Emeric Bouin

Jean Dolbeault

Christian Schmeiser

DIFFUSION WITH VERY WEAK CONFINEMENT

Keywords: 2010 Mathematics Subject Classification. Primary: 35B40, 35Q84, Secondary: 82C40, 76P05, 26D10 Nash's inequality, Caffarelli-Kohn-Nirenberg inequalities, decay rates, semigroup, weak Poincar inequality, unbounded invariant measure, rate of convergence, Fokker-Planck operator, kinetic equations, scattering operator, transport operator, hypocoercivity

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Diffusion and kinetic transport with very weak confinement

1. Introduction. This paper addresses the large time behavior of the solutions to the macroscopic Fokker-Planck equation and to kinetic equations with Fokker-Planck or scattering collision operators.

The first part of this paper deals with the macroscopic Fokker-Planck equation

∂u ∂t = ∆ x u + ∇ x • (∇ x V u) = ∇ x e -V ∇ x e V u (1)
where x ∈ R d , d ≥ 3, and V is a potential such that e -V ∈ L 1 (R d ), that is, e -V dx is an unbounded invariant measure. We shall investigate the two following examples V 1 (x) = γ log |x| and V 2 (x) = γ log x with γ < d and x := 1 + |x| 2 for any x ∈ R d . These two potentials share the same asymptotic behavior as |x| → ∞. The potential V 1 is invariant under scalings, whereas V 2 is smooth at the origin. In both cases, the only integrable equilibrium state is 0. Thus, if the initial datum u 0 is such that u 0 ∈ L 1 (R d ), we expect that the solution to (1) converges to 0 as t → +∞. When γ > 0, the potential V is very weakly confining in the sense that, even if it eventually slows down the decay rate, it is not strong enough to produce a stationary state of finite mass: the diffusion wins over the drift. Our goal to establish the rate of convergence in suitable norms. We shall use the notation • p := • L p (dx) in case of Lebesgue's measure and specify the measure otherwise.

Theorem 1.1. Assume that d ≥ 3, γ < (d -2)/2 and V = V 1 or V = V 2 . Then any solution u of (1) with initial datum u 0 ∈ L 1 + ∩ L 2 (R d ) satisfies, for all t ≥ 0, u(t, •) .

(

) 2
Here C Nash denotes the optimal constant in Nash's inequality [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF][START_REF] Carlen | Sharp constant in Nash's inequality[END_REF] u

2+ 4 d 2 ≤ C Nash u 4 d 1 ∇u 2 2 ∀ u ∈ L 1 ∩ H 1 (R d ) . (3)
Note that the rate of decay is independent of γ and we recover the classical estimate due to J. Nash when V = 0 (here γ = 0). The proof of Theorem 1.1 and further considerations on optimality are collected in Section 2.1. Theorem 1.1 does not cover the interval (d -2)/2 < γ < d. This range is covered by employing the natural setting of L 2 e V and by requiring additional moment bounds.

Theorem 1.2. Let d ≥ 3, γ < d, V = V 1 or V = V 2 ,
and u 0 ∈ L 1 + ∩ L 2 e V . If γ > 0, let us assume that |x| k u 0 1 < ∞ for some k ≥ max{2, γ/2}. Then any solution of (1) with initial datum u 0 satisfies

∀ t ≥ 0 , u(t, •) 2 L 2 (e V dx) ≤ u 0 2 L 2 (e V dx) (1 + c t) -d-γ 2 .
The constant c depends on d, γ, k, u 0 L 2 (e V dx) , u 0 1 , and |x| k u 0 1 .

The proof of Theorem 1.2 is done in Section 2.2. Although this is a side result, let us notice that the case in which the potential contributes to the decay, i.e., when γ < 0, is also covered in Theorem 1.2. The scale invariance of (1) with V = V 1 can be exploited to obtain intermediate asymptotics in self-similar variables. Let us define

u ⋆ (t, x) = c ⋆ (1 + 2 t) d-γ 2 |x| -γ exp - |x| 2 2 (1 + 2 t) , (4)
The following result on intermediate asymptotics allows us to identify the leading order term of the solution of (1) as t → +∞. It is the strongest of our results on [START_REF] Bakry | Weighted Nash inequalities[END_REF] but initial data need to have a sufficient decay as |x| → ∞.

Theorem 1.3. Let d ≥ 1, γ ∈ (0, d) and V = V 1 .
If for some constant K > 1, the function u 0 is such that

∀ x ∈ R d , 0 ≤ u 0 (x) ≤ K u ⋆ (0, x)
where c ⋆ is chosen such that u ⋆ 1 = u 0 1 then the solution u of (1) with initial datum u 0 satisfies

∀ t ≥ 0 , u(t, •) -u ⋆ (t, •) p ≤ K c 1-1 p ⋆ u 0 1 p 1 e 2 |γ| γ 2 1- 1 p (1 + 2 t) -ζp for any p ∈ [1, +∞), where ζ p := d 2 1 -1 p + 1 2 p min 4, 4 (d -γ), d -1 .
More detailed results will be stated in Section 2.3. Let us quote some relevant papers for [START_REF] Bakry | Weighted Nash inequalities[END_REF]. In the case without potential, the decay rates of the heat equation is known for more than a century and goes back to [START_REF] Fourier | Théorie analytique de la chaleur[END_REF]. Standard techniques use the Fourier transform, Green kernel estimates and integral representations: see for instance [START_REF] Evans | Partial differential equations[END_REF]. There are many other parabolic methods which provide decay rates and will not be reviewed here like, for instance, the Maximum Principle, Harnack inequalities and the parabolic regularity theory: see for instance [START_REF] Vázquez | The mathematical theories of diffusion: nonlinear and fractional diffusion[END_REF].

In his celebrated paper [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF], J. Nash was able to reduce the question of the decay rates for the heat equation to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]: see [START_REF] Bouin | A variational proof of Nash's inequality[END_REF] for detailed comments on the optimality of such a method. Entropy methods have raised a considerable interest in the recent years, but the most classical approach based on the so-called carré du champ method applies to (1) only for potentials V with convexity properties and a sufficient growth at infinity: typically, if V (x) = |x| α , then α ≥ 1 is required for obtaining a Poincaré inequality and the rate of convergence to a unique stationary solution is then exponential, when measured in the appropriate norms; see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] for a general overview. An interesting family of weakly confining potentials is made of functions V with an intermediate growth, such that e -V is integrable but lim |x|→∞ V (x)/|x| = 0: all solutions of (1) are attracted by a unique stationary solution, but the rate is expected to be algebraic rather than exponential. A typical example is V (x) = |x| α with α ∈ (0, 1). The underlying functional inequality is a weak Poincaré inequality: see [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF][START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF], and [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré[END_REF] for related Lyapunov type methods à la Meyn and Tweedie or [START_REF] Ben-Artzi | Weak Poincaré and Nash-type inequalities via density of states estimates[END_REF] for recent spectral considerations. We refer to [START_REF] Bakry | Weighted Nash inequalities[END_REF] and [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF][START_REF]Functional inequalities and spectrum estimates: the infinite measure case[END_REF][START_REF] Wang | Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures[END_REF] for further considerations on, respectively, weighted Nash inequalities and spectral properties of the diffusion operator.

The second part of this paper is devoted to kinetic equations involving a degenerate diffusion operator acting only on the velocity variable or scattering operators, for very weak potentials like V 1 or V 2 . Various hypocoercivity methods have been developed over the years in, e.g., [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation[END_REF][START_REF] Hérau | Short and long time behavior of the Fokker-Planck equation in a confining potential and applications[END_REF][START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF][START_REF] Villani | Hypocoercivity[END_REF][START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF], in order to prove exponential rates in appropriate norms, in presence of a strongly confining potential. In that case, the growth of the potential at infinity has to be fast enough not only to guarantee the existence of a stationary solution but also to provide macroscopic coercivity properties which typically amount to a Poincaré inequality. A popular simplification is to assume that the position variable is limited to a compact set, for example a torus. Such results are the counterpart in kinetic theory of diffusions covered by the carré du champ method, as emphasized in [START_REF] Baudoin | [END_REF].

Recently, hypocoercivity methods have been extended in [START_REF] Bouin | Hypocoercivity without confinement[END_REF] to the case without any external potential by replacing the Poincaré inequality by Nash type estimates. The sub-exponential regime or the regime with weak confinement, i.e., of a potential V such that a weak Poincaré inequality holds, has also been studied in [START_REF] Cao | The kinetic Fokker-Planck equation with weak confinement force[END_REF][START_REF] Hu | Subexponential decay in kinetic Fokker-Planck equation: Weak hypocoercivity[END_REF]. What we will study next is the range of very weak potentials V , which have a growth at infinity which is below the range of weak Poincaré inequalities, but are still such that lim |x|→∞ V (x) = +∞. This regime is the counterpart at kinetic level of the results of Theorems 1.1, 1.2 and 1.3. As in the case of (1) when γ ≥ 0, the drift is opposed to the diffusion, but it is not strong enough to prevent that the solution locally vanishes.

Let us consider the kinetic equation

∂ t f + v • ∇ x f -∇ x V • ∇ v f = Lf (5)
where Lf is one of the two following collision operators:

(a) a Fokker-Planck operator

Lf = ∇ v • M ∇ v M -1 f , (b) a scattering collision operator Lf = R d σ(•, v ′ ) f (v ′ ) M (•) -f (•) M (v ′ ) dv ′ .
We consider the case of a global equilibrium of the form

∀ (x, v) ∈ R d × R d , M(x, v) = M (v) e -V (x) where M (v) = (2π) -d 2 e -1 2 |v| 2 .
We shall say that the gaussian function M (v) is the local equilibrium and assume that the scattering rate σ(v, v ′ ) satisfies

(H1) 1 ≤ σ(v, v ′ ) ≤ σ , ∀ v , v ′ ∈ R d , for some σ ≥ 1 , (H2) R d σ(v, v ′ ) -σ(v ′ , v) M (v ′ ) dv ′ = 0 ∀ v ∈ R d . Notice that M ∈ L 1 (R d ×R d ) if V = V 1 or V = V 2
, so that the space L 2 M -1 dx dv is defined with respect to an unbounded measure. As in the case of (1), the only integrable equilibrium state is 0. Thus, if the initial datum f 0 is such that f 0 ∈ L 1 (dx dv), we expect that the solution to (5) converges to 0 locally as t → +∞ and look for the rate of convergence in suitable norms. When V = 0, the optimal rate of convergence of a solution f of (5) with initial datum f 0 is known. In [START_REF] Bouin | Hypocoercivity without confinement[END_REF], it has been proved that there exists a constant C > 0 such that

R d ×R d |f (t, •, •)| 2 dµ ≤ C (1 + t) -d 2 R d ×R d |f 0 | 2 dµ ∀ t ≥ 0 ,
where dµ = M -1 dx dv and by factorization, the result is extended with same rate for an arbitrary ℓ > d to the measure v ℓ dx dv if

f 0 ∈ L 2 R d × R d , v ℓ dx dv ∩ L 2 + R d , v ℓ dv; L 1 R d , dx .
Our main result on ( 5) is a decay rate in the presence of a very weak potential. It is an extension of the results of Theorem 1.2 to the framework of kinetic equations.

Theorem 1.4. Let d ≥ 3, V = V 2 with γ ∈ [0, d) and k > max {2, γ/2}. We assume that (H1)-(H2) hold and consider a solution f of (5) with initial datum

f 0 ∈ L 2 (M -1 dx dv) such that R d ×R d x k f 0 dx dv + R d ×R d |v| k f 0 dx dv < +∞.
Then there exists C > 0 such that

∀ t ≥ 0 , f (t, •, •) 2 L 2 (M -1 dx dv) ≤ C (1 + t) -d-γ 2 .
Standard methods of kinetic theory can be used to establish the existence of solutions of (5) when V = V 2 . We will not give details here. At formal level, similar results can be expected when V = V 1 but the singularity at x = 0 raises difficulties which are definitely outside of the scope of this paper.

The expression of the constant C is explicit. However, due to the method, we cannot claim optimality in the estimate of Theorem 1.4, but at least the asymptotic rate is expected to be optimal by consistency with the diffusion limit, as it is the case when V = 0 studied in [START_REF] Bouin | Hypocoercivity without confinement[END_REF]. The strategy of the proof and further relevant references will be detailed in Section 3.

2. Decay estimates for the macroscopic Fokker-Planck equation. In this section, we establish decay rates for (1) and discuss the optimal range of the parameters.

2.1. Decay in L 2 (R d ). We prove Theorem 1.1. By testing (1) with u, we obtain

d dt R d u 2 dx = -2 R d |∇u| 2 dx + R d ∆V |u| 2 dx , with ∆V 1 (x) = γ d -2 |x| 2 and ∆V 2 (x) = γ d -2 1 + |x| 2 + 2 γ (1 + |x| 2 ) 2 .
For γ ≤ 0 we deduce d dt u

2 2 ≤ -2 ∇u 2 2 ≤ - 2 C Nash u 0 -4/d 1 u 2+4/d 2
, from Nash's inequality [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Integration completes the proof of (2). For the case 0 < γ < (d -2)/2 we use the following Hardy-Nash inequalities.

Lemma 2.1.

Let d ≥ 3 and δ < (d -2) 2 /4. Then u 2+ 4 d 2 ≤ C δ ∇u 2 2 -δ R d u 2 |x| 2 dx u 4 d 1 ∀ u ∈ L 1 ∩ H 1 (R d ) , (6)
with

C δ = C Nash 1 -4 δ (d-2) 2 -1
.

Let additionally η < (d 2 -4)/4. Then, for any u ∈ L 1 ∩ H 1 (R d ), u 2+ 4 d 2 ≤ C δ,η ∇u 2 2 -δ R d u 2 x 2 dx -η R d u 2 x 4 dx u 4 d 1 (7)
with

C δ,η = C Nash min 1 -4 δ (d-2) 2 , 1 -4 η d 2 -4 -1
.

The proof of Lemma 2.1 is given in Appendix C. We use Lemma 2.1 with δ = γ (d -2)/2 and with η = γ (for V = V 2 ), and proceed as for γ ≤ 0 to complete the proof of Theorem 1.1.

Remark 1. The condition δ < (d -2) 2 /4 in Lemma 2.1 is optimal for (6) and [START_REF] Bouin | A variational proof of Nash's inequality[END_REF]. The restriction on γ in Theorem 1.1 is also optimal.

Let d ≥ 3, γ > (d -2)/2 and V = V 1 or V = V 2 . Then there exists u ∈ L 1 ∩ H 1 (R d ) such that u 2 = 1 and -2 R d |∇u| 2 dx + R d ∆V |u| 2 dx > 0 . In the case V = V 1 , it is indeed enough to observe that (d -2) 2 /4 is the optimal constant in Hardy's inequality (see Appendix C). The case V = V 2 follows from the case V = V 1 by an appropriate scaling.
2.2. Decay in L 2 (e V dx). We prove Theorem 1.2. By testing [START_REF] Bakry | Weighted Nash inequalities[END_REF] with u e V , we obtain 1 2

d dt R d u 2 e V dx = - R d e -V ∇ u e V 2 dx . (8)
In the case V = V 1 , we have e V = |x| γ and (8) takes the form 1 2

d dt R d |x| γ u 2 dx = - R d |x| -γ |∇ (|x| γ u)| 2 dx .
If γ ≤ 0 and a = d-γ d+2-γ , the inequality

R d |x| γ u 2 dx ≤ C R d |x| -γ |∇ (|x| γ u)| 2 dx a R d |u| dx 2(1-a) (9)
follows from the Caffarelli-Kohn-Nirenberg inequalities (see Appendix A, Ineq. [START_REF] Villani | Hypocoercivity[END_REF] applied with k = 0 to v = |x| γ u). The conservation of the L 1 norm of u gives

d dt R d |u| 2 |x| γ dx ≤ -2 C -(1+ 2 d-γ ) u 0 -4 d-γ 1 R d |u| 2 |x| γ dx 1+ 2 d-γ
.

The conclusion of Theorem 1.2 follows by integration. An analogous argument based on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality

R d |u| 2 x γ dx ≤ K R d x -γ |∇ ( x γ u)| 2 dx a R d |u| dx 2(1-a) with a = d -γ d + 2 -γ applies to the case γ ≤ 0, V = V 2 (see Appendix B, Ineq. (30) applied with k = 0 and v = x γ u).
Without additional assumptions, it is not possible to expect a similar result for γ > 0. Let us explain why. In the case V = V 1 and with v = |x| γ u, consider the quotient

Q[v] := R d |x| -γ |∇v| 2 dx a R d |x| -γ |v| dx 2(1-a) R d |x| -γ v 2 dx As a consequence of (9), Q[v] is bounded from below by a positive constant if γ ≤ 0 and a = (d -γ)/(d -γ + 2). Let us consider the case γ > 0. Lemma 2.2. Let d ≥ 1, γ ∈ (0, d) and a = (d-γ)/(d-γ+2).
Then there exists a sequence (v n ) n∈N of smooth, compactly supported functions such that

lim n→∞ Q[v n ] = 0.
Proof. Let us take a smooth function v and consider v n (x) = v(x + n e) for some

e ∈ S d-1 . Then Q[v n ] = O n -(1-a) γ . With γ > 0, we know that a is in the range 0 < a < 1 if and only if γ ∈ (0, d).
For the proof of Theorem 1.2 in the case 0 < γ < d, V = V 1 , we start by estimating the growth of the moment

M k (t) := R d |x| k u dx ,
which evolves according to

M ′ k = k d + k -2 -γ R d u |x| k-2-γ dx ≤ k d + k -2 -γ M 2 k 0 M 1-2 k k ,
where we have used Hölder's inequality and M 0 (t) = M 0 (0) = u 0 1 . Integration gives

M k (t) ≤ M k (0) 2/k + 2 d + k -2 -γ M 2/k 0 t k/2
.

If γ ∈ (0, d) and a = d+2k-γ d+2k+2-γ , by inserting the Caffarelli-Kohn-Nirenberg inequality

R d |x| γ u 2 dx ≤ C R d |x| -γ |∇ (|x| γ u)| 2 dx a R d |x| k |u| dx 2(1-a) (see Appendix A, Ineq. (26) applied to v = |x| γ u) in (8), we observe that the function z = R d u 2 |x| γ dx solves dz dt ≤ -2 C -1 z 1+ 2 d+2k-γ M k (t) -4 d+2k-γ ,
and, after integration,

z(t) ≤ z(0) 1 + a 1 + b t 1-2k d+2k-γ -1 -d+2k-γ 2
with a and b depend only on the quantities entering into the constant c of Theorem 1.2. Let θ = 2k/(d + 2k -γ) and observe that

1 + a 1 + b t 1-θ -1 ≥ 1 + c t 1-θ ∀ t ≥ 0 , if c = b min a, a 1/(1-θ) . Our estimate becomes z(t) ≤ z(0) 1 + a 1 + b t 1-θ -1 -k/θ ≤ z(0) 1 + c t -k (1-θ)/θ = z(0) 1 + c t -d-γ 2 .
In the case V = V 2 we can adopt the same strategy, based on a moment now defined as

M k (t) := R d x k u dx ,
and on the inhomogeneous Caffarelli-Kohn-Nirenberg inequality

R d x γ u 2 dx ≤ K R d x -γ |∇( x γ u)| 2 dx a M 2(1-a) k with a = d + 2k -γ d + 2 + 2k -γ (see Appendix B, Ineq. (30) applied to v = x γ u).
This completes the proof of Theorem 1.2.

Decay in self-similar variables and intermediate asymptotics.

We prove Theorem 1.3. With the parabolic change of variables

u(t, x) = (1 + 2 t) -d/2 v(τ, ξ) , τ = 1 2 log(1 + 2 t) , ξ = x √ 1 + 2 t , (10)
which preserves mass and initial data, ( 1) is changed into

∂v ∂τ = ∆ ξ v + ∇ ξ • (v ∇ ξ Φ) , (11)
where Φ(τ, ξ) = V (e τ ξ) + 1 2 |ξ| 2 . We investigate the long-time behavior of solutions of (1) by considering quasiequilibria v ⋆ (τ, ξ) := M (τ ) e -Φ(τ,ξ) , (12) of ( 11) with an appropriately chosen M (τ ).

For the scale invariant case 11) can be replaced by the time independent potential

V = V 1 , the potential Φ 1 (τ, ξ) = γ log |ξ|+τ + 1 2 |ξ| 2 in (
φ 1 (x) = γ log |ξ| + 1 2 |ξ| 2 . With M (τ ) = c ⋆ e γ τ , time independent equilibria v ⋆,1 (ξ) := c ⋆ |ξ| -γ e -|ξ| 2 /2 , ( 13
)
are available. For the second case V = V 2 with potential

Φ 2 (τ, ξ) := γ 2 log 1 + e 2τ |ξ| 2 + 1 2 |ξ| 2 , we shall use v ⋆,2 (τ, ξ) := c ⋆ e -2τ + |ξ| 2 -γ/2 e -|ξ| 2 /2 , (14)
so that v ⋆,2 is asymptotically equivalent to v ⋆,1 as τ → ∞.

If a quasi-equilibrium of the form [START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF] satisfies

∂v ⋆ ∂τ ≥ 0 ,
which holds for both examples ( 13) and ( 14) if γ > 0, then v ⋆ is obviously a supersolution of [START_REF] Dolbeault | Hypocoercivity for kinetic equations with linear relaxation terms[END_REF], thus proving the following result on uniform decay estimates.

Proposition 1. Let γ ∈ (0, d) and u(t, x) be a solution of (1) with initial datum such that, for some constant c ⋆ > 0,

0 ≤ u(0, x) ≤ c ⋆ σ + |x| 2 -γ/2 exp - |x| 2 2 ∀ x ∈ R d , with σ = 0 if V = V 1 and σ = 1 if V = V 2 . Then 0 ≤ u(t, x) ≤ c ⋆ (1 + 2 t) d-γ 2 σ + |x| 2 -γ/2 exp - |x| 2 2 (1 + 2 t) ∀ x ∈ R d , t ≥ 0 .
For 0 < γ < d, we obtain a pointwise decay: the attracting potential is too weak for confinement (no stationary state can exist, at least among L 1 (R d ) solutions) but it slows down the decay compared to solutions of the heat equation (that is, solutions corresponding to V = 0).

The result of Proposition 1 is also true for γ

≤ 0 if V = V 1 .
In that case, a repulsive potential with γ < 0 accelerates the pointwise decay, but does not change the uniform decay rate as t → +∞ because

∀ t > 0 , max r>0 r -γ exp - r 2 4 t = e 2 |γ| t γ/2 . ( 15
)
In order to obtain an estimate in L 2 e V dx , let us state a result on a Poincaré inequality. We introduce the notations

Φ γ,σ (ξ) := 1 2 |ξ| 2 + γ 2 log σ + |ξ| 2 , Z γ,σ := R d
e -Φγ,σ(ξ) dξ and dµ γ,σ := Z -1 γ,σ e -Φγ,σ dξ .

Lemma 2.3. Assume that d ≥ 1, γ ∈ (0, d) and σ ∈ R + . With the above notations, there is a positive constant λ γ,σ such that

R d |∇w| 2 dµ γ,σ ≥ λ γ,σ R d |w -w| 2 dµ γ,σ ∀ w ∈ H 1 (R d , dµ γ,σ ) such that w = R d w dµ γ,σ . (16)
Moreover, for any γ ∈ (0, d), min σ∈[0,1] λ γ,σ > 0.

Proof. Let us consider a potential ψ on R d . We assume that ψ is a measurable function such that With w = f e Φγ,σ /2 and ψ

ℓ = lim r→+∞ inf f ∈D(B c r )\{0} R d |∇f | 2 + ψ |f | 2 dξ R d |f | 2 dξ > 0 ,
= 1 4 |∇Φ γ,σ | 2 -1 2 ∆Φ γ,σ , λ γ,σ is either ℓ = 4 if
-∆+ ψ has no eigenvalue in the interval (0, 4), or the lowest positive eigenvalue of -∆ + ψ in the interval (0, 4), since the kernel is generated by constant functions. This proves that 0 < λ γ,σ ≤ 4. An elementary computation shows that

4 ψ(ξ) = X -2 d + σ -2 γ -γ 2 d + 2 σ -γ -4 X -1 -γ (γ + 4) X -2
with X = |ξ| 2 + σ, which allows in principle for an explicit computation of λ γ,σ and shows that it is continuous with respect to σ on R + .

In the special case σ = 0, it is possible to compute λ γ,0 as follows.

Lemma 2.4. Assume that d ≥ 1 and γ ∈ (0, d). With the above notations, we have

λ γ,0 = 4 (1 -γ) if d = 1 and λ γ,0 = min {4, 4 (d -γ), d -1} if d ≥ 2.
Proof. A decomposition in spherical harmonics shows that the lowest eigenvalue associated with a non-radial eigenfunction (in dimension d ≥ 2) is of the form

f (ξ) = g(r) Y (ω) with r = |ξ|, ω = ξ/r and -∆ S d-1 Y = k (k + d -2) Y , k ∈ N. If k = 0, g ≡ 1 is
optimal and the eigenvalue is k (k+d-2) with k = 1. Otherwise k = 0 and g is the lowest non-trivial Hermite polynomial with zero average on R + ∋ r in dimension n = d -γ, that is g(r) = r 2 -n and the corresponding eigenvalue is 4n. Notice that n is not necessarily an integer, but can be considered as a real parameter. All other eigenvalues are larger. We conclude by taking the minimum of the two eigenvalues. If d = 1, a similar conclusion holds with f (ξ) = ξ.

An interesting consequence of Lemma 2.4 is a result of intermediate asymptotics, which allows to identify the leading order term of the solution of (1) as t → +∞.

Corollary 1. Assume that d ≥ 1, γ ∈ (0, d) and V = V 1 . With the above notations, if u solves (1) with an initial datum u 0 ∈ L 1 + (R d ) such that u ⋆ (0, x) -1 u 2 0 ∈ L 1 + (R d )
, with u ⋆ defined by (4), and if we choose c ⋆ in (4) such that u ⋆ (0,

•) 1 = u 0 1 , then R d u(t, x) -u ⋆ (t, x) 2 u ⋆ (t, x) dx ≤ (1 + 2 t) -λγ,0 R d u(0, x) -u ⋆ (0, x) 2 u ⋆ (0, x) dx .
Proof. By definition of u ⋆ , we have

R d v ⋆,1 dξ = R d v(0, ξ) dξ = R d u 0 dx .
Then, using the Poincaré inequality (16) and Lemma 2.4, we know that

d dτ R d (v -v 1,⋆ ) 2 e φ1 dξ = -2 R d ∇ ξ e φ1 (v -v 1,⋆ ) 2 e -φ1 dξ ≤ -2 λ γ,0 R d (v -v 1,⋆ ) 2 e φ1 dξ ,
from which we deduce that

R d (v -v 1,⋆ ) 2 e φ1 dξ ≤ e -2 λγ,0τ R d (u(0, x) -v 1,⋆ ) 2 e φ1 dx .
This concludes the proof using the parabolic change of variables [START_REF] Carlen | Sharp constant in Nash's inequality[END_REF].

Proof of Theorem 1.3. A Cauchy-Schwarz inequality shows that

R d u(t, x) -u ⋆ (t, x) dx 2 ≤ R d u ⋆ (t, x) dx R d u(t, x) -u ⋆ (t, x) 2 u ⋆ (t, x) dx ≤ (1 + 2 t) -λγ,0 R d u 0 dx R d u(0, x) -u ⋆ (0, x) 2 u ⋆ (0, x) dx .
The Hölder interpolation inequality

u(t, •) -u ⋆ (t, •) p ≤ u(t, •) -u ⋆ (t, •) 1 p 1 u(t, •) -u ⋆ (t, •) 1-1 p ∞
combined with the results of Proposition 1 and Corollary 1 concludes the proof after taking [START_REF] Fourier | Théorie analytique de la chaleur[END_REF] and the expression of λ γ,0 stated in Lemma 2.4 into account.

3. Decay estimate for the kinetic equation with weak confinement. In this section, we prove Theorem 1.4 by revisiting the L 2 approach of [START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF] in the spirit of [START_REF] Bouin | Hypocoercivity without confinement[END_REF].

3.1. Notations and elementary computations. On the space L 2 (M -1 dx dv), we define the scalar product

f, g = R d ×R d
f g e V M -1 dx dv and the norm f = f, f 1/2 . Let Π be the orthogonal projection operator on Ker(L) given by Πf := M ρ[f ], where ρ[f ] := R d f (v) dv, and T be the transport operator such that Tf

= v • ∇ x f -∇ x V • ∇ v f . We assume that M (v) = (2π) -d 2 e -1 2 |v| 2 ∀ v ∈ R d .
Let us use the notation u[f ] := e V ρ[f ] and observe that

T Πf = M e -V v • ∇ x u[f ] , (T Π) * f = -M ∇ x • ρ [v f ] , (TΠ) * (TΠ)f = -M ∇ x • e -V ∇ x u[f ] ,
where the last identity follows from

R d M (v) v ⊗ v dv = Id.
To build a suitable Lyapunov functional, as in [START_REF] Dolbeault | Hypocoercivity for kinetic equations with linear relaxation terms[END_REF][START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Bouin | Hypocoercivity without confinement[END_REF] we introduce the operator A defined by

A := Id + (TΠ) * (TΠ) -1 (TΠ) * .
As in [START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF] we define the Lyapunov functional H by

H[f ] := 1 2 f 2 + ε Af , f
and obtain by a direct computation that

d dt H[f ] = -D[f ] with D[f ] := -Lf , f + ε ATΠf , Πf + ε AT(Id -Π)f , Πf -ε TA(Id -Π)f , (Id -Π)f -ε AL(Id -Π)f , f , (17)
where we have used that Af , Lf = 0. For the first term in D[f ], we rely on the microscopic coercivity estimate (see [START_REF]Hypocoercivity for linear kinetic equations conserving mass[END_REF])

-Lf , f ≥ λ m (Id -Π)f 2 .
The second term ATΠf , Πf is expected to control the macroscopic contribution Πf . In Section 3.2 the remaining terms will be estimated to show that for ε small enough D[f ] controls (Id -Π)f 2 + ATΠf , Πf . As in Section 2.2, estimates on moments are needed, which will be proved in Section 3.3 and used in Section 3.4 to show a Nash type estimate and to complete the proof of Theorem 1.4 by relating the entropy dissipation D[f ] to H[f ] and by solving the resulting differential inequality.

Proof of the Lyapunov functional property of H[f ]. Let us define the notations

u 1 , u 2 V := R d u 1 u 2 e -V dx and u 2 V := u, u V associated with the norm L 2 (e -V dx). Unless it is specified, ∇ means ∇ x .
Lemma 3.1. With the above notations, we have

Af ≤ 1 2 (Id -Π)f , TAf ≤ (Id -Π)f and | TA(Id -Π)f , (Id -Π)f | ≤ (Id -Π)f 2 .
Proof. We already know from [12, Lemma 1] that the operator TA is bounded. Let us give a short proof for completeness. The equation Af = g is equivalent to

(TΠ) * f = g + (TΠ) * (TΠ) g . (18)
Multiplying ( 18) by g M -1 e V , we get that

g 2 + TΠg 2 = f, TΠg = (Id -Π)f , TΠg ≤ (Id -Π)f TΠg ≤ 1 4 (Id -Π)f 2 + TΠg 2
from which we deduce that Af = g ≤ 1 2 (Id -Π)f . Since A = ΠA, because ( 18) can be rewritten as g = ΠT 2 Πg -ΠTf using (TΠ) * = -ΠT, we also have that TAf = TΠg and obtain that TAf = TΠg ≤ (Id -Π)f . The estimate on

| TA(Id -Π)f , (Id -Π)f | follows.
The term ATΠf , Πf is the one which gives the macroscopic decay rate. Let w[f ] be such that Id + (TΠ) * (TΠ)

-1 Πf = w M e -V . Then w solves

w -Lw = u[f ] where Lw := e V ∇ • e -V ∇w . ( 19
)
Lemma 3.2. With the above notations, if u = u[f ] and w = w[f ] solves (19), we have

ATΠf , Πf = ∇w 2 V + Lw 2 V ≤ 5 4 u 2 V .
Proof. Let w be a solution of [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF]. Since

ATΠf = Id + (TΠ) * (TΠ) -1 (TΠ) * (TΠ) Πf = Id + (TΠ) * (TΠ) -1 Id + (TΠ) * TΠ -Id Πf = Πf -Id + (TΠ) * (TΠ) -1 Πf = Πf -w M e -V ,
we obtain that ATΠf = (u -w) M e -V .

Using [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF] and integrating on R d after multiplying by Πf = u M e -V , we obtain that

ATΠf , Πf = u, u -w V = w -Lw, -Lw V = ∇w 2 V + Lw 2 V .
On the other hand, we can also write that

ATΠf , Πf = u, u -w V = -u, Lw V and obtain that ∇w 2 V + Lw 2 V = -u, Lw V ≤ u V Lw V ≤ 1 4 u 2 V + Lw 2 V ,
using the Cauchy-Schwarz inequality. As a consequence, we obtain that

∇w 2 V ≤ 1 4 u 2 V and Lw V ≤ u V ,
which concludes the proof.

Lemma 3.3. With the above notations, if u = u[f ] and w solves (19), we have

Hess(w) 2 V ≤ max{1, γ} ATΠf , Πf . Proof. The operator L = ∆ -∇V • ∇ is such that [L, ∇] w = L(∇w) -∇(L w) = L ∂w ∂xi -∂ ∂xi (L w) d i=1 = -Hess(V ) • ∇w
and it is self-adjoint on L 2 (e V dx) so that

Lw 1 , w 2 V = -∇w 1 , ∇w 2 V = w 1 , Lw 2 V
for any w 1 and w 2 . Applied first with w 1 = w and w 2 = Lw and then with w 1 = w 2 = ∇w, this shows that

Lw 2 V = -∇w, ∇Lw V = -∇w, L∇w V - R d ∇w • [L, ∇] w e -V dx = Hess(w) 2 V + R d Hess(V ) : (∇w ⊗ ∇w)e -V dx
where Hess(w)

2 2 = R d |Hess(w)| 2 e -V dx = d i,j=1 R d ∂ 2 w ∂xi∂xj 2 e -V dx. In the case V = V 2 , we deduce from ∂ 2 V ∂x i ∂x j = γ x 2 δ ij -2 x i x j x 2 that Hess(V ) ≥ -γ Id . Hence max{1, γ} ATΠf , Πf ≥ Lw 2 V + max{1, γ} ∇w 2 V ≥ Hess(w) 2 V -γ ∇w 2 V + max{1, γ} ∇w 2
V , which concludes the proof. Lemma 3.4. With the above notations and with m γ := 3 max{1, γ}, we have

| AT(Id -Π)f , Πf | ≤ m γ ATΠf , Πf 1/2 (Id -Π)f .
Proof. Assume that u = u[f ] and w solves [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF]. Using g = Id + (TΠ) * (TΠ)

-1 f so that Id + (TΠ) * (TΠ) g = f means g -(Lw) M e -V = f , let us compute AT(Id -Π)f , Πf = T(Id -Π)f , A * Πf = T(Id -Π)f , TΠg = R d ×R d √ M v ⊗ v (Id -Π)f √ M : Hessw dx dv = R d ×R d √ M v ⊗ v -1 d Id (Id -Π)f √ M : Hessw dx dv
We conclude using a Cauchy-Schwarz inequality, Lemma 3.2 and Lemma 3.3.

In order to have unified notations, we adopt the convention that σ = 1/ √ 2 if L is the Fokker-Planck operator. Lemma 3.5. With the above notations, we have

AL(Id -Π)f , Πf ≤ √ 2 σ ATΠf , Πf 1/2 (Id -Π)f .
Proof. We use duality to write

AL(Id -Π)f , Πf = L(Id -Π)f , h
where h = A * f = (TΠ)g and g = Id + (TΠ) * (TΠ) -1 f so that

Id + (TΠ) * (TΠ) g = f and h = v • ∇w M e -V .
Here w solves [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF]

with u = u[f ]. • If L is the Fokker-Planck operator, then R d v Lf dv = -j and | AL(Id -Π)f , f | = | j, ∇w 2 | ≤ j V ∇w V ≤ (Id -Π)f ∇w V .
We conclude using Lemma 3.2 and an estimate on j = |j| e where e ∈ S d-1 , that goes as follows: by computing

|j| = R d v f dv = R d v (Id -Π)f dv ≤ R d (Id -Π)f M -1/2 |v • e| M 1/2 dv ≤ R d Id -Π)f 2 M -1 dv R d |v • e| 2 M dv 1 2 = R d Id -Π)f 2 M -1 dv 1 2
, we know that

j e V 2 V = R d |j| 2 e V dx ≤ R d ×R d Id -Π)f 2 M -1 e V dx dv = (Id -Π)f 2 .
• If L is the scattering operator, then

L(Id -Π)f 2 ≤ σ 2 R d 1 M R d M M ′ f ′ M ′ - f M dv ′ 2 dv ≤ σ 2 R d M R d √ M ′ √ M ′ f ′ M ′ - f M dv ′ 2 dv ≤ σ 2 R d ×R d M M ′ f ′ M ′ - f M 2 dv dv ′ ≤ 4 σ 2 R d f 2 M -1 dv and h = v • ∇w M e -V = ∇w 2 so that AL(Id -Π)f , f ≤ √ 2 σ ATΠf , Πf 1/2 (Id -Π)f .
Notice that for a nonnegative function f , we have the improved bounds

L(Id-Π) ≤ σ (Id -Π)f and AL(Id -Π)f , Πf ≤ σ ATΠf , Πf 1/2 (Id -Π)f .
Finally, we apply the results of Lemmas 3.1, 3.4, 3.5 to the right hand side of ( 17):

Lemma 3.6. With the above notations, we have

D[f ] ≥ λ ε (Id -Π)f 2 + ATΠf , Πf with λ ε := 1 2 λ m -(λ m -2 ε) 2 + ε 2 m γ + √ 2 σ 2 and λ ε > 0, if ε > 0 is small enough.
D[f ] ≥ (λ m -ε) (Id -Π)f 2 + ε ATΠf , Πf -ε m γ + √ 2 σ (Id -Π)f ATΠf , Πf 1/2 .
The Lyapunov function property is a consequence of ( 19) and Lemma 3.2.

Moment estimates.

Let us consider the case V = V 2 and define the k th order moments in x and v by

J k (t) := x k f (t, •, •) 1 and K k (t) := |v| k f (t, •, •) 1 .
Our goal is to prove estimates on J k and K k . Notice that

J 0 = K 0 = f 0 L 1 (R d ×R d ) is constant if f solves (5). Lemma 3.7. Let γ ∈ (0, d), k ∈ N with k ≥ 2, V = V 2 and assume that f ∈ C R + , L 2 (M -1 dx dv) is a nonnegative solution of (5) with initial datum f 0 such that R d ×R d x k f 0 dx dv < +∞ and R d ×R d |v| k f 0 dx dv < +∞. There exist constants C 2 , . . . , C k such that J ℓ (t) ≤ C ℓ (1 + t) ℓ/2 and K ℓ (t) ≤ C ℓ ∀ t ≥ 0 , ℓ = 2, . . . , k . (20)
Proof. We present the proof for a Fokker-Planck operator, the case of a scattering operator follows the same steps. A direct computation shows that

dK ℓ dt ≤ ℓ γ R d ×R d |x • v| x 2 |v| ℓ-2 f (t, x, v) dx dv + ℓ (ℓ + d -2) K ℓ-2 -ℓ K ℓ .
A bound C ℓ for K ℓ , ℓ ∈ N, follows after observing that

R d ×R d |x • v| x 2 |v| ℓ-2 f (t, x, v) dx dv ≤ K ℓ-1 ≤ K 1/ℓ 0 K 1-1/ℓ ℓ and K ℓ-2 ≤ K 2/k 0 K 1-2/ℓ ℓ using Hölder's inequality twice. Next, let us compute dJ ℓ dt = ℓ R d ×R d x ℓ-2 x • v f (t, x, v) dx dv =: ℓ L ℓ , and
dL ℓ dt = R d ×R d x ℓ-2 |v| 2 f dx dv + (ℓ -2) R d ×R d x ℓ-4 (x • v) 2 f dx dv -γ R d ×R d x ℓ-4 |x| 2 f dx dv -L ℓ ≤ (ℓ -1) R d ×R d x ℓ-2 |v| 2 f dx dv -L ℓ . (21)
Note that, again by Hölder's inequality,

|L ℓ | ≤ J 1-1/ℓ ℓ K 1/ℓ ℓ , ℓ = 2, . . . , k. We prove the bound on J ℓ (t) by induction. If ℓ = 2, (21) implies L 2 (t) ≤ max L 2 (0), C 2 and, thus, J 2 (t) ≤ C 2 (1 + t), up to a redefinition of C 2 .
Now let ℓ > 2 and assume that

J ℓ-1 (t) ≤ C ℓ-1 (1 + t) ℓ- 1 2
.

We use Hölder's inequality once more for the right hand side of ( 21):

dL ℓ dt ≤ (ℓ -1) J ℓ-2 ℓ-1 ℓ-1 K 1 ℓ-1 2(ℓ-1) -L ℓ ≤ (ℓ -1) C ℓ-2 ℓ-1 ℓ-1 C 1 ℓ-1 2(ℓ-1) (1 + t) ℓ 2 -1 -L ℓ , which implies L ℓ ≤ C (1 + t) ℓ 2 -1
, and one more integration with respect to t establishes the estimate for J ℓ in [START_REF] Kavian | The Fokker-Planck equation with subcritical confinement force[END_REF], up to an eventual redefinition of C ℓ .

Lemma 3.8. Let γ ∈ (0, d), k ∈ N with k > 2, V = V 2 and assume that f ∈ C R + , L 2 (M -1 dx dv) is a nonnegative solution of (5) with initial datum f 0 such that R d ×R d x k f 0 dx dv < +∞ and R d ×R d |v| k f 0 dx dv < +∞. Let w = w[f ] be determined by (19) in terms of u = u[f ]. Then there exists a positive constant C k such that 0 ≤ M k (t) := R d w x k-γ dx ≤ C k (1 + t) k/2 ∀ t ≥ 0 .
Proof. The solution w of ( 19) is positive by the maximum principle. In what follows we use the definition of M ℓ for arbitrary integers ℓ and note that for ℓ ≤ 0,

M ℓ ≤ M 0 = R d w e -V dx = R d u e -V dx = f 0 1 . (22)
Multiplication of ( 19) by x ℓ-γ and integration over R d gives

M ℓ = ℓ (ℓ -2 + d -γ) M ℓ-2 -(ℓ -2 -γ) M ℓ-4 + J ℓ , (23)
where J ℓ has been estimated in Lemma 3.7. Then, with ℓ = 2 and ( 22), we obtain M 2 (t) ≤ C 2 (1 + t). This implies by the Hölder inequality that

M 1 (t) ≤ M 0 M 2 (t) ≤ C 1 (1 + t) 1/2 . For 2 < ℓ ≤ k the estimate M ℓ (t) ≤ C ℓ (1 + t) ℓ/2
follows recursively from (23).

3.4.

Decay estimate for the kinetic equation (proof of Theorem 1.4). Lemma 3.9. Let γ ∈ (0, d), k ≥ max{2, γ/2}, V = V 2 and assume that f ∈ C R + , L 2 (M -1 dx dv) is a nonnegative solution of (5) with initial datum f 0 such that R d ×R d x k f 0 dx dv < +∞ and R d ×R d |v| k f 0 dx dv < +∞. Assume the above notations, in particular with M k defined as in Lemma 3.8, with the constant K from (30), and with a = d+2k-γ d+2+2k-γ . Then

Πf 2 ≤ 2 ATΠf , Πf +K M 2(1-a) k ATΠf , Πf a =: Φ ( ATΠf , Πf ; M k ) ∀ t ≥ 0 .
Proof. If u = u[f ] and w solves [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF], we recall that

ATΠf , Πf = ∇w 2 V + Lw 2
V by Lemma 3.2. From [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF], we also deduce that

u 2 V = u, w -Lw V ≤ u V w 2 V + 2 ∇w 2 V + Lw 2 V 1/2 .
By inequality (30), we have that

w 2 V ≤ K ∇w 2a V M 2(1-a) k
.

Combining these inequalities gives

u 2 V ≤ K ∇w 2a V M 2(1-a) k + ∇w 2 V + ATΠf , Πf , which, noting that Πf = u V , implies the result.
As a consequence of Lemmas 3.1, 3.6, 3.9 and of the properties of Φ we have

H[f ] = 1 2 f 2 + ε Af , f ≤ 1 + ε 2 f 2 ≤ 1 + ε 2 (Id -Π)f 2 + Φ ( ATΠf , Πf ; M k ) ≤ 1 + ε 2 Φ (Id -Π)f 2 + ATΠf , Πf ; M k ≤ 1 + ε 2 Φ D[f ] λ ε ; M k , implying, with Lemma 3.8, dH[f ] dt = -D[f ] ≤ -λ ε Φ -1 2 1 + ε H[f ]; C k (1 + t) k/2 .
The decay of H[f ] can be estimated by the solution z of the corresponding ODE problem

dz dt = -λ ε Φ -1 2 1 + ε z; C k (1 + t) k/2 , z(0) = H[f 0 ] .
By the properties of Φ it is obvious that z(t) → 0 monotonically as t → +∞, which implies that the same is true for dz dt . Therefore, there exists t 0 > 0 such that, in the rewritten ODE

- 2 λ ε dz dt + K C 2(1-a) k (1 + t) k(1-a) - 1 λ ε dz dt a = 2z 1 + ε ,
the first term is smaller than the second for t ≥ t 0 , implying the differential inequality dz dt ≤ -κ z 1/a (1 + t) k(1-1/a) for t ≥ t 0 , with an appropriately defined positive constant κ. Integration and estimation as in Section 2.2 gives

z(t) ≤ C (1 + t) 1+k(1-1/a) 1-1/a = C (1 + t) γ-d 2 ,
thus completing the proof of Theorem 1.4.

Appendix A. Homogeneous Caffarelli-Kohn-Nirenberg inequalities of Nash type.

A.1. The general Caffarelli-Kohn-Nirenberg inequalities. The main result of [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] goes as follows. Assume that p ≥ 1, q ≥ 1, r > 0, 0 ≤ a ≤ 1 and

1 p + α ⋆ d > 0 , 1 q + β ⋆ d > 0 , 1 r + γ ⋆ d > 0 , 1 r + γ ⋆ d = a 1 p + α ⋆ -1 d + (1 -a) 1 q + β ⋆ d and, with σ such that γ ⋆ = a σ + (1 -a)β ⋆ , 0 ≤ α ⋆ -σ if a > 0 . Assume moreover that α ⋆ -σ ≤ 1 if a > 0 and 1 p + α ⋆ -1 d = 1 q + β ⋆ d .
Then there exists a positive constant C such that the inequality

|x| γ⋆ v r ≤ C |x| α⋆ ∇v a p |x| β⋆ v 1-a q (24)
holds for any v ∈ C ∞ 0 (R d ). These interpolation inequalities are known in the literature as the Caffarelli-Kohn-Nirenberg inequalities according to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] but were introduced earlier by V.P. Il'in in [START_REF] Il'in | Some integral inequalities and their applications in the theory of differentiable functions of several variables[END_REF]. Next we specialize Ineq. [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] to various cases of Nash type corresponding to q = 1.

A.2. Weighted Nash type inequalities. We consider special cases corresponding to r = p = 2 and q = 1.

• Ineq. [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] with α ⋆ = β/2, β ⋆ = β/2, and γ ⋆ = β/2 can be written under the condition β > -d as

R d |x| β v 2 dx ≤ C R d |x| β |∇v| 2 dx a R d |x| β/2 |v| dx 2(1-a) with a = d d + 2 . ( 25
)
We can indeed check that α ⋆ -σ = 0 for any β ≤ 0 and 1 p + α⋆ d > 0, 1 q + β⋆ d > 0, and 1 r + γ⋆ d > 0 if and only if β > -d. • Ineq. [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] with α ⋆ = -γ/2, β ⋆ = k -γ and γ ⋆ = -γ/2 can be written as

R d |x| -γ v 2 dx ≤ C R d |x| -γ |∇v| 2 dx a R d |x| k-γ |v| dx 2(1-a) with a = d + 2k -γ d + 2 + 2k -γ (26)
under the condition γ < d and k ≥ γ/2. We can indeed check that α ⋆ -σ = 2k-γ 2k-γ+d ≥ 0. In that case, we have α ⋆ -σ < 1 for any γ ≤ 0 and the conditions 1 p + α⋆ d > 0, 1 q + β⋆ d > 0, and 1 r + γ⋆ d > 0 are always satisfied. A.3. A weighted Nash inequality on balls. We adapt the proof of E. Carlen and M. Loss in [START_REF] Carlen | Sharp constant in Nash's inequality[END_REF] to the case of homogeneous weights. With g = |x| -γ/2 v, (26) is equivalent to

R d g 2 dx ≤ C R d |∇g| 2 dx - γ 4 (2 d -γ -4) R d g 2 |x| 2 dx a • R d |x| k-γ 2 |g| dx 2(1-a)
.

Without loss of generality, we can assume that the function g is nonnegative and radial, by spherically non-increasing rearrangements. From now on, we will only consider nonnegative, radial, non-increasing functions g and the corresponding functions v(x) = |x| γ/2 g(x). For any R > 0, let

g R := g ½ BR and v R (x) = |x| γ/2 g R (x) . We observe that g -g R is supported in R d \ B R and g -g R ≤ g(R) ≤ g R := R d g R |x| k-γ/2 dx BR |x| k-γ/2 dx = R d v R |x| k-γ dx BR |x| k-γ/2 dx because g is radial non-increasing, so that R d |v -v R | 2 |x| -γ dx = g -g R 2 2 ≤ g R R d |g -g R | dx = g R R γ 2 -k R d |v -v R | |x| k-γ dx , i.e., R d |v -v R | 2 |x| -γ dx ≤ R d v R |x| k-γ dx BR |x| k-γ/2 dx R γ 2 -k R d |v -v R | |x| k-γ dx . (27)
On the other hand, let us define

v R := R d vR |x| k-γ dx B R |x| 2k-γ dx and observe that R d |v R | 2 |x| k-γ dx = R d v R -v R |x| k 2 |x| -γ dx + v 2 R BR |x| 2k-γ dx .
Let us consider the weighted inequality

BR |w| 2 |x| -γ dx ≤ 1 λ R 1 BR |∇w| 2 |x| -γ dx ∀ w ∈ H 1 (B R , |x| -γ dx) such that BR w |x| k-γ dx = 0 . ( 28
)
The existence of a positive, finite constant λ R 1 can be deduced from elementary variational techniques as in [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF]. We infer from the definition of v R that this inequality is equivalent to

R d |v R | 2 |x| -γ dx ≤ 1 λ R 1 BR |∇v| 2 |x| -γ dx + R d v R |x| k-γ dx 2 BR |x| 2k-γ dx . (29)
With

λ 1 := λ 1 1 , a simple scaling shows that λ R 1 = λ 1 R -2 . Let us come back to the estimation of R d v 2 |x| -γ dx. By definition of v R , we know that R d v 2 |x| -γ dx ≤ R d |v R | 2 |x| -γ dx + R d |v -v R | 2 |x| -γ dx .
After summing [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF] and [START_REF] Wang | Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures[END_REF], we arrive at

R d v 2 |x| -γ dx ≤ R 2 λ 1 BR |∇v| 2 |x| -γ dx + R d v R |x| k-γ dx 2 BR |x| 2k-γ dx + R γ 2 -k BR |x| k-γ/2 dx R d v R |x| k-γ dx R d |v -v R | |x| k-γ dx and notice that R d v R |x| k-γ dx 2 BR |x| 2k-γ dx + R γ 2 -k BR |x| k-γ/2 dx R d v R |x| k-γ dx R d |v -v R | |x| k-γ dx ≤ R d v R |x| k-γ dx R d v R |x| k-γ dx BR |x| 2k-γ dx + R γ 2 -k BR |x| k-γ/2 dx R d |v -v R | |x| k-γ dx ≤ R d v |x| k-γ dx 2 max 1 BR |x| 2k-γ dx , R γ 2 -k BR |x| k-γ/2 dx = R d v |x| k-γ dx 2 c R γ-d-2k
using k > 0 and v R ≤ v, for some numerical constant c which depends only on d and γ. Collecting terms, we have found that

R d v 2 |x| -γ dx ≤ R 2 λ 1 BR |∇v| 2 |x| -γ dx + c R γ-d-2k R d v |x| k-γ dx 2 .
We can summarize our observations as follows.

Proposition 2. Let d ≥ 3, γ ∈ (0, d), k ≥ γ/2 and a = d+2k-γ d+2+2k-γ . If C denotes the optimal constant in (26), then (28) holds with a constant λ R 1 = λ 1 R -2 for any R > 0, where λ 1 is a positive constant such that λ 1 ≤ κ C -1/a for some explicit positive constant κ depending only on γ and d.

The numerical value of κ can be deduced from the expression of c and from the coefficients that arise from the optimization with respect to R > 0.

Appendix B. Inhomogeneous Caffarelli-Kohn-Nirenberg inequalities of Nash type. Our goal is to establish an extension of (26) adapted to the inhomogeneous case. with a = d+2k-γ d+2+2k-γ holds for some optimal constant K > 0.

Proof. Again we rely on the method of E. Carlen and M. Loss in [START_REF] Carlen | Sharp constant in Nash's inequality[END_REF]. The computations are similar to the ones of Proposition 2 except that |x| has to be replaced by x . With g = x -γ/2 v, (30) is equivalent to

R d g 2 dx ≤ K R d |∇g| 2 dx - γ 4 (2 d -γ -4) R d g 2 x -2 dx - γ 4 (γ + 4) R d g 2 x -4 dx a • R d x k-γ 2 |g| dx 2(1-a)
.

Without loss of generality, we assume that the function g is nonnegative, radial by spherically non-increasing rearrangements, and nonnegative. Let v(x) = x γ/2 g(x) and g R := g ½ BR and v R (x) = x γ/2 g R (x) for any R > 0. We observe that g -g R is supported in R d \ B R and

g -g R ≤ g(R) ≤ g R := R d g R x k-γ/2 dx BR x k-γ/2 dx = R d v R x k-γ dx BR x k-γ/2 dx because g is radial non-increasing, so that R d |v -v R | 2 x -γ dx = g -g R 2 2 ≤ g R R d |g -g R | dx = g R R γ 2 -k R d |v -v R | x k-γ dx ,
that is,

R d |v -v R | 2 x -γ dx ≤ R d v R x k-γ dx BR x k-γ/2 dx R γ 2 -k R d |v -v R | x k-γ dx . ( 31
)
On the other hand, using

R d |v R | 2 x -γ dx = R d v R -v R x k 2 x -γ dx + v 2 R BR
x 2k-γ dx where v R := R d v R x k-γ dx BR x 2k-γ dx , we deduce from the weighted Poincaré inequality

BR |w| 2 x -γ dx ≤ 1 λ R 1 BR |∇w| 2 x -γ dx ∀ w ∈ H 1 (B R ) such that BR w x 2k-γ dx = 0
and from the definition of v R that

R d |v R | 2 x -γ dx ≤ 1 λ R 1 BR |∇v| 2 x -γ dx + R d v R x k-γ dx 2 BR x 2k-γ dx . ( 32
)
By definition of v R , we also know that

R d v 2 x -γ dx ≤ R d |v R | 2 x -γ dx + R d |v -v R | 2 x -γ dx .
After summing (31) and (32), we arrive at and such that lim R→0+ R d b(R) ∈ (0, +∞), lim R→+∞ R d+2k-γ b(R) ∈ (0, +∞), lim R→+∞ R -2 a(R) = 1/λ 1 where λ 1 is the optimal constant in Proposition 2 while lim R→0+ R -2 a(R) = 1/λ is related with Nash's inequality as in [START_REF] Carlen | Sharp constant in Nash's inequality[END_REF] and such that

R d v 2 x -γ dx ≤ 1 λ R 1 BR |∇v| 2 x -γ dx + R d v R x k-γ dx 2 BR x 2k-γ dx + R γ 2 -k BR x k-γ/2 dx R d v R x k-γ dx R d |v -v R | x k-γ dx ≤ a(R) BR |∇v| 2 x -γ dx + b(R) R d v x k-γ dx
B1 |w| 2 dx ≤ 1 λ B1 |∇w| 2 dx ∀ w ∈ H 1 (B 1 ) such that B1 w dx = 0 .
In order to prove (30), we can use the homogeneity of the inequality and assume that R d x -γ v 2 dx = 1. What we shown so far is that we get that there exists a constant K > 0 such that a(R) X + b(R) < K X a . This proves (30) with K ≤ K.

∀ R > 0 , 1 ≤ R d x k-γ |v| dx
Appendix C. Hardy-Nash inequalities.

C.1. Proof of Lemma 2.1. We start with the proof of ( 7) by first showing a Hardy type inequality. For some α ∈ R to be fixed later we compute

0 ≤ R d ∇u + α x 1 + |x| 2 u 2 dx = R d |∇u| 2 dx + α 2 R d |x| 2 u 2 (1 + |x| 2 ) 2 dx + α R d ∇ u 2 • x 1 + |x| 2 dx .
We deduce that

R d |∇u| 2 dx+α 2 R d |x| 2 u 2 (1 + |x| 2 ) 2 dx-α d R d u 2 1 + |x| 2 dx+2 α R d |x| 2 u 2 (1 + |x| 2 )
2 dx ≥ 0 , so that, by writing |x| 2 = x 2 -1, we obtain

  where B c r := x ∈ R d : |x| > r and D(B c r ) denotes the space of smooth functions on R d with compact support in B c r . According to Persson's result [23, Theorem 2.1], the lower end of the continuous spectrum of the Schrödinger operator -∆ + ψ is ℓ.

  The functional H[f ] is a Lyapunov function in the sense that D[f ] ≥ 0 and the equation D[f ] = 0 has a unique solution f = 0. Proof. The above mentioned Lemmas imply

Theorem B. 1 .

 1 If d ≥ 3, γ ∈ (0, d) and k ≥ γ/2, then R d x -γ v 2 dx ≤ K R d x -γ |∇v| 2 dx a R d x k-γ |v| dx 2(1-a) (30)

2 where a

 2a and b are two positive continuous functions on (0, +∞) defined bya(R) := 1/λ R 1 and b(R) := max 1 BR x 2k-γ dx , R γ 2 -k BR x k-γ/2 dx

2 a

 2 (R) X + b(R) where X = R d x -γ |∇v| 2 dx/ R d x k-γ |v| dx 2 . With the choice R = X -(1-a)/2 ,

  ∇u

2 2 + 2 ( 1 +

 221 α (α -d + 2) R d u 2 1 + |x| 2 dx -α (α + 2) R d u |x| 2 ) 2 dx ≥ 0 . (33)

Acknowledgments: This work has been partially supported by the Projects EFI ANR-17-CE40-0030 (E.B., J.D.) of the French National Research Agency. The work of C.S. has been supported by the Austrian Science Foundation (grants no. F65 and W1245), by the Fondation Sciences Mathématiques de Paris, and by Paris Sciences et Lettres. All authors are part of the Amadeus project Hypocoercivity no. 39453PH.

c 2019 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Concerning the second term, we choose the optimal value α = (d -2)/2 in (33), producing the optimal upper bound for δ. It is now straightforward to show

∇u 2 2 , whence the proof of ( 7) is completed by an application of Nash's inequality [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF].

The result ( 6) is shown analogously by using the standard Hardy inequality

instead of (33). This completes the proof of Lemma 2.1.

C.2. Hardy-Nash vs. Caffarelli-Kohn-Nirenberg inequalities. The values for C δ and C δ,η given in Lemma 2.1 cannot be expected to be optimal, since the Hardy and Nash inequalities used in the proof have different optimizing functions.

Here we shall present an alternative proof of ( 6), showing that the optimal value for C δ can be given in terms of the optimal constant of an appropriately chosen Caffarelli-Kohn-Nirenberg inequality of Nash type. We start by rewriting [START_REF] Vázquez | The mathematical theories of diffusion: nonlinear and fractional diffusion[END_REF] with optimal constant C = C CKN as


G Rosaz

B Salem

N Pauc

P Gentile

A Potié

T Baron

Electrical characteristics of a vertically integrated field-effect transistor using non-intentionally doped Si nanowires

In this letter, we report the fabrication and the electrical characterization of Vertical Gate All Around Field-Effect Transistors (GAA-FET) using non intentionally doped Silicon NanoWires (SiNWs) grown by Chemical Vapor Deposition (CVD) using the Vapor-Liquid-Solid (VLS) mechanism as conduction channel. The SiNWs GAA-FET devices exhibited nchannel type semiconductor behavior with a threshold voltage at room temperature around -1.15 V, a high ION/IOFF ratio up to 10 6 with a low IOFF current about 1 pA, a maximum transconductance (gm,max ~ 0.9 µS at VGS = -0.65 V and VDS = 1 V) and a minimum inverse subthreshold slope around 145 mV/decade. In light of these characteristics, these devices can be suitable for high performance, low power consumption components and especially for high density integration in integrated circuits (ICs) interconnections regarding to their 3D architecture.

Introduction

Bottom-up approach has opened a new route for building blocks of future device applications such as field effect transistors, laser diodes, photovoltaic devices [1][2] . The properties of nanowires have been widely studied mainly in planar configuration 3,4 but recently it has been demonstrated that it is possible to use them as the conduction channel of vertical FET with the bulk-nanowire epitaxial interface as a source or drain contact. Different materials have been investigated such as ZnO 5 , InAs 6,7,8,9, and Si 10,11,12,13 . One benefit of vertical integration is the possibility to build a wrapping gate with a cylindrical geometry which enhances the electrostatic gate coupling and improves the control of the gate potential in the wire body compared with planar devices. This configuration allows to build high performance and low power FET combined with a 3D integration providing a very promising way to increase the integration density on microelectronic circuits. Indeed the vertical integration offers a new option to integrate much more devices than in planar approach for the same surface area which is one of the mainstreams of microelectronics technology 14 .

In this letter we report the fabrication and the electrical performance of a non-intentionally doped vertical SiNWs GAA FET using a generic process, which is also described. This process requires non critical lithographic steps and allows the interconnection of many nanowires at the same time. Electrical performances are discussed and technological improvements are proposed in order to obtain better devices characteristics.

Experimental

The starting material for NW growth was <111> -oriented n-type Si wafers (1 m.cm). In order to open windows for the localized deposition of NW catalyst, substrates were patterned using photolithography, resist wastes were cleaned with oxygen plasma at 450 W for 60 s and native oxide was removed by dipping into HF (10%) for 10 s. Gold colloids acting as catalysts for the VLS growth were then deposited onto the patterned substrates by drop casting a droplet of colloidal solution mixed with another droplet of HF (10%) to increase the colloid precipitation onto the surface. After 30 s the substrate was rinsed with deionised water and dried under nitrogen. Finally lift-off was used to remove the resist from the substrates and leave localized gold colloids (Fig. 1. (a)). SiNWs growth was performed in a CVD furnace at 600°C using a 30 mTorr silane partial-pressure with H2 as carrier gas and 100 sccm HCl to inhibit the gold diffusion and 2D growth [START_REF] Oehler | [END_REF] . The as-grown vertically standing <111>-oriented NWs are typically 2.5 µm in length and 100 nm in diameter. Immediately after growth, gold catalyst was removed from the top of the wires using KI: I2 solution followed by a thermal oxidation in a Rapid Thermal Process (RTP) furnace at 800 °C under 2000 sccm of oxygen to create an uniform SiO2 layer leading to a SiO2 shell dielectric layer around 10 nm.

As gate electrode a 70 nm thick Al film was deposited by sputtering. A photoresist was spincoated and a photolithography step was used to define gate pads. The photoresist was then etched-back to define the gate length. Using the resist as a mask, the Al was then etched from the top of the wire by wet etching as shown on Fig. 1(b). The wafer was coated with a polymer layer (Fig. 1 (d)), which was again etched-back to reveal the top of the wires. The drain contact was defined by photolithography, then thermal oxide was removed and the drain was metalized with Ti/Au. Gate contact was reached by a last oxygen plasma etch step using drain contact as a mask (Fig. 1 (c)). Finally, the gate length (LG) of all the devices was around 1.5 µm.

Discussion

Typical output characteristic of a SiNWs GAA FET device is displayed in Fig. 2 (b). The mean diameter of the oxidized wires is about 115 nm. The FET has n-type behavior as verified from the conductance increase when we increase the gate voltage bias (Fig. 2 (a)). We show that the subthreshold slope, which is a key parameter for low power switching applications, is around 145 mV/decade but is still more than twice the theoretical room temperature limit of 60 mV/decade. In fact, the thermal oxide being quite thin the explanation of this high value can be given by a poor Si/SiO2 interface and would be improved by a thermal annealing under forming gas. The ION/IOFF ratio which is the ratio of IDS at current saturation (ION) to IDS at depletion (IOFF) is close to 10 6 (Fig. 2 (a)). We would mention that this ION/IOFF ratio is the highest obtained for vertical GAA FET using un-doped silicon nanowires as a channel elaborated by CVD. Moreover the hysteresis effect during gate voltage sweep being very small gives to the device reproducible electrical characteristics. This hysteresis value does not vary with gate voltage sweep rate contrary to planar devices 16,17 which is also relevant of the efficient passivation of the wires and most of all of their embedding in polymer. Despite of this good passivation this polymer could be the source of a highest hysteresis than air if the surface of the wire was not oxidized. Finally as shown on Fig. 2 (a) the threshold voltage which is around -1.15 V, close to the silicon gap, with a maximum deviation of 15% does not change either with gate voltage sweep direction or rate as previously shown by 13 . Contrary to what has been observed in III/V nanowires 12 the threshold voltage does not vary with VDS which is more suitable for reproducible behavior of the devices.

The transconductance defined as gm=dIDS/dVGS is calculated to be close to 0.9 µS as shown in Fig. 3. The electron mobility is µ=gmLG²/(CoxNVDS), where LG is the gate length, N is the number of nanowires, and Cox is the gate capacitance for a single nanowire. The gate capacitance is expressed as Cox=20oxL/ln(rg/rnw), where 0 is the vacuum permittivity, ox is the dielectric constant of the gate oxide, rg is the inner radius of the gate electrode, and rnw is the nanowire radius. This calculus gives us an electron field-effect mobility from 1 to 11 cm²/V.s with an average value of 7 cm²/V.s depending on the number of connected nanowires which is far from the bulk value (~1350 cm²/V.s). The main reason explaining the small value of µe is probably the high access resistance at the un-gated top part of the nanowires just below the drain contact which is about one third of the total channel length and this effect has already been pointed out by Björk et al 12 and recently simulated by Karmalkar et al 18 . Such a resistive part in our devices could also explain the low current level observed in the ON state. This problem can be solved either using doped nanowires on their top part or using a silicided contact in order to control the length of the un-gated region of the wires. This point is under study.

In summary we have shown that it is possible to build vertical SiNWs based GAA-FET with good electrical characteristics such as a low subthreshold slope of 145mV/decade, a high ION/IOFF ratio up to 10 6 which is the highest value reported for VLS grown undoped silicon nanowires vertical FET. Despite the fact this ratio is remarkably good it is still possible to improve it by playing on the contacts quality in order to obtain better carriers injection in the channel. Our devices exhibited low hysteresis effect demonstrating a good surface passivation of the nanowires. We confirm that improvements should be performed to obtain a better carriers mobility using either doped or silicided contacts in order to reduce the access resistance of the drain and also using forming gas annealing step to improve the Si/SiO2 interface.

Figure 1 :

 1 Figure 1 : Vertical SiNW transistor process flow illustration. (a) Si nanowire localized in a

Figure 2 :

 2 Figure 2 : (a) Drain-source current IDS versus gate voltage VGS at VDS = 1 V. The device exhibits an ION-IOFF current ratio of 10 6 and a threshold voltage of -1.15 V as shown in inset.

Figure 3 :Figure 1 Figure 2

 312 Figure 3 : Transconductance of the device plotted versus the gate voltage for a VDS = 1 V. The

Figure 3

 3 Figure 3


M Syntfeld-Kazuch

D Szawlowski

Wolski

Pawel Sibczynski

Andrzej Dziedzic

Krystian Grodzicki

Joanna Iwanowska-Hanke

Zuzanna Mianowska

Marek Moszyĕski

Lukasz Swiderski

Agnieszka Syntfeld-Kaīuch

Marek Szawáowski

Dariusz Wolski

Aleksandra Doáċbska

Wojciech Gċsikowski

Jerzy Godlewski

Frédérick Carrel

Amélie Grabowski

Frederic Laine

Guillaume Sannie

Adrien Sari

Sandra Moretto

Cristiano Fontana

Felix Pino

Bertrand Perot

Alix Sardet

Cedric Carasco

C-BORD -an overview of efficient toolbox for high-volume freight inspection

C-BORD -an overview of efficient toolbox for high-volume freight inspection

I. INTRODUCTION

ETECTION of various threats or illicit materials on country borders plays important role in Homeland Security over the world. Every year the amount of transported goods is increasing, thus, effective and efficient solutions for cargo containers inspection are required in a day-to-day operations by Custom Offices. The higher throughput, increased sensitivity, improved image quality are highly demanded by end-users in order to detect possible illegal goods and, finally, execute a seizure. The non-intrusive inspection (NII) using new or upgraded technology would avoid the cargo physical examination, resulting in time lost and cost increase. Unfortunately, the NII used today cannot cope with all targets under all circumstances with equivalent efficiency.

The whole system, as a complete toolbox, would be able to efficiently detect the illegal and dangerous goods, which Custom Offices can face with, such as:

• Tobacco, alcohol, drugs, chemicals • Explosives, weapons • Radioactive and nuclear materials.

In the frame of the project not only the devices are developed, but also end-users requirements will be taken into account and tests in real conditions will be performed. The systems will be tested in three locations, where local Custom Offices requirements differ from each other:

• Large seaport -Rotterdam -stationary solutions • Medium seaport -Deepwater Container Terminal (DCT) GdaĔsk -relocatable solutions • Land Border -Hungary -mobile solutions. In this paper, we would like to provide general overview of the devices developed in the frame of C-BORD project [START_REF]C-BORD Project -Effective Container Inspection at BORDer Control Points[END_REF] , supported by EU Horizon 2020 program.

II. TECHNOLOGIES DEVELOPED IN THE FRAME OF C-BORD PROJECT

A. Radiation Portal Monitors

The first technology is based on advanced Radiation Portal Monitors (RPM), designed and commercially available from Symetrica LTD [START_REF]Symetrica LTD[END_REF]. The detectors, based on organic scintillators, will be used for gamma ray detection. Specialized software is used for classifying and identifying the radiation source. High Speed Naturally Occurring Radioactive Materials (NORM) management was implemented to the RPM systems, which can be provided as a fixed, relocatable or mobile solution.

B. X-ray system

The Smiths Detection HCVM-T ViZual 6032 towable X-ray system, which is a commercially available device [START_REF] Detection | HCVM-T towable X-ray system[END_REF], will be upgraded in the frame of C-BORD project. The HCVM-T trailer is equipped with dual energy linear accelerator, currently offering the throughput of up to 25 trucks per hour in scan mode and up to 150 trucks per hour in pass through mode. Main improvement will cover the classification of D organic materials as well as global image quality. Moreover, the modernization will also cover the stationary solution, working currently at Rotterdam test site.

C. Rapidly relocatable tagged neutron inspection system (RRTNIS)

The RRTNIS system is based on neutron generator with associated particles, allowing for choosing the coincidence with neutrons emitted in a specific direction in form of cone. Precursor of this device was primarily developed in EURITRACK project [START_REF] Perot | Development of the EURITRACK tagged neutron inspection system[END_REF]. Similarly, the improved system will be used for searching tobacco, drugs and chemicals, performing the identification of chemical elements composition. The intention is to improve efficiency of the system and optimize the system weight in comparison with the EURITRACK project. Due to use of 14 MeV neutrons and the fact that the space at border crossing or seaport is often limited, well optimized PE and Pb shielding were introduced.

The device design allows to dismantle in 48 hours and transport to another location after placing in a standard transportation container. The RRTNIS data analysis principle is presented in Figure 2.

. Figure 2. The RRTNIS data analysis principle.

Figure 3. Fully operational detectors and data readout for photofission system.

Detectors were designed at NCBJ, the electronics were produced by CAEN.

D. Photofission system

The photofission system for detection of special nuclear materials (SNM) and fertile materials, will be able to detect three main photofission signatures [START_REF] Gozani | Fission Signatures for Nuclear Material Detection[END_REF]:

• High energy delayed gamma-rays detection with use of large plastic scintillators [START_REF] Carrel | Detection of highenergy delayed gammas for nuclear waste packages characterization[END_REF], [START_REF] Sibczynski | Decay chains and photofission investigation based on nuclear spectroscopy of highly enriched uranium sample[END_REF],

• Delayed neutrons [START_REF] Gmar | Assessment of actinide mass embedded in large concrete waste packages by photon interrogation and photofission[END_REF] • Prompt neutrons by means of Threshold Activation Detection (TAD) [START_REF] Gozani | Neutron threshold activation detectors (TAD) for the detection of fissions[END_REF]-[12] Moreover, a system for detection of early-delayed gamma rays is also developed, which will be useful for immediate classification of smuggled nuclear materials. For prompt neutron detection CaF2 detectors, designed at NCBJ, was tested at CEA LIST SAPHIR facility and will be used during test at Rotterdam site. Detailed study of the CaF2 scintillator as the TAD detector was done in [START_REF] Sibczynski | New perspectives for undoped CaF2 scintillator as a threshold activation neutron detector[END_REF]. The fully operational detectors with associated electronics from CAEN are shown in Figure 3. The detection is based on neutron activation through 19 F(n,Į) 16 N reaction in the scintillator medium, resulting in emission of characteristic radiation -both gamma rays and beta particles -up to 10.4 MeV.

E. Evaporation based detectors (SNIFFER)

The evaporation based detection system [START_REF] Persaud | Electronic Nose'-New Condition Monitoring Devices for Environmental Applications[END_REF], complementary to X-ray system, sucks vapor from the inspected container in order to find characteristic agents ascribed to tobacco, drugs or explosives or other illicit chemicals. The picture of a part of the analyzing device is presented in Figure 4.

III. SUMMARY

The document presents the overview of the systems developed in the C-BORD project. All these systems will be integrated into common software, providing information required by end-user from each system in a one frame. Finally, all devices will be tested at real conditions in a field trial at three border crossing -two seaports and one land border crossing. The main achievement of the C-BORD project is to provide complementary toolbox for high volume freight NII, resulting in better monitoring of goods transported to Europe in the effective and cost efficient way, and further, to increase the safety level in Europe..

Figure 1 .

 1 Figure 1. The HCVM-T towable X-ray system upgraded in C-BORD project.

Figure 4 .

 4 Figure 4. A part of the analyzing device of the SNIFFER unit.

  Manuscript received November 10, 2017. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 653323. Disclaimer: This text reflects only the author's views and the Commission is not liable for any use that may be made of the information contained therein.

	P. Sibczynski, A. Dziedzic, K. Grodzicki, J. Iwanowska-Hanke, Z.
	Mianowska, M. MoszyĔski, L. Swiderski, A. Syntfeld-KaĪuch, M.
	Szawáowski, D. Wolski are with the National Centre for Nuclear Research
	(NCBJ), Otwock, 05-400 Poland (telephone: +48 22 273 1440, e-mail:
	pawel.sibczynski@ncbj.gov.pl, pawel.sibczynski@gmail.com)
	A. DoáĊbska, W. GĊsikowski, J. Godlewski are with the Custom Office
	Gdynia, ul. Trzy Lipy 3, GdaĔsk, Poland
	F. Carrel, A. Grabowski, F. Laine, G. Sannie, A. Sari are with the CEA
	LIST, F-91191 Gif-Sur-Yvette, France
	S. Moretto, C. Fontana, F. Pino are with the Istituto Nazionale di Fisica
	Nucleare, Via Marzolo 8, 35131 Padova, Italy
	B. Perot, A. Sardet, C. Carasco are with the CEA, DEN, Cadarache,
	Nuclear Measurement Laboratory, F-13108 Saint-Paul-lez-Durance, France

ACKNOWLEDGMENT

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 653323. Disclaimer: This text reflects only the author's views and the Commission is not liable for any use that may be made of the information contained therein.


Yves Dauvilliers
email: ydauvilliers@yahoo.fr

Philippe Peigneux

Yves Pr

Dauvilliers

Ictal SPECT in patients with rapid eye movement sleep behaviour disorder

Sir,

We have read with great interest the article entitled 'Ictal SPECT in patients with rapid eye movement sleep behaviour disorder' published March 2015 by [START_REF] Mayer | Ictal SPECT in patients with rapid eye movement sleep behaviour disorder[END_REF] in Brain. In this article, the authors evidence brain regions activated during an episode of rapid eye movement (REM) sleep behaviour disorder (RBD) using ictal single photon emission tomography (SPECT).Four patients were included in this study: one with idiopathic RBD, one with RBD and Parkinson's disease, and two with RBD and narcolepsy. Ictal SPECT in an RBD episode as compared to wakefulness revealed RBD-related activations in the bilateral premotor areas, the interhemispheric cleft, the periaqueductal area, the dorsal and ventral pons and the anterior lobe of the cerebellum in all patients, thus bypassing the basal ganglia. The authors also claim this to be the first study to report results with ictal SPECT in patients with idiopathic RBD, or associated with Parkinson's disease or narcolepsy.We here dispute this claim.

Indeed, in 2011 we published the first SPECT recording during a well-documented and controlled RBD episode associated with multiple system atrophy, in which we demonstrated a selectively increased perfusion in the supplementary motor area [START_REF] Dauvilliers | Increased perfusion in supplementary motor area during a REM sleep behaviour episode[END_REF]. We used the same technique as [START_REF] Mayer | Ictal SPECT in patients with rapid eye movement sleep behaviour disorder[END_REF], i.e. ictal SPECT immediately (14 s) after the onset of a long (2-min duration) and complex RBD episode in a 54-year-old right-handed man with a 3-year history of probable multiple system atrophy-cerebellar (predominance of cerebellar ataxia) [START_REF] Dauvilliers | Increased perfusion in supplementary motor area during a REM sleep behaviour episode[END_REF]. Dream-enacting behaviours were particularly frequent and violent in this patient, and confirmed by video-polysomnography that documented profound REM sleep disturbances with only 8% of REM sleep with atonia. Additionally, and at variance with the study of Mayer et al., we scanned in similar conditions two healthy control subjects matched for age and gender in both normal REM sleep (i.e. without RBD) and during wakefulness. In these controls, we found no perfusion changes in the supplementary motor area in REM sleep as compared to wakefulness. However, a higher perfusion was found in the anterior cingulate cortex both in the RBD and normal REM sleep states, a finding mostly related to the decreased perfusion in this area during wakefulness time. No perfusion changes were found in the brainstem. To sum up, both our own and the recent Mayer et al. studies have underlined a neuralfingerprint (i.e. the activation of the cortical premotor supplementary area) associated with the generation of movements during an RBD episode, whether idiopathic or comorbid with Parkinson's disease, multiple system atrophy or narcolepsy [START_REF] Dauvilliers | Increased perfusion in supplementary motor area during a REM sleep behaviour episode[END_REF][START_REF] Mayer | Ictal SPECT in patients with rapid eye movement sleep behaviour disorder[END_REF]. Noticeably, another study previously reported decreased perfusion in the frontal lobe and in the pons in patients with idiopathic RBD scanned during symptomatic, but not EEG-controlled RBD episodes, and without providing details on the methodology used [START_REF] Shirakawa | Study of image findings in rapid eye movement sleep behavioural disorder[END_REF].

Other SPECT studies recording brain activity at wake time in idiopathic RBD reported increased perfusion in the pons, putamen and right hippocampus, together with decreased perfusion in fronto-temporo-parietal cortices [START_REF] Mazza | Assessing whole brain perfusion changes in patients with REM sleep behavior disorder[END_REF]. Moreover, abnormal perfusion in the hippocampus was a predictor for the conversion risk to neurodegenerative disorders such as Parkinson's disease in these patients [START_REF] Dang-Vu | Hippocampal perfusion predicts impending neurodegeneration in REM sleep behavior disorder[END_REF]. Increased activity in a Parkinson's disease-related metabolic network was found in idiopathic RBD, especially in patients who converted to a progressive neurodegenerative syndrome during follow-up [START_REF] Holtbernd | Abnormal metabolic network activity in REM sleep behavior disorder[END_REF]. Finally, a specific RBDrelated metabolic network was recently characterized during wakefulness using PET, showing increased activity in the pons, thalamus, medial frontal and sensorimotor areas, hippocampus, supramarginal and inferior temporal gyri, and posterior cerebellum, and decreased activity in occipital and superior temporal regions [START_REF] Wu | Consistent abnormalities in metabolic network activity in idiopathic rapid eye movement sleep behaviour disorder[END_REF]. These abnormal metabolic networks provided valuable markers of idiopathic RBD during wakefulness to identify patients at higher risk to develop neurodegenerative parkinsonism.

The current [START_REF] Mayer | Ictal SPECT in patients with rapid eye movement sleep behaviour disorder[END_REF] and our preceding [START_REF] Dauvilliers | Increased perfusion in supplementary motor area during a REM sleep behaviour episode[END_REF] studies have emphasized a common motor pathway in RBD. They also both localized the motor generator responsible for dream-enacting behaviour in the supplementary motor area by bypassing the basal ganglia. Although the muscle atonia during REM sleep is induced by brainstem centres and more precisely by the sublaterodorsal tegmentalnucleus REM-on neurons,recent evidence also suggests thatdescending projections ofneurons from the motor cortex are involved in phasic activation (i.e. muscle twitches) and potentially complex and severe movements (i.e. RBD) during REM sleep [START_REF] Beau | Neurotransmitter phenotypes of descending systems in the rat lumbar spinal cord[END_REF][START_REF] Luppi | New aspects in the pathophysiology of rapid eye movement sleep behavior disorder: the potential role of glutamate, gamma-aminobutyric acid, and glycine[END_REF]. The supplementary motor area is a key structure for behavioural planning, coordination, execution and speech production, and may subtend complex actions that are under internal control, such as the performance for a sequence of movements from memory, as opposed to movements guided by visualcues [START_REF] Picard | Activation of the supplementary motor area (SMA) during performance of visually guided movements[END_REF].

To conclude, involvement of the supplementary motor area as one of the key neural generators for abnormal movement production during RBD, whether comorbid or not, is turning out to be a major finding. This highlights a functional role of high-order motor cortical areas in generating movements during enacted dreams in RBD, when tonus inhibition is no longer efficient, and we are glad to see here with this new publication a confirmation of our previously published results [START_REF] Dauvilliers | Increased perfusion in supplementary motor area during a REM sleep behaviour episode[END_REF].


Diego Pescarini°

Giulia Donzelli

Concealed pseudo-clefts? Evidence from a Lombard dialect *

Keywords: clefts, pseudo-clefts, northern Italian dialects, interrogatives

This paper focuses on the syntax of clefts in the Lombard dialect of Comun Nuovo (Bergamo). In this dialect, clefts are highly constrained (in particular, they are ungrammatical in questions) and, in the contexts where clefts and pseudo-clefts alternate, the distinction between the two is often blurred. We argue that Comunuovese clefts are better analysed as concealed pseudo-clefts (Paul 2001 a.o.).

Introduction

The paper investigates the syntax of clefts and pseudo-clefts in the dialect of Comun Nuovo (hereafter, CN). 1 Since in CN clefts are highly constrained, we argue that they are better analysed as (a kind of) pseudo-clefts (in the spirit of [START_REF] Percus | Prying open the cleft[END_REF]Paul 2001 a.o.).

The structure of the paper is the following: §2 introduces a simplified taxonomy of clefts and pseudo-clefts and, through a comparison of CN and Italian data, highlights some peculiarities of CN clefts; §3 deals with interrogation in CN, while §4 focuses on cleft interrogatives; §5 elaborates on some theoretical consequences; §6 concludes.

Types of clefts

Clefts are focus constructions in which the copula introduces a focalised XP (on Italian, see [START_REF] Sornicola | It-clefts and wh-clefts: two awkward sentence types[END_REF][START_REF] Salvi | Le frasi copulative[END_REF]. Following [START_REF] Belletti | The CP of clefts[END_REF], we assume that the XP is extracted from a (small) CP, see (1). Pseudo-clefts, by contrast, are copular constructions in which the subordinate clause is a Relative Clause (RC), either headless or headed. In the former case, we obtain so-called wh-(pseudo)-clefts of the type what he likes more is money (in Italian, wh-pseudo-clefts are rare as headless RCs are seldom introduced by wh elements). In the latter case, see (2), the RC is introduced by a D element or a generic noun, e.g. the thing that, the person that, etc., giving rise to socalled th-(pseudo)-clefts [START_REF] Collins | Cleft and Pseudo-cleft Constructions in English[END_REF].

( In what follows, we focus on the comparison between clefts and thpseudo-clefts. It is worth noting that in many Romance languages the distinction between the two is almost blurred because both may exhibit the same word order, e.g. copula XP CP/RC. Despite their similarity, however, clefts and pseudo-clefts differ under several respects. First, in Italian, pseudoclefts are not necessarily corrective and, in fact, can be used as answers, see (3a). By contrast, clefts often yield a corrective interpretationsave for cases in which the subject is focalisedand, therefore, are slightly degraded as answers to a question, see (3b):

(3) -chi hai salutato?

'who did you greet The ban on PP-clefts in CN is quite puzzling under the usual focus-analysis of clefts, in which an XP is focus-fronted to an A' position, whichby definitionis supposed to be category-neutral.

The third puzzle about CN clefts regards interrogatives, as it turns out that cleft interrogatives are often ruled out in CN, while pseudo-clefts are always grammatical. This restriction is rather surprising as clefting is normally regarded as a common interrogation strategy in northern Italo-Romance (Poletto & Vanelli 1997 a.o.).

(7) a Chi é-l *(chel) che (a)l dorma?

Who is=it the.one that he= sleeps b Chi é-l *(chel) che t-é vest?

Who is=it the.one that you=have seen

The following sections illustrate some properties of CN interrogatives and focus on the interplay of clefting and interrogation.

An aside on interrogatives in CN

CN interrogatives are characterised by residual subject clitic inversion. Inversion is forbidden in yes/no questions and, in wh questions, is restricted to present-tense clauses with a deictic temporal interpretation such as (8)a. Conversely, enclisis cannot co-occur with present forms with a futurate or habitual interpretation such as (8)b or with past/future tenses, see (8)c.

(8) a ndo core-l? where running=he 'where is he running?' b ndo al cor a nedal/töcc i martedè sira? where he= runs at Christmas/every Tuesday evening 'where does he run?' c ndo al-ha corìt/corerà? where he=has run/will.run 'where did/will he run?'

The second main feature of CN interrogatives is the occurrence of different classes of wh elements, each exhibiting a peculiar syntactic behaviour (see [START_REF] Manzini | wh-in situ & wh-doubling in Northern Italian Varieties: Against Remnant Movement Linguistic Analysis[END_REF] The interplay between clitic inversion, in/ex situ placement, and various kinds of wh items gives rise to several patterns of interrogation that, for space limits, cannot be discussed here (see [START_REF] Donzelli | Standard and special questions in Comunuovese. Talk given at Romance Interrogatives[END_REF].

Cleft interrogatives

This section deals with the distribution of clefts among different types of interrogative clauses. For the sake of clarity, we distinguish three types of interrogatives: what/who interrogatives, where a bare DP is interrogated; temporal interrogatives, where a Measure Phrase (MP) is interrogated, and other types of interrogatives.

who/what interrogatives

CN permits the interrogation of pseudo-clefts, whereas cleft interrogatives are not possible. It is worth noting that pseudo-clefts do not behave like plain wh interrogatives. First, they do not allow wh in situ and always exhibit inversion, even if the fronted wh element is not clitic, cf. cos in (9)c. With past or future tenses, where inversion is not permitted, pseudo-clefts are degraded, see ( 11).

(10) Chi *l'è/é-l chèl che l è dre a durmì? Who he=is/is=he that that he=is sleeping (11) ? Cosa l éra chèl che ta séret dre a mangià? What it=was that that you= were eating

Temporal interrogatives

Temporal clefts exhibit a peculiar behaviour as the temporal Measure Phrase can occur as either a DP or a PP [START_REF] Benincà | Sono tre ore che ti aspetto[END_REF]). In the former case, the copula may agree in number with the MP. In these respects, Italian and CN do not differ, although it is worth noting that in CN, as well as in most northern Italian dialects, the third and sixth person of the verb are identical; number agreement is marked on the subject clitic: (a)l (sg) vs i/j (pl), see (12)b:

(12) a l'è (da) tre ure che ta spete/so dre a spetat it=is (for) three hours that you= I.wait/have.been.waiting b a j'è (*da) tre ure che ta spete a=they=are (for) three hours that you= I.have.been.waiting Simple wh elements such as quand(o) 'when' and quat 'how long' always exhibit clitic inversion, even in the contexts in which inversion is normally banned, i.e. with past/future tenses, cf. (13)c:

(13) a Quand é-l/*l è che ta egnet a troam? When is=it/it=is that you= come to meet=me b Quat é-l/*a l è che ta lauret in svisera? How.long is=it/a=it=is that you= work in Switzerland c Quat ere-l/*a l era che ta lauraet in svisera? How.long was=it/a=it=was that you= worked in Switzerland Conversely, complex wh elements such as quace agn 'how many years' do not trigger inversion.

Other interrogatives

The remaining wh elements, e.g. when, how, why, etc. cannot occur in clefts.

The following examples illustrate the contrast between CN and Italian:

Summary and theoretical implications

The following table summarises the distribution of clefts in CN. Recall that, although clefts are permitted in declarative clauses, they do not yield a corrective interpretation and do not allow the focalisation of PPs. Temporal clefts, by contrast, are always permitted, even in questions. It is worth noting that the behaviour of inversion sets apart temporal clefts from other types of clefts. The occurrence of inversion with any tense in combination with (simple) temporal wh elements points towards a separate analysis of temporal pseudo-clefts. In a nutshell, we are going to argue that temporal clefts are in fact the only true clefts of CN, while other prima facie clefts are better analysed as concealed pseudo-clefts.

Leaving temporal clefts aside, the data introduced so far challenge an analysis of clefts in terms of focus-movement. If clefting was a focalisation strategy, all kinds of XP would be expected to occur in clefts and no declarative/interrogative asymmetry should occur, contra evidence. To account for the observed restrictions, we argue for an alternative analysis, according to which in some languages clefts and pseudo-clefts have the same structure (Paul 2001 a.o.; see also [START_REF] Percus | Prying open the cleft[END_REF].

Let us assume that pseudo-clefts are equative copular constructions, i.e. a Small Clause (Heycock & Kroch 1999: 381-382) headed by an equative head (=) and taking a headed Relative Clause as its complement.

(17) [SC DP = [RC DP …]]

The equative head establishes an identity relation between the subject of the SC and the head of the RC, e.g. ( 18) è [SC Luca = [RC quello che ho baciato quello ]] 'Luca is the boy who I kissed'

The equative relation holds if and only if the subject of the SC and the head of the RC have the same categorial features, thus ruling out cases like ( 19):

(19) [SC *PP = [RC DP …]]
This explains why PPs cannot occur in pseudo-clefts with a specificational reading (recall that, when locative PPs are allowed in a copular construction as in (20)c, they can have only a locative interpretation, i.e. 'the place is in Milan', and not 'the Place is Milan', see fn 2):

(20) a *è con Luca il ragazzo con cui mi sposo it.is with L.

the boy that I marry b *è a matita il modo in cui disegno it.is with pencil the way in which I draw c *è a Milano il posto in cui vado it.is in Milan the place where I go 'the place were I go is Milan' With this in mind, let us suppose that in CN clefts are concealed pseudoclefts: they do not result from extraction from a (small) CP à la Belletti, see (1), but from an equative SC like (17). Under this analysis, we can easily account for the ungrammaticality of PP clefts, which in CN are as ungrammatical as pseudoclefts because of the constraint in (19) (the same holds for wh elements corresponding to PP arguments/adjuncts, cf. §4.3).

By the same token, prima facie clefts such as ( 21) are therefore supposed to derive from a copular structure like ( 22), once the head of the RC (chel 'that') is deleted:

(21) A l'è ol Luca che l'ha mangiat la turta a=it=is the L. that he=ate the cake (22) A l'è [SC ol Luca = [RC (chel) che l'ha mangiat la turta]] a=it=is the L. that that ate the cake To derive ( 21) from ( 22), one has to resort to a mechanism of deletion (not dissimilar from the one invoked for the analysis of relative clauses since Chomsky 1977), in which the head of the RC is deleted under identity with the subject of the SC (for an alternative machinery, see [START_REF] Percus | Prying open the cleft[END_REF].

However, following this analysis, one wonders why deletion does not take place in interrogatives, where only pseudo-clefts are allowed (the relevant examples are repeated below for the sake of clarity). We contend that deletion is blocked as the featural specifications of the equated XPs do not correspond: if the subject of the SC bears a [+WH] specification, we hypothesise that deletion cannot take place, thus giving rise to the observed asymmetry between declaratives, where pseudo-clefts are finally turned into prima facie clefts, and interrogatives, where the head of the RC cannot be omitted:

(24) Chi é-l [SC chi [RC *(chèl) che l'è dre a durmì]]?
who is=he the.one that he=is sleeping

The above solution might sound rather ad hoc. Nonetheless, it is worth noting that similar declarative/interrogative asymmetries are found in Italian as well and, to the best of our knowledge, have remained unnoticed so far. Let us start with the minimal pair in (25): the two sentences have the same meaning and the same information structure; in the former, the PP is extracted from the subordinate clause, whereas in the latter a DP element co-occurs along with an oblique wh element (con cui 'with whom'). ( 25 (28) Chi è _ quello con cui vuoi scappare? Who is the.one with whom you want to escape

In the light of the analysis of CN, we argue that the sentences in ( 25) and ( 27

Conclusions

This contribution has focused on the syntax of clefts in the Lombard dialect of Comun Nuovo (Bergamo). We have shown that in this dialect clefts are highly constrained and, in the contexts where clefts and pseudo-clefts alternate, the distinction between the two is often blurred.

In particular, clefts are ungrammatical in questions, save for temporal clefts which, however, exhibit a puzzling behaviour with respect to inversion (inversion normally occurs with certain types of wh elements and with deictic present tense).

Leaving temporal clefts aside, we contend that CN clefts are better analysed as concealed pseudo-clefts, an analysis put forth for other non-European languages (see Paul 2001 and references therein). We do not claim that the usual raising analysis of clefts must be always replaced by a pseudocleft analysis, but notice that the divide between clefts and pseudo-clefts is far less straightforward than usually thought and that the boundary between the two analyses is ultimately an empirical matter.

  for similar data on Lombard dialects). For instance, the wh corresponding to 'what' has three possible forms: [sa], [ˈkɔza], and [koˈzɛ]. [sa] belongs to the class of clitic wh elements (in the terms of Poletto 2000; Poletto & Pollock 2006; but see Manzini & Savoia 2011 and Manzini 2014); clitic wh elements in CN can co-occur with subject clitic inversion; [ˈkɔza] never triggers inversion and occurs either in or ex situ; [koˈzɛ], by contrast, cannot be fronted.

  (9) a Chi é-l *(chèl) che l è dre a durmì? who-S who is=he that that he=is sleeping b Chi é-l *(chèl) che ta set dre ad ardà? who-O who is=he that that you= are watching c Cos é-l *(chèl) che ta manget? what-O what is=it that that you= eat

(

  23) a Chi é-l *(chèl) che l è dre a durmì?

  ) are not of the same kind: (25)a and (27)a are fully-fledged clefts, whereas (25)b and (27)b are concealed pseudo-clefts of the CN type. Underlyingly, they are copular constructions in which the head of the RC has been deleted: (29) È [SC Giorgio = [RC quello con cui voglio scappare]] Like in CN, the head of the RC cannot be deleted if the subject of the SC is interrogated: (30) Chi è [SC _ = [RC *(quello) con cui voglio scappare]] This is why only fully-fledged pseudo-clefts such as (25)a can occur in questions, while concealed pseudo-clefts such as (25)b cannot.

  1) a It is money [small CP that he wants money] b è il denaro [small CP che vuole il denaro] (2) a Money is [headed RC the thing he wants the thing] b Il denaro è [headed RC ciò che vuole ciò]

  Interestingly, the slight asymmetry noticed by Benincà becomes a full contrast once the sentences in (25) are turned into questions: the former remains grammatical, while the latter results in severe ungrammaticality.

	(27) a con chi	è _ che vuoi scappare?
	with whom is	that you want to escape
	b *Chi è _ con cui	vuoi scappare?
	Who is	with whom you want to escape
	Again, (27)b is fine if we resort to a pseudo-cleft construction:
	) a È	con Giorgio che	voglio scappare
	it.is with G.		that	I want to escape
	b ? È	Giorgio		con cui	voglio scappare.
	it.is G.			with whom I want to escape
	Benincà 1978: fn. 2 points out that sentences like (25)b are slightly
	degraded and, instead of (25)b, a pseudo-cleft such as (26) is normally
	preferred.
	(26) È	Giorgio quello con cui	voglio scappare
	it.is G.	the.one with whom I want to escape

In this respect, locative PPs are more complicated than other PPs since they can in fact occur in a copular construction along with a RC, e.g. (i) È (a) Milano il posto che amo Is (in) Milan the place that I.love However, (i) has only a locative interpretation, i.e. 'the place is in Milan', not the identificational interpretation characterising pseudo-clefts, i.e. 'the place is Milan'.

The behaviour of the particle a in Lomberd dialects is a well-known puzzle that cannot be addressed here for space limits. For the sake of clarity, in the glosses we try to distinguish the particle a, which occurs before subject clitics under certain pragmatic conditions, from the prosthetic vowel a-that syllabifies the 3 rd person subject clitic. Crucially, the two are in complementary distribution (*a al); our glosses reflect the intuition of speakers. The same particle, which arguably derives from a pronominal form (Lat. EGO 'I'?) is attested in several northern Italian dialects with various functions/interpretations, some of which are related to information structure; for a syntactic analysis of the particle a in Paduan, see[START_REF] Benincà | II clitico a nel dialetto padovano[END_REF]Benincà /1994).