Gilles Lebeau

DPMMS Iván Án Moyano

SPECTRAL INEQUALITIES FOR THE SCHR ÖDINGER OPERATOR

In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in R d

where ∆g is the Laplace-Beltrami operator with respect to an analytic metric g, which is a perturbation of the Euclidean metric, and V (x) a real valued analytic potential vanishing at infinity.

In this paper we prove spectral inequalities for the Schrödinger operator

H g,V := -∆ g + V (x), in R d , d ≥ 1.
Our approach relies on interpolation inequalities, in the spirit of the works [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF], but adapted to the unbounded case. We will use spectral projectors, holomorphic extension arguments, and suitable interpolation estimates for holomorphic functions.

In the case V = 0, we recover some classical quantifications of the uncertainty principle dues to Zigmund [23, pp.202-208], Logvenenko and Svereda [START_REF] Logvinenko | Equivalent norms in spaces of entire functions fo exponential type[END_REF] and Kovrojkine [START_REF] Kovrojkine | Some results related to the Logvinenko-Sereda inequality[END_REF][START_REF] Kovrojkine | The Uncertainty Principle for relatively dense sets and lacunary spectra[END_REF], among other references (see Section 2.1.2 for further details).

1.1. Geometric conditions for the observability sets. Given R > 0 and x ∈ R d , we denote by B g R (x) the ball of radius R, with respect to the metric g centered in x. When x = 0, we simply write B g R and if moreover g = Id, one can simply write B R as usual. We shall work with Lebesgue measurable sets ω ⊂ R d satisfying the condition

(1.1) ∃R, δ > 0 such that mes(ω ∩ B g R (x)) mes(B g R (x)) ≥ δ, ∀x ∈ R d .
Since in the present work the metric g will be asymptotically the flat metric, (1.1) can be replaced by the same condition with the Euclidean metric:

(1.2) ∃R, δ > 0, such that inf

x∈R d mes {t ∈ ω, |x -t| < R} ≥ δ.
1.2. Main result: spectral inequality for the Schrödinger operator on R d . Let g = g ij (x) be a Riemannian metric in R d and consider the Laplace-Beltrami operator associated to g, i.e., (1.3) ∆ g u = 1 √ det g div det gg -1 ∇u , where g -1 (x) = (g ij )(x) denotes as usual the inverse metric of g. Given V = V (x) a real-valued potential function, one defines the associated Schrödinger operator (1.4) H g,V := -∆ g + V (x), on D(H g,V ), where D(H g,V ) = u ∈ L 2 (R d ; det g dx); H g,V (u) ∈ L 2 (R d ; det g dx) .

We will assume that the metric g and the real-valued potential V satisfy the following hypothesis: the metric g is analytic, of the form g = Id + g with lim |x|→∞ g(x) = 0, (1.5) the potential V is analytic, real valued, with lim |x|→∞ V (x) = 0, (1.6) ∃a > 0 such that g and V extend holomorphically in the (1.7) complex open set defined by U a := z ∈ C d ; | Im(z)| < a , ∃ǫ ∈ (0, 1) such that g and V satisfy for |α| ≤ 2 (1.8)

|∂ α g(z)| + |∂ α V (z)| ≤ C α (1 + |z|) -ǫ-|α| , ∀z ∈ U a .
Under these hypothesis one can check (see Proposition 4.46) that the Schrödinger operator H g,V is an unbounded self-adjoint operator. In section 4 we will recall some basic facts on the functional calculus of self-adjoint operators. In particular, the spectral projectors Π µ (g, V ) are defined in (4.55). In this article we will prove that the family of specral projectors Π µ (g, V ) enjoy a spectral inequality, i.e., an observability inequality on a set ω ⊂ R d for low frequencies as long as the observability set satisfies (1.1).

Given µ ∈ R, let us introduce (1.9) µ

1 2 ± := √ µ, µ ≥ 0, µ 1 2
± := ±i |µ|, µ < 0. Our main result is the following: THEOREM 1.1. Let ω ⊂ R be a measurable set satisfying the geometric condition (1.2) and let (g, V ) satisfy hypothesis (1.5)-(1.8). Then, there exist constants A = A(ω, g, V ) > 0 and C = C(ω, g, V ) > 0 such that for all µ ∈ R and all f ∈ L 2 (R d ), one has

(1.10) Π µ (g, V )f L 2 (R d ; √ det g dx) ≤ A |e Cµ 1 2 ± | Π µ (g, V )f L 2 (ω; √ det g dx) .
1.3. A special case: spectral inequality for the Laplacian operator in R d . When g = Id and V = 0 as H 0 = -∆ x is the usual flat Laplacian, it is wellknown that that σ(Id, 0) = [0, ∞) is a purely absolutely continuous spectrum. Furthermore, the spectral projectors are explicitly determined through the Fourier transform, i.e., Π(Id, 0

) µ f = 1 (2π) d |ξ|<µ f (ξ)e ix•ξ dx,
recalling that the classical Fourier transform is defined by

ĝ(ξ) := R d g(x)e -ix•ξ dx, ∀ξ ∈ R d , ∀g ∈ L 2 (R d ).
As a result, one can recast (1.10) as the following familiar spectral inequality.

THEOREM 1.2. Let ω ⊂ R be a Lebesgue measurable set satisfying the geometric condition (1.2). Then, there exist constants

A = A(ω) > 0, C = C(ω) > 0 such that for all µ ∈ R + and all f ∈ L 2 (R d ), one has (1.11) f L 2 (R) ≤ A(ω)e C(ω)µ f L 2 (ω) , whenever supp f ⊂ B µ .

Context of our results and previous works

2.1. Spectral inequalities, Logvenenko-Sereda inequalities and the uncertainty principle.

2.1.1. Spectral Inequalities. Given a compact Riemannian manifold M equipped with a metric g the spectral inequalities for V = 0 have been introduced by the first author and D. Jerison in [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF] and also in [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF], among other works (see [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] and the references therein).

In the non-compact case when M = R d with the usual Euclidean metric, the methods of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] have been extended in [START_REF] Rousseau | Null-controllability of the Kolmogorov equation in the whole phase space[END_REF] in such a way that inequality (1.11) holds whenever ω ⊂ R d is an open set satisfying the geometric condition

(2.12) ∃R > 0, δ > 0 s.t. ∀y ∈ R d , ∃y ′ ∈ R d with B(y ′ , R) ⊂ ω and d(y, y ′ ) < δ.
This condition is thus proven to be sufficient for the null-controllability of the heat equation to hold, as well as for some hypoellipitic equations, like the Kolmogorov equation, arising in kinetic theory [START_REF] Moyano | Contrôlabilité de quelques équations cinétiques collisionnelles et non collisionnelles : Fokker-Planck et Vlasov-Navier-Stokes[END_REF]. This condition is however far from being necessary (see Section 2.2 for details).

2.1.2. Quantification of the uncertainity principle and the Logvenenko-Sereda inequality. The classical uncertainty principle in harmonic analysis accounts for the fact that a function cannot be localised both in space and in the frequency variable (cf. [17, Prop. 10.2, p. 270]) :

∃C > 0, R x 2 |f (x)| 2 dx 1 2 R ξ 2 | f (ξ)| 2 dξ 1 2 ≥ C f 2 L 2 (R) , ∀f ∈ L 2 (R).
A version of this inequality due to Amrein and Berthier (cf. [START_REF] Amrein | On support properties of L p functions and their Fourier transforms[END_REF]) guarantees that

R\ω |f (x)| 2 dx + R\Σ | f (ξ)| 2 dξ ≥ C f 2 L 2 (R) , mes(ω) + mes(Σ) < ∞.
In the aim of giving more quantitative versions of the uncertainty principle in R d , one can find very significant literature (see [START_REF] Kovrojkine | The Uncertainty Principle for relatively dense sets and lacunary spectra[END_REF] and the references therein for further details). In particular, Logvinenko and Sereda proved in [START_REF] Logvinenko | Equivalent norms in spaces of entire functions fo exponential type[END_REF] that the condition

(2.13) E ⊂ R measurable s.t. ∃γ > 0, a > 0 s.t. mes(E ∩ I) mes(I) ≥ γ
whenever I is an interval of length a is sufficient to ensure that (2.14)

∀b ≥ 0, ∃C = C(γ, a, b) > 0 s.t. E |f (x)| 2 dx ≥ C f 2 L 2 (R) , if supp( f ) ⊂ (-b, b).
On the other hand, the authors do not get a sharp estimate of C with respect to the parameters a, b, γ. This was achieved by Kovrijkine [START_REF] Kovrojkine | Some results related to the Logvinenko-Sereda inequality[END_REF], where the author proves that

∃K > 0 such that C(γ, a, b) = γ K K(ab+1)
, with γ < K. Moreover, the Logvenenko-Sereda inequality (2.14) also holds in any L p (R) space with p ∈ [1, ∞]. This is possible by combining the Bernstein's inequality, a suitable Remez-type inequality and some previous results by A. Zigmund in lacunary series [23, pp. 202-208]. The same results were obtained by Nazarov in [START_REF] Nazarov | Local estimates of exponential polynomials and their application to inequalities of uncertainty principle type[END_REF]. We refer to [START_REF] Egidi | Null-controllability and control cost estimates for the heat equation on unbounded and large domains[END_REF] and the references therein for more details on this subject. Thus our Theorem 1.2 (when V = 0 and g = Id) recovers the Logvinenko-Sereda inequality in R d . 2.2. Controllability of the parabolic equations in the whole space. Some recent work has been concerned with the problem of characterising the sets having "good observability properties" in the whole space in the context of the controllability of the heat equation and some other parabolic problems. More precisely, given an open set ω ⊂ R d , for d ≥ 1, let us consider the heat equation

(2.15) (∂ t -∆ x ) u = χ ω g, (t, x) ∈ R + × R d ,
where χ ω is the characteristic function of ω ⊂ R d and g is a forcing term, that we call a control, supported in (0, T ) × ω. The small-time null-controllability of (2.15) in an L 2 setting is equivalent to the following property

(2.16) ∀T > 0, ∀u 0 ∈ L 2 (R d ), ∃g ∈ L 2 ((0, T ) × ω) s.t u| t=T = 0,
where u is the solution of (2.15) with u| t=0 = u 0 . According to the classical HUM method, the null controllability of (2.15) is equivalent to the observability for the adjoint system

(2.17)

(∂ t -∆ x ) ψ = 0, (t, x) ∈ R + × R d ,
which is equivalent to the following observability inequality:

(2.18)

∃C obs > 0 s.t. ∀ψ ∈ L 2 (R d ), R d |ψ| t=T | 2 dx ≤ T 0 ω |ψ(t, x)| 2 dt dx,
where ψ solves (2.17) with ψ| t=0 = ψ 0 .

In the recent works [START_REF] Wang | Observable sets, observability, interpolation inequality and spectral inequality for the heat equation in R n[END_REF][START_REF] Egidi | Sharp geometric condition for null-controllability of the heat equation on R d and consistent estimates on the control cost[END_REF][START_REF] Nakić | Scale-free unique continuation principle for spectral projectors, eigenvalue lifting and Wegner estimates for random Schrödinger operators[END_REF], the authors use the Logvinenko-Sereda inequality to show that (2.18) holds if and only if ω ⊂ R d is a measurable set satisfying (1.1) for some R, δ > 0.

2.3. Outline of the paper. In Section 3 we give a proof of Proposition 3.1 which is a basic interpolation estimate for holomorphic functions defined in a tubular neighborhood of R d . Although one can obtain this type of result like in [START_REF] Kovrojkine | Some results related to the Logvinenko-Sereda inequality[END_REF] using the Remez inequality for polynomials as a starting point, we choose instead to present a proof which uses a Carleman estimate for functions of one complex variable (see Lemma 3.2).

In section 4 we recall some facts on the spectral theory of the Schrödinger operator and we introduce the Poisson kernel. In section 5, we prove estimates on the holomorphic extension of solutions to the Poisson equation. Finally, in section 6, we show that Theorem 1.1 is an easy consequence of the previous holomorphic extension estimates.

Carleman estimates and interpolation inequalities

Recall that for a > 0 we denote by U a the tubular neighborhood of

R d in C d , U a = {z ∈ C d , |Im(z)| < a}.
Let H a be the Banach space of holomorphic functions f (x + iy) in U a such that

f Ha := sup |y|<a f (. + iy) L 2 (R d ) < ∞.
By the Paley-Wiener Theorem, H a is the space of Fourier Transforms of functions

g ∈ L 2 (R d ) such that sup |η|<a e η.y g(y) L 2 (R d ) < ∞.
The goal of this section is to prove the following result

PROPOSITION 3.1. Let ω ⊂ R d satisfying the density condition (1.2). Then, there exist constants C = C(a, ω) > 0 and ν = ν(a, ω) ∈]0, 1[ such that (3.19) R d |f | 2 dx ≤ C ω |f | 2 dx ν Ua |f | 2 | dz| 1-ν , for any f ∈ H a .
Observe that Theorem 1.2 is an easy consequence of Proposition 3.1. In fact, if f ∈ L 2 (R d ) is such that its Fourier transform f (ξ) is supported in the ball |ξ| ≤ µ, by the Fourier inversion formula one has

f (x) = (2π) -d |ξ|≤µ e ixξ f (ξ)dξ, ∀x ∈ R d .
Therefore f is the restriction to R d of the holomorphic function f (z) defined on

C d by f (z) = (2π) -d
|ξ|≤µ e izξ f (ξ)dξ, and one has by Plancherel theorem

f (. + iy) 2 L 2 (R d ) = (2π) -d |ξ|≤µ |e -yξ f (ξ)| 2 dξ ≤ e 2µ|y| f 2 L 2 (R d ) .
Therefore, from (3.19) we get

f 2 L 2 (R d ) ≤ C ω |f | 2 dx ν c d a d e 2µa f 2 L 2 (R d ) 1-ν
, where c d is the volume of the unit sphere in R d , and this implies (1.11).

We will prove Proposition 3.1 in several steps. First we prove suitable Carleman estimates for holomorphic functions of one complex variable. In particular, we prove the interpolation estimate given in Lemma 3.3 below. We then deduce the multidimensional interpolation inequality given in Proposition 3.7. Finally, we get the proof of Proposition 3.1 by a simple covering argument.

3.1.

Carleman estimates for ∂ in C. In this section, we prove basic estimates for holomorphic functions of one complex variable. We denote by dλ the Lebesgue measure on C ≃ R 2 .

Let X ⊂ C be an open bounded connected domain with regular boundary. We start with the following classical Carleman inequality. LEMMA 3.2. Let ϕ(x) be a continuous function on X such that △ϕ = ν is a Borel measure on X. For all f ∈ C ∞ 0 (X) and all h > 0, the following inequality holds true

(3.20) 4h 2 X e 2 ϕ(x) h |∂f (x)| 2 dλ ≥ h X e 2 ϕ(x) h |f (x)| 2 dν.
Proof. Let f ∈ C ∞ 0 (X) be given and let Q ⊂ X be a compact set such that the support of f is contained in Q. We first assume that ϕ is smooth in a neighborhood of Q. Let P := 2h i ∂. We define the conjugate operator

P ϕ := e ϕ h P e -ϕ h = A + iB, A = h i ∂ x -ϕ ′ y = A * , B = h i ∂ y + ϕ ′ x = B * . Set g = e ϕ h f ∈ C ∞ 0 (Q).
One has P ϕ g = e ϕ h P f and by integration by part we get

(3.21) P ϕ g 2 L 2 (X) = Ag 2 L 2 (X) + Bg 2 L 2 (X) + i(Bg, Ag) L 2 (X) -i(Ag, Bg) L 2 (X) = Ag 2 L 2 (X) + Bg 2 L 2 (X) + i([A, B]g, g) L 2 (X) Since [A, B] = AB -BA = -ih△ϕ, we get (3.22) 4h 2 X e 2 ϕ(x) h |∂f (x)| 2 dλ = P ϕ g 2 L 2 (X) ≥ h X e 2 ϕ(x) h |f (x)| 2 △ϕ(x)dλ,
and thus (3.20) holds true. Let now ϕ(x) be a continuous function on X such that △ϕ = ν is a Borel measure on X. Let χ ε be a smooth approximation of the identity. Then ϕ ε = ϕ * χ ε and ν ε = ν * χ ε are well defined in a neighborhood V of Q for ε small and one has △ϕ ε dλ = ν ε on V . The inequality (3.22) holds true for ϕ ε . Since ϕ ε converge uniformly to ϕ on Q, it remains to verify

lim ε→0 X e 2 ϕε(x) h |f (x)| 2 △ϕ ε (x)dλ = X e 2 ϕ(x) h |f (x)| 2 dν .
Since the total variation on Q of the measures

ν ε is bounded, i.e sup ε Q d|ν ε | < ∞, one has lim ε→0 X |e 2 ϕε(x) h -e 2 ϕ(x) h ||f (x)| 2 dν ε = 0 .
Then, the result follows from the convergence of the measures ν ε to ν, i.e lim ε→0 X g(x)dν ε = X gdν for any continuous function g with support in Q. The proof of Lemma 3.2 is complete.

Let K ⊂ X a compact subset of X, and µ a positive Borel measure with support in K such that µ(K) = 1. We will assume that µ satisfies the following hypothesis:

(3.23) The potential function W (x) = - 1 2π log |x -y|dµ(y), solution in D ′ (R 2 ) of the equation -△W = µ, is continuous on C ≃ R 2 .
For y ∈ X, we denote by G(x, y) the Green function of the Dirichlet problem

(3.24) -∆ x G(x, y) = δ x=y in X, G(., y)| ∂X = 0.
Recall that one has G(x, y) > 0 for all (x, y) ∈ X × X with x = y, and that there exist constants 0 < c 1 < c 2 such that for all x ∈ X with dist(x, ∂X) small one has

(3.25) c 1 dist(x, ∂X) ≤ inf y∈K G(x, y), sup y∈K G(x, y) ≤ c 2 dist(x, ∂X)
Moreover, G(x, y) is analytic in x ∈ X \ {y}, and more precisely one has

(3.26)    G(x, y) = -1 2π log |x -y| + H(x, y) -∆ x H(x, y) = 0, in X, H(x, y)| x∈∂X = 1 2π log |x -y|. Let Φ µ (x) = G(x, y)dµ(y). Then Φ µ satisfies (3.27) -∆Φ µ = µ, in X, Φ µ | ∂X = 0.
By assumption (3.23), the function Φ µ (x) is continuous on X. Moreover, Φ µ is smooth on X \ K, one has Φ µ (x) > 0 for all x ∈ X, and

(3.28) Φ µ (x) = W (x) + h(x) -∆h = 0, in X, h| ∂X = -W | ∂X .
We will denote by C µ > 0 the constant

(3.29) C µ = sup x∈X Φ µ (x) = sup x∈K Φ µ (x) .
Observe that from (3.25), one has for all x ∈ X with dist(x, ∂X) small

(3.30) Φ µ (x) ≤ c 2 dist(x, ∂X).
Let Y ⊂ X an open subset of X with regular boundary such that K ⊂ Y and Y ⊂ X. We will denote by c Y > 0 the constant

(3.31) c Y = inf x∈Y,y∈K G(x, y) .
From Φ µ (x) = G(x, y)dµ(y) we get

(3.32) c Y ≤ inf x∈Y Φ µ (x) . Let Ψ Y (x) = -Y G(x, y)dλ(y) be the solution of (3.33) △Ψ Y = 1 x∈Y , Ψ Y | ∂X = 0 .
The function Ψ Y is continuous on X and one has Ψ Y (x) ≤ 0 for all x ∈ X. We denote by C Y the constant

(3.34) C Y = sup x∈Y |Ψ Y (x)| .
Let ρ > 0 such that 2ρC Y ≤ c Y and define ϕ by the formula

(3.35) ϕ(x) = Φ µ (x) + ρΨ Y (x) .
Then ϕ is continuous on X and one has △ϕ = -µ + ρ1 x∈Y . Thus we can apply (3.20) and we get for all f ∈ C ∞ 0 (X)

(3.36) 4h 2 X e 2 ϕ(x) h |∂f (x)| 2 dλ + h K e 2 ϕ(x) h |f (x)| 2 dµ ≥ hρ Y e 2 ϕ(x) h |f (x)| 2 dλ.
Let g be an holomorphic function in X such that g ∈ L 2 (X). We will apply (3.36) to ψg, with ψ ∈ C ∞ 0 (X) such that ψ is equal to 1 in a neighborhood of Y , and ∇ψ is supported in dist(x, ∂X) ≤ r with r > 0 small enough to have 4c 2 r ≤ c Y . By our choice of the constants ρ and r, the following inequalities hold true:

(3.37) sup x∈K ϕ(x) ≤ sup x∈K Φ µ (x) = C µ , inf x∈Y ϕ(x) ≥ inf x∈Y Φ µ (x) -ρ sup x∈Y |Ψ Y (x)| ≥ c Y /2, sup x∈support(∇ψ) ϕ(x) ≤ sup dist(x,∂X)<r Φ µ (x) ≤ c Y /4 .
Since ∂(ψg) = g∂ψ, we get that the following Lemma holds true, with M = ∂ψ L ∞ .

LEMMA 3.3. For every holomorphic function g ∈ L 2 (X) and all h > 0, the following inequality holds true PROPOSITION 3.5. There exists a constant C, depending only on X, Y such that for all holomorphic function g ∈ L 2 (X), the following interpolation inequality holds true

(3.38) 4h 2 M e c Y 2h X |g(x)| 2 dλ + he 2Cµ h K |g(x)| 2 dµ ≥ hρe c Y h Y |g(x)| 2 dλ.
(3.39) Y |g(x)| 2 dλ ≤ C K |g(x)| 2 dµ δ X |g(x)| 2 dλ 1-δ , δ = c Y 4C µ -c Y . Proof. Set X = X |g(x)| 2 dλ, O = K |g(x)| 2 dµ and Y = Y |g(x)| 2 dλ. Inequality (3.38) reads (3.40) 4hM e -c Y 2h X + e 2Cµ-c Y h O ≥ ρY .
If O = 0, by taking the limit h → 0, we get Y = 0, hence we may assume O > 0. Let h * be such that

X O = F (h * ), F (h) = 1 4hM e 4Cµ -c Y 2h . If h * ≥ 1, we use F (h * ) ≤ F (1), hence X ≤ F (1)
O, and we write

Y ≤ X ≤ F (1) δ O δ X 1-δ .
Observe that F (1) δ is independent of C µ , hence depends only on X, Y .

If h * ≤ 1, we use e 4Cµ -c Y 2h * ≤ 4M X
O , and we write

4h * M e -c Y 2h * X + e 2Cµ -c Y h * O = 2e 2Cµ-c Y h * O ≤ (4M ) 1-δ O δ X 1-δ .
The proof of Proposition 3.5 is complete.

From Proposition 3.5 we deduce the following Lemma.

LEMMA 3.6. Let X ⊂ C be a complex neighborhood of [0, 1]. There exists constants C, c depending only on X such that the following holds true. If E ⊂ [0, 1] is a measurable set with positive measure |E| > 0, and g a holomorphic and bounded function on X, one has:

(3.41) sup x∈[0,1] |g(x)| ≤ C |E| δ/2 E |g(x)| 2 dx δ/2 sup x∈X |g(x)| 1-δ , δ = c 1 + | log |E|| .
Proof. We may assume X bounded with regular boundary. We apply Proposition 3.5 with K = [0, 1] and the measure µ defined by gdµ = |E| -1 E g(x)dx. By Cauchy integral formula, if Y ⊂⊂ X is a complex neighborhood of [0, 1], there exists a constant C such that sup

x∈[0,1] |g(x)| ≤ C( Y |g(x)| 2 dλ) 1/2 .
Thus, from (3.39) it just remains to verify the lower bound on δ. By formula (3.39), this is equivalent to get a upper bound on C µ . From formula (3.28) we get

C µ ≤ C + sup x∈[0,1] 1 |E| E | log |x -t||dt ≤ C(1 + | log |E||),
with C independent of E. The proof of Lemma 3.3 is complete.

3.2. Interpolation estimates in R d . Let R > 0 be given. Let X ⊂ C d be a bounded complex neighborhood of the closed Euclidean ball B R = {x ∈ R d , |x| ≤ R}. Let E ⊂ B R be a measurable set with positive Lebesgue measure, |E| > 0.
The goal of this section is to prove the following interpolation inequality.

PROPOSITION 3.7. There exists constants C > 0, δ ∈]0, 1], depending only on X, R and |E|, such that for all holomorphic function g ∈ L 2 (X), the following interpolation inequality holds true

(3.42) BR |g(x)| 2 dx ≤ C E |g(x)| 2 dx δ X |g(z)| 2 |dz| 1-δ .
Proof. The proof of Proposition 3.7 is an usual consequence of Lemma 3.6. We recall it for the reader's convenience. We may assume R = 1. If X ⊂⊂ X is a complex neighborhood of B = B 1 , by Cauchy integral formula one has

sup x∈ X |g(x)| ≤ C( X |g(z)| 2 |dz|) 1/2 .
Therefore, replacing X by X, we just have to prove

( r∈Eω r d-1 dr)dσ(ω) ≤ 2 d-1 S d-1 |E ω |dσ(ω). Set V = {ω ∈ S d-1 such that |E ω | ≥ | Ẽ| 2 d c d } . By (3.44) one has | Ẽ| ≤ | Ẽ|/2 + 2 d V ol(V ), hence V ol(V ) ≥ | Ẽ|/2 d+1 .
Observe that there exists a > 0 such that X is a complex neighborhood of B 1+a . Therefore, for each ω ∈ V , the function of one complex variable z,

g ω (z) = g(x 0 + z r ω + a ω)
is defined in a complex neighborhood Z of the interval [0, 1] independent of ω and one has x 0 + z rω+a ω ∈ X for z ∈ Z. Therefore, we can apply Lemma 3.3, and we get for all ω ∈ V ,

|g(x 0 )| 2/δ = |g ω (0)| 2/δ ≤ C Eω |g(x 0 + rω)| 2 dr sup x∈X |g(x)| 2(1-δ) δ
, with C, δ depending only on X and |E| since with have the lower bound

|E ω | ≥ |E| c d 2 d+1 . By integration in ω ∈ V , using E ω ⊂]ρ, 2[, we get V ol(V )|g(x 0 )| 2/δ ≤ C ρ d-1 ω∈V Eω |r d-1 g(x 0 + rω)| 2 drdσ(ω) sup x∈X |g(x)| 2(1-δ) δ . Since ω∈V Eω |r d-1 g(x 0 + rω)| 2 drdσ(ω) ≤ E |g| 2 dx
R d |f | 2 dx ≤ k∈Z d BR(k) |f | 2 dx .
Let X ⊂ U a/2 be a complex neighborhood of B R (0) and for any k

∈ Z d set ω k = ω ∩ B R (k).
By assumption, one has |ω k | ≥ δ for all k. By Proposition 3.7, there exists constants C, ν > 0 independent of k ∈ Z d such that (3.45)

BR(k) |f (x)| 2 dx ≤ C ω k |f (x)| 2 dx ν X+k |f (z)| 2 |dz| 1-ν . Set c k = BR(k) |f (x)| 2 dx, a k = ω k |f (x)| 2 dx, b k = X+k |f (z)| 2 |dz|
By Hölder's inequality with 1/p = ν, 1/q = 1ν, we get

k c k ≤ C k a ν k b 1-ν k ≤ C( k a k ) ν ( k b k ) 1-ν .
It remains to observe that one has

k a k ≤ C ω |f (x)| 2 dx, k b k ≤ C Ua |f (z)| 2 |dz| .
The proof of Proposition 3.1 is complete.

Spectral analysis

4.1. Description of the spectrum. The goal of this section is to give a description of the spectrum of the operator H g,V defined by (1.4). To do this, we apply the long-range scattering theory developed in [9, Chap. 30], which yields the following result.

PROPOSITION 4.1. The operator H g,V defined by (1.4) is self-adjoint in the space L 2 (R d ; √ det g dx). The spectrum of H g,V is of the form

(4.46) Σ(H g,V ) = Λ ∪ {0} ∪ H ac ,
where H ac = (0, +∞) is the absolutely continuous spectrum, and Λ is the set of non zero eigenvalues. Moreover, there exists E 0 > 0 such that Λ ⊂ [-E 0 , 0) and any eigenvalue λ ∈ Λ is isolated with finite multiplicity.

REMARK 4.2. The set Λ may be empty, or finite, or countable, and 0 is its only possible accumulation point.

Proof. According to [9, Sec. 30.2], let us split the operator H g,V in the following manner

(4.47) H g,V = -∆ + Ṽ (x, ∂),
where ∆ is the flat Laplacian in R d and

Ṽ (x, ∂) = V (x) -div (g -1 -Id)∇ - g -1 √ det g ∇( det g) • ∇.
Observe that we may further write

Ṽ (x, ∂) = ∂ i (δ ij -g ij )∂ j - g -1 √ det g ∇( det g) • ∇ + V (x) (4.48) = V L (x, ∂) + V S (x, ∂), with V L (x, ∂) := (δ ij -g ij )∂ 2 i,j -∂ i (g ij )∂ j + V (x), (4.49) V S (x, ∂) := - 1 √ det g g ij ∂ j ( det g)∂ i , (4.50)
where we have used the expression in components and Einstein's convention. We have to verify that the operator Ṽ = V S + V L is a 1-admissible perturbation of the flat Laplacian , i.e., the short-range

V S (x, ∂) = d i=1 V S i (x)∂ i satisfies (4.51) V S i (x) ≤ C S (1 + |x|) -1-ǫ , and the long-range part V L (x, ∂) = |α|≤2 V L α (x)∂ α satisfies ∃ǫ > 0, ∀|α| ≤ 2, |∂ β V L α (x)| ≤ C α,β (1 + |x|) -|β|-ǫ , |β| = 0, 1, (4.52)
The estimates (4.51) and (4.52) follows from (1.8) and the fact that one can write g -1 = Id + ĝ and det g = 1 + f where ĝ and f satisfy (1.8). Thus the operator H g,V is a 1-admissible perturbation of the flat Laplacian. As a consequence of [9, Theorem 30.2.10, p.295] we get that the eigenvalues of the operator H g,V in R \ {0} are isolated with finite multiplicity. Furthermore, applying [20, Theorem 1, p. 530] ensures that H g,V does not have eigenvalues in R + . Finally, since for f ∈ D(H V,g ) one has

H g,V f, f L 2 (R d ; √ det g dx) ≥ R d V (x)|f | 2 det g dx ≥ inf x∈R d V (x) f 2 L 2 (R d ; √ g dx) , one get Σ(H g,V ) ⊂ [-E 0 , 0) for some E 0 > 0.
4.2. Spectral projectors and Poisson kernels. According to the spectral theorem ( cf. [4, Section 2.5]) applied to the self-adjoint operator H g,V , there exist a measure dν on R × N, supported in Σ(g, V ) × N, and a unitary operator

U : L 2 (R d ; det g dx) -→ L 2 (R × N; dν) such that (4.53) U H g,V U -1 (h) = σh(σ, n), h ∈ L 2 (R × N; dν).
If F is a bounded Borel measurable function on R, the operator F (H g,V ) is defined by the formula

(4.54) U F (H g,V )U -1 = F (σ).
In particular, for λ ∈ R, the spectral projector Π λ (g, V ), associated with the function F (σ) = 1 σ<λ , is defined by

(4.55) Π λ (g, V )(f ) = U -1 1 σ<λ U (f ) .
For s ≥ 0, we define the Poisson operator P s,± , associated with the function F (σ) = e -sσ 1/2 ± , by the formula

(4.56) P s,± (f ) = U -1 e -sσ 1/2 ± U (f ) ,
where the function σ

1/2 ± is defined in (1.9). PROPOSITION 4.3. Let f ∈ L 2 (R d ; √ det g dx) and set u ± (s, x) = P s,± (f )(x)
. Then u ± (s, x) satisfies the following elliptic boundary value problem

(4.57) (-∂ 2 s + H g,V )u = 0, in (0, ∞) × R d , lim s→0 + u(s, x) = f (x), in L 2 (R d ; √ det g dx). Proof. Setting v(s, σ, n) = U (u(s, .)), g(σ, n) = U f (x),
the boundary value problem (4.57) writes

(4.58) (-∂ 2 s + σ)v(s, σ, n) = 0, in (0, ∞) × R × N, lim s→0 + v(s, σ, n) = g(σ, n), in L 2 (R × N; dν).
which is true since by construction one has

v(s, σ, n) = e -sσ 1 2 ± g(σ, n).
5. Analytic estimates for second order elliptic operators 5.1. Holomorphic extensions estimates. Throughout this section we use N ∈ N instead of d to denote the dimension of the space. Let B R = {x ∈ R N , |x| < R} and let X ⊂ C N a complex neighborhood of the closed ball B R . Let D 0 > 0, d 0 > 0 be given. We denote by Q = Q X,D0,d0 the family of second order differential operators Q(x, ∂ x ) of the form

(5.59) Q(x, ∂ x ) = |α|≤2 q α (x)∂ α
where the functions q α (x) are holomorphic in X, and such that (5.60)

sup α q α L ∞ (X) ≤ D 0 , |α|=2 Re(q α (x))ξ α ≥ d 0 |ξ| 2 , ∀x ∈ X, ∀ξ ∈ R N .
Let R ′ ∈]0, R[. The goal of this section is to prove the two following propositions.

In this subsection, we will denote by C j various constants independent of Q ∈ Q and of a particular solution u ∈ L 2 (B R ) of the equation Qu = 0.

PROPOSITION 5.1. There exists a constant C such that for any Q ∈ Q and u ∈ L 2 (B R ) such that Qu = 0, the following inequality holds true

(5.61) u L ∞ (B R ′ ) ≤ C u L 2 (BR\B R ′ )
PROPOSITION 5.2. There exists constants C j > 0 such that for any Q ∈ Q and u ∈ L 2 (B R ) such that Qu = 0, the function u extends as a holomorphic function in the set

Y = {z ∈ C N , |Re(z)| < R ′ , |Im(z)| < C 1 (R ′ -|Re(z)|)}
and the following inequality holds true

(5.62) sup z∈Y |u(z)| ≤ C 2 u L 2 (BR)
REMARK 5.3. It will be essential in the proof of Proposition 5.5 below that the constants C j can be chosen independent of Q ∈ Q.

Proof. The proof of Proposition 5.1 is classical. Let ϕ ∈ C ∞ 0 (B R ) equal to 1 in a neighborhood of B R ′ and ψ ∈ C ∞ 0 (B R ) equal to 1 in a neighborhood of the support of ϕ. Let s > N/2. By classical pseudo-differential calculus, since Q is elliptic, there exist a pseudo-differential operator E of degree -2 such that EQ = ϕ + ψT ψ.

where T is a pseudo-differential operator of degree -s such that

ψT (ψf ) H s ≤ C 2 f L 2 (BR)
Since the construction of E, T involves only a finite number of derivatives of the coefficients of Q, from (5.60), the constant C 2 is independent of Q ∈ Q. From Qu = 0, we get ϕu = -ψT ψu and therefore (5.63) ϕu

H s = ψT (ψu) H s ≤ C 2 u L 2 (BR) .
Let us now prove (5.61) by a contradiction argument. If (5.61) is untrue, one can find a sequence

Q n ∈ Q and a sequence u n ∈ L 2 (B R ) such that Q n u n = 0, u n L ∞ (B R ′ ) = 1 and u n L 2 (BR\B R ′ ) → 0. The sequence u n is bounded in L 2 (B R )
and from (5.63), ϕu n is bounded in H s . Thus we may assume that u n weakly converge in L 2 to some u ∈ L 2 (B R ) and, since s > N/2, that u n converge strongly to u in L ∞ (B R ′ ). Then we have u

L ∞ (B R ′ ) = 1 and u = 0 on B R \ B R ′ . Let X ′
an open neighborhood of B R such that X ′ ⊂ X. We may also assume that Q n converge in Q(X ′ , D 0 , d 0 ) to some Q ∈ Q(X ′ , D 0 , d 0 ). Then u satisfies Qu = 0 and since Q is elliptic with analytic coefficient and

u = 0 on B R \ B R ′ we get u = 0 on B R , in contradiction with u L ∞ (B R ′ ) = 1.
In order to prove Proposition 5.2, we will use complex deformation arguments.

We first prove that for C 1 > 0 small enough u extends as a holomorphic function in Y . For r ∈ [0, R ′ ] let us define the non negative function ψ r (t):

ψ r (t) := max(R ′ -r 2 + t 2 , 0), t ∈ R.
The function ψ r is Lipschitz with |ψ ′ (r)| ≤ 1, and supp

ψ r = |t| ≤ √ R ′2 -r 2 . Observe that ψ r (t) is decreasing in r, ψ R ′ (t) = 0, ψ 0 (t) = max(R ′ -|t|, 0). Take C 1 > 0 small. For r ∈ [0, R ′ ], let K r be the compact set in C N K r := z ∈ C N ; | Im(z)| 2 ≤ C 1 ψ 2 r (| Re(z)|), | Re z| ≤ R ′2 -r 2 . The interior Ω r of K r , is given by Ω r := z ∈ C N ; | Im(z)| 2 < C 1 ψ 2 r (| Re(z)|), | Re z| < R ′2 -r 2 .
The open sets Ω r are decreasing in r and one has Ω R ′ = ∅, Ω 0 = Y . Let I = {r ∈ [0, R ′ [ such that u extends as a holomorphic function in Ω r }. Since Qu = 0, the function u is analytic near Im(z) = 0. Thus for r close to R ′ one has r ∈ I. From ∪ r>ρ Ω r = Ω ρ , we get that I is of the form [r 0 , R ′ [. In order to prove r 0 = 0, it is sufficient to prove that if 0 < r ∈ I, u extends near any point

z 0 ∈ K r \ Ω r . If Im(z 0 ) = 0, this is true since u is analytic in B R . If Im(z 0 ) = 0, one has | Re(z 0 )| < √ R ′2 -r 2 , | Im(z 0 )| 2 = C 1 ψ 2 r (| Re z 0 |), and locally near z 0 , Ω r is defined by f < 0 with f (z) = | Im(z)| 2 -C 1 ψ 2 r (| Re(z)|). Let ∂ = 1 2 (∂ x -i∂ y ). At z 0 = a + ib, one has b 2 = C 1 ψ 2 r (|a|) and ∂f (z 0 ) = ζ 0 = ξ 0 + iη 0 , ξ 0 = C 1 ψ r (|a|) a √ r 2 + a 2 , η 0 = -b . This implies |ξ 0 | ≤ √ C 1 |η 0 |. Therefore, if q(z, ζ) = |α|=2 q α (z)ζ α is the principal symbol of Q, one finds, using |b| = √ C 1 ψ r (|a|) ≤ √ C 1 R ′ Re q(z 0 , ζ 0 ) = Re q(a, ζ 0 ) + O(|b||ζ 0 | 2 ) = -Re q(a, b) + O( C 1 b 2 ) .
By the second line of (5.60) this implies q(z 0 , ζ 0 ) = 0 for C 1 small. Then the result follows from the Zerner Lemma that we recall for the reader's convenience.

LEMMA 5.4 (M.Zerner). Let Q(z, ∂) = α,|α|≤m q α (z)∂ α z be a linear differential operator with holomorphic coefficients defined in a neighborhood U of 0 in C N and let q(z, ζ) = |α|=m q α (z)ζ α be its principal symbol. Let f ∈ C 1 (C N ; R) be a real function such that f (0) = 0 and ∂f (0) = 0. Let u be a holomorphic function defined in U ∩ {f < 0}, such that Qu extends holomorphically to U . Then, if q(0, ∂f (0)) = 0, u extends holomorphically near 0.

Finally, let us verify that (5.62) holds true. Let R

′ < R 1 < R 2 < R. Let ϕ ∈ C ∞ 0 (B R2 ), equal to 1 on B R1 . Let δ > 0 small and D = {w ∈ R N , |w| ≤ δ}. For w ∈ D, we deform the real ball B R into the countour Σ w Σ w = {z ∈ C d , ∃x ∈ B R , z = x + iwϕ(x)} .
By the first part of the proof of Proposition 5.2, that we apply with some R ′ ∈ ]R 2 , R[, if δ is small enough, the function u extends holomorphically near any z ∈ Σ w , w ∈ D. Let u w (x) = u(x + iwϕ(x)). Then u w ∈ L 2 (B R ) and one has By taking C 1 small enough, this implies that (5.61) holds true. Proof. Recall that u ± (s, x) is a solution of the elliptic equation (5.65) (-∂ 2 s + H g,V )u ± = 0, in (0, ∞) × R d . We first choose R ∈]0, s 0 /2]. We denote here by B R ⊂ R d+1 the ball

Q w (u w ) = 0 where Q w is the operator induced by Q on Σ w . One has Q w ∈ Q(X ′ , D ′ 0 , d ′ 0 ) with (X ′ , D ′ 0 , d ′ 0 )
B R = {(σ, x) ∈ R d+1 , σ 2 + |x| 2 < R 2 } .
For w ∈ R d , we define the function u w (σ, x) by the formula u w (σ, x) = u ± (s 0 + σ, w + x) .

Then one has u w ∈ L 2 (B R ), and u w satisfies the equation Q w (u w ) = 0 on B R with -Q w = τ (-s0,-w) (-∂ 2 s +H g,V )τ (s0,w) where τ (s0,w) is the translation by (s 0 , w). By hypothesis (1.7) and (1.8) there exist (X, D 0 , d 0 ) such that Q w ∈ Q(X, D 0 , d 0 ) for all w. Let R ′ < R. By Proposition 5. This proves u ± (s 0 , z) ∈ H b and (5.64) follows from u ± (s, .) L 2 (R d ) ≤ f L 2 (R d ) for all s > 0.

6. Proof of Theorem 1.2

Let µ ∈ R and let f ∈ L 2 (R d ; √ det g dx) be such that f = Π µ (g, V )f . Take s 0 > 0. Since the support of U (f ) is contained in σ < µ, we can define the function h = U -1 e s0σ 1/2 ± U (f ) which satisfies h L 2 ≤ |e s0µ 1/2 ± | f L 2 . The function u ± (s, .) = P s,± (h) is solution of the Poisson equation with data h on s = 0 and one has by construction (6.68) f = u ± (s 0 , .) .

By Proposition 5.5, there exists b > 0 such that f ∈ H b and one has (6.69)

z∈U b |f (z)| 2 |dz| ≤ C h 2 L 2 ≤ C|e 2s0µ 1/2 ± | f 2 L 2 .
By Proposition 3.1, we get that there exists ν > 0 such that (6.70)

R d |f | 2 dx ≤ C ω |f | 2 dx ν U b |f | 2 | dz| 1-ν
, From (6.69) and (6.70) we get

R d |f | 2 dx ≤ C|e 2(1-ν)s 0 µ 1/2 ± ν | ω |f | 2 dx .
The proof of Theorem 1.2 is complete.

REMARK 3 . 4 .

 34 Observe that from the definition (3.29) of C µ , one has obviously C µ ≥ c Y .

  Let x 0 ∈ B such that |g(x 0 )| = sup x∈B |g(x)|. Let ρ ∈]0, 1[ such that |B ρ | ≤ |E|/2. Set Ẽ = E ∩ {ρ < |x| < 1}. One has | Ẽ| ≥ |E|/2. Let S d-1 be the unit sphere in R d and let c d be its volume. For ω ∈ S d-1 , define r ω ∈ [0, 2] as the largest value of r ≥ 0 such that x 0 + rω ∈ B 1 and set E ω = {r ∈]ρ, r ω ], such that x 0 + rω ∈ E} . We denote by |E ω | ∈ [0, 2] the Lebesgue measure of E ω ⊂ [0, 2]. One has (3.44) | Ẽ| = S d-1

  close to (X, D 0 , d 0 ) if δ is small enough. Now, we apply Proposition 5.1 with R ′ = R 2 . We get in particular sup w∈D supx∈BR 1 |u(x + iw)| ≤ C u L 2 (BR\BR 2 ) .

5. 2 .

 2 Estimates on solutions of the Poisson equation. Let f ∈ L 2 (R d ; √ det g dx) and u ± (s, x) = P s,± (f )(x) the solution of the Poisson equation (4.57). Let s 0 > 0. The goal of this subsection is to prove the following proposition PROPOSITION 5.5. There exists constants b > 0, C > 0 independent of f such that, u ± (s 0 , .) extends as an holomorphic function in the set U b = {z ∈ C d , |Im(z)| < b}. Moreover u(s 0 , z) ∈ H b and one has (5.64) z∈U b |u ± (s 0 , z)| 2 |dz| ≤ C f 2 L 2 .

  2, there exist b > 0 and C > 0 independent of w ∈ R d such that sup |x|<R ′ ,|y|<b|u w (0, x + iy)| 2 ≤ C BR |u w (σ, x)| 2 dσdx .This implies(5.66) sup|y|<b B R ′ |u ± (s 0 , w + x + iy)| 2 dx ≤ C BR |u ± (s 0 + σ, w + x)| 2 dσdx.Applying (5.66) at points w k = hk, k ∈ Z d with h small enough and adding all these inequalities, we get with a different constant C(5.67)sup |y|<b R d |u ± (s 0 , x + iy)| 2 dx ≤ C R d s∈[s0-R ′ ,s0+R ′ ] |u ± (s, x)| 2 dsdx.


Marion Girod-Roux

Thomas Hueber

Diandra Fabre

Silvain Gerber

Mélanie Canault

Nathalie Bedoin

Audrey Acher

Nicolas Béziaud

Eric Truy

Pierre Badin

Rehabilitation of speech disorders following glossectomy, based on ultrasound visual illustration and feedback

Keywords: Speech therapy, tongue ultrasound imaging, biofeedback, articulation disorder, assistive technologies

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Speech perception and production mechanisms are concurrently developed during the language acquisition process and remain closely related in its maintenance. Various studies demonstrate how critical the perception of others and of oneself is for language acquisition and maintenance (e.g. Joseph S [START_REF] Perkell | Movement goals and feedback and feedforward control mechanisms in speech production[END_REF] or [START_REF] Turgeon | Exploring consequences of short-and long-term deafness on speech production: A lip-tube perturbation study[END_REF] for auditory perception, and [START_REF] Mills | The development of phonology in the blind child[END_REF] for visual perception). Joseph S. [START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF] develop a theory of the segmental component of motor control which considers that the programming of the articulatory movements to reach auditory targets (i.e. phonemes or segments) relies on an internal model. This model, also called efferent copy, predicts the acoustic consequences of articulatory movements and adjusts the motor commands to minimize the discrepancy between the prediction and both the auditory feedback and somatosensory feedback.

The link between perception and production, that is to say the perceptuo-motor nature of speech as described by [START_REF] Schwartz | The Perception-for-Action-Control Theory (PACT): A perceptuo-motor theory of speech perception[END_REF], is made explicit in various studies that propose internal models of speech representation. For instance, [START_REF] Treille | Inside speech: Multisensory and modality-specific processing of tongue and lip speech actions[END_REF] suggest that the visual processing of audible but not visible movements (such as tongue movements) induces 'motor and visual mental simulation of the perceived actions to facilitate their recognition'.

Numerous studies have highlighted and quantified the contribution of the vision of visible articulators (such as lips, jaw, face, tongue tip, teeth) to the perception of speech (e.g. [START_REF] Erber | Auditory-visual perception of speech[END_REF], [START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF], [START_REF] Benoît | Audio-visual speech synthesis from French text: Eight years of models, designs and evaluation at the ICP[END_REF]; see [START_REF] Badin | Can you 'read' tongue movements? Evaluation of the contribution of tongue display to speech understanding[END_REF] for a review). Interestingly, [START_REF] Badin | Can you 'read' tongue movements? Evaluation of the contribution of tongue display to speech understanding[END_REF] demonstrate that, in addition to the well documented lip reading ability, humans possess a tongue reading ability as well, i.e. they are able to use the vision of tongue inside the oral cavity for the recognition of phonemes in adverse (e.g. noisy) conditions. This theoretical and experimental work has demonstrated the links between speech production and perception, as well as the 'naive' capacities to exploit visual information from an inside articulator whose natural representation, almost exclusively based on proprioception, remains partial. This framework served as the starting point for several studies focusing on the interest of an articulatory visual representation that would help the rehabilitation of speech disorders. Indeed, speech disorders may be attributed to a deterioration of the efficiency of the sensory channels, especially the proprioceptive one, necessary for the acquisition or maintenance of speech production capabilities, or to a lack of internal representations of the speaker. These deficiencies may be related to deafness, motor disability, developmental disorders or orofacial surgery. Using the vision of articulators as a complement of sensory feedback may thus offer an interesting opportunity for the rehabilitation of speech disorders.

Articulatory visual illustration and feedback

In this article we distinguish two articulatory visualisation paradigms. The first one, called 'articulatory visual illustration', aims to provide the patient with an intuitive visualisation of a targeted lingual movement, recorded on a reference speaker (typically someone with no speech disorders such as the speech therapist). Through various collaborations between engineers, researchers and speech-language pathologists (SLPs) various software applications have emerged. These applications are based either on stylized movements such as Diadolab [START_REF] Menin-Sicard | Intérêt de la visualisation de la position et du mouvement des articulateurs pour améliorer l'intelligibilité : la plate-forme Diadolab[END_REF]), 'The vocal tract in action' (Canault, 2010) 1 , or on real articulatory data such as [START_REF] Massaro | Using visible speech to train perception and production of speech for individuals with hearing loss[END_REF], [START_REF] Engwall | Can audio-visual instructions help learners improve their articulation? -An ultrasound study of short term changes[END_REF], [START_REF] Fagel | A 3-D virtual head as a tool for speech therapy for children[END_REF], [START_REF] Badin | Can you 'read' tongue movements? Evaluation of the contribution of tongue display to speech understanding[END_REF] or [START_REF] Hueber | Ultraspeech-player: intuitive visualization of ultrasound articulatory data for speech therapy and pronunciation training[END_REF]. [START_REF] Chen | Systematic review of virtual speech therapists for speech disorders[END_REF] review the literature on the contribution 1 This software can be tested at http://anatomie3d.univ-lyon1.fr/webapp/website/website.html?id=3346735&pageId=223201 of these visual illustration techniques in various populations: hearing-impaired people with cochlear implants, children with stigmatism or adults with aphasia. In each case, visual illustration seems to accelerate the learning process. It also appears to be easily assimilated by the patient who manages to associate what (s)he sees on the screen -which stands in the articulatory internal space of another speaker -with his/her own articulation.

In contrast, the second paradigm, called 'articulatory visual feedback'2 , aims to provide the patient with a visual representation of his/her own tongue movements, in his/her own articulatory space and with his/her own morphology. The goal is to help him/her better understand and correct his/her gestures, in particular by providing him/her with his/her own complementary visual articulatory information, along with the practitioner's instructions. The two main techniques actually used for visual biofeedback in speech therapy are electropalatography (EPG)3 and ultrasound imaging (US).

EPG has been used for three decades as biofeedback in speech therapy (see [START_REF] Gibbon | Bibliography of electropalatographic (EPG) studies in English[END_REF] for an exhaustive review). All studies conclude to some positive effects of EPG, though most of them do not involve control subjects and statistical evaluation. Drawbacks of EPG are the need to build a customized palate for each patient as well as the invasiveness of the palate and connecting wires.

Ultrasound imaging is a very interesting technique to observe tongue movements during speech production in real time, as it is harmless and minimally invasive for speakers (e.g. [START_REF] Epstein | Ultrasound and the IRB[END_REF] and recently Jonathan L. [START_REF] Preston | Do participants report any undesired effects in ultrasound speech therapy?[END_REF]). The reader is referred to [START_REF] Stone | A guide to analysing tongue motion from ultrasound images[END_REF] for a complete description of the use of ultrasound imaging in phonetic research, experimental setup, analysis techniques, etc. More than three decades ago, [START_REF] Shawker | Ultrasound biofeedback for speech training: Instrumentation and preliminary results[END_REF] suggested to use ultrasound images of the tongue for the rehabilitation of English /ɹ/ for a nine year old patient. Since this first case study, English /ɹ/ rehabilitation with ultrasound remains a widely studied topic (Adler-Bock, Bernhardt et al. (2007), Jonathan L Preston & Leaman (2014), [START_REF] Byun | Retroflex versus bunched in treatment for rhotic misarticulation: Evidence from ultrasound biofeedback intervention[END_REF], [START_REF] Cavin | The use of ultrasound biofeedback for improving English /r[END_REF], Jonathan L Preston, McAllister et al. (2018)). [START_REF] Modha | Case study using ultrasound to treat /r[END_REF], Jonathan L Preston, Brick et al. (2013), Joanne Cleland, James M. Scobbie et al. (2015), Jonathan L Preston, Maas et al. (2016) or [START_REF] Cleland | Covert contrast and covert errors in persistent velar fronting[END_REF] studied the use of ultrasound in speech sound disorders or apraxia of speech. Studies have also been conducted in cases such as severe hearing impairment [START_REF] Adler-Bock | The use of ultrasound in remediation of north American English /r/ in 2 adolescents[END_REF], [START_REF] Bernhardt | Speech habilitation of hard of hearing adolescents using electropalatography and ultrasound as evaluated by trained listeners[END_REF] or [START_REF] Gallagher | Critical Review: The effectiveness of ultrasound technology as a visual biofeedback tool on the productive speech intelligibility of adolescents and young adults with a hearing impairment[END_REF]), Down syndrome [START_REF] Fawcett | Ultrasound as visual feedback in speech therapy for/r/with adults with Down Syndrome[END_REF]), cleft palate [START_REF] Roxburgh | Multiple phonetically trained-listener comparisons of speech before and after articulatory intervention in two children with repaired submucous cleft palate[END_REF]), acquired apraxia of speech post stroke (Jonathan L [START_REF] Preston | Ultrasound visual feedback for acquired apraxia of speech: A case report[END_REF], [START_REF] Acher | Retour visuel en aphasiologie : résultats comportementaux, acoustiques et en neuroimagerie[END_REF] and [START_REF] Haldin | Speech recovery and language plasticity can be facilitated by Sensori-Motor Fusion training in chronic non-fluent aphasia. A case report study[END_REF]) or partial glossectomy [START_REF] Blyth | Ultrasound visual feedback in articulation therapy following partial glossectomy[END_REF]).

In summary, the literature on ultrasound visual feedback reflects a great diversity of experimental designs. Thus, depending on speech disorders and studies, various protocols regarding number, duration and/or content of sessions, or number and age of participants, have been set up. Importantly, most studies have been conducted in English-speaking countries. Moreover, the performance was mostly evaluated through speech assessments although acoustic (audio recordings) or articulatory (contours segmentation of images) analyses sometimes supplement these reports (Joanne Cleland, James M Scobbie et al. (2015)). Besides, the experimental designs differed from one study to another: while [START_REF] Bernhardt | Ultrasound as visual feedback in speech habilitation: Exploring consultative use in rural British Columbia, Canada[END_REF] alternated traditional rehabilitation and ultrasound visual feedback, [START_REF] Roxburgh | Multiple phonetically trained-listener comparisons of speech before and after articulatory intervention in two children with repaired submucous cleft palate[END_REF] proposed a comparative study between the ultrasound visual feedback and the illustration by an articulatory talking head, whereas [START_REF] Cleland | Tongue reading: Comparing the interpretation of visual information from inside the mouth, from electropalatographic and ultrasound displays of speech sounds[END_REF] and [START_REF] Bacsfalvi | Electropalatography and ultrasound in vowel remediation for adolescents with hearing impairment[END_REF] aimed to compare two main visual feedback devices, EPG and ultrasound. As a matter of fact, the recent study by Jonathan L Preston et al. (2018), aiming to compare two types of ultrasound biofeedback, is the only one that led to a statistical analysis of the outcomes. Finally, [START_REF] Blyth | Ultrasound visual feedback in articulation therapy following partial glossectomy[END_REF] led the only study of ultrasound biofeedback for two glossectomized patients.

Despite their great diversity, all these studies converge towards the same conclusion: the use of ultrasound seems to have a therapeutic benefit in speech rehabilitation as it enhances the positive impact of speech therapy, especially in the early stages (e.g. [START_REF] Blyth | Ultrasound visual feedback in articulation therapy following partial glossectomy[END_REF], Jonathan L Preston et al. (2018)). This contribution, yet, remains subtle and difficult to measure. It is therefore necessary to conduct larger studies focused on specific disorders to hope to gather convincing arguments. This is the main motivation of the present study which focusses on speech disorders occurring after oral surgery.

Speech disorders related to oral surgery with partial glossectomy

When a cancerous lesion appears on the tongue, it can be removed by a surgical procedure called partial glossectomy, which can be extended to the mouth floor and/or the mandible. If the resected specimen is too large and the functional outcomes are expected to be too severe, the surgeon may be required to replace the resected specimen by a flap made from a patient's piece of muscle, bone and/or skin. These surgical gestures are often supplemented by chemoradiotherapy (CRT) which severely alters the remaining tissues and the patients' quality of life (QoL). Because of the intraoral resection and the CRT adjuvant treatment, patients may often experience difficulties in both eating (since tongue in involved in oral preparation, propulsion and swallowing of the bolus) and communicating (as tongue is deeply involved in speech production). The vocal tract has been modified in terms of shape and volume, strength and sensibility of the remaining tissues, as well as in amplitude and velocity of their movements. Subsequently, speech is often affected by a loss of tone, precision and speed, which can alter the production of certain phonemes as described by [START_REF] Perrier | Speech production after tongue surgery and tongue reconstruction[END_REF]. [START_REF] Acher | Speech production after glossectomy: Methodological aspects[END_REF] report possible alterations in French patients after partial glossectomy. Thus, consonants involving tongue movements in the point and the manner of articulation, i.e. alveolar, post-alveolar and velar consonants, are naturally affected by these anatomical changes. For instance, a significant noise can be noticeable during the holding phase of these obstruent consonants. Speech therapy is then necessary to help the patient to accurately produce speech and master his/her new articulatory space.

Goal of the present study

The purpose of the present study was thus to evaluate -in a clinical environment -the efficiency of visual tongue illustration and/or feedback in speech rehabilitation after intra-oral surgery including partial glossectomy. For this purpose, ten patients who have undergone intra-oral surgery have been included in this study over a period of two years (2016 -2018). They followed speech therapy sessions based either on visual illustration or on a combination of visual illustration and feedback, as detailed in Section 2.1, and were regularly assessed.

Materials and methods

This study was carried out at the Rocheplane rehabilitation centre in Saint Martin d'Hères, France, with the agreement of the ethical committee Lyon Sud Est II (69HCL15_0736). This rehabilitation centre receives patients when they are discharged from the hospital after their surgery and provides them with intensive speech and swallowing therapy.

Three SLPs were involved in this project. Two of them (MGR and BG) were alternatively responsible for the patients' rehabilitation, according to their schedules, whereas two of them (MGR and OC) were in charge of monitoring the patients' progress.

Therapy sessions

This study aimed to compare the progress achieved with two speech rehabilitation protocols.

Two protocols were followed alternatively. In the protocol providing visual illustration only (IL protocol), the patient visualized the target/gesture produced by a reference speaker but relied only on his/her acoustic and somatosensory feedback to understand why his/her gesture was wrong and how (s)he should correct it. In the protocol providing both visual illustration and feedback (FB+I protocol), the addition of a real time visual feedback of his/her own gesture is expected to fasten the recalibration process.

Thirty minutes therapy sessions were conducted once or twice a day, weekend excluded. The patients were divided in two cohorts (FI and IF). The patients of the FI cohort followed a first series (S1) of 10 therapy sessions with the FB+I protocol, and then a second series (S2) of 10 sessions with the IL protocol. The order of the series in the IF cohort was reversed. A speech assessment was performed by the SLP at baseline, prior to the first series (at time T0) and then at the end of each series S1 and S2 (times T1 and T2). Speech therapy sessions were usually conducted at a mean frequency of about 5 sessions per week (2 to 10 depending on patient overall condition, but constant for each patient, cf. Table 1). Patients were assigned to the FI or IF cohorts using stratified random sampling based on their T0 assessment scores in order to obtain similar means for both cohorts.

Patients

Twelve adult patients were initially included in the study: eight males and four females. One male left before the end of the study whereas one female experienced problems of concentration during the T1 assessment and did not read the right words; their scores are thus not reported nor analysed in this study. Finally, ten patients (seven males and three females) were retained (average age of 59.4 years for the FI cohort and 58.6 years for the IF cohort). Details of surgery and treatment for each patient are presented in Table 1. The patients included in this study did not report any speech disorder or speech therapy prior to surgery. Note that all of them were explained the protocol and signed an informed consent before being included in the study, as described in the protocol allowed by the ethical committee.

Experimental setup

Visual illustration software

The Ultraspeech-player software [START_REF] Hueber | Ultraspeech-player: intuitive visualization of ultrasound articulatory data for speech therapy and pronunciation training[END_REF]) was used in this study for visual tongue illustration. This software is dedicated to the visualization of high-speed ultrasound and video sequences of the tongue and lips, recorded synchronously with the speech audio signal, using the Ultraspeech software [START_REF] Hueber | Acquisition of ultrasound, video and acoustic speech data for a silent-speech interface application[END_REF]). The software aims to display natural tongue movements acquired on a reference speaker, for different kinds of sequences (isolated vowels, VCV, swallowing, etc.), as illustrated in Figure 1 (top). Ultraspeech-player embeds a mechanism of real-time processing of audio and video streams, allowing the user to control the speed of the illustrated articulatory gesture, and that of the associated sound signal, in order to better observe it. The use of articulatory data acquired on a real speaker allows to restore realistic coarticulatory patterns. This software is free to download at http://www.ultraspeech.com.

For the present study, a new database was recorded and added to Ultraspeech-player 4 . One of the SLTs involved in the study was instructed to pronounce all the training sequences used during the therapy (see 2.4 for more details). Her tongue movements were recorded at 60fps using the Telemed Echoblaster system configured with a typical parameter set for tongue imaging (e.g. 7cm depth, with all image post-processing techniques such as frame averaging or speckle reduction disabled), with a 3-5MHz microconvex probe, maintained fixed with respect to the speaker's head (and below the chin) thanks to a slightly modified version of the helmet developed by Articulate Instrument5 . Lip movements were also recorded (but not used in the therapy session) at 60fps (640480 pixels) using an industrial CMOS camera (ImagingSource DFM22BUC03ML). The acoustic speech signal was acquired at 44.1kHz / 32bits with the lowlatency (ASIO compatible) RME Fireface soundcard and a cardioid AKG C-1000S microphone. All recordings were performed in a semi-anechoic room using the Ultraspeech 1.3 recording software.

One possible issue with ultrasound tongue images is the difficulty for the patient to make sense of them. Actually, the tongue is displayed out of any spatial context since the palate, teeth and pharynx are not visible in ultrasound images. In order to make the image more intuitive, the ultrasound image was supplemented with a schematic contour of the oral cavity (in red on Figure 1). This contour was extracted from a static MRI scan available for a female speaker. A set of transformations (rotation, shift, scaling) was then manually estimated to register the ultrasound image with the MRI contours. Using a 'trial and error' approach, the coarse-grained location of tongue-palate contacts for alveolar and velar stops was empirically checked, as well as the position of the tongue for some front and back vowels (note that the goal was not to register perfectly the two modalities together, which is a research problem on its own, but simply to make the ultrasound tongue image easier to interpret).

Visual feedback hardware and software

The same ultrasound system was used during the therapy sessions as a biofeedback tool. However, the probe stabilization helmet was not used since it may have been too uncomfortable in case of post-surgery facial oedemas. The probe was thus hold manually by the SLT, as patients had a partial loss of sensitivity in the region below the chin and tongue, and were not able to maintain the ultrasound probe in the adequate position.

In order to ensure a consistent display between illustration and feedback, a new software was developed, called Ultraspeech-biofeedback6 . It allows to superimpose in real-time the same schematic contour delimiting the oral cavity mentioned in section 2.3.2 on the ultrasound image (see Figure 1, bottom). Just before the first session for each new patient, the SLT took a minute to adjust the display to the intraoral morphology of the patient using the trial-and-error procedure mentioned above and stored the settings in a preset file which was reloaded at each following session and possibly updated.

Rehabilitation material

For the first half of the therapy sessions, the patients were instructed to perform Oral Motor Exercices (OME) including lips, mouth, cheeks and tongue range of movements. For the second part of the sessions, articulation of selected phonemes involving tongue (/t/, /k/, /kt/, /kl/, /st/, /sk/, /tʃ/ and /tl/) was trained using previously recorded words and sentences as described in Section 2.3.1.

Speech assessment tools

Speech assessment was carried out to evaluate patients' progress by means of tests extracted from the French BECD test battery [START_REF] Auzou | BECD: batterie d'évaluation clinique de la dysarthrie[END_REF]). Inspired by the Frenchay Dysarthria Assessment [START_REF] Enderby | Frenchay dysarthria assessment[END_REF]), it assesses the degree of dysarthria with different approaches. In the present study, three tests addressing isolated phonemes and words / sentences / conversation were used:

 'Phonetic Intelligibility test' (TPI): the patient is instructed to read from cards a list of 52 words randomly chosen from 13 groups of four words. In each group, one word differs from the three others by one or two phonetic features (for instance 'début' [deby], 'débute'

[debyt], 'des bouts' [debu], 'déboute ' [debut]). The SLP selects the closest word to what (s)he hears (without seeing it) from a list of four words. In each of the 13 series, the phonetic feature errors are scored on between 0 and 8 (since two features may simultaneously be erroneous for each of the four words, e.g.

[deby] vs.

[debut] in the previous example). Note that the three random lists (for T0, T1 and T2) were the same for all patients.

 'Intelligibility test' (Intell): it is made of three parts, based on a set of words, a set of sentences and a spontaneous conversation. The patient is instructed to read twelve words and twelve sentences randomly picked in a pile of cards. The SLP reports what (s)he hears, without seeing it, and compares it with the expected content. Finally, the patient has a short conversation with the SLP. A global score between 0 and 8 is attributed by the SLP for each part.

 'Phonetic Analysis test' (PhAn): the patient is instructed to repeat a list of 33 isolated phonemes, 37 words including simple phonemes and 25 words with complex phonemes pronounced by the SLP. The SLP scores the altered phonemes for each production (33 isolated, 88 simple and 30 complex phonemes), leading to a total of 151 scores on 0-1.

The rehabilitation program is conducted by two SLPs who alternatively educate the patients. Initially, all assessments were performed at T0, T1 and T2 by only one of them (MGR) for the sake of evaluation homogeneity. However, as this evaluation was not blinded, a blind SLP (OC) was asked to rate all the patients again, based on the audio records, with the same protocol.

Results

In order to evaluate statistically the results of each test, it is needed to model the relation between the response variable, i.e. the score obtained by the patient, and several explanatory variables as well as their interactions, i.e. cohort (qualitative variable, 2 categories, FI and IF) and time (qualitative variable, 3 categories, T0, T1 and T2).

The score variables are integer values between 0 and 8 for TPI and between 0 and 8 for Intell, and can thus be considered as ordered categorical variables. Besides, as each patient was tested three times, the patient variable (10 participants) is considered a random effect for the test with repeated data. We have therefore employed ordinal regression with random effects for the TPI and Intell tests, using the clmm procedure of the R package (R Development Core Team ( 2008)) based on [START_REF] Tutz | Random effects in ordinal regression models[END_REF].

Then, it was checked whether the models adjusted the data correctly, i.e. if the empirical probabilities (i.e. relative frequency) were close to the probabilities estimated by the model and if they were within the range of 95% prediction. Finally, four contrasts have been estimated by means of the lsmeans procedure of the R package. For each test, the following comparative contrasts have been evaluated (note that 'FI or IF(Ti-Tj)' denotes the difference in scores in the FI or IF cohort between times Ti and Tj): For test PhAn, the score variable is the sum of the scores obtained for each of the 151 assessed phonemes, divided by 151. This, consequently, takes a nearly continuous value between 0 and 1. The patient variable is again considered as a random effect. For this type of scores, beta regression with random effects has been employed using the glmmadmb procedure of the R package glmmADMB. The four contrasts were estimated with the glht procedure of the R multcomp package based on [START_REF] Hothorn | Simultaneous inference in general parametric models[END_REF]. The whole analysis has been performed for both raters (MGR and OC) separately.

The analysis was first applied, for the three tests, to the scores of the ten patients by the two raters: no difference was found significant at p=0.05. This means that it is impossible to claim that any patient significantly benefited better from the FB+I sessions than from the IL sessions.

A cohort analysis was then performed for each cohort of patients (FI and IF), for each test (TPI, Intell, PhAn) and each SLP (MGR, OC). The first result is that none of the differences were significant for the Intell test. This was not unexpected for two reasons: as the number of words in the list was small, the SLP tended to un-volitionally learn them and eventually to answer in a forced choice way; moreover, each score level grouped several possible errors, which reduced the chance to observe small progress. No detailed results are thus reported for this test which appeared to be less relevant as expected. The results for the other two tests (TPI and PhAn) are reported in Table 2.

The first important observation is that, for the TPI, the FB+I sessions are significantly more efficient than the IL sessions, on average over both S1 and S2 series, for both raters. This is a positive indication that patients' overall progress tends to be better with the FB+I sessions than with the IL sessions. A similar observation can be made from the results of the S2 series, though the difference related to rater MGR has a p-value slightly above 0.05. More surprising are the results for the S1 series, where only rater OC finds that the IL sessions were significantly more efficient than FB+I sessions, though the difference remains small. Finally, Table 2 shows that the PhAn test does not lead to any significant differences, except for rater MGR in session S1.

Discussion, conclusion and perspectives

Our literature review showed the importance of multiple sensory feedback channels in motor actions such as speech, including speech in glossectomized speakers. These multiple channels help maintain or correct internal models of coherent speech production and perception. We presented two paradigms of articulatory visual representation (for the rehabilitation of articulatory disorders) that were expected to improve the patients' speech ability by providing them with additional information on the defective phonemes: 'illustration' and 'articulatory visual feedback'. Based on the 10 glossectomized patients included in the present study, taking care to counterbalance the order and treatment conditions to avoid possible bias, the results of this research support the idea that the use of articulatory visual feedback significantly improves speech outcomes.

Baseline (T0) and tests (T1 and T2) have only assessed non-treated words (which included treated and non-treated phonemes) in order to observe generalization in conversational speech. The present study tends to show that speech was significantly improved in both cohorts with the use of US biofeedback. Indeed, in the early stage of the rehabilitation (2-3 first weeks) and especially within the S2 series, both cohorts show better speech outcomes (p<0.05) at the TPI test with the combination of US visual feedback and illustration, than with the illustration only.

Regarding S1 series and according to rater OC, results seem to be in favour of the illustration only. At this point, we have not found any interpretation. One hypothesis is that this rater was less accustomed to glossectomized speech than the other rater MGR.

Intra-patient analysis has not been conducted as only cohort analysis led to statistically significant results, and as inter-rater agreement was low. The analysis of the TPI-test shows that all patients improved their speech during the study (from T0 to T2), except patient FI-008; this patient had been particularly more affected by chemoradiation, leading to a lot more pain, fatigue and lack of motivation, than the other patients. Moreover, according to both raters, the speech quality decreased for one patient (patient IF-004) within the first series (IL); this can be explained by the occurrence of intercurrent medical events (in this case, a flap alteration). Furthermore, a qualitative analysis of the errors in the TPI shows that errors on aperture for vowels and place of articulation for consonants benefited from the use of the US visual feedback. Lastly, previous single-case studies of biofeedback intervention, including the latest of Jonathan L Preston et al. (2018) reported the existence of non-responders, which could be the case of patient FI-007: according to rater OC, this patient produced more errors at T1 assessment after the US visual feedback sessions.

Finally, the patients of this study have informally reported benefitsat least for the first sessions of the US real-time feedback: they felt that the feedback helped them better imagine their tongue position, movement or shapeof whose they were not aware due to their naïve condition and their lack of tongue sensitivity following surgery. During the therapy sessions, the SLPs hold the probe under the patients' chin. Indeed, participants were unable to adequately maintain the probe due to the lack of sensitivity in the chin's area. This is consistent with Jonathan L. Preston et al. (2018)'s study of the undesired effects in ultrasound speech therapy, where 21% of the responses were related to the difficulties in positioning the probe, which was found uncomfortable or annoying. In agreement with [START_REF] Roxburgh | Articulation therapy for children with cleft palate using visual articulatory models and ultrasound biofeedback[END_REF], both SLPs also reported that the US biofeedback enabled them to provide positive reinforcement to their patient on their skills.

Note that, since the 10 subjects of the study were patients following intensive therapy and complementary treatment after surgery (chemotherapy and/or radiation), the frequency of the therapy sessions had to be modulated and adapted to the participants (overall condition, medical complication, pain, fatigue, medical consultation appointment, chemoradiation sessions, public holidays, etc.). Thus, even if they all have followed the study design (10 sessions of each protocol I and F), they did not have the same number of sessions per week (mean = 5 sessions / week, range = 2-10 sessions / week): this modulation could have had an impact on the results. Moreover, the heterogeneity of the cohort in terms of tumour stage, type of the surgery, need for chemoradiation, number of remaining teeth, etc., may also have a potential influence on the results, for instance by increasing the dispersion of the scores.

Another possible bias may arise from the heterogeneity of the therapist contribution to the rehabilitation and assessment tasks. Indeed, two SLPs were in charge of the patients' rehabilitation with or without US visual feedback, depending on their work load and schedule, though only one of them was in charge of the assessment, for the sake of homogeneity. A third blind-SLP was in charge of the reassessment of the 10 patients based on the audio records, so that the inter-rater reliability could be evaluated. The results show various degrees of agreement between the two raters, consistent with the findings of [START_REF] Roxburgh | Multiple phonetically trained-listener comparisons of speech before and after articulatory intervention in two children with repaired submucous cleft palate[END_REF] who reported a statistically weak agreement between the scores of 22 phonetically-trained listeners for the same patients.

Regarding the assessment material, [START_REF] Woisard | From intelligibility to comprehension, which measurement in practice?[END_REF] showed that the standard BECD test battery [START_REF] Auzou | BECD: batterie d'évaluation clinique de la dysarthrie[END_REF]) suffers from limitation, in particular due to its use of real words: the raters are more likely to restore distorted sequences for real words, which may induce them to under-or over-estimate the errors, depending on phonemes.

In the future, it could be more efficient to use the new intelligibility test proposed by [START_REF] Ghio | Une mesure d'intelligibilité par décodage acoustico-phonétique de pseudo-mots dans le cas de parole atypique[END_REF], based on the pronunciation of pseudo-words that neutralizes unwanted lexical and learning effects of items by the raters. However, such a test would not allow therapists to assess speech in its functional dimension of communication.

To conclude, the present study is, to the best of our knowledge, the first cohort study demonstrating the effectiveness of ultrasound visual feedback jointly used with illustration for rehabilitating glossectomized patients.

The generalisability of these results is however subject to certain limitations. For instance, the sample size remains limited and should be increased in future studies (a multicentre trial could be an efficient way to recruit more patients while maintaining strict inclusion criteria). The speech assessment method could also be improved, for instance by relying on intelligibility tests recently proposed in the literature (e.g. [START_REF] Ghio | Une mesure d'intelligibilité par décodage acoustico-phonétique de pseudo-mots dans le cas de parole atypique[END_REF]); it could also involve a more extended set of raters (though [START_REF] Roxburgh | Multiple phonetically trained-listener comparisons of speech before and after articulatory intervention in two children with repaired submucous cleft palate[END_REF] have shown that the agreement remains low). In addition, future work could also exploit recent technological development aiming at improving the displayed ultrasound tongue image. Indeed, such image may remain difficult to interpret for some patient since the tongue is displayed out of any spatial context (the palate, teeth and pharynx are not visible in an ultrasound image of the vocal tract). Fabre, Hueber et al. (2017), have recently proposed a machine learning-based method for animating automatically and in real-time the 3D tongue model of an articulatory talking head from ultrasound images. Validating this technique in a clinical context is part of our roadmap.

Figure 1 .

 1 Figure 1. Top: illustration of the control interface and midsagittal ultrasound image display with sketchy vocal tract outline provided by Ultraspeech-player and used in the IL protocol. Bottom: illustration of the double interface, including the real time display of the speaker's tongue by Ultraspeech-biofeedback, used in the FB+I protocol.

FI_IF_S1=

  FI(T0-T1) vs. IF(T0-T1): difference in progress achieved after the first series S1, i.e. difference between FI cohort (FB+I sessions) and IF cohort (IL sessions) after the first series; FI_IF_S2 = FI(T1-T2) vs. IF(T1-T2): difference in progress achieved after the second series S2, i.e. difference between FI cohort (FB+I protocolsessions) and IF cohort (IL sessions) after the second series S2; FI_IF = ½ [FI(T0-T1) + IF(T1-T2)] vs. ½ [FI(T1-T2) + IF(T0-T1)]: difference in progress between FB+I protocol and IL sessions averaged over series S1 and S2.

Table 1 . General information on patients and their cohorts. IF-refers to the cohort IL / FB+I, while FI-refers to FB+I / IL.

 1

	Code patient	IF-001	FI-002	FI-003	IF-004	IF-005	FI-007	FI-008	IF-009	FI-010	IF-012
	Age (years)	47	54	52	58	58	53	75	70	63	60
	Gender	M	M	F	F	M	M	M	M	F	M
	Inclusion
	day after	J+20	J+21	J+14	J+28	J+24	J+26	J+18	J+13	J+23	J+31
	surgery
		T4	T2	T1	T3	T4	T4	T3	T3	T2
	Type of cancer	Floor of mouth + mandible	Junction base / body of tongue	Floor of mouth	Lateral border of the tongue	Anterior left floor of mouth	Right oropharynx	Amygdalog lossal sulcus + base of tongue	Lateral border + hemibase of tongue + floor of	Floor of mouth / Tongue anterior right	T4 Floor of mouth
									mouth	sulcus
							Oral cavity
	Surgery	FOM + Partial glossectom ie + Mandibule ctomy (transoral + transcervic al)	Oropharynge ctomy + Partial glossectomy (transcervica l + transoral)	FOM excisi on	Lateral glossect omy	FOM excision + anterior glossectom y + mandibulec tomy	/ transcervic Oropharyn (transoral + geal resection / Mandibule ctomy + BOT resection	Oral cavity l) / transcervica Oropharyng eal resection / Mandibulec tomy (transoral +	Lateral resection oropharynge ctomy + Partial partial + BOT glossectomy	Glossect omy + resection FOM anterior	FOM
							al)
		Flap	Flap	Local Flap	Free Flap	Flap (AB) Flap (GP)	Flap (GP)	Local FAMM Flap	Local flap	Tongu e
											CRT
	Chemoradi ation (CRT)	CRT during study	CRT during study	Not planne d	CRT during study	CRT during study	CRT during study	RT during study	Not planned	Not planned	planne d after the
											study
	Dental status	Toothless on the inferior left arch	Toothless (superior arch)	Toothl ess	Complet e e dental applianc	Toothless arch on the inferior left	Toothless	Toothless	Toothless	Healthy e and complet	Toothl ess
	Frequency
	of sessions	5-6	6-7	6-8	2-4	4-6	4-6	3-6	6-7	10	3-7
	/ week

Table 2 . Cohort score progress averages and standard deviations (SD); p-values of their comparisons (significant differences are marked in bold). The 'S1 + S2' columns correspond to the merging of the FB+I scores and of the IL scores from S1 and S2.

 2

				S1			S2			S1 + S2
			Cohort FI	Cohort IF		Cohort IF	Cohort FI
	Rater Test	FI+I	IL	p-val	FI+I	IL	p-val	FI+I	IL	p-val
	MGR TPI	4.4 SD 2.7 0.8 SD 3.9	0.1154 5.0 SD 2.3 0.0 SD 2.2	0.0578 4.7 SD 2.4 0.4 SD 3.0 0.0437
	OC	TPI	0.4 SD 2.8 0.8 SD 2.6	0.0001 4.6 SD 3.6 0.8 SD 1.3	0.0001 2.5 SD 3.7 0.8 SD 1.9 0.0001
	MGR PhAn 7.4 SD 5.1 3.2 SD 1.9	0.0372 2.4 SD 2.1 2.4 SD 3.2	0.8638 4.9 SD 4.5 2.8 SD 2.5	0.7602
	OC	PhAn 2.6 SD 14.3 9.8 SD 11.6 0.4330 -4.8 SD 10.0 6.4 SD 12.8 0.4140 -1.1 SD 12.3 8.1 SD 11.6 0.3710

As the feedback or illustration used in the present study are of an articulatory nature only, the term « articulatory » will not be specified in the rest of the article.

Electromagnetic articulography (EMA) has also been used by[START_REF] Katz | Studies of articulatory feedback treatment for apraxia of speech based on electromagnetic articulography[END_REF] and[START_REF] Mental | Measuring progress during practice: Motion analysis throughout visual biofeedback treatment for residual speech sound errors[END_REF] but remains too invasive and not convenient enough to be concretely used in a clinical environment.

This database will be made publicly available if the paper is accepted.

http://www.articulateinstruments.com/

This software will be made publicly available if the paper is accepted.

Acknowledgements

This work was funded by the Rhône-Alpes Region ARC6 program 'Information and Communication Technologies and Innovative IT Practices', as part of Diandra Fabre's doctoral thesis. Authors are very indebted to SLP Bérengère Gal who shared the rehabilitation work and to SLP Ondine Champavère who performed the second assessment based on audio recordings, as well as to Dr Thomas Dell'Accio who participated in the follow-up examinations.

Statement of interest

The authors report no declarations of interest.