabashed socialist critique of ideological systems of domination and of cultural hegemony under capitalism. It identified the necessity for struggles in culture and for cultural pluralism, against the cultural exclusion of alternative voices and traditions. It advocated for forms of cultural production understood as collective not individualistic, free as opposed to commercial and profit-driven, active and not passive: "in a genuine democracy people make their culture rather than have it made for them -locally, nationally and internationally" (Another Standard, 39).

In spite of the enduring commitment to these values on the part of the community arts organisations which managed to survive cuts in their funding in the 1980s, the ideological sea-change of New Right politics did take its toll on the radicalism of the movement. Also, there were divisions within the movement, and the radical views expressed by the Shelton Trust were not shared by all practitioners. The middle of the 1980s corresponds to a moment when the movement lost its politically radical bite. François Matarasso, a participant in and a historian of the community arts movement describes the way community arts practice and theory evidence a gradual depoliticisation in the 1990s, a process which he dates back to the Conservative governments of the preceding decade. A telling sign of this, he points out, is the shift in the choice of terms and designation of the practice:

The path from "community art" to "participatory art", whilst seen as merely pragmatic by those who made it, marked and allowed a transition from the politicised and collectivist action of the seventies, towards the depoliticised, individual-focused arts programmes supported by public funds in Britain today. […] The trend of the past forty years had been from radicalism to remedialism. (Matarasso,216) 22 "[Art] is a vital part of our civilisation, of our vision, and our heritage.

[…] The health of society depends as much on the discouragement of rubbish as on the fostering of excellence." Margaret Thatcher, « Speech at the Royal Academy Banquet », 22 nd March 1980. [Online] http://www.margaretthatcher.org/document/104370 Accessed on 27 th April 2016.

Conclusion

A survey of the origins and development of the community arts movement in the 1970s and 1980s, through the prism of leisure gives insights into the structuration of the cultural field in Britain in that period. It highlights the way specific contestations were shaped, at the grassroots level of communities, from a struggle for access to leisure and creative practices to more radical demands for a recognition of cultural pluralism.

From its emergence in the late 1960s, the movement embodied a cultural opposition to dominant systems: an opposition to elitist conceptions of art and the effects of social distinction they entail; to a top-down conception and practice of politics; to commodified forms of leisure, based on a passive logic of consumption. The community arts movement played a central role in the definition of a socialist cultural politics, based on concepts of empowerment, diversity, collective authorship, democratic access to and control of the means of cultural production. At stake in the practice and theory of community arts, was the struggle to establish the conditions for a genuine participatory democracy, in which the arts would play a central role, as a vehicle for the expression of collective meanings and the definition of cultural alternatives. The movement belonged to the radical activism of the late 1960s and 1970s, and embraced many of its perspectives: the recognition that social change should come from the people themselves, the strength of the collective, the importance of local relevance, and the exercise of democratic control of institutions.

Yet this type of radical discourse was jeopardised in the changed political and ideological landscape of the 1980s: the notion of public support for the arts was seriously undermined by the Conservative government's assaults on state provision, while the notion of empowerment was absorbed in a neoliberal discourse stressing individual responsibility and redefined in a way which neutralised their radical potential (Bacqué & Biewener). The versatile notion of "community" was itself reclaimed in a conservative rhetoric for its connotations of traditional social order and moral regulation (Buckler,40).

However, the community arts movement did bear a mark on British society and on people's expectations about what a cultural policy should comprise. A telling sign of its various successes can be found in the fact that all local authorities today provide leisure and creative activities and are expected to deliver on making these resources accessible. Similarly, the principle of consultation of residents and the voluntary sector on local policy issues, has become part of normal community development processes. On another level, the struggle of the movement in favour of cultural pluralism has been vindicated since 2001 at an international level with the ratification by the UNESCO of the Universal Declaration on Cultural Diversity, which recognises the notion of cultural rights and the necessity to secure universal access to cultural diversity (UNESCO, 2001).

  Culture and Democracy : The Manifesto. London: Comedia: 1986. Augustin, Jean-Pierre. L'animation professionnelle: histoire, acteurs, enjeux. Paris : L'Harmattan, 2000. Arts Council of Great Britain. The Arts Council of Great Britain and the Regional Arts Associations: Towards a new relationship, published in May 1980. Arts Council Archives, ACGB/21/58, London, Blythe House.

  They opened spaces, scraped together what grants they could get from with some success. See Paul Carter, British Image 1, London: Arts Council of Great Britain, 1975; Bootle Art in Action, Art in Action, A Community Photographic Project on Merseyside. Bootle Arts and Action, 1980. 8 Tellingly, Birmingham and the West Midlands have consistently been a very active region for community arts. It is worthy of note that one founder member of Jubilee Arts, Stephen Lacey held a Masters' degree in Contemporary Cultural Studies.

  . Largely due to the pressure exerted by the newly created Association of Community Artists (created in 1974), and thanks to the supportive conclusion of the report of the Community Arts Working Party (formed in 1974 and chaired by Harold Baldry), the Arts 16 See Stuart Hall, "Introduction to Media Studies at the Centre", Culture, Media, Language. Working Papers in Cultural Studies, 1972-79. London : Routledge, 1992; "The Emergence of Cultural Studies and the Crisis of the Humanities", October. Vol. 53, Summer 1990, p. 11-23. Also: Centre for Contemporary Cultural Studies (Coll.). Working Papers in Cultural Studies, Spring 1971. University of Birmingham: Centre for Contemporary Cultural Studies, 1971.

The aim was "to expand the creative expressions of people". Association of Community Artists, Press Release, c. Spring 1977. from the Arts Council Archive, Blythe House, London.

"As drama students we also knew the role of play in make believe and imagination, drawing on the possibilities of placing children in roles that could give them a voice and a window onto other worlds." Kate Organ, Jubilee Arts, from "Dangerous Play", Ania Bas, 2014. http://www.brendanjackson.co.uk/wpcontent/uploads/Dangerous-Play-Web.pdf

"Investing in leisure. Out of the waste land", Municipal and Public Services Journal, 12 September 1975, p.1161. Inter-Action papers, Arts Council Archives, Blythe House, London.

A geological landmark five miles West of Telford, in Shropshire.

The "demystification" of art was a byword of many practitioners, that is, the challenge to conceptions of creativity as something reserved to a specific category of people, artists, but inaccessible to laypersons.

"The arts world must come to terms with the fact that Government policy in general has decisively tilted away from the expansion of the public to the private sector. The Government fully intends to honour its pledge to maintain support for the arts as a major feature of its policy, but we look to the private sphere to meet any shortfall and to provide immediate means of increase." Norman St John Stevas, Minister of State for the Arts between 1979 and 1981, quoted bySinclair, 248.