Mourad

Jérôme Le Rousseau

J Le

PROBABILITÉS Rousseau Laboratoire De Mathématiques -Analyse

CARLEMAN ESTIMATES FOR ELLIPTIC OPERATORS WITH COMPLEX COEFFICIENTS PART II: TRANSMISSION PROBLEMS

Keywords: Carleman estimate, elliptic operators, transmission problem and condition, unique continuation 35B45, 35J30, 35J40

published or not. The documents may come

Carleman estimates for elliptic operators with complex coefficients Part II: transmission problems

The second author acknowledges support from Agence Nationale de la Recherche (grant ANR-13-JS01-0006 -iproblems -Problèmes Inverses).

INTRODUCTION AND MAIN RESULT

Let Ω be an open subset of R n with a smooth boundary and let Ω 1 be an open subset of Ω such that Ω 1 ⋐ Ω and such that S = ∂Ω 1 is smooth. We set Ω 2 = Ω \ Ω 1 . We thus have ∂Ω 2 = S ∪ ∂Ω.

Points in R n are denoted by x = (x 1 , . . . , x n ) and we write D j = -i∂/∂x j where i = √ -1. Let us consider two linear partial differential operators P k , k = 1, 2 of respective order m k = 2µ k , with µ k ≥ 1, (1.1)

P k = |α|≤m k a k α (x)D α , k = 1, 2,
where the coefficients a k α (x) are bounded measurable complex-valued functions defined in Ω. The higherorder coefficients a k α (x) with |α| = m k are required to be C ∞ in Ω k . In what follows, we assume that boths operators P k , k = 1, 2 are elliptic.

In addition, we consider a system of m 1 + m 2 linear transmission operators (1.2)

T j k = |α|≤β j k t j k,α (x)D α , k = 1, 2, j = 1, . . . , m = µ 1 + µ 2 ,
with 0 ≤ β j k < m k , and where the coefficients t j k,α (x) are C ∞ complex-valued functions defined in some neighborhood of S. Setting β j = (β j 1 + β j 2 )/2 we assume that m 1β j 1 = m 2β j 2 = mβ j , j = 1, . . . , m = µ 1 + µ 2 . (1.3) We consider a system of µ 2 = m 2 /2 linear boundary operators of order less than m 2 (1.4)

B j = |α|≤β j ∂ b j α (x)D α , j = 1, . . . , µ 2 ,
where the coefficients b j α (x) are C ∞ complex-valued functions defined in some neighborhood of ∂Ω. We can then consider the following elliptic boundary-value transmission problem

     P k u k = f k in Ω k , , k = 1, 2
T j 1 u 1 + T j 2 u 2 = g j , in S, j = 1, . . . , m. B j u = h j , in ∂Ω, j = 1, . . . , µ 2 .

The aim of the present article is to derive a Carleman estimate for this transmission problem. Carleman estimates are weighted a priori inequalities for the solutions of a partial differential equation (PDE), where the weight is of exponential type. For the partial differential operator P away from the boundary and from the interface, say for w ∈ C ∞ c (Ω 1 ) or C ∞ c (Ω 2 ), it takes the form: (1.5) e τ ϕ w L 2 ≤ C e τ ϕ P w L 2 , τ ≥ τ 0 .

The exponential weight involves a parameter τ that can be taken as large as desired. The weight function ϕ needs to be chosen carefully. Additional terms in the l.h.s., involving derivatives of u, can be obtained depending on the order of P and on the joint properties of P and ϕ. For instance for a second-order operator P such an estimate can take the form τ 3 e τ ϕ w 2 L 2 + τ e τ ϕ ∇ x w 2 L 2 ≤ C e τ ϕ P w 2 L 2 , τ ≥ τ 0 , w ∈ C ∞ c (Ω 1 ) or C ∞ c (Ω 2 ). (1.6) This type of estimate was used for the first time by T. Carleman [START_REF] Carleman | Sur une problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes[END_REF] to achieve uniqueness properties for the Cauchy problem of an elliptic operator. Later, A.-P. Calderón and L. Hörmander further developed Carleman's method [START_REF] Calderón | Uniqueness in the Cauchy problem for partial differential equations[END_REF][START_REF] Hörmander | On the uniqueness of the Cauchy problem[END_REF]. To this day, Carleman estimates remain an essential method to prove unique continuation properties; see for instance [START_REF] Zuily | Uniqueness and Non Uniqueness in the Cauchy Problem[END_REF] for manifold results. On such questions more recent advances have been concerned with differential operators with singular potentials, starting with the contribution of D. Jerison and C. Kenig [START_REF] Jerison | Unique continuation and absence of positive eigenvalues for Schrödinger operators[END_REF]. The reader is also referred to [START_REF] Sogge | Oscillatory integrals and unique continuation for second order elliptic differential equations[END_REF][START_REF] Koch | Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients[END_REF][START_REF]Sharp counterexamples in unique continuation for second order elliptic equations[END_REF]. In more recent years, the field of applications of Carleman estimates has gone beyond the original domain; they are also used in the study of:

• Inverse problems, where Carleman estimates are used to obtain stability estimates for the unknown sought quantity (e.g. coefficient, source term) with respect to norms on measurements performed on the solution of the PDE, see e.g. [START_REF] Bukhgeim | Global uniqueness of class of multidimensional inverse problems[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Kubo | Uniqueness in inverse hyperbolic problems-Carleman estimate for boundary value problems[END_REF][START_REF] Yu | An inverse problem for the dynamical Lamé system with two sets of boundary data[END_REF]; Carleman estimates are also fundamental in the construction of complex geometrical optic solutions that lead to the resolution of inverse problems such as the Calderón problem with partial data [START_REF] Kenig | The Calderón problem with partial data[END_REF][START_REF] Santos Ferreira | Limiting carleman weights and anisotropic inverse problems[END_REF]. • Control theory for PDEs; through unique continuation properties, Carleman estimates are used for the exact controllability of hyperbolic equations [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. They also yield the null controllability of linear parabolic equations [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and the null controllability of classes of semi-linear parabolic equations [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. For general elliptic operators, Carleman estimates away from boundaries and interfaces can be found in [START_REF]Linear partial differential operators[END_REF]Chapter 8]. The essential condition for the derivation of such an estimate is a compatibility property between the elliptic operator P and the weight function ϕ, the so-called sub-ellipticity condition which is known to be necessary and sufficient for the estimate to hold in the case of an elliptic operator. At the boundary ∂Ω, a Lopatinskii-type compatibility condition involving P , ϕ, and the operators B k can be put forward yielding a Carleman estimate in conjunction with the sub-ellipticity condition [START_REF] Tataru | Carleman estimates and unique continuation for solutions to boundary value problems[END_REF][START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF]. The main goal of the present article is the extension of this analysis to transmission problems.

Note that Carleman estimates of the form given here are local. Yet, they can be patched together to form global estimates. Our goal here is to derive such an estimate in the neighborhood of a point of the interface S. Derivation of Carleman estimates away for the interface can be found in the aforementionned references. Then, the patching procedure allows one to obtain a global estimate in the whole Ω, following for instance [START_REF]Linear partial differential operators[END_REF]Lemma 8.3.1] and [START_REF] Rousseau | On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations[END_REF]. We do not cover this issue here.

Here the weight function ϕ will be chosen continuous and piecewise smooth, that is,

ϕ k = ϕ |Ω k ∈ C ∞ (Ω k ).
The estimate we shall obtain will exhibit additional terms that account for the transmission conditions given by the operators T j k , k = 1, 2, j = 1, . . . , µ 1 + µ 2 . The key conditions for the derivation of the present Carleman estimate are compatibility properties between the elliptic operator P , the weight function ϕ, and the transmission operators T j k . Those are the sub-ellipticity condition described above that expresses compatibility between P and ϕ, and in addition a condition that connects them to T j k at the interface; we shall refer to this latter condition as to the transmission condition. This condition is an extension of the condition presented in [START_REF] Schechter | A generalization of the problem of transmission[END_REF] in the case of a conjugated operator. There, it was introduced towards to understanding of the well-posedness of the elliptic transmission problem. The condition we use is very close in its formulation to the Lopatinskii type boundary condition used in the first part of this work [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF]. In [START_REF] Tataru | Carleman estimates and unique continuation for solutions to boundary value problems[END_REF][START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] the derivation of Carleman estimates at a boundary is based on the study of interior and boundary differential quadratic forms, an approach that originates in the work of [START_REF]Linear partial differential operators[END_REF] for estimates away from boundaries and in [START_REF] Sakamoto | Mixed problems for hyperbolic equations. I. Energy inequalities[END_REF][START_REF]Hyperbolic boundary value problems[END_REF][START_REF] Miyatake | Mixed problems for hyperbolic equations of second order with first order complex boundary operators[END_REF] for the treatment of boundaries. This approach is here extended to interface tranmission problems. By proper (tangential) microlocalizations at the interface we show the precise action of our transmission condition. These microlocalizations are important as the transmission condition is function of the sign of the imaginary parts of the roots of1 p k,ϕ (x, ξ ′ , τ, ξ n ) = p k (x, ξ + iτ ϕ ′ (x)), k = 1, 2, viewed as a polynomial in ξ n . Of course the configuration of the roots changes as the other parameters (x ′ , ξ ′ , τ ) are modified. Roots can for instance cross the real axis. Each configuration needs to be addressed separately through a microlocalization procedure. For the Laplace operator at a boundary this was exploited to obtain a Carleman estimate in [START_REF]Stabilisation de l'équation des ondes par le bord[END_REF] for the purpose of proving a stabilization result for the wave equation. This approach was used for the study of an interface problem in [START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF][START_REF] Rousseau | Carleman estimates for anisotropic elliptic operators with jumps at an interface[END_REF] in the case of second-order elliptic operators. The present article provides a generalization of these earlier works, both with respect to the order of the operators and with respect to the generality of the transmission operators used.

The Carleman estimate we prove here is of the form, with

u k = u |Ω k , k=1,2 e τ ϕ k u k 2 + k=1,2 |e τ ϕ tr(u k )| 2 ≤ C k=1,2 e τ ϕ k P (x, D)u k 2 + µ 1 +µ 2 j=1 |e τ ϕ |S (T j 1 (x, D)u 1|S + T j 2 (x, D)u 2|S )| 2 ,
for u supported near a point at the interface, where tr(u k ) stands for the trace of (u k , D ν u k , . . . , D m-1 ν u k ), the successive normal derivatives of u k , at the interface S. In this form, the estimate is incorrect as norms needs to be made precise. For a correct statement please refer to Theorem 1.6 below.

For Carleman estimates, one is often inclined to choose a weight function of the form ϕ = exp(γψ), with the parameter γ > 0 chosen large. Several authors have derived Carleman estimates for some operators in which the dependency upon the second parameters γ is kept explicit. See for instance [START_REF] Fursikov | Controllability of evolution equations[END_REF]. Such results can be very useful to address systems of PDEs, in particular for the purpose of solving inverse problems. On such questions see for instance [START_REF] Eller | Carleman estimates with a second large parameter[END_REF][START_REF] Eller | Carleman estimates with two large parameters and applications[END_REF][START_REF] Isakov | Carleman estimates with second large parameter and applications to elasticity with residual stress[END_REF][START_REF] Bellassoued | Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases[END_REF].

Compatibility conditions need to be introduced between the operator P and the weight ψ. Those are the so-called strong pseudo-convexity conditions introduced by L. Hörmander [START_REF]Linear partial differential operators[END_REF][START_REF]The analysis of linear partial differential operators[END_REF]. With the weight function ϕ of the form ϕ = exp(γψ), the parameter γ can be viewed as a convexification parameter. As shown in Proposition 28.3.3 in [START_REF]The analysis of linear partial differential operators[END_REF] the strong pseudo-convexity of the function ψ with respect to P implies the sub-ellipticity condition for ϕ mentioned above 2 for γ chosen sufficiently large. Away from any boundary and interface, for a second-order estimate the resulting Carleman estimate can take the form (compare with (1.6)):

(γτ ) 3 ϕ 3/2 e τ ϕ u 2 L 2 + γτ ϕ 1/2 e τ ϕ ∇ x u 2 L 2 e τ ϕ P u 2 L 2 , τ ≥ τ 0 , γ ≥ γ 0 , u ∈ C ∞ c (Ω 1 ) or C ∞ c (Ω 2 ).
(1.7)

We aim to extend such estimate in the neighborhood of the interface S. We then assume that the transmission condition holds for the operators P , T j k , k = 1, 2, j = 1, . . . , µ 1 + µ 2 , and the weight ψ. The work [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] provides a general framework for the analysis and the derivation of Carleman estimates with two large parameters away from boundaries. For that purpose it introduces a pseudo-differential calculus of the Weyl-Hörmander type that resembles the semi-classical calculus and takes into account the two large parameters τ and γ as well as the weight function ϕ = exp(γψ). Here, following the first part of this article [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF], the analysis of [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] is adapted to the case of an estimate at the interface. Estimates with the two large parameters τ and γ are derived in the case of general elliptic operators.

If we strengthen strong pseudo-convexity condition of ψ and P , assuming the so-called simple characteristic property, sharper estimates can be obtained [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF]. We also derive such estimates at the interface.

With the different Carleman estimate that we obtain here she shall be able to achieve unique continuation properties at an interface across some hypersurface for some classes of elliptic operators and some products of such operators.

1.1. Setting. Now, we give the precise setting of the problem we consider. For x = (x 1 , . . . , x n ) ∈ R n , we denote by ξ = (ξ 1 , . . . , ξ n ) the corresponding Fourier variables. Moreover, for every ξ ∈ R n and α ∈ N n we define ξ α = ξ α 1 1 • • • ξ αn n . We denote by

p k (x, ξ) = |α|=m k a k α (x)ξ α
the principal symbol of the operator P k given in (1.1), k = 1, 2. The operators P k are assumed to be elliptic, viz., p k (x, ξ) = 0, ∀x ∈ Ω k , ∀ξ ∈ R n \ {0} . With m = µ 1 + µ 2 = (m 1 + m 2 )/2, we denote by

t j k (x, ξ) = |α|=β j k t j k,α (x)ξ α , k = 1, 2 j = 1, . . . , m,
the principal symbol of the transmission operator T j k defined in (1.2). Each set T j 1 1≤j≤m

and T j 2 1≤j≤m

is assumed normal, that is

0 ≤ β 1 k ≤ β 2 k ≤ • • • ≤ β m k < m k
, and for all x ∈ S the conormal vector ν(x) is non characteristic, i.e., t j k (x, ν(x)) = 0. We recall that we assume

m 1 -β j 1 = m 2 -β j 2 = m -β j , j = 1, . . . , m = µ 1 + µ 2 .
We now review the definition of two inportant notions that will be used in what follows: 2 The terminology for the strong pseudo-convexity condition and the sub-ellipticity condition are often confused by authors.

Here we make a clear distinction of the two notions.

• the sub-ellipticity condition between the operators P k and the weight function ϕ;

• the transmission condition stating the compatibility between the transmission operators T j k , the operators P k , and the weight function ϕ at a point of the interface.

1.2. Sub-ellipticity condition. For any two functions f (x, ξ) and g

(x, ξ) in C ∞ (Ω k × R n ) we denote their Poisson bracket in phase-space by {f, g} = n j=0 ∂f ∂ξ j ∂g ∂x j - ∂f ∂ξ j ∂g ∂x j .
It is to be connected with the commutator of two (pseudo-)differential operators. In fact, if f and g are polynomials in ξ, then the principal symbol of the commutator [f (x, D), g(x, D)] is precisely -i{f, g}(x, ξ).

The sub-ellipticity condition connecting the symbol p k and a weight function ϕ is the following (See [START_REF]Linear partial differential operators[END_REF]Chapter 8] and [START_REF]The analysis of linear partial differential operators[END_REF]).

p k (x, ξ + iτ ϕ ′ (x)) = 0 ⇒ 1 2i p k (x, ξ -iτ ϕ ′ (x)), p k (x, ξ + iτ ϕ ′ (x)) > 0,
for all x ∈ U and all non-zero ξ ∈ R n , τ > 0.

For an elliptic operator the sub-ellipticity condition is necessary and sufficient for a Carleman estimate of the form of (1.5) to hold away from the boundary [START_REF]The analysis of linear partial differential operators[END_REF]Section 28.2]. For a simple exposition of the derivation of Carleman estimates for second-order elliptic operators under the sub-ellipticity condition we refer to [START_REF] Rousseau | On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations[END_REF].

Note also that the sub-ellipticity condition is invariant under changes of coordinates. This is an important fact here as we shall work in local coordinates in what follows.

Remark 1.2. Note that here, as the operator P k are elliptic, we have p k (x, ξ) = 0 for each ξ ∈ R n , ξ = 0. The sub-ellipticity condition thus holds naturally at τ = 0.

Remark 1.3. Setting p k,ϕ (x, ξ, τ ) = p k (x, ξ + iτ ϕ ′ k )
and writing p k,ϕ = a + ib with a and b real, we have

1 2i p k (x, ξ -iτ ϕ ′ k ), p k (x, ξ + iτ ϕ ′ k ) = 1 2i {p k,ϕ , p k,ϕ }(x, ξ, τ ) = {a, b}(x, ξ, τ ).
Below, we shall use the sub-ellipticity condition in the form

p k (x, ξ + iτ ϕ ′ k ) = 0 ⇒ {a, b}(x, ξ, τ ) > 0, for all x ∈ U and all non-zero ξ ∈ R n , τ > 0.
In connection with the symbol interpretation of the Poisson bracket given above, we see that the subellipticity condition guarantees some positivity for the operator i

[a(x, D, τ ), b(x, D, τ )] on the characteristic set of p k (x, D + iτ ϕ ′ k ) = a(x, D, τ ) + ib(x, D, τ ). A proper combination of a(x, D, τ ) * a(x, D, τ ) + b(x, D, τ ) * b(x, D, τ ) and i[a(x, D, τ ), b(x, D, τ )]
thus leads to a positive operator. This is the heart of the proof of Carleman estimates. 1.3. Transmission conditions. We consider a neighborhood X of a point of the interface S, chosen sufficiently small, so that there exists a smooth function θ(x) such that dθ(x) = 0 in X and

(1.8) {x ∈ X; θ(x) < 0} = Ω 1 ∩ X, {x ∈ X; θ(x) = 0} = S ∩ X, {x ∈ X; θ(x) > 0} = Ω 2 ∩ X.
For x ∈ S, we denote by N *

x (S) the conormal space above x given by

N * x (S) = {ν ∈ T * x (Ω); ν(Z) = 0, ∀Z ∈ T x (S)} .
The conormal bundle of S is given by

N * (S) = {(x, ν) ∈ T * (Ω); x ∈ S, ν ∈ N * x (S)} . In fact, if x ∈ X ∩ S and (x, ν) ∈ N * (S) then ν = t dθ(x) for some t ∈ R.
By an interface quadruple ω = (x, Y, ν, τ ) we shall mean

x ∈ S ∩ X, Y ∈ T * x (S), ν = tdθ(x) ∈ N *
x (S) with t > 0, and τ ≥ 0. In particular ν "points" from Ω 1 into Ω 2 . For an interface quadruple ω and λ ∈ C, we set, for k = 1, 2, pk,ϕ (ω, λ)

:= p k (x, Y + λν k + iτ dϕ k (x)), with ϕ k = ϕ |Ω k , and ν k = (-1) k ν ∈ N * x (S). (1.9)
Note that for Ω k the covector ν k points inward and we have ν 1 = -ν 2 = -ν.

For a fixed interface quadruple ω 0 = (x 0 , Y 0 , ν 0 , τ 0 ), we denote by σ j k , k = 1, 2, the roots of pk,ϕ (ω 0 , λ) with multiplicity µ j k , viewed as a polynomial of degree m in λ, with leading-order coefficient c k,0 . We can then factorize this polynomial as follows:

pk,ϕ (ω 0 , λ) = c k,0 p+ k,ϕ (ω 0 , λ)p - k,ϕ (ω 0 , λ)p 0 k,ϕ (ω 0 , λ), with p± k,ϕ (ω 0 , λ) = ± Im σ j k >0 (λ -σ j k ) µ j k , p0 k,ϕ (ω 0 , λ) = Im σ j k =0 (λ -σ j k ) µ j k .
We define the polynomial κ k,ϕ (ω 0 , λ) by

(1.10) κ k,ϕ (ω 0 , λ) = p+ k,ϕ (ω 0 , λ)p 0 k,ϕ (ω 0 , λ).
Similarly, for the set of transmission operators, T j k k=1,2;j=1,...,m , m = µ 1 + µ 2 , and their principal symbols, t j k (x, ξ), for an interface quadruple ω = (x, Y, ν, τ ) we set (1.11)

tj k,ϕ (ω, λ) = t j k (x, Y + λν k + iτ dϕ k (x)
), with ν k = (-1) k ν as above.

Definition 1.4. We say that P k , T j k , ϕ, k = 1, 2; j = 1, . . . , m satisfies the transmission condition at an interface quadruple ω 0 = (x 0 , Y 0 , ν 0 , τ 0 ) if for all pairs of polynomials, q k (λ), k = 1, 2, there exist U k , k = 1, 2, polynomials and c j ∈ C, j = 1, . . . , m, such that:

q 1 (λ) = m j=1 c j tj 1,ϕ (ω 0 , λ) + U 1 (λ)κ 1,ϕ (ω 0 , λ) and q 2 (λ) = m j=1 c j tj 2,ϕ (ω 0 , λ) + U 2 (λ)κ 2,ϕ (ω 0 , λ),
where the polynomials κ k,ϕ (ω 0 , λ) are those defined by (1.10).

Additionally, for x 0 ∈ S, we say that P k , T j k , ϕ, k = 1, 2; j = 1, . . . , m satisfies the transmission condition at x 0 if the above property holds for all interace quadruples ω = (x 0 , Y 0 , ν 0 , τ 0 ) with Y ∈ T *

x 0 (S), ν = t dθ(x 0 ) with t > 0, and τ ≥ 0.

It should be noted that the same coefficients c j are used in both decompositions.

Remark 1.5.

(1) There is a strong similarity in the form between Definition 1.4 and the strong Lopatinskii condition that is used in the derivation of a Carleman estimate at the boundary. The latter condition connects the elliptic operator, the weight functions and the boundary operators given in (1.4) in [START_REF] Tataru | Carleman estimates and unique continuation for solutions to boundary value problems[END_REF][START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF].

(2) Note that we did not choose any particular co-normal vectors ν k connected with the function θ that locally defines S, apart from their orientation. In fact, for any t > 0 replacing ν k by tν k does not affect the transmission condition of Definition 1.4. We could for instance use normalized conormal vectors, yet keeping the directions of ν k .

1.3.1. Alternative formulation. Setting n k = d • κ k,ϕ we have n k = m k -m - k with m - k = d • p- k,ϕ . We have n k ≤ m k .
Hence, it is sufficient to consider the polynomials q k , k = 1, 2, to be of degree less than m k -1 respectively. Then both polynomials U k are of degree less than or equal to

m k -n k -1 = m - k -1, recalling that tj k,ϕ is of degree β j k < m k . We then set ẽj 1 = tj 1,ϕ if j = 1, . . . , m, λ j-(m+1) κ 1,ϕ if j = m + 1, . . . , m + m - 1 , ẽj 2 = tj 2,ϕ if j = 1, . . . , m, λ j-(m+1) κ 2,ϕ if j = m + 1, . . . , m + m - 2
, and the linear map

Φ : C m × C m - 1 × C m - 2 → C m 1 -1 [λ] × C m 2 -1 [λ], (1.12) (c, γ 1 , γ 2 ) → m j=1 c j ẽj 1 + m - 1 j=1 γ 1,j ẽj+m 1 , m j=1 c j ẽj 2 + m - 2 j=1 γ 2,j ẽj+m 2 .
The transmission condition of Definition 1.4 means precisely that the map Φ is surjective. In particular this implies that

m ′ = m + m - 1 + m - 2 ≥ m 1 + m 2 = 2m. 1.3.2.
Transmission condition and well-posedness. Transmission conditions across an interface for elliptic problems can be found in [START_REF] Schechter | A generalization of the problem of transmission[END_REF] to prove well-posedness of the transmission elliptic problem. In [START_REF] Schechter | A generalization of the problem of transmission[END_REF] it corresponds to the case ϕ = 0, as no conjugation with a weight function is performed and, there, the transmission condition reads as follows: let U k , k = 1, 2, polynomials and c j ∈ C, j = 1, . . . , m, be such that

0 = m j=1 c j tj 1,ϕ=0 (ω 0 , λ) + U 1 (λ)κ 1,ϕ=0 (ω 0 , λ) and 0 = m j=1 c j tj 2,ϕ=0 (ω 0 , λ) + U 2 (λ)κ 2,ϕ=0 (ω 0 , λ),
then U k ≡ 0, k = 1, 2, and c j = 0, j = 1, . . . , m. This condition precisely means that the map Φ introduced in (1.12) is injective in the case ϕ = 0. Above we gave a surjectivity condition that suits the purpose of the present article. We now explain how the two conditions coincide in the cases studied in [START_REF] Schechter | A generalization of the problem of transmission[END_REF].

In fact if ϕ = 0 none of the roots can be real as the operator is elliptic. If #{Im σ j k > 0} = #{Im σ j k < 0}, such an operator is called properly elliptic by Schechter [START_REF] Schechter | A generalization of the problem of transmission[END_REF] and he only studies this type of elliptic operators. In fact, if the dimension n ≥ 3 then every elliptic operator is properly elliptic (see e.g. the proof of Proposition 1.1 in [40, Chapter 2, Section 1]). In such a case we have m - k = m k /2 = µ k . Hence for the map Φ defined in (1.12) we have

dim C m × C m - 1 × C m - 2 = dim C m 1 -1 [λ] × C m 2 -1 [λ], as m + m - 1 + m - 2 = 2m = m 1 + m 2 ,
meaning that in this case the surjectivity of Φ is equivalent to its injectivity. In the case of a properly elliptic operator and ϕ = 0 the transmission condition of [START_REF] Schechter | A generalization of the problem of transmission[END_REF] thus coincides with Definition 1.4 in the case ϕ = 0.

Note that the injectivity of the map Φ may be lost for ϕ = 0 and τ > 0, as the conjugated operator P ϕ may not be elliptic, yet the transmission condition we give here precisely states that Φ is surjective.

1.3.3.

Invariance by change of local coordinates. We finish the presentation of the transmission condition by observing that this definition is of geometrical nature, independent of the choice of coordinates. This fact is important as we shall make use of local coordinates at the interface S in what follows.

In fact, for a point x ∈ S we consider an open neighborhood X ⊂ Ω of x, chosen sufficiently small so that there exists θ as in (1.8).

We consider two coordinate systems (X (i) , ψ (i) ), i = 1, 2, that is ψ (i) : X → X (i) is a diffeomorphism and X (i) is an open set in R n . We set x (i) = ψ (i) (x). We moreover set

θ (i) = θ • (ψ (i) ) -1 , X (i) k = {x ∈ X (i) ; (-1) k θ (i) (x) > 0} = ψ (i) (X ∩ Ω k ).
We then introduce the diffeomorphism κ : X (1) → X (2) given by κ = ψ (2) • (ψ (1) ) -1 and we have κ(x (1) ) = x (2) , θ (1) = θ (2) • κ, yielding κ(X

(1) k ) = X (2)
k . We also define ϕ (i) = ϕ•ψ (i) , i = 1, 2, the local versions of the weight function in the coordinate patches and we set ϕ

(i) k = ϕ (i) |X (i) k . Let Y (i) , ν (i) k , k = 1, 2, i = 1, 2,
be the local versions of Y and ν k in the two coordinate systems. With standard differential geometry arguments we have the following relations:

Y (1) = t κ ′ (x (1) )Y (2) , ν (1)
k = t κ ′ (x (1) )ν (2) k , dϕ (1)
k (x (1) ) = t κ ′ (x (1) )dϕ (2) k (x (2) ), Similarly let p (i) k and t j(i) k , k = 1, 2, j = 1, . . . , m, i = 1, 2,
be the local versions of the principal symbols of the differential operators P k and T j k . We have

p (1) k (x, ξ) = p (2) k (κ(x), t κ ′ (x) -1 ξ), t j(1) k (x, ξ) = t j(2) k (κ(x), t κ ′ (x) -1 ξ). If we set f (i) k (λ) = p (i) k (x (i) , Y (i) + λν (i) k + iτ dϕ (i) k (x (i) )), i = 1, 2, we find f (1) k (λ) = p (1)
k (x (1) , Y (1) + λν

(1) k + iτ dϕ (1) k (x (1) )) = p (2) k κ(x (2) ), t κ ′ (x (1) ) -1 (Y (1) + λν (1) k + iτ dϕ (1) k (x (1) )) = p (2)
k (x (2) , Y (2) + λν

(2) k + iτ dϕ (2) k (x (2) )) = f (2) k (λ),
which simply means that the polynomial function pk,ϕ defined in (1.9) does not depend on the coordinate system chosen. The same holds for the polynomial function tj k,ϕ defined in (1.11), which allows one to conclude that the transmission condition of Definition 1.4 can be stated (and checked) in any coordinate system.

1.4. Sobolev norms with a parameter. For non-negative integer m and a real number τ ≥ 0, we introduce the Sobolev spaces H m τ (Ω k ) and H m τ (S) defined by the following norms respectively:

(1.13) u 2 m,τ = m k=0 τ 2(m-k) u 2 H k (Ω k ) and |u| 2 m,τ = m k=0 τ 2(m-k) |u| 2 H k (S) ,
where we denote the usual Sobolev norms on Ω k and S by . H s (Ω k ) and |.| H s (S) . The L 2 inner-products on Ω k and S will be denoted by (., .) and (., .) ∂ respectively. Observe that for the norm . m,τ on H m τ (Ω k ) we do not specify explicitly the integer k that refers to which side of the interface we consider, Ω 1 or Ω 2 . In the main text there should never be any ambiguity as the norm will be used for functions that are clearly defined on one of the open sets. If u ∈ C ∞ (Ω k ) we set tr m (u) = (tr 0 (u), . . . , tr m (u)) where tr j (u) = ( 1 i ∂ ν ) j u |S , with ν conormal to S, is the sectional trace of u of order j and, in accordance with (1.14), we define

|tr m (u)| 2 m,s,τ = m j=0 |tr j (u)| 2 m-j+s,τ .
In what follows we shall write tr(u) in place of tr m (u) for concision. We shall also write norms of the form |e τ ϕ tr(u)| 2 m,s,τ actually meaning

|e τ ϕ tr m (u)| 2 m,s,τ = m j=0 |e τ ϕ tr j (u)| 2 m-j+s,τ .
1.5. Statement of the main result. We can now state the local Carleman estimate that we prove in the neighborhood of a point of the interface, with the sub-ellipticity and transmission conditions.

Theorem 1.6. Let x 0 ∈ S and let ϕ ∈ C 0 (Ω) be such that

ϕ k = ϕ |Ω k ∈ C ∞ (Ω k ) for k = 1, 2
and such that the pairs {P k , ϕ k } have the sub-ellipticity property of Definition 1.1 in a neighborhood of x 0 in Ω k . Moreover, assume that P k , ϕ, T j k , k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Then there exist a neighborhood W of x 0 in R n and two constants C and τ * > 0 such that

(1.15) k=1,2 τ -1 e τ ϕ k u k 2 m k ,τ + |e τ ϕ |S tr(u k )| 2 m k -1,1/2,τ ≤ C k=1,2 e τ ϕ k P k (x, D)u k 2 L 2 (Ω k ) + m j=1 e τ ϕ |S T j 1 (x, D)u 1 + T j 2 (x, D)u 2 |S 2 m-1/2-β j ,τ ,
for all u k = w k|Ω k with w k ∈ C ∞ c (W ) and τ ≥ τ * . First, this results will be established microlocally: at an interface point x 0 we shall assume that the transmission condition holds for some interface quadruple (x 0 , Y 0 , ν 0 , τ 0 ) and we shall prove that a Carleman estimate of the form above holds in a conic neighborhood of this interface quadruple in phase-space; localization in phase-space will be done by means of cut-off functions and associated pseudo-differential operators. We refer the reader to Section 4.3. Second, we will deduce Theorem 1.6 from such microlocal estimates.

Estimate (1.15) concerns function located near the interface and vanishing near the boundary ∂ Ω . Hence this estimate involves the transmission operators T j k and not the boundary operators B j . Estimates of the form of (1.15) are local. Yet, such estimates and their counterpart estimates at the boundary proven in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] can be patched together to form global estimates. We do not cover such details here and we refer to [START_REF] Rousseau | Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces[END_REF] where this is done in the case of a transmission problem.

In Section 6 we shall prove Carleman estimates with a weight function of the form ϕ(x) = exp(γψ(x)) as is usually done in practice with the parameter γ chosen as large as desired. We shall provide the precise dependency of the Carleman estimate with respect to this second large parameter.

Examples of elliptic transmission problem and weight function for which the above result applies, and other for which it does not, will be given in Section 1.7 below. In fact, as the condition for the Carleman estimate to hold are geometrical, that is, coordinate invariant, we shall postpone the exposition of example after we introduce local variables that ease the writting of the transmission condition. 1.6. Local reduction of the problem near the interface. All the different aspects of the problem we consider -operators, sub-ellipticity condition, transmission condition-are coordinate invariant as we saw above. We shall thus work locally, in the neighborhood of a point of the interface and choose coordinates that allow use to ease the subsequent analysis and derivation of the Carleman estimate. 1.6.1. Choice of local coordinates. Let x 0 ∈ S. There exists a neighborhood V of x 0 and a local system of coordinates

x = (x 1 , . . . , x n ) where V ∩ Ω 1 ⊂ {x n > 0}, V ∩ Ω 2 ⊂ {x n < 0} and x ′ = (x 1 , . . . , x n-1 )
parametrizes the interfaceV ∩ S ⊂ {x n = 0}. We assume that V ∩ ∂Ω = ∅, that is, we focus our analysis on the interface and remain away from the boudnary.

We denote by R n ± the half space {±x n > 0} and V ± = V ∩ R n ± . For our purpose here, without any loss of generality, we may assume that V ± is bounded.

In such local coordinates, in V ± , the differential operator3 P k of order m with complex coefficients takes the form

P k = P k (x, D) = m k j=1 P k,j (x, D ′ )D j n , D n = 1 i ∂ n , k = 1, 2,
where P k,j (x, D ′ ), j = 1, . . . , m k , k = 1, 2, are tangential differential operators with complex coefficients of order m kj. Similarly the transmission operators take the form

T j k = T j k (x, D) = β j k i=0 T j k,i (x, D ′ )D i n , 1 ≤ j ≤ m, k = 1, 2
where T j k,i (x, D ′ ), i = 0, . . . , β j k , are tangential differential operators of order β j ki. The local transmission problem we study thus takes the form

     P 1 u 1 = f 1 in V 1 = {x n < 0}, P 2 u 2 = f 2 in V 2 = {x n > 0},
T j 1 u 1 + T j 2 u 2 = g j , in S, j = 1, . . . , m.

(1.16)

We have P k,m = P k,m (x) = 0. Upon dividing the functions f k by P k,m (x) we may assume that P k,m = 1.

Calling (ξ ′ , ξ n ) the Fourier variables corresponding to (x ′ , x n ) we have, for the principal symbol of P k ,

p k (x, ξ) = m k j=0 p k,j (x, ξ ′ )ξ j n ,
which is a polynomial homogeneous of degree m k in the n variables (ξ ′ , ξ n ). We introduce

p k,ϕ (x, ξ, τ ) := p k (x, ξ + iτ ϕ ′ k (x)
), which is the principal symbol of the operator e τ ϕ k P k e -τ ϕ k viewed in the class of (pseudo-)differential operators with a large parameter presented in Section 2.

Setting ̺ ′ = (x, ξ ′ , τ ) and ̺ = (̺ ′ , ξ n ), for simplicity we shall write p k,ϕ (̺) in place of p k,ϕ (x, ξ, τ ) and often p k,ϕ (̺ ′ , ξ n ) to emphasize that the symbol is polynomial in ξ n . Similarly we introduce t j k,ϕ (x, ξ, τ

) := t j k (x, ξ + iτ ϕ ′ k (x)) = t j k,ϕ (̺) = t j k,ϕ (̺ ′ , ξ n ).
In the present local coordinate, we have θ = x n and thus ν k = (-1) k dx n (see Section 1.3). For a fixed point ̺ ′ = (x, ξ ′ , τ ) with x ∈ S, setting Y = n-1 j=1 ξ j dx j and ω = (x, Y, τ ) this gives pk,ϕ (ω, (-1)

k ξ n ) = p k (x, Y + (-1) k ξ n ν k + iτ dϕ k (x)) (1.17) = p k (x, ξ + iτ dϕ k (x)) = p k,ϕ (x, ξ, τ ),
with ξ = (ξ ′ , ξ n ) and pk,ϕ as defined in (1.9). Similarly, we have

tj k,ϕ (ω, (-1) k ξ n ) = t j k,ϕ (x, ξ), (1.18)
with tj k,ϕ as defined in (1.11). 1.6.2. A system formulation. To avoid the (-1) k terms in the two previous equations and to simplify a large part of the analysis that will follow we shall write the local transmission problem as a system of equation in V + .

Without any loss of generality we can choose the open neighborhood V to be of the form V ′ × (-ε, ε) with V ′ an open set of S and ε > 0.

In V -= V ′ × (ε, 0) we apply the change of variables

(x ′ , x n ) → σ(x ′ , x n ) = (x ′ , -x n ).
We denote by P ℓ and T j ℓ the operators obtained from P 1 and T j 1 through this change of variable. For the principal symbols we have

p ℓ (x, ξ) = p 1 σ(x); σ(ξ) , t ℓ (x, ξ) = t 1 σ(x); σ(ξ) , for x n > 0, using that t σ ′ (x) -1 ξ = (ξ ′ , -ξ n ).
We also define ϕ ℓ = ϕ 1 • σ for x n > 0. In V + we do not apply any change of variable and for the benefit of readibility we set P r = P 2 , p r = p 2 , and ϕ r = ϕ 2 . The subscripts ℓ and r are chosen to keep in mind that part of the system we shall write comes from the left-hand side of the interface and the second part from the right-hand side.

The transmission problems now reads as the following system

P ℓ u ℓ = f ℓ , P r u r = f r in V + = {x n > 0}, T j ℓ u ℓ + T j r u r = g j , in S = {x n = 0}, j = 1, . . . , m, (1.19)
where

u ℓ = u 1 • σ, f ℓ = f 1 • σ, u r = u 2 , f r = f 2 .
We set Pr / ℓ ,ϕ = e τ ϕr / ℓ Pr / ℓ e -τ ϕr / ℓ and T j r / ℓ ,ϕ = e τ ϕr / ℓ T j r / ℓ e -τ ϕr / ℓ . They have for respective principal symbols (in the calculus with a large parameter of Section 2)

pr / ℓ ,ϕ (x, ξ, τ ) = pr / ℓ (x, ξ + iτ dϕr / ℓ (x)), t j r / ℓ ,ϕ (x, ξ, τ ) = t j r / ℓ (x, ξ + iτ dϕr / ℓ (x)). We have p ℓ,ϕ (x, ξ, τ ) = p 1 (σ(x), σ(ξ + iτ dϕ ℓ (x))) = p 1 σ(x), σ(ξ) + iτ dϕ 1 (σ(x))) Now, as t σ ′ (x) -1 ν 1 = ν 2 = ν, for a fixed point ̺ ′ = (x, ξ ′ , τ ) with x ∈ S, i.e., x n = 0 giving σ(x) = x, if we set Y = n-1
j=1 ξ j dx j and ω = (x, Y, τ ) we have, for p1,ϕ as defined in (1.9), p1,ϕ (ω,

ξ n ) = p 1 (x, Y -ξ n ν + iτ dϕ 1 (x)) (1.20) = p ℓ (x, Y + ξ n ν + iτ dϕ ℓ (x)) = p ℓ,ϕ (x, ξ, τ ), with ξ = (ξ ′ , ξ n ). Similarly we have tj 1,ϕ (ω, ξ n ) = t j ℓ,ϕ (x, ξ, τ
pr / ℓ ,ϕ (̺ ′ , ξ n ) = p + r / ℓ ,ϕ (̺ ′ , ξ n ) p - r / ℓ ,ϕ (̺ ′ , ξ n ) p 0 r / ℓ ,ϕ (̺ ′ , ξ n ), ̺ ′ ∈ U , ξ n ∈ R, with p ±
r / ℓ ,ϕ and p 0 r / ℓ ,ϕ , polynomials in ξ n of constant degrees in U , smooth and homogeneous; in U the imaginary parts of the roots of p + r / ℓ ,ϕ (̺ ′ , ξ n ) (resp. p - r / ℓ ,ϕ (̺ ′ , ξ n )) are all positive (resp. negative) and we have

p ± r / ℓ ,ϕ (̺ ′ 0 , ξ n ) = ± Im αr / ℓ ,j >0 (ξ n -αr / ℓ ,j ) µr / ℓ ,j , p 0 r / ℓ ,ϕ (̺ ′ 0 , ξ n ) = Im αr / ℓ ,j =0
(ξ nαr / ℓ ,j ) µr / ℓ ,j .

The polynomials pr / ℓ ,ϕ are thus decomposed into three factors in the neighborhood U of ̺ ′ 0 . For p ± r / ℓ ,ϕ the sign of the imaginary part of their roots remain constant equal to ± respectively; for p 0 r / ℓ ,ϕ this sign may change and the roots are precisely real at ̺ ′ = ̺ ′ 0 . We then define the polynomial κr / ℓ ,ϕ (̺ ′ , ξ n ) by (1.23) κr

/ ℓ ,ϕ (̺ ′ , ξ n ) = p + r / ℓ ,ϕ (̺ ′ , ξ n ) p 0 r / ℓ ,ϕ (̺ ′ , ξ n ).
As above, for the principal symbols of the conjugated transmission operators T j r / ℓ ,ϕ , j = 1, . . . , m, we write t j r / ℓ ,ϕ (x, ξ, τ ) = t j r / ℓ ,ϕ (̺ ′ , ξ n ) where ̺ ′ = (x, ξ ′ , τ ) to emphasize that the symbol is polynomial in ξ n .

Remark 1.7. Observe that the factorizations in (1.22) depends quite significantly on the point ̺ ′ 0 . They may actually be different even for point ̺ ′ in the neighborhood U introduced above. We should rather write something like

pr / ℓ ,ϕ (̺ ′ , ξ n ) = p + r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) p - r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) p 0 r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ), ̺ ′ ∈ U , ξ n ∈ R, in place of (1.22) and set κr / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) = p + r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) p 0 r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ).
For ̺ ′ 1 ∈ U we may very well have

p + r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) = p + r / ℓ ,ϕ,̺ ′ 1 (̺ ′ , ξ n ), or p - r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) = p - r / ℓ ,ϕ,̺ ′ 1 (̺ ′ , ξ n ), or p 0 r / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) = p 0 r / ℓ ,ϕ,̺ ′ 1 (̺ ′ , ξ n
). Yet, we shall see below that the notation in (1.22) is sufficiently clear for our purpose.

Still, if we denote by M ± r / ℓ (̺ ′ ) the number of roots (counted with their multiplicities) with postive (resp. negative) imaginary parts of pr / ℓ ,ϕ (̺ ′ , ξ n ) for ̺ ′ ∈ U we may have

M ± r / ℓ (̺ ′ 0 ) = M ± r / ℓ (̺ ′ ) for some ̺ ′ ∈ U . Note that in such case we have M ± r / ℓ (̺ ′ 0 ) ≤ M ± r / ℓ (̺ ′
) from the construction of the neighborhood U given in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF]Lemma A.2]. Arguing as in the proof of [4, Lemma A.2], using the continuity of the roots w.r.t. ̺ ′ we can in fact prove that for ̺ ′ 1 ∈ U there exists a conic neighborhood

U ′ ⊂ U of ̺ ′ 1 such that κr / ℓ ,ϕ,̺ ′ 0 (̺ ′ , ξ n ) = hr / ℓ (̺ ′ , ξ n )κr / ℓ ,ϕ,̺ ′ 1 (̺ ′ , ξ n ), ̺ ′ ∈ U ′ , (1.24)
where hr / ℓ (̺ ′ , ξ n ) is polynomial in ξ n with coefficients that are smooth w.r.t. ̺ ′ ∈ U ′ . 1.6.4. The transmission conditions in the local coordinates. In the present coordinate system (x ′ , x n ) in V + , a conormal vector ν pointing from Ω 1 to Ω 2 is given by (0, . . . , 0, ν n ) with ν n > 0. For the statement of the transmission condition we may choose ν = (0, . . . , 0, 1). A boundary quadruple ω = (x, Y, ν, τ ), with Y = (ξ ′ , 0) can thus be identified with ̺ ′ = (x, ξ ′ , τ ).

The transmission condition of Definition 1.4 being invariant under change of variables as seen in Section 1.3.3, because of (1.20) and (1.21), we may use the polynomials pr / ℓ ,ϕ and t j r / ℓ ,ϕ to state locally this condition for Pr / ℓ , T j r / ℓ , ϕr / ℓ , j = 1, . . . , m at ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ). It reads: For all pairs of polynomials, qr / ℓ (ξ n ), there exist Ur / ℓ , polynomials, and c j ∈ C, j = 1, . . . , m, such that

q ℓ (ξ n ) = m j=1 c j t j ℓ,ϕ (̺ ′ , ξ n ) + U ℓ (ξ n )κ ℓ,ϕ (̺ ′ , ξ n ), (1.25) and q r (ξ n ) = m j=1 c j t j r,ϕ (̺ ′ , ξ n ) + U r (ξ n )κ r,ϕ (̺ ′ , ξ n ), (1.26) for ̺ ′ = ̺ ′ 0 . We set m - r / ℓ = d • p - r / ℓ ,ϕ (̺ ′ , .) , that is independent of ̺ ′ ∈ U
, with the open conic neighborhood U as introduced above, and we let κr / ℓ ,ϕ (̺ ′ , ξ n ) be the polynomial function given in (1.23). It takes the form

κr / ℓ ,ϕ (̺ ′ , ξ n ) = mr / ℓ -m - r / ℓ i=0 κr / ℓ ,i (̺ ′ )ξ i n , ̺ ′ ∈ U , ξ n ∈ R,
where κr / ℓ ,i is homogeneous of degree mr / ℓm - r / ℓi w.r.t. (ξ ′ , τ ). Similarly we write

t j r / ℓ ,ϕ (̺ ′ , ξ n ) = β j r / ℓ i=0 t j r / ℓ ,i (̺ ′ )ξ i n , ̺ ′ ∈ U , ξ n ∈ R, j = 1, 2,
with β j ℓ = β j 1 and β j r = β j 2 , and where t j r / ℓ ,i is homogeneous of degree β j r / ℓi w.r.t. (ξ ′ , τ ). We recall that we have (see (1.3))

m ℓ -β j ℓ = m r -β j r = m -β j , j = 1, . . . , m. (1.27)
Now, we introduce two famillies of polynomial functions, e j r / ℓ (̺ ′ , .), of degree less than or equal to mr

/ ℓ - 1, for j = 1, . . . , m ′ r / ℓ , with m ′ r / ℓ = m + m - r / ℓ . We recall that m = (m ℓ + m r )/2.
We set

e j r / ℓ (̺ ′ , ξ n ) = t j r / ℓ ,ϕ (̺ ′ , ξ n ) for 1 ≤ j ≤ m, ξ j-(m+1) n κr / ℓ ,ϕ (̺ ′ , ξ n ) for m + 1 ≤ j ≤ m ′ r / ℓ , if m ′ r / ℓ > m. Observe that m ′ r / ℓ > m if m - r / ℓ > 0. If we write e j r / ℓ (̺ ′ , ξ n ) = mr / ℓ -1 i=0 e j r / ℓ ,i (̺ ′ )ξ i n ,
we thus obtain

• for 1 ≤ j ≤ m, e j r / ℓ ,i = t j r / ℓ ,i if 0 ≤ i ≤ β j r / ℓ , 0 otherwise. • for m + 1 ≤ j ≤ m ′ r / ℓ , e j r / ℓ ,i = κr / ℓ ,i-j+m+1 if j -(m + 1) ≤ i ≤ j -(m + 1) + mr / ℓ -m - r / ℓ , 0 otherwise, if m ′ r / ℓ > m. For 1 ≤ j ≤ m, e j r / ℓ ,i is homogeneous of degree β j r / ℓ -i w.r.t. (ξ ′ , τ ). If m ′ r / ℓ > m, setting (1.28) β j r / ℓ = j + mr / ℓ -(m + m - r / ℓ + 1), j = m + 1, . . . , m ′ r / ℓ , we see that, for m + 1 ≤ j ≤ m ′ r / ℓ , the tangential symbol e j r / ℓ ,i is homogeneous of degree β j r / ℓ -i w.r.t. (ξ ′ , τ
) and the symbol e j r / ℓ is homogeneous of degree β j r / ℓ w.r.t. (ξ, τ ). Introducing the matrices

T 1 r / ℓ (̺ ′ ) = e j r / ℓ ,i-1 1≤i≤m r / ℓ 1≤j≤m , T 2 r / ℓ (̺ ′ ) = e j+m r / ℓ ,i-1 1≤i≤m r / ℓ 1≤j≤m - r / ℓ
, by Section 1.3.1, we see that the transmission condition of Definition 1.4 for [START_REF] Jerison | Unique continuation and absence of positive eigenvalues for Schrödinger operators[END_REF] with the local setting introduced here, reads as follows

Pr / ℓ , T j r / ℓ , ϕr / ℓ , j = 1, . . . , m at ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) ∈ S * T,τ (V ), with x 0 ∈ S, also stated in (1.25)-(1.
rank T (̺ ′ ) = m ℓ + m r = 2m, with T (̺ ′ ) =   T 1 ℓ (̺ ′ ) T 2 ℓ (̺ ′ ) 0 T 1 r (̺ ′ ) 0 T 2 r (̺ ′ )   , (1.29) for ̺ ′ = ̺ ′ 0 .
Note that 2m is the number of rows in T (̺ ′ ). We find again that m ′ = m + m - ℓ + m - r ≥ 2m. Moreover, there exists a 2m × 2m sub-matrix T 0 (̺ ′ 0 ) such that det T 0 (̺ ′ 0 ) = 0. As the coefficients of T 1 r / ℓ (̺ ′ ) and T 2 r / ℓ (̺ ′ ) are continuous and homogeneous of degree

β j r / ℓ -i + 1 and β j+m r / ℓ -i + 1 w.r.t. (ξ ′ , τ
) respectively, where j is the column number and i is the line number, we then have det T

0 (̺ ′ ) = 0 homogenous w.r.t. (ξ ′ , τ ). It follows that det T 0 (̺ ′ ) = 0 for ̺ ′ in a small conic neighborhood V ⊂ U of ̺ ′ 0 .
Note that the homogeneity of the coefficients is important for V to be chosen conic since det T 0 (̺ ′ ) is itself homogeneous w.r.t. (ξ ′ , τ ). The rank of T (̺ ′ ) thus remains equal to 2m in V , meaning that condition (1.29) is valid for ̺ ′ in the whole V .

We have thus reached the following result.

Proposition 1.8. Let the transmission condition Pr

/ ℓ , T j r / ℓ , ϕr / ℓ , j = 1, . . . , m hold at ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ). Then we have m - ℓ + m - r ≥ m. Moreover there exists a conic neighborhood V of ̺ ′ 0 such that (1.29) is valid for ̺ ′ ∈ V .
Remark 1.9. Observe that the result of Proposition 1.8 implies the condition (1.25)-(1.26) holds for ̺ ′ ∈ V ⊂ U , yet with κr / ℓ ,ϕ defined by the symbol factorizations at ̺ ′ 0 , that is κr / ℓ ,ϕ = κr / ℓ ,ϕ,̺ ′ 0 using the notation of Remark 1.7. Now using (1.24) we see that this implies that the transmission condition also holds at ̺ ′ 1 . We thus see that the transmission condition remains valid in a conic neighborhood of ̺ ′ 0 . However, we shall not use this aspect here. The importance aspect we shall use is the local persistence of condition (1.25)-(1.26) for ̺ ′ in a conic neighborhood of ̺ ′ 0 as stated in Proposition 1.8 (of course the two are very related). This explains why we do not use the "more precise" notation of Remark 1.7 throughout the article.

With the locally presistant decomposition of the conjugated opeators pr / ℓ ,ϕ = p - r / ℓ ,ϕ κr / ℓ ,ϕ in the neighborhood U of ̺ ′ 0 , the following states roughly the proof strategy we shall adopt: (1) The factors p - r / ℓ ,ϕ associated with roots with negative imaginary parts yields two perfect elliptic estimates at the interface and no transmission condition is needed.

(2) Each factors κr / ℓ ,ϕ yields an estimate at the interface that involves trace terms. These terms will be estimated via the actions of the transmission operators T j r / ℓ ,ϕ by means of to the transmission condition. Note that we have 2mm - ℓm - r ≤ m by Proposition 1.8. The number of trace relations available at the interface in (1. [START_REF]Linear partial differential operators[END_REF]) is precisely m. This indicates that we shall have at hand a sufficiently large number of transmission relation to control the terms originating from the estimate with the factors κr / ℓ ,ϕ that are of degree mr

/ ℓ -m - r / ℓ whose sum is m ℓ -m - ℓ + m r -m - r = 2m -m - ℓ -m - r .
1.7. Examples. We provide several examples that illustrate the generality of the elliptic transmission problems that can be addressed through the results of the present article.

1.7.1. Second-order elliptic operators. We start by providing fairly classical cases of transmission problems, where the operators are of second order on both sides of the interface. Consider the operators

P k = 1≤i,j≤n D i a (k) ij (x)D j , k = 1, 2,
with real coefficients, assumed to be ellitpic, that is, a

(k) ij = a (k)
ji and (a

(k) ij (x)) ≥ C > 0 uniformly for x ∈ Ω k . Let x 0 ∈ S. As above, local coordinates are chosen so that Ω 1 = {x n < 0} and Ω 2 = {x n > 0} in a neighborhood of x 0 .
A. A natural transmission problem for second-order elliptic operators. A natural and classical transmission problem can be stated with the following interface operators

T 1 k = (-1) k , T 2 k = (-1) k 1≤j≤n a (k) nj (x)D j , k = 1, 2.
The transmission problem thus reads

P 1 u 1 = f 1 in Ω 1 , P 2 u 2 = f 2 in Ω 2 ,
and

u 1|S = u 2|S and 1≤j≤n a (1) nj (x)D j u 1 |S = 1≤j≤n a (2) nj (x)D j u 2 |S ,
that is, we impose, at the interface, the continuity of the solution, as well as that of the normal flux (in the sense of the anisotropic diffusion matrices (a

(1) ij ) and (a (2) ij ))
. This is physically very natural, and, mathematically, it implies that piecewise smooth functions satisfying these two conditions are in the domain of the self-adjoint operator

P = ∇ • A∇, where A = (a ij (x)) with a ij (x) = a (k) ij (x) if x ∈ Ω k , k = 1, 2.
This configuration was treated in [START_REF] Rousseau | Carleman estimates for anisotropic elliptic operators with jumps at an interface[END_REF] following some works on some particular "conformal" cases [START_REF] Bellassoued | Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization[END_REF][START_REF] Rousseau | Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF]. Since this example has been extensively studied, it is natural to question if the results of the present article generalize those provided in these references.

We choose a weight function ϕ(x) that is smooth on both sides of S and continuous across S. Then for an interface quadruple ω = (x 0 , Y, ν, τ ), with here ν = (0, . . . , 0, 1),

Y = (ξ ′ , 0) = (ξ 1 , . . . , ξ n-1 , 0), we have, for k = 1, 2, t1 k,ϕ (ω, λ) = (-1) k , and t2 k,ϕ (ω, λ) = (-1) k a (k) nn (x 0 ) (-1) k λ + iτ ∂ xn ϕ k (x 0 ) + (-1) k 1≤j≤n-1 a (k) nj (x 0 ) ξ j + iτ ∂ x j ϕ k (x 0 ) .
As a (k) nn = 0 because of the ellipticity of P k we see that t2 k,ϕ (ω, λ) are exactly of degree 1 in λ. Observe that we can write the principal symbol of P k , k = 1, 2, as

p k (x, ξ) = a (k) nn ξ n + n-1 j=1 a (k) nj /a (k) nn ξ j 2 + b k (x, ξ ′ ) ,
with the quadratic form

b k (x, ξ ′ ) = a (k) nn -2 n-1 i,j=1 a (k) ij a (k) nn -a (k) ni a (k) nj ξ i ξ j ,
which is positive definite in ξ ′ . We thus find

pk,ϕ (ω, λ) = a (k) nn (x 0 ) (-1) k λ + iτ ∂ xn ϕ k (x 0 ) + n-1 j=1 a (k) nj (x 0 )/a (k) nn (x 0 )(ξ j + iτ ∂ x j ϕ k (x 0 )) 2 + b k (x 0 , ξ ′ + iτ d x ′ ϕ k (x 0 ))
In fact, we may write b

k (x, ξ ′ + iτ d x ′ ϕ k ) = (A k -iB k ) 2
, where A k and B k are functions of x, ξ ′ and τ , homogeneous of degree one in (ξ ′ , τ ), with A k ≥ 0, and we thus find

pk,ϕ (ω, λ) = a (k) nn k ′ =1,2 (-1) k λ + iτ ∂ xn ϕ k (x 0 ) + n-1 j=1 a (k) nj /a (k) nn (x 0 )(ξ j + iτ ∂ x j ϕ k (x 0 )) + (-1) k ′ (B k + iA k ) . Writing pk,ϕ (ω, λ) = a (k) nn (λ -σ 1 k (ω))(λ -σ 2 k (ω))
, several cases can occur depending on the signs of the imaginary part of the roots σ j k , k = 1, 2, j = 1, 2.

Case 1. Either p1,ϕ (ω, λ) or p2,ϕ (ω, λ) has two roots in {Im z < 0}. Assume that for instance p1,ϕ (ω, λ) has its two roots in {Im z < 0}. Then κ 1,ϕ = 1 while κ 2,ϕ (ω, λ) is of degree 0, 1, or 2. Let then q 1 (λ), q 2 (λ) be two polynomial functions. As t1 2,ϕ (ω, λ) = 1, and t2 2,ϕ (ω, λ) is exaclty of degree 1, we may write

q 2 (λ) = c 1 t1 2,ϕ (ω, λ) + c 2 t2 2,ϕ (ω, λ) + U 2 (λ)κ 2,ϕ (ω, λ).
by means of a Euclidean division by κ 2,ϕ (ω, λ), writing the remainder polynomial as a linear combination of t1 2,ϕ (ω, λ) and t2 2,ϕ (ω, λ). We then have

q 1 (λ) = c 1 t1 1,ϕ (ω, λ) + c 2 t2 1,ϕ (ω, λ) + U 1 (λ)κ 1,ϕ (ω, λ), by simply choosing U 1 (λ) = q 1 (λ) -c 1 t1 1,ϕ (ω, λ) -c 2 t2 1,ϕ (ω, λ).
The transmission condition of Definition 1.4 thus holds in this case.

Case 2. Each symbol pk,ϕ (ω, λ), k = 1, 2 has only one roots in {Im z < 0}. In this case, both polynomials κ 1,ϕ (ω, λ) and κ 2,ϕ (ω, λ) are of degree 1 in λ. As we chose A k ≥ 0, the root of pk,ϕ (ω, λ) with non-negative imaginary part is given by σ

+ k = -ζ k /a (k) nn (x 0 ) + B k + iA k with ζ k = (-1) k n-1 j=1 a (k) nj (x 0 )(ξ j + iτ ∂ x j ϕ j (x 0 )) + iτ a (k) nn (x 0 )∂ xn ϕ k (x 0 ) .
To understand whether the transmission condition holds or not, it is handy to use the matrix T introduced in (1.29). It is a 4 × 4 matrix here. Using (1.20) and (1.21), we find

T =         -1 ζ 1 -σ + 1 0 0 a (1) nn 1 0 1 ζ 2 0 -σ + 2 0 a (2) nn 0 1         .
Observe that the rank of T is 4 if and only if

ζ 1 + ζ 2 = -a (1)
nn σ + 1 -a (2)
nn σ + 2 . This yields the condition a (1) nn

(B 1 + iA 1 ) + a (2) nn (B 2 + iA 2 ) = 0. As we have A 1 ≥ 0 and A 2 ≥ 0 this condition holds if A 1 > 0 or A 2 > 0. In fact if A k = 0 this mean that Im σ 1 k = Im σ 2
k which is exluded here. Hence we have A 1 > 0 and A 2 > 0 and the transmission condition holds in this case.

Case 3. One symbol pk,ϕ (ω, λ) has two roots in {Im z ≥ 0} and the second one has at most one root in {Im z < 0}. Assume, for instance, that p1,ϕ (ω, λ) has two roots in {Im z ≥ 0}. Then, κ 1,ϕ is of degree 2. With the assumption on p2,ϕ (ω, λ), then, κ 1,ϕ is at least of degree 1. In this case, we find that the matrix T (̺ ′ ) has 4 lines and, at most, 3 columns. It cannot be of rank 4. Hence, the transmission condition cannot hold in this case.

From the three exhaustive cases studied above we thus conclude that the derivation of a Carleman estimate in a neighborhood of the point x 0 can be achieved, according to Theorem 1.6, if one chooses the weight function ϕ so that Case 3 does not occur.

Using the computation made above we write

pk,ϕ (ω, λ) = a (k) nn k ′ =1,2 (-1) k λ + n-1 j=1 a (k) nj /a (k) nn (x 0 )ξ j + iτ γ k + (-1) k ′ (B k + iA k ) .
where

γ k = ∂ xn ϕ k (x 0 ) + n-1 j=1 a (k) nj /a (k) nn (x 0 )∂ x j ϕ k (x 0 ) (1.30)
The values of γ k are fixed by the choice of the weight function ϕ. The imaginary parts of the roots are given by

Im α k ′ k = -(-1) k τ γ k + (-1) k ′ A k . (1.31) If both b k (x, ξ ′ + iτ d x ′ ϕ k ), k = 1, 2, are nonpositive real numbers, that is A k = 0, in particular if ξ ′ = 0
and τ > 0, then, the imaginary parts of the roots of pk,ϕ (ω, λ) coincide:

Im α k ′ k = -(-1) k τ γ k , k ′ = 1, 2.
In particular, this requires

γ 2 > 0 if γ 1 ≥ 0, and γ 1 < 0 if γ 2 ≤ 0, (1.32)
as, otherwise, we face the occurence of Case 3. The case γ 1 ≥ 0 and γ 2 ≤ 0 is thus excluded.

Let us assume that γ 1 ≥ 0 and γ 2 > 0. On the one hand, as we have assumed A k ≥ 0, the root α 2 1 remains in {Im z ≥ 0} and the root α 2 2 remains in {Im z < 0}. On the other hand, we have

Im α 1 1 = τ γ 1 -A 1 , Im α 1 2 = -τ γ 2 + A 2 . Hence, Case 3 does not occur if and only if Im α 1 2 ≥ 0 ⇒ Im α 1 1 < 0, that is, 0 < τ γ 2 ≤ A 2 ⇒ 0 ≤ τ γ 1 < A 1 . A sufficient condition is then γ 1 γ 2 < A 1 A 2 if A 2 = 0. (1.33)
Observe that in the case where ϕ k are only functions of x n , then A 1 and A 2 are independent of τ and (1.33) becomes a necessary and sufficient condition for the transmission condition to hold.

The case γ 1 < 0 and γ 2 ≤ 0 leads similarly to the sufficient condition

γ 2 γ 1 < A 2 A 1 if A 1 = 0.
Finally, we consider the case γ 1 < 0 and γ 2 > 0. Because of (1.31) we then find that Case 3 cannot occur with this choice of γ 1 and γ 2 . This particular choice, is however not very interesting, as it somehow corresponds to an observation 4 of the transmission problem both from Ω 2 and Ω 1 . The choice γ 1 ≥ 0 and γ 2 > 0 correspond to an observation of the transmission problem from Ω 2 only. This is relevant for pratical applications, for instance, in unique continuation problems, as one may want to find uniqueness across an interface, having information on the solution on one side of the interface only. Similarly, the choice γ 1 < 0 and γ 2 ≤ 0 corresponds to an observation from Ω 1 only.

Remark and open question. Note that with (1.33) we recover the condition stated in [START_REF] Rousseau | Carleman estimates for anisotropic elliptic operators with jumps at an interface[END_REF]. There, in the case ϕ k = ϕ k (x n ) it is proven to be sharp for a Carleman estimate to hold. This raises the following question: is the transmission condition presented here necessary and sufficient for the Carleman estimate to hold? Sufficiency is the subject of the present article. Necessity is clear in particular cases as shown in [START_REF] Rousseau | Carleman estimates for anisotropic elliptic operators with jumps at an interface[END_REF] but it is not clear in general. Second-order transmission problems, in the case where ϕ k depend on x n and also x ′ would be a natural field of investigation, but the question extends to higher order transmission problems.

B. Two "non communicating" Dirichlet problems. If we consider

T 1 1 = T 1 2 = 1, T 2 1 = -T 2 2 = 1
, observe, then, that the transmission problem

P 1 u 1 = f 1 in Ω 1 , P 2 u 2 = f 2 in Ω 2 , and
T 1 1 u 1 |S + T 1 2 u 2 |S = g 1 , T 2 1 u 1 |S + T 2 2 u 2 |S = g 2 ,
corresponds two having the following two problems

P 1 u 1 = f 1 , u 1|S = (g 1 + g 2 )/2, and
P 2 u 2 = f 2 , u 2|S = (g 1 -g 2 )/2,
that is, two well-posed elliptic equations with independent Dirichlet boundary conditions.

Using the notation and symbol computations of the previous example, let us assume that one of the symbols pk,ϕ (ω, λ), say for k = 1, has two roots in {Im z ≥ 0}. Then, the matrix T introduced in (1.29) takes the form

              1 0 1 0 1 0 -1 0 0 ⊗ T =
and thus cannot be of rank 4. Hence, to derive a Carleman estimate, according to Theorem 1.6, we need to avoid an occurance of two roots associated with the same symbol pk,ϕ (ω, λ) in {Im z ≥ 0}. With γ k , as defined in (1.30), and the imaginary parts of the roots given in (1.31), we note that we then need to impose γ 1 < 0 and γ 2 > 0. As explained in the previous example, this corresponds to observing the transmission problem from both sides of the interface S. This is totally sensible here, as the example we consider represents two totally decoupled ellitpic problems: observing from one side of the interface cannot yield any information about the system on the other side. As Theorem 1.6 implies unique continuation properties (see Section 7) we see that it is very natural that the Carleman estimate cannot be derived, unless observations are made on both sides.

1.7.2. Higher-order elliptic operators. Here, we consider an example that involves both a second-and a fourth-order elliptic operator. In R 2 , we consider the operators P k (x, D), k = 1, 2, such that in the local coordinates as above, the principal symbols are given by

p 1 (x, ξ 1 , ξ 2 ) = ξ 2 2 + b 1 (x, ξ 1 ), p 2 (x, ξ 1 , ξ 2 ) = ξ 4 2 + b 2 2 (x, ξ 1 )
, where b k (x, .), k = 1, 2, are two positive definite quadratic forms. We assume that the principal symbols of the transmission operators are given by

t 1 1 (x, ξ 1 , ξ 2 ) = -1, t 1 2 (x, ξ 1 , ξ 2 ) = 1, t 2 1 (x, ξ 1 , ξ 2 ) = -ξ 2 , t 2 2 (x, ξ 1 , ξ 2 ) = ξ 3 2 , t 3 1 (x, ξ 1 , ξ 2 ) = 0, t 3 2 (x, ξ 1 , ξ 2 ) = ξ 2 2 .
We choose a weight function ϕ(x) = ϕ(x 2 ) that is smooth on both sides of S and continous across S. Then for an interface quadruple ω = (x 0 , Y, ν, τ ), with ν = (0, 1), Y = (ξ 1 , 0), we have for

µ k = -iτ ϕ ′ k , t1 1,ϕ (ω, λ) = -1, t1 2,ϕ (ω, λ) = 1, t2 1,ϕ (ω, λ) = (λ + µ 1 ), t2 2,ϕ (ω, λ) = (λ -µ 2 ) 3 , t3 1,ϕ (ω, λ) = 0, t3 2,ϕ (ω, λ) = (λ -µ 2 ) 2 . For smooth function ϕ k , k = 1, 2, such that ϕ k is only a function of x 2 , we assume that ∂ x 2 ϕ 1 (x 0 ) > 0 and ∂ x 2 ϕ 2 (x 0 ) > 0. We obtain p1,ϕ (ω, λ) = (λ -α 1 )(λ -α 2 ), where α 1 = iτ ∂ x 2 ϕ 1 (x 0 ) + i b 1 (x 0 , ξ 1 ), α 2 = iτ ∂ x 2 ϕ 1 (x 0 ) -i b 1 (x 0 , ξ 1
). We thus have Im α 1 > 0. The sign of Im α 2 may however vary. We also have

p2,ϕ (ω, λ) = 4 j=1 (λ -β j )
where

β j = -iτ ∂ x 2 ϕ 2 (x 0 ) -e iπ(2j-1)/4 b 2 (x 0 , ξ 1 ), j = 1, 2, 3, 4.
As ∂ x 2 ϕ 2 (x 0 ) > 0, this forbid all the roots to be in the upper complex half plane. Then Im β 1 < 0 and Imβ 2 < 0. Yet, the signs of Im β 3 and Im β 4 are equal and may vary. We have

Im β 3 = Im β 4 = -τ ∂ x 2 ϕ 2 (x 0 ) + b 2 (x 0 , ξ 1 )/2.
According to Section 1.6.4, using (1.20) and (1.21), we have,

T 1 ℓ =   -1 µ 1 0 0 1 0   , T 1 r =         1 -µ 3 2 µ 2 2 0 3µ 2 2 -2µ 2 0 -3µ 2 1 0 1 0         . Case 1. Im α 2 < 0 and Im β 3 = Im β 4 < 0.
In this case, we have κ1,ϕ (ω, λ) = (λ-α 1 ) and κ2,ϕ (ω, λ) = 1. We then have

T 2 ℓ =   -α 1 1   and T 2 r = Id 4 .
Recalling the form of T in (1.29), we have

              T 1 ℓ T 1 r 0 Id 4 T = -α 1 1 0
whose rank is 6 as rankT 1 ℓ = 2. Hence, the transmission condition holds in this case, by its formulation given in (1.29). Case 2. Im α 2 ≥ 0 and Im β 3 = Im β 4 < 0. In such case, we have κ1,ϕ (ω, λ) = (λα 1 )(λα 2 ) and κ2,ϕ (ω, λ) = 1. As m - ℓ = 0 then no matrix T 2 ℓ enters in the composition of T . Still, this matrix has the same form as in Case 1 with the fourth column removed and, in this case, the rank is 6 implying that the transmission condition holds in this case. Case 3. Im α 2 < 0 and Im β 3 = Im β 4 ≥ 0. In such case, we have κ1,ϕ (ω, λ) = (λα 1 ) and κ2,ϕ (ω, λ) = (λβ 3 )(λβ 4 ). We thus have

T 2 ℓ =   -α 1 1   , T 2 r =         β 3 β 4 0 -(β 3 + β 4 ) β 3 β 4 1 -(β 3 + β 4 ) 0 1        
.

Thus, the matrix T reads

T =               -1 µ 1 0 -α 1 0 0 0 1 0 1 0 0 1 -µ 3 2 µ 2 2 0 β 3 β 4 0 0 3µ 2 2 -2µ 2 0 -(β 3 + β 4 ) β 3 β 4 0 -3µ 2 1 0 1 -(β 3 + β 4 ) 0 1 0 0 0 1              
.

Computing its determinant we find det(T ) = b 2 (x 0 , ξ 1 ) 2 that does not vanish as here ξ 1 = 0. In fact, ξ 1 = 0 yields Im α 2 = τ ∂ x 2 ϕ 1 (x 0 ) > 0 in contradiction with the assumption Im α 2 < 0 made here.

Case 4. Im α 2 ≥ 0 and Im

β 3 = Im β 4 ≥ 0. In this case we have κ1,ϕ (ω, λ) = (λ -α 1 )(λ -α 2 ) and κ2,ϕ (ω, λ) = (λ -α 3 )(λ -α 4
). In such case the matrix T is a 6 × 5 matrix. Its rank cannot be 6. The transmission condition cannot hold in this case.

The four exhaustive cases studied above reveal that the weight function ϕ needs to be chosen so that Case 4 does not occur. Hence, the following condition needs to be fulfilled:

Im β 3 = Im β 4 ≥ 0 ⇒ Im α 2 < 0.
Recalling the forms of the roots derived above this reads

0 < τ ϕ ′ 1 (x 0 ) ≤ b 2 (x 0 , ξ 1 )/2 ⇒ 0 < τ ϕ ′ 2 (x 0 ) < b 1 (x 0 , ξ 1 )
. A necessary and sufficient condition is then

ϕ ′ 1 (x 0 ) ϕ ′ 2 (x 0 ) < 2b 1 (x 0 , ξ 1 ) b 2 (x 0 , ξ 1 ) .
Since b 1 (x, ξ 1 )/b 2 (x, ξ 1 ) is bounded from below, for any ξ 1 , locally in x, we see that this yields a precise condition on the weight function ϕ. The condition prescribes a minimal relative jump of the normal derivative of the weight function across the interface (going from {x 2 < 0} to {x 2 > 0}). Note that one can also provide a sufficient condition in the case ϕ depends also on the x 1 variable, as in Example 1.7.1-A.

1.8. Notation. If V ⊂ R n +
we denote the semi-classical unit half cosphere bundle over V (in the cotangential direction ξ ′ ) by

S * T,τ (V ) = {(x, ξ ′ , τ ); x ∈ V, ξ ′ ∈ R n-1 , τ ∈ R + , |ξ ′ | 2 + τ 2 = 1}. The canonical inner product in C m is denoted by (z, z ′ ) C m = m-1 j=0 z j z ′ j , for z = (z 0 , . . . , z m-1 ), z ′ = (z ′ 0 , . . . , z ′ m-1 ) ∈ C m .
The associated norm will be denoted

|z| 2 C m = m-1 j=0 |z j | 2 .
We shall use some spaces of smooth functions in the closed half space. We set

S (R n + ) = {u |R n + ; u ∈ S (R n )}. For two u, v ∈ S (R n + ) we set (u, v) + = (u, v) L 2 (R n + ) u |xn=0 + , v |xn=0 + ∂ = u |xn=0 + , v |xn=0 + L 2 (R n-1 ) .
We also set

u + = u L 2 (R n + ) u |xn=0 + ∂ = u |xn=0 + L 2 (R n-1 ) .
In this article, when the constant C is used, it refers to a constant that is independent of the large parameter τ . Its value may however change from one line to another. If we want to keep track of the value of a constant we shall use another letter.

In what follows, for concision, we shall sometimes use the notation for ≤ C, with a constant C > 0. We shall write a ≍ b to denote a b a.

PSEUDO-DIFFERENTIAL OPERATORS WITH A LARGE PARAMETER

Parameter-dependent pseudo-differential operators have proven to be important tools for the derivation of Carleman estimates. The general aim is to obtain a pseudo-differential calculus with a large parameter, and then to derive estimates with constants that are independent of the parameter. Often such a pseudodifferential calculus is referred to as a semi-classical calculus.

2.1. Classes of symbols. We first introduce symbols that depend on a parameter.

Definition 2.1. Let a(̺) ∈ C ∞ (R n × R n ), ̺ = (x, ξ, τ ),
with τ as a parameter in [τ min , +∞), τ min > 0, and m ∈ R, be such that for all multi-indices α, β ∈ N n we have

(2.1) ∂ α x ∂ β ξ a(̺) ≤ C α,β λ m-|β| , x ∈ R n , ξ ∈ R n , τ ∈ [τ min , +∞)
,

where λ = |(ξ, τ )| = |ξ| 2 + τ 2 1 2
. Thus differentiation with respect to ξ improves the decay in ξ and τ simultaneously. We write a ∈ S m τ (R n × R n ) or simply S m τ . For a ∈ S m τ we denote by σ(a) its principal part, that is, its equivalence class in S m τ /S m-1 τ . We also introduce tangential symbols. Let

a(̺ ′ ) ∈ C ∞ (R n + × R n-1 ), ̺ ′ = (x, ξ ′ , τ ), with τ as a parameter in [τ min , +∞), τ min > 0, and m ∈ R, be such that for all multi-indices α ∈ N n , β ∈ N n-1 we have ∂ α x ∂ β ξ ′ a(̺ ′ ) ≤ C α,β λ m-|β| T , x ∈ R n + , ξ ′ ∈ R n-1 , τ ∈ [τ min , +∞)
,

where λ T = |(ξ ′ , τ )| = |ξ ′ | 2 + τ 2 1 2 . We write a ∈ S m T,τ (R n + × R n-1
) or simply S m T,τ . For a ∈ S m T,τ we denote by σ(a) its principal part, that is, its equivalence class in S m T,τ /S m-1 T,τ . We also introduce symbol classes that behave polynomially in the

ξ n variable. Let a(̺) ∈ C ∞ (R n + ×R n ),
with τ as a parameter in [τ min , +∞), τ min > 0, and m ∈ N and r ∈ R, be such that

a(̺) = m j=0 a j (̺ ′ )ξ j n , a j ∈ S m-j+r T,τ , ̺ = (̺ ′ , ξ n ), ̺ ′ = (x, ξ ′ , τ ), with x ∈ R n + , ξ ∈ R n , τ ≥ τ min , and ξ n ∈ R. We write a(̺) ∈ S m,r τ (R n + × R n ) or simply S m,r τ . Note that we have S m,r τ ⊂ S m+m ′ ,r-m ′ τ , if m, m ′ ∈ N and r ∈ R. We shall call the principal symbol of a the symbol σ(a)(̺) = m j=0 σ(a j )(̺ ′ )ξ j n ,
which is a representative of the class of a in S m,r τ /S m,r-1 τ . Note that S m,r τ ⊂ S m+r τ . For example consider a(x, ξ, τ

) = |(ξ ′ , τ )|ξ n for |(ξ ′ , τ )| ≥ 1. We have a ∈ S 2,0 τ ∩ S 1,1
τ and yet a / ∈ S 2 τ . In fact observe that differentiating with respect to ξ ′ yields

|∂ α ξ ′ a(x, ξ, τ )| ≤ C α |(ξ ′ , τ )| 1-|α| |ξ n |.
An estimate of the form of (2.1) is however not achieved for |α| ≥ 2. A microlocalization is required to repair this flaw and to use the two different symbol classes in a pseudo-differential calculus (See [START_REF]The analysis of linear partial differential operators[END_REF]Theorem 18.1.35]).

Finally, we define the corresponding spaces of poly-homogeneous symbols. Such symbols are often referred to as classical symbols; they are characterized by an asymptotic expansion where each term is positively homogeneous with respect to (ξ, τ ) (resp. (ξ ′ , τ )):

Definition 2.2. We shall say a ∈ S m τ,cl (R n × R n ) or simply S m τ,cl (resp. S m T,τ,cl (R n + × R n-1 ) or simply S m T,τ,cl ) if there exists a (j) ∈ S m-j τ (resp. S m-j T,τ ), homogeneous of degree m -j in (ξ, τ ) for |(ξ, τ )| ≥ r 0 , (resp. (ξ ′ , τ ) for |(ξ ′ , τ )| ≥ r 0 ), with r 0 ≥ 0, such that (2.2) a ∼ j≥0 a (j) , in the sense that a - N j=0 a (j) ∈ S m-N -1 τ (resp. S m-N -1 T,τ
).

A representative of the principal part is then given by the first term in the expansion.

Finally for m ∈ N and r ∈ R, we shall say that a(̺) ∈ S m,r τ,cl (R

n + × R n ) or simply S m,r τ,cl , if a(̺) = m j=0 a j (̺ ′ )ξ j n , with a j ∈ S m-j+r T,τ,cl , ̺ = (̺ ′ , ξ n ).
The principal part is given by m j=0 σ(a j )(̺ ′ )ξ j n and is homogeneous of degree m in (ξ, τ ).

Classes of semi-classical pseudo

-differential operators. For a ∈ S m τ (R n × R n ) (resp. S m τ,cl (R n × R n )) we define the following pseudo-differential operator in R n : a(x, D, τ )u(x) = Op(a)u(x) = (2π) -n ∫ R n e i(x,ξ) a(x, ξ, τ )û(ξ) dξ, u ∈ S (R n ), (2.3)
where û is the Fourier transform of u. In the sense of oscillatory integrals we have a(x, D, τ )u(x) = Op(a)u(x) = (2π) -n ∫∫ R 2n e i(x-y,ξ) a(x, ξ, τ )u(y) dξ dy.

We write

+ × R n ) (resp. S m T,τ,cl (R n + × R n )) we set a(x, D ′ , τ )u(x) = Op(a)u(x) = (2π) -(n-1) ∫∫ R 2n-2 e i(x ′ -y ′ ,ξ ′ ) a(x, ξ ′ , τ )u(y ′ , x n ) dξ ′ dy ′ , (2.4) for u ∈ S (R n + ), where x ∈ R n + . Here D ′ denotes D x ′ . We write A = Op(a) ∈ Ψ m T,τ (R n + ) or simply Ψ m T,τ (resp. Ψ m T,τ,cl (R n + ) or simply Ψ m T,τ,cl ). The principal symbol of A = Op(a) is σ(A) = σ(a) ∈ S m T,τ /S m-1 T,τ (resp. S m T,τ,cl /S m-1 T,τ,cl ). Finally for m ∈ N, r ∈ R, and a ∈ S m,r τ (resp. S m,r τ,cl ) with a(̺) = m j=0 a j (̺ ′ )ξ j n , a j ∈ S m-j+r T,τ (resp. S m-j+r T,τ,cl ), ̺ = (̺ ′ , ξ n ), we set a(x, D, τ ) = Op(a) = m j=0 a j (x, D ′ , τ )D j n ,
and we write

A = Op(a) ∈ Ψ m,r τ (R n + ) or simply Ψ m,r τ (resp. Ψ m,r τ,cl (R n + ) or simply Ψ m,r τ,cl ). The principal symbol of A is σ(A)(̺) = σ(a)(̺) = m j=0 σ(a j )(̺ ′ )ξ j n in S m,r τ /S m,r-1 τ (resp. S m,r τ,cl /S m,r-1 τ,cl
).

We provide some basic calculus rules in the case of tangential operators. (resp. S m+m ′ T,τ,cl ) defined by the (oscillatory) integral:

c(̺ ′ ) = (a # b)(̺ ′ ) = (2π) -(n-1) ∫∫ e -i(y ′ ,η ′ ) a(x, ξ ′ + η ′ , τ ) b(x ′ + y ′ , x n , ξ ′ , τ ) dy ′ dη ′ = |α|<N (-i) |α| α! ∂ α ξ ′ a(̺ ′ ) ∂ α x ′ b(̺ ′ ) + r N , where r N ∈ S m+m ′ -N T,τ (resp. S m+m ′ -N T,τ,cl
) is given by

r N = (-i) N (2π) (n-1) |α|=N 1 ∫ 0 N (1 -s) N -1 α! ∫∫ e -i(y ′ ,η ′ ) ∂ α ξ ′ a(x, ξ ′ + η ′ , τ )∂ α x ′ b(x ′ + sy ′ , x n , ξ ′ , τ ) dy ′ dη ′ ds.
Proposition 2.4 (formal adjoint). Let a ∈ S m T,τ (resp. S m T,τ,cl ). There exists a * ∈ S m T,τ (resp. S m T,τ,cl ) such that

(Op(a)u, v) + = (u, Op(a * )v) + , u, v ∈ S (R n + )
. and a * is given be the following asymptotic expansion

a * (̺ ′ ) = (2π) -(n-1) ∫∫ e -i(y ′ ,η ′ ) a(x ′ + y ′ , x n , ξ ′ + η ′ , τ ) dy ′ dη ′ = |α|<N (-i) |α| α! ∂ α ξ ′ ∂ α x ′ a(̺ ′ ) + r N , r N ∈ S m-N T,τ (resp. S m-N T,τ,cl ),
where

r N = (-i) N (2π) (n-1) |α|=N 1 ∫ 0 N (1 -s) N -1 α! ∫∫ e -i(y ′ ,η ′ ) ∂ α ξ ′ ∂ α x ′ a(x ′ + sy ′ , x n , ξ ′ + η ′ , τ ) dy ′ dη ′ ds.
We denote Op(a) * = Op(a * ). We refer to Op(a) * as to the formal adjoint of Op(a).

A consequence of the previous calculus results is the following proposition.

Proposition 2.5. Let a(̺ ′ ) ∈ S m T,τ (resp. S m T,τ,cl ) and b(̺ ′ ) ∈ S m ′ T,τ (resp. S m ′ T,τ,cl ), with m, m ′ ∈ R. Define h(̺ ′ ) = D x ′ (b∂ ξ ′ a)(̺ ′ ) ∈ S m+m ′ -1 τ . Then we have Op(a) * Op(b) -Op(ab + h) ∈ Ψ m+m ′ -2 T,τ (resp. Ψ m+m ′ -2 T,τ,cl ), or equivalently a * #b -ab -h ∈ S m+m ′ -2 T,τ (resp. S m+m ′ -2 T,τ,cl
).

For semi-classical operators in the half space with symbols that are polynomial in ξ n we also provide a notion of formal adjoint. In other words, in this definition we ignore the possible occurrence of boundary terms when performing the operator transposition.

Note that for a ∈ S m T,τ,cl we have [D n , Op(a)] = Op(D n a) ∈ Ψ m T,τ,cl and more generally, for j ≥ 1, we have

D j n , Op(a) = j-1 k=0 Op(α k )D k n , α k ∈ S m T,τ,cl ,
where the symbols α k involve various derivatives of a in the x n -direction. As an application we see that if we consider a j ∈ S m-j+r T,τ,cl then we have

m j=0 D j n a j (x, D ′ , τ ) = m j=0 ãj (x, D ′ , τ )D j n ,
where ãj ∈ S m-j+r T,τ,cl and its principal part satisfies σ(ã j ) ≡ a j in S m-j+r

T,τ /S m-j+r-1 T,τ
. Hence

σ m j=0 D j n a j (x, D ′ , τ ) = m j=0 a j (x, ξ ′ , τ )ξ j n mod S m,r-1 τ .
From the calculus rules given above for the tangential operators and the above observation we have the following results on the principal symbols. ). ).

Proposition 2.7. Let a ∈ S m,r τ (resp. S m,r τ,cl ) and b ∈ S m ′ ,r ′ τ (resp. S m ′ ,r ′ τ,cl ) with a(̺) = m j=0 a j (̺ ′ )ξ j n , b(̺) = m ′ j=0 b j (̺ ′ )ξ j n , ̺ = (̺ ′ , ξ n ), ̺ ′ = (x, ξ ′ , τ ). (1

Moreover we have

(2) a(x, D, τ )b(x, D, τ ) ∈ Ψ m+m ′ ,r+r ′ τ (resp. Ψ m+m ′ ,r+r ′ τ,cl
) and

σ a(x, D, τ )b(x, D, τ ) ≡ 0≤j≤m 0≤k≤m ′ a j (̺ ′ )b k (̺ ′ )ξ j+k n ∈ S m+m ′ ,r+r ′ τ /S m+m ′ ,r+r ′ -1 τ (resp. S m+m ′ ,r+r ′ τ,cl /S m+m ′ ,r+r ′ -1 τ,cl ). We have Op(a) Op(b)u -Op(ab)u ∈ Ψ m+m ′ ,r+r ′ -1 τ (resp. Ψ m+m ′ ,r+r ′ -1 τ,cl
).

Sobolev continuity results.

Here we state continuity results for the operators defined above using the Sobolev norms with parameters introduced in Section 1.4. Such results can be obtained from their standard counterparts.

Let λ T (ξ ′ , τ ) = τ 2 + |ξ ′ | 2 1/2 and Λ T := Op(λ T ). For a given real number s, the boundary norm given by (1.14) is equivalent to the following norms (see (1.13) for the definition of |.| p,τ ):

(2.5) |u| 2 m,s,τ = m k=0 |Λ s T u k | 2 m-k,τ , u = (u 0 , . . . , u m ) ∈ S (R n-1 ) m+1 .
Moreover, we define the following semi-classical interior norm

(2.6) u 2 m,s,τ = Λ s T u 2 m,τ , u ∈ S (R n + ).
Proposition 2.8. If a(̺) ∈ S m,r τ , with m ∈ N and r ∈ R, then for m ′ ∈ N and r ′ ∈ R there exists C > 0 such that

Op(a)u m ′ ,r ′ ,τ ≤ C u m+m ′ ,r+r ′ ,τ , u ∈ S (R n +
). A consequence of this results and Proposition 2.7 is the following property.

Corollary 2.9. Let a ∈ S m,r τ and m ′ ∈ N and s ∈ R. We have

a(x, D, τ ) * u -a(x, D, τ )u m ′ ,s,τ ≤ C u m+m ′ ,r+s-1,τ , u ∈ S (R n + ).
The following simple inequality will be used implicitly at many places in what follows when we invoke the parameter τ to be chosen sufficiently large. This will then allow us to absorb semi-classical norms of lower order.

Corollary 2.10. Let m ∈ N and s ∈ R and ℓ ≥ 0. For some C > 0, we have

u m,s,τ ≤ Cτ -ℓ u m,s+ℓ,τ , u ∈ S (R n +
). This implies that u m,s,τ ≪ u m,s+ℓ,τ for τ sufficiently large.

INTERFACE QUADRATIC FORMS

For a(̺) ∈ S p,σ τ,cl (R

n + × R n ), we have a(̺) = p j=0 a j (̺ ′ )ξ j n , with a j ∈ S p-j+σ T,τ,cl (R n + × R n-1 ),
and for z = (z 0 , . . . , z p ) ∈ C p+1 we set

Σ a (̺ ′ , z) = p j=0 a j (̺ ′ )z j . (3.1)
We let m ℓ and m r be two integers. For applications of the results of this section we shall use the values of mr / ℓ that come with the elliptic transmission problem we consider in the present article. 2 . We say that

G (w) = N s=1 A s ℓ w ℓ|x n=0 + + A s r w r|x n=0 + , B s ℓ w ℓ|x n=0 + + B s r w r|x n=0 + ∂ , with A s r / ℓ = a s r / ℓ (x, D, τ ) and B s r / ℓ = b s r / ℓ (x, D, τ ), is an interface quadratic form of type (m ℓ -1, m r -1, σ) with C ∞ coefficients, if for each s = 1, . . . , N , we have a s r / ℓ (̺), b s r / ℓ (̺) ∈ S mr / ℓ -1,σr / ℓ τ,cl (R n + × R n ), with σ ℓ + σ r = 2σ, ̺ = (̺ ′ , ξ n ) with ̺ ′ = (x, ξ ′ , τ ). For w = (z ℓ , z r ), w = (z ℓ , zr ) ∈ C m ℓ × C mr , z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ), zr / ℓ = (z r / ℓ 0 , . . . , zr / ℓ mr / ℓ -1 ) ∈ C
mr / ℓ with the interface quadratic form G we associate the following bilinear symbol Definition 3.2. Let W be an open conic set in R n-1 × R n-1 × R + and let G be an interface quadratic form of type (m ℓ -1, m r -1, σ) associated with the bilinear symbol Σ G (̺ ′ , w, w). We say that G is positive definite in W if there exists C > 0 and R > 0 such that

Σ G (̺ ′ , w, w) = N s=1 Σ a s ℓ (̺ ′ , z ℓ ) + Σ a s r (̺ ′ , z r ) Σ b s ℓ (̺ ′ , zℓ ) + Σ b s r (̺ ′ , zr ) .
Re Σ G (̺ ′′ , x n = 0 + , w, w) ≥ C m ℓ -1 j=0 λ 2(m ℓ -1-j+σ ℓ ) T z ℓ j 2 + mr -1 j=0 λ 2(mr -1-j+σr) T z r j 2 ,
for any w = (z ℓ , z r ), z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ) ∈ C mr / ℓ , and

̺ ′′ = (x ′ , ξ ′ , τ ) ∈ W , τ ≥ 0, such that λ T = |(ξ ′ , τ )| ≥ R.
Then we have the following Lemma

Lemma 3.3. Let W be an open conic set in R n-1 × R n-1 × R + and let G be an interface quadratic form of type (m ℓ -1, m r -1, σ) that is positive definite in W . Let χ ∈ S 0 T,τ be homogeneous of degree 0, with supp(χ |xn=0 + ) ⊂ W and let N ∈ N. Then there exist τ * ≥ 1, C > 0, C N > 0 such that Re G (Op(χ)u) ≥ C |tr(Op(χ)u ℓ )| 2 m ℓ -1,σ ℓ ,τ + |tr(Op(χ)u r )| 2 mr-1,σr,τ -C N |tr(u ℓ )| 2 m ℓ -1,σ ℓ -N,τ + |tr(u r )| 2 mr-1,σr-N,τ for u = (u ℓ , u r ) ∈ S (R n + ) 2 and τ ≥ τ * .
Proof. The interface quadratic form can be written as

G (u) = m ℓ -1 j,k=0 G ℓℓ kj Λ m ℓ -1-j+σ ℓ T D j n u ℓ |xn=0 + , Λ m ℓ -1-k+σ ℓ T D k n u ℓ |xn=0 + ∂ + mr-1 j,k=0
G rr kj Λ mr -1-j+σr

T D j n u r |xn=0 + , Λ mr -1-k+σr T D k n u r |xn=0 + ∂ + m ℓ -1 j=0 mr-1 k=0 G rℓ kj Λ m ℓ -1-j+σ ℓ T D j n u ℓ |xn=0 + , Λ mr-1-k+σr T D k n u r |xn=0 + ∂ + mr-1 j=0 m ℓ -1 k=0 G ℓr kj Λ mr-1-j+σr T D j n u r |xn=0 + , Λ m ℓ -1-k+σ ℓ T D k n u ℓ |xn=0 + ∂ ,
where

G ii ′ jk = Op(g ii ′ jk ) ∈ Ψ 0 T,τ,cl , with i, i ′ = r / ℓ . We set the 2m × 2m-matrix tangential symbol g(̺ ′ ) =   g ℓℓ g ℓr g rℓ g rr   (̺ ′ ), g ii ′ (̺ ′ ) = g ii ′ jk (̺ ′ ) 0≤j≤m i -1 0≤k≤m i ′ -1 , i, i ′ = r / ℓ .
We introduce χ ∈ S 0 T,τ that has the same properties as χ with moreover 0 ≤ χ ≤ 1 and χ = 1 in a neighborhood of supp χ. We then set

g = χg + (1 -χ)I 2m ,
where I 2m is the 2m × 2m identity matrix.

As G is positive definite in W we have, for some C > 0,

Re g(̺ ′′ , x n = 0 + )w, w ≥ C |w| 2 C 2m , ̺ ′′ ∈ W , w ∈ C 2m . Therefore we have, for some C ′ > 0, Re g(̺ ′′ , x n = 0 + )w, w ≥ C ′ |w| 2 C 2m , ̺ ′′ ∈ R n-1 × R n-1 × R + , w ∈ C 2m . (3.2)
For a function v we define the mr / ℓ -tuple functions Vr / ℓ = (vr / ℓ ,0 , . . . , vr / ℓ ,mr / ℓ -1 ) by

vr / ℓ ,k = Λ mr / ℓ +σr / ℓ -1-k T D k n vr / ℓ |xn=0 + , k = 0, . . . , mr / ℓ -1. We then have, for N ∈ Z, Vr / ℓ 2 N,τ = mr / ℓ -1 k=0 vr / ℓ ,k 2
N,τ = mr / ℓ -1 k=0 Λ mr / ℓ +σr / ℓ -1-k T D k n vr / ℓ |xn=0 + 2 N,τ (3.3)
≍ mr / ℓ -1 k=0 Λ σr / ℓ +N T D k n vr / ℓ 2 mr / ℓ -1-k,τ = tr(vr / ℓ ) 2 mr / ℓ -1,σr / ℓ +N,τ .
We set ur / ℓ = Op(χ)ur / ℓ and introduce Ur / ℓ = (ur / ℓ ,0 , . . . , ur / ℓ ,mr / ℓ -1 ) and U r / ℓ = (ur / ℓ ,0 , . . . , ur / ℓ ,m-1 ) as above:

ur / ℓ ,k = Λ mr / ℓ +σr / ℓ -1-k T D k n ur / ℓ |xn=0 + , ur / ℓ ,k = Λ mr / ℓ +σr / ℓ -1-k T D k n ur / ℓ |xn=0 + , k = 0, . . . , mr / ℓ -1.
Setting t U = (U ℓ , U r ) we obtain

G (u) = Op(g |xn=0 + )U , U ∂ . Writing g = g + r with r = (g -I 2m )(1 -χ) we find G (u) = Op(g |xn=0 + )U , U ∂ + Op(r |xn=0 + )U , U ∂
As the supports of 1χ and χ are disjoint, with the pseudo-differential calculus, for any N ∈ N we have for some

C N > 0 Op(r |xn=0 + )U , U ∂ ≤ C N |U | 2 -N,τ . (3.4)
Next, from (3.2) with the Gårding inequality in the tangential direction we deduce that for some C > 0 we have

Re Op(g |xn=0 + )U , U ∂ ≥ C |U | 2 ∂ , (3.5
′ 0 in V + × R n-1 × R + such that m j=1 λ 2(m-1/2-β j ) T Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + Σ t j r,ϕ (̺ ′ , z r ) 2 + m ′ ℓ j=m+1 λ 2(m ℓ -1/2-β j ℓ ) T Σ e j ℓ,ϕ (̺ ′ , z ℓ ) 2 + m ′ r j=m+1 λ 2(mr-1/2-β j r ) T Σ e j r,ϕ (̺ ′ , z r ) 2 ≥ C m ℓ -1 j=0 λ 2(m ℓ -1/2-j) T |z ℓ j | 2 + mr -1 j=0 λ 2(mr -1/2-j) T |z r j | 2 , for ̺ ∈ U 1 and z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ) ∈ C mr / ℓ .
We recall that β j = (β j ℓ + β j r )/2 for j = 1, . . . , m, and that we have (1.27).

Proof. With the transmission condition holding at ̺ ′ 0 , by Proposition 1.8 there exists a conic open set U 1 neighborhood of ̺ ′ 0 where condition (1.29) is valid. Observe that K = U 1 ∩ S * T,τ (V ) is compact, recalling that V + is bounded.

Let ̺ ′ 1 ∈ K and T (̺ ′ 1 ) be as in (1.29). We have

rank T (̺ ′ 1 ) = rank T (̺ ′ 1 ) t T (̺ ′ 1 ) = 2m, For w = (z ℓ , z r ) ∈ C 2m , z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1
) ∈ C mr / ℓ , recalling that 2m = m ℓ + m r , we thus have

T (̺ ′ 1 ) t T (̺ ′ 1 )w, w ≥ C|w| 2 C 2m
, for some C > 0. Observe that we have

T (̺ ′ 1 ) t T (̺ ′ 1 ) =   T 1 ℓ (̺ ′ 1 ) t T 1 ℓ (̺ ′ 1 ) T 1 ℓ (̺ ′ 1 ) t T 1 r (̺ ′ 1 ) T 1 r (̺ ′ 1 ) t T 1 ℓ (̺ ′ 1 ) T 1 r (̺ ′ 1 ) t T 1 r (̺ ′ 1 )   +   T 2 ℓ (̺ ′ 1 ) t T 2 ℓ (̺ ′ 1 ) 0 0 T 2 r (̺ ′ 1 ) t T 2 r (̺ ′ 1 )   .
For w = (z ℓ , z r ), z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ) ∈ C mr / ℓ , we have

T (̺ ′ 1 ) t T (̺ ′ 1 )w, w = t T 1 ℓ (̺ ′ 1 )z ℓ + t T 1 r (̺ ′ 1 )z r 2 C m + t T 2 ℓ (̺ ′ 1 )z ℓ 2 C m - ℓ + t T 2 r (̺ ′ 1 )z r 2 C m - r = m j=1 m ℓ -1 i=0 t j ℓ,i (̺ ′ 1 )z ℓ i + mr-1 i=0 t j r,i (̺ ′ 1 )z r i 2 + m - ℓ j=1 m ℓ -1 i=0 e j+m ℓ,i (̺ ′ 1 )z ℓ i 2 + m - r j=1 mr-1 i=0 e j+m r,i (̺ ′ 1 )z r i 2 = m j=1 Σ t j ℓ,ϕ (̺ ′ 1 , z ℓ ) + Σ t j r,ϕ (̺ ′ 1 , z r ) 2 + m - ℓ j=1 Σ e j+m ℓ,ϕ (̺ ′ 1 , z ℓ ) 2 + m ′- r j=1 Σ e j+m r,ϕ (̺ ′ 1 , z r ) 2 .
We thus obtain

m j=1 Σ t j ℓ,ϕ (̺ ′ 1 , z ℓ ) + Σ t j r,ϕ (̺ ′ 1 , z r ) 2 + m - ℓ j=1 Σ e j+m ℓ,ϕ (̺ ′ 1 , z ℓ ) 2 + m - r j=1 Σ e j+m r,ϕ (̺ ′ 1 , z r ) 2 |(z ℓ , z r )| 2 .
By continuity this inequality remains true in a small neighborhood of ̺ ′ 1 in K. Using the compactness of K we thus find

m j=1 Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + Σ t j r,ϕ (̺ ′ , z r ) 2 + m - ℓ j=1 Σ e j+m ℓ,ϕ (̺ ′ , z ℓ ) 2 + m - r j=1 Σ e j+m r,ϕ (̺ ′ , z r ) 2 |(z ℓ , z r )| 2 , for ̺ ′ ∈ K and z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ) ∈ C mr / ℓ . Introducing the map M t ̺ ′ = (x, tη), ̺ ′ = (x, η) ∈ R n + × R n-1 × R + , t > 0, as we have U 1 = {M t ̺ ′ ; t > 0, ̺ ′ ∈ K}, we find m j=1 Σ t j ℓ,ϕ (M t ̺ ′ , zℓ ) + Σ t j r,ϕ (M t ̺ ′ , zr ) 2 + m - ℓ j=1 Σ e j+m ℓ,ϕ (M t ̺ ′ , zℓ ) 2 + m - r j=1 Σ e j+m r,ϕ (M t ̺ ′ , zr ) 2 |(z ℓ , zr )| 2 , where t = λ -1 T = |(ξ ′ , τ )| -1 and zr / ℓ = (z r / ℓ 0 , . . . , zr / ℓ mr / ℓ -1 ) ∈ C mr / ℓ , with zr / ℓ j = t -mr / ℓ +1/2+j z r / ℓ j , yielding m j=1 λ -(β j ℓ -m ℓ +1/2) T Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + λ -(β j r -mr+1/2) T Σ t j r,ϕ (̺ ′ , z r ) 2 + m - ℓ j=1 λ -(β j ℓ -m ℓ +1/2) T Σ e j+m ℓ,ϕ (̺ ′ , z ℓ ) 2 + m - r j=1 λ -(β j r -mr+1/2) T Σ e j+m r,ϕ (̺ ′ , z r ) 2 m ℓ -1 j=0 |λ m ℓ -1/2-j T z ℓ j | 2 + mr-1 j=0 |λ mr -1/2-j T z r j | 2 .
With (1.27) we obtain the sought result.

PROOF OF THE CARLEMAN ESTIMATE

As is usual in the proof of Carleman estimates we consider the following conjugated operators Pr / ℓ ,ϕ = e τ ϕr / ℓ Pr / ℓ e -τ ϕr / ℓ .

As e τ ϕr / ℓ D j e -τ ϕr / ℓ = D j + iτ ∂ j ϕr / ℓ ∈ Ψ 1,0 τ,cl , we see that Pr / ℓ ,ϕ ∈ Ψ 2m,0 τ,cl . Their principal symbols are given by pr / ℓ ,ϕ (̺) = pr / ℓ (x, ξ + iτ ϕ ′ r / ℓ (x)) ∈ S 2m,0 τ,cl . Similarly we recall that we set

T j r / ℓ ,ϕ = e τ ϕr / ℓ T j r / ℓ e -τ ϕr / ℓ ∈ Ψ β j r / ℓ ,0 τ,cl , j = 1, . . . , m,
with principal symbols tr / ℓ ,ϕ (̺) = t j r / ℓ (x, ξ + iτ ϕ ′ r / ℓ (x)) ∈ S β j r / ℓ ,0 τ,cl . We start the proof of the main theorem with a microlocal estimate that exploits the transmission condition.

4.1. Let h(̺ ′ , ξ n ) ∈ S k,0
τ , ̺ ′ = (x, ξ ′ , τ ), with k ≥ 1, be polynomial in ξ n with homogeneous coefficients in (ξ ′ , τ ) and H = h(x, D, τ ). When viewed as a polynomial in ξ n the leading coefficient is

1. Let U be a conic open subset of V + × R n-1 × R + .
We assume that all the roots of h(̺ ′ , ξ n ) = 0 have negative imaginary part for ̺ ′ = (x, ξ ′ , τ ) ∈ U . Letting χ(̺ ′ ) ∈ S 0 T,τ be homogeneous of degree 0 and such that supp(χ) ⊂ U , and N ∈ N, there exist C > 0, C N > 0, and τ * > 0 such that

Op(χ)w 2 k,τ + |tr(Op(χ)w)| 2 k-1,1/2,τ ≤ C H Op(χ)w 2 + + C N w 2 k,-N,τ + |tr(w)| 2 k-1,-N,τ , for w ∈ S (R n + ) and τ ≥ τ * .
Now, we consider a point in the cotangent bundle, at the interface where the transmission condition holds. We then obtain an estimate of an interface norm. Proposition 4.2. Assume that the transmission condition of Definition 1.4 is satisfied at

̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) ∈ S * T,τ (V ) with x 0 ∈ S ∩ V . Then there exists U , a conic open neighborhood of ̺ ′ 0 in V + × R n-1 × R + , such
that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp(χ) ⊂ U , there exist C > 0 and τ * > 0 such that

C |tr(Op(χ)v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ)v r )| 2 mr-1,1/2,τ ≤ m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr ,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ , for τ ≥ τ * , v ℓ , v r ∈ S (R n + ).
Proof. As the transmission condition holds at ̺ ′ 0 the local smooth symbol factorizations of Section 1.6.3,

pr / ℓ ,ϕ (̺ ′ , ξ n ) = p - r / ℓ ,ϕ (̺ ′ , ξ n ) κr / ℓ ,ϕ (̺ ′ , ξ n ), is such that condition (1.29) is valid for ̺ ′ ∈ U 0 with U 0 a conic neighborhood of ̺ ′ 0 in V + × R n-1 × R + . Moreover, by Proposition 3.4 there exists U 1 ⊂ U 0 , a conic open neighborhood of ̺ ′ 0 , such that m ′ ℓ j=m+1 λ 2(m ℓ -1/2-β j ℓ ) T Σ e j ℓ,ϕ (̺ ′ , z ℓ ) 2 + m ′ r j=m+1 λ 2(mr -1/2-β j r ) T Σ e j r,ϕ (̺ ′ , z r ) 2 + m j=1 λ 2(m-1/2-β j ) T Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + Σ t j r,ϕ (̺ ′ , z r ) 2 m ℓ -1 j=0 λ 2(m ℓ -1/2-j) T |z ℓ j | 2 + mr-1 j=0 λ 2(mr -1/2-j) T |z r j | 2 ,
for ̺ ′ ∈ U 1 and z r / ℓ ∈ C mr / ℓ . We now choose U a conic open subset, neighborhood of ̺ ′ 0 , such that U ⊂ U 1 . We let χ be as in the statement and we also choose χ ∈ S 0 T,τ homogeneous of degree 0 with supp( χ) ⊂ U 1 and χ = 1 in a neighborhood of U . Then, (4.1)

m ′ ℓ j=m+1 λ 2(m ℓ -1/2-β j ℓ ) T χ(̺ ′ )Σ e j ℓ,ϕ (̺ ′ , z ℓ ) 2 + m ′ r j=m+1 λ 2(mr -1/2-β j r ) T χ(̺ ′ )Σ e j r,ϕ (̺ ′ , z r ) 2 + m j=1 λ 2(m-1/2-β j ) T Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + Σ t j r,ϕ (̺ ′ , z r ) 2 m ℓ -1 j=0 λ 2(m ℓ -1/2-j) T |z ℓ j | 2 + mr-1 j=0 λ 2(mr -1/2-j) T |z r j | 2 ,
for ̺ ′ ∈ U and z r / ℓ ∈ C mr / ℓ . We set 5 Er / ℓ ,ϕ = Op( χer / ℓ ,ϕ ) and we define the following interface quadratic form (see Definition 3.1):

G S (u) = m j=1 T j ℓ,ϕ u ℓ|x n=0 + + T j r,ϕ u r|x n=0 + 2 m-1/2-β j ,τ + m ′ ℓ j=m+1 E j ℓ,ϕ u ℓ|x n=0 + 2 m+m - ℓ +1/2-j,τ + m ′ r j=m+1 E j r,ϕ u ℓ|x n=0 + 2 m+m - r +1/2-j,τ
. 5 The introduction of χ is made so that χe k ϕ is defined on the whole tangential phase-space.

Observe that with (1.28) we have m + m - r / ℓ + 1/2j = mr / ℓ -1/2β j r / ℓ for j = m + 1, . . . , m ′ r / ℓ . We thus find that G S is of type (m ℓ -1, m r -1, 1 2 ) and its bilinear symbol is given by

Σ G S (̺ ′ , w, w) = m j=1 λ 2(m-1/2-β j ) T Σ t j ℓ,ϕ (̺ ′ , z ℓ ) + Σ t j r,ϕ (̺ ′ , z r ) Σ t j ℓ,ϕ (̺ ′ , zℓ ) + Σ t j r,ϕ (̺ ′ , zr ) + | χ(̺ ′ )| 2 m ′ ℓ j=m+1 λ 2(m ℓ -1/2-β j ℓ ) T Σ e j ℓ,ϕ (̺ ′ , z ℓ )Σ e j ℓ,ϕ (̺ ′ , zℓ ) + | χ(̺ ′ )| 2 m ′ r j=m+1 λ 2(mr-1/2-β j r )
T Σ e j r,ϕ (̺ ′ , zr )Σ e j r,ϕ (̺ ′ , zr ),

with w = (z ℓ , z r ), w = (z ℓ , zr ) ∈ C m ℓ × C mr . Hence (4.1)
gives

Σ G S (̺ ′ , w, w) m ℓ -1 j=0 λ 2(m ℓ -1/2-j) T |z ℓ j | 2 + mr-1 j=0 λ 2(mr -1/2-j) T |z r j | 2 ,
for ̺ ′ ∈ U . For any N ∈ N, by Lemma 3.3 there exists

τ * ≥ 1, C > 0, C N > 0 such that G S (v) = Re G S (v) ≥ C |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ (4.2)
-C N |tr(v ℓ )| 2 m ℓ -1,1/2-N,τ + |tr(v r )| 2 mr -1,1/2-N,τ , with v = (v ℓ , v r ) and vr / ℓ = Op(χ)vr / ℓ , for v = (v ℓ , v r ) ∈ S (R n + ) 2 and τ ≥ τ * .
The functions p - r / ℓ ,ϕ (̺ ′ , ξ n ) and κr / ℓ ,ϕ (̺ ′ , ξ n ) in the symbol factorization recalled at the begining of the proof are polynomial in ξ n with homogeneous coefficients in ̺ ′ ∈ U 0 and the leading coefficient of p - r / ℓ ,ϕ (̺ ′ , ξ n ) is equal to 1. For ̺ ′ ∈ U 0 , their degrees are constant and equal to m - r / ℓ and mr / ℓm - r / ℓ respectively. We smoothly extend p - r / ℓ ,ϕ (̺ ′ , ξ n ) for ̺ ′ outside of U 0 keeping the leading coefficient equal to 1 and we denote this extension by pr / ℓ ,ϕ . In fact we have χpr / ℓ ,ϕ = χκr / ℓ ,ϕ p - r / ℓ ,ϕ = χ χκr / ℓ ,ϕ pr / ℓ ,ϕ . We thus obtain Op(χ)Pr / ℓ ,ϕ = Op(p - r / ℓ ,ϕ ) Op(χ) Op( χκr / ℓ ,ϕ ) + Rr / ℓ with Rr / ℓ in Ψ m,-1 τ by the last point of Proposition 2.7. Observe that χκr / ℓ ,ϕ is a well defined symbol.

Applying Lemma 4.1 to Op(p - r / ℓ ,ϕ ) and wr / ℓ = Op( χκr / ℓ ,ϕ )vr / ℓ we obtain

Op(χ)w ℓ 2 m - ℓ ,τ + Op(χ)w r 2 m - r ,τ + |tr(Op(χ)w ℓ )| 2 m - ℓ -1,1/2,τ + |tr(Op(χ)w r )| 2 m - r -1,1/2,τ Op(p - ℓ,ϕ ) Op(χ)w ℓ 2 + + Op(p - r,ϕ ) Op(χ)w r 2 + + w ℓ 2 m - ℓ ,-N,τ + w r 2 m - r ,-N,τ + |tr(w ℓ )| 2 m - ℓ -1,-N,τ + |tr(w r )| 2 m - r -1,-N,τ Op(χ)P ℓ,ϕ v ℓ 2 + + Op(χ)P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + v ℓ 2 m ℓ ,-N,τ + v r 2 mr,-N,τ + |tr(v ℓ )| 2 m ℓ -1,-N,τ + |tr(v r )| 2 mr -1,-N,τ P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-N,τ + |tr(v r )| 2 mr-1,-N,τ , yielding m - ℓ -1 j=0 D j n Op(χ)w ℓ|x n=0 + 2 m - ℓ -1/2-j,τ + m - r -1 j=0 D j n Op(χ)w r|x n=0 + 2 m - r -1/2-j,τ P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-N,τ + |tr(v r )| 2 mr-1,-N,τ , Recalling that e j+m+1 r / ℓ ,ϕ = κr / ℓ ,ϕ ξ j n , j = 0, . . . , m - r / ℓ -1 in U 1 we have D j n Op(χ) Op( χκr / ℓ ,ϕ )vr / ℓ = E j+m+1 r / ℓ ,ϕ vr / ℓ + Rr / ℓ ,j vr / ℓ with Rr / ℓ ,j ∈ Ψ mr / ℓ -m - r / ℓ +j,-1 τ
by the last point of Proposition 2.7. We then obtain, for τ chosen sufficiently large

m - ℓ -1 j=0 E j+m+1 ℓ,ϕ v ℓ|x n=0 + 2 m - ℓ -1/2-j,τ + m - r -1 j=0 E j+m+1 r,ϕ v r|x n=0 + 2 m - r -1/2-j,τ P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2
,τ , which we write, by a shift of indices,

m ′ ℓ j=m+1 E j ℓ,ϕ v ℓ|x n=0 + 2 m+m - ℓ +1/2-j,τ + m ′ r j=m+1 E j r,ϕ v r|x n=0 + 2 m+m - r +1/2-j,τ (4.3)
P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,
|tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ . Writing T j ℓ,ϕ Op(χ) = Op(χ)T j ℓ,ϕ + [T j ℓ,ϕ , Op(χ)] we observe that (using that m ℓ -β j ℓ = m r -β j r = m -β j ) T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + m-1/2-β j ,τ T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + m-1/2-β j ,τ + |tr(v ℓ )| β j ℓ ,m ℓ -1/2-β j ℓ -1,τ + |tr(v r )| β j r ,mr-1/2-β j r -1,τ T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + m-1/2-β j ,τ + |tr(v ℓ )| m ℓ -1,-1/2,τ + |tr(v r )| mr-1,-1/2,τ .
This concludes the proof. We assume that a(̺) = b(̺) = 0 ⇒ {a, b} > 0, ̺ = (x, ξ, τ ), for x ∈ U , (ξ, τ ) = (0, 0). Then there exist C > 0, C ′ > 0, and τ * > 0 such that

C v 2 m,τ ≤ C ′ Av 2 + + Bv 2 + + |tr(v)| 2 m-1,1/2,τ + τ Q a,b (v) -Re B a,b (v) , for τ > τ * and for v ∈ C ∞ (R n + ) with supp(v) ⊂ U , with B a,b satisfying B a,b (v) ≤ C ′ tr(v) 2
m-1,1/2,τ . We refer to [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] for a proof.

A microlocal Carleman estimate.

With the results of Sections 4.1 and 4.2, if the transmission condition holds at one point of the cotangent bundle at the interface and if the sub-ellipticity property also holds we can then derive a Carleman estimate that holds microlocally, that is, with a cut-off in phase-space applied through a tangential pseudo-differential operator.

Theorem 4.4. Let x 0 ∈ S ∩ V . Assume that {Pr / ℓ , ϕr / ℓ } satisfies the sub-ellipticity condition on a neighborhood of x 0 in V + . Assume moreover that {Pr / ℓ , T j r / ℓ , ϕr / ℓ , j = 1, . . . , m} satisfies the transmission condition at

̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) ∈ S * T,τ (V + ). Then there exists U a conic open neighborhood of ̺ ′ 0 in V + × R n-1 × R + such that for χ ∈ S 0
T,τ , homogeneous of degree 0, with supp(χ) ⊂ U , there exist C > 0 and τ * > 0 such that (4.4)

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ ≥ C τ -1 Op(χ)v ℓ 2 m ℓ ,τ + τ -1 Op(χ)v r 2 mr,τ + |tr(Op(χ)v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ)v r )| 2 mr -1,1/2,τ , for τ ≥ τ * , v ℓ , v r ∈ S (R n +
). Note that there are remainder terms, viz.

vr / ℓ 2 mr / ℓ ,-1,τ , tr(vr / ℓ ) 2 mr / ℓ -1,-1/2,τ
that concern the unknown functions vr / ℓ everywhere and not only in the microlocal region U we consider here. The norms of these remainder terms are weaker that those in the r.h.s. of the estimates. When patching microlocal estimates of the form of (4.4) together these remainder terms can be dealt with; see Section 4.4 below.

Proof. Let U 0 be a open neighborhood of x 0 in V + with the sub-ellipticity condition holding in U 0 .

In the local coordinates we have chosen we have

Pr / ℓ = Pr / ℓ (x, D) = m j=1 Pr / ℓ ,j (x, D ′ )D j n ,
with Pr / ℓ ,m = 1 (see Section

pr / ℓ ,ϕ (x, ξ, τ ) = 0 ⇒ {ar / ℓ , br / ℓ }(x, ξ, τ ) > 0,
for x ∈ U 0 and (ξ, τ ) = (0, 0). Note that the case τ = 0 is achieved because of the ellipticity of P (see Definition 1.1 and Remark 1.2).

Let now U be as given by Proposition 4.2, possibly reduced so that U ⊂ U 0 × R n-1 × R + , and let χ be as in the statement of the theorem. By Lemma 4.3 we then have, for vr

/ ℓ = Op(χ)vr / ℓ , (4.6) Q r / ℓ a,b (vr / ℓ ) -Re B ar / ℓ ,br / ℓ (vr / ℓ ) ≥ Cτ -1 vr / ℓ 2 mr / ℓ ,τ -C ′ τ -
Re B a ℓ ,b ℓ (v ℓ ) + Re B ar,br (v r ) + M m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ (4.7) ≥ C |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ -C ′ v ℓ 2 m ℓ ,-1,τ + v r 2 mr ,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ + P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2
+ . Summing (4.6) ℓ , (4.6) r , and (4.7) we find, by taking τ sufficiently large,

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + Q ℓ a,b (v ℓ ) + Q r a,b (v r ) + τ -1 ( A ℓ v 2 + + B ℓ v ℓ 2 + + A r v r 2 + + B r v r 2 + ) + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ τ -1 v ℓ 2 m ℓ ,τ + v r 2 mr,τ + |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ . Finally, noting that τ -1 Ar / ℓ vr / ℓ 2 + + Br / ℓ vr / ℓ 2 + + Q r / ℓ a,b (vr / ℓ ) ≤ (Ar / ℓ + iBr / ℓ )vr / ℓ 2 + Pr / ℓ ,ϕ vr / ℓ 2 + + vr / ℓ 2 mr / ℓ ,-1,τ Pr / ℓ ,ϕ vr / ℓ 2 + + vr / ℓ 2 mr / ℓ
,-1,τ , by (4.5) and pseudo-differential calculus (last point of Proposition 2.7), we obtain the sought microlocal estimate. 4.4. Proof of Theorem 1.6. We shall patch together estimates of the form given in Theorem 4.4.

With x 0 as in the statement of Theorem 1.6 the transmission condition holds for all boundary quadruples ω = (x 0 , Y, ν 0 , τ ) with Y ∈ T *

x 0 (∂Ω), ν 0 ∈ N * x 0 (S), and τ ≥ 0. In the local coordinates that we use here this means that this property is satisfied for ν = d xn equal to the (oriented) unit conormal to {x n = 0} and all ̺ ′ = (x 0 , ξ ′ , τ ) with ξ ′ ∈ R n-1 and τ ≥ 0. (See Section 1.6.4.) It is fact sufficient to consider

(ξ ′ , τ ) ∈ S n-1 + = {(ξ ′ , τ ) ∈ R n , τ ≥ 0, |(ξ ′ , τ )| = 1}. By Theorem 4.4 for all (ξ ′ 0 , τ 0 ) ∈ S n-1
+ there exists a conic openneighborhood U ̺ ′ 0 of ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) in V + ×R n-1 ×R + such that the estimate (4.4) holds. In fact by reducing U ̺ ′ 0 we can choose

U ̺ ′ 0 = O ̺ ′ 0 ×Γ ̺ ′ 0 where O ̺ ′ 0 is an open set in V + and Γ ̺ ′ 0 is a conic open set in R n-1 × R + .
With the compactness of S n-1

+
we can thus find finitely many such open sets

U j = O j × Γ j , j ∈ J, such that S n-1 + ⊂ ∪ j∈J Γ j . We then set O = ∩ j∈J O j that is an open neighborhood of x 0 in V + and we set V j = O × Γ j ⊂ U j . We also choose an open neighborhood W of x 0 in R n such that W + = W ∩ V + ⋐ O.
We then choose a partition of unity, χ j ∈ S 0 T,τ , j ∈ J, on W + × R n-1 × R + subordinated by the covering by the open sets V j :

j∈J χ j (̺ ′ ) = 1, for ̺ ′ = (x, ξ ′ , τ ) ∈ W + × R n-1 × R + and |(ξ ′ , τ )| ≥ r 0 > 0, supp(χ j ) ⊂ V j .
The symbols χ j are chosen homogeneous of degree 0 for |(ξ ′ , τ )| ≥ r 0 > 0. We set χ = 1 -j∈J χ j and have χ ∈ ∩ N ∈N S -N T,τ . As supp(χ j ) ⊂ U j , we can apply the microlocal estimate of Theorem 4.4: (4.8)

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ τ -1 v ℓ 2 m ℓ ,τ + v r 2 mr,τ + |tr(Op(χ j )v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ j ))v r )|
/ ℓ ) m-1,1/2,τ ≤ j∈J tr(Op(χ j )vr / ℓ ) m-1,1/2,τ + tr(Op(χ)vr / ℓ ) m-1,1/2,τ j∈J tr(Op(χ j )vr / ℓ ) m-1,1/2,τ + tr(vr / ℓ ) m-1,-N,τ .
Summing estimates (4.8) for each χ j we thus obtain

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + v ℓ 2 m ℓ ,-1,τ + v r 2 mr,-1,τ + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ τ -1 v ℓ 2 m ℓ ,τ + v r 2 mr,τ + |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ ,
Choosing now τ sufficiently large we obtain

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ τ -1 v ℓ 2 m ℓ ,τ + v r 2 mr,τ + |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ , Setting vr / ℓ = e
τ ϕr / ℓ ur / ℓ the conclusion of the proof of Theorem 1.6 is then classical.

4.5. Shifted estimates. As in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] it may be interesting to consider shifted estimates in the Sobolev scales. Namely we may wish to have an estimate of the following form.

Corollary 4.5. Let x 0 ∈ S and let ϕ ∈ C 0 (Ω) be such that

ϕ k = ϕ |Ω k ∈ C ∞ (Ω k ) for k = 1, 2
and such that the pairs {P k , ϕ k } have the sub-ellipticity property of Definition 1.1 in a neighborhood of x 0 in Ω k . Moreover, assume that P k , ϕ, T j k , k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Let ℓ ∈ N. Then there exist a neighborhood W of x 0 in R n and two constants C and τ * > 0 such that (4.9)

k=1,2 τ -1 e τ ϕ k u k 2 ℓ+m k ,τ + |e τ ϕ |S tr(u k )| 2 ℓ+m k -1,1/2,τ ≤ C k=1,2 e τ ϕ k P k (x, D)u k 2 ℓ,τ + m j=1 e τ ϕ |S T j 1 (x, D)u 1 + T j 2 (x, D)u 2 |S 2 ℓ,m-1/2-β j ,τ , for all u k = w k|Ω k with w k ∈ C ∞ c (W ) and τ ≥ τ * .
The proof of this corollary can be adapted from that of its counterpart at a boundary, namely Corollary 4.5 in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF]. 4.6. Interior-eigenvalue transmission problems. Interior-eigenvalue transmission problems are very related to the transmission problem we have considered. In fact, for Ω, a bounded open set in R n , we consider two elliptic operators P 1 and P 2 of respective orders m 1 and m 2 , as in Section 1, yet both defined on Ω.

In addition, we consider 2m = m 1 + m 2 boundary operators operators (4.10)

T j k = |α|≤β j k t j k,α (x)D α , k = 1, 2, j = 1, . . . , m,
with 0 ≤ β j k < m k , and where the coefficients t j k,α (x) are C ∞ complex-valued functions defined in some neighborhood of ∂Ω. Setting

β j = (β j 1 + β j 2 )/2 we assume that m 1 -β j 1 = m 2 -β j 2 = m -β j , j = 1, . . . , m. (4.11)
The interior-eigenvalue transmission problem consist in resolving a system of the form

(P k -τ m k )u k = f k in Ω, , k = 1, 2 T j 1 u 1 + T j 2 u 2 = g j , in Γ, j = 1, .
. . , m. We refer to [START_REF] Colton | The interior transmission problem[END_REF][START_REF] Cakoni | Transmission eigenvalues and the nondestructive testing of dielectrics[END_REF][START_REF] Päivärinta | Transmission eigenvalues[END_REF][START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF][START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory, Inverse problems and applications: inside out[END_REF][START_REF] Robbiano | Spectral analysis of the interior transmission eigenvalue problem[END_REF][START_REF] Petkov | Asymptotics of the number of interior transmission eigenvalues[END_REF] and the reference therein for more details on this very active field of research.

In the analysis of such problems, resolvant estimates are central. In the proof of such resolvent estimates, a Carleman inequality at the boundary can be a very efficient tool. Here, we provide such an estimate in a neighborhood of a point of ∂Ω, as the proof is in fact given by the analysis of the previous section, in particular, as we used the system formulation of Section 1.6.2, which yield a formulation close to that of the interior-eigenvalue transmission problem.

Let x 0 ∈ ∂Ω and V be a neighborhood of x 0 where Ω = {x n > 0}. We consider two smooth weight functions ϕ 1 and ϕ 2 in V such that ϕ 1|x n=0 + = ϕ 2|x n=0 + .

With the notation of Sections 1.6.2-1.6.3, with the letter ℓ replaced by 1 and the letter r replaced by 2, and moreover P ℓ (resp. P r ) replaced by P 1τ m 1 (resp. P 2τ m 2 ), we say that the transmission condition holds at x 0 for {P kτ m k , T j k , ϕ k , k = 1, 2, j = 1, . . . , m} if for all ̺ ′ = (x 0 , ξ ′ , τ ) and for all pairs of polynomials, q 1 (ξ n ), q 2 (ξ n ), there exist U 1 , U 2 , polynomials, and c j ∈ C, j = 1, . . . , m, such that

q 1 (ξ n ) = m j=1 c j t j ℓ,ϕ (̺ ′ , ξ n ) + U 1 (ξ n )κ 1 (̺ ′ , ξ n ), (4.12)
and

q 2 (ξ n ) = m j=1 c j t j r,ϕ (̺ ′ , ξ n ) + U 2 (ξ n )κ 2 (̺ ′ , ξ n ). (4.13)
Then the proof of the following local Carleman estimate is the same as that of Theorem 1.6.

Theorem 4.6. Let x 0 ∈ ∂Ω and let ϕ k ∈ C ∞ (Ω), k = 1, 2, as above, such that the pairs {P kτ m k , ϕ k } satisfy the sub-ellipticity property of Definition 1.1 in a neighborhood of x 0 in Ω. Moreover, assume that

P k -τ m k , T j k , ϕ k , k = 1, 2, j = 1, .
. . , m satisfies the transmission condition at x 0 . Then there exist a neighborhood W of x 0 in R n and two constants C and τ * > 0 such that

(4.14) k=1,2 τ -1 e τ ϕ k u k 2 m k ,τ + |e τ ϕ k γ(u k )| 2 m k -1,1/2,τ ≤ C k=1,2 e τ ϕ k (P k (x, D) -τ m k )u k 2 L 2 + m j=1 |e τ ϕ |∂Ω (T j 1 (x, D)u 1 + T j 2 (x, D)u 2 ) |∂Ω | 2 m-1/2-β j ,τ , for all u k = w k|Ω with w k ∈ C ∞ c (W ) and τ ≥ τ * .

A PSEUDO-DIFFERENTIAL CALCULUS WITH TWO LARGE PARAMETERS

The weight function we shall consider below is of the form ϕ(x) = exp(γψ(x)). The function ψ is assumed to be C 0 , piecewise smooth, and to satisfy

0 < C ≤ ψ and ψ (k) L ∞ < ∞, k ∈ N. We take γ ≥ 1.
The goal of what follows is to achieve estimates as in Theorem 1.6 with the explicit dependency upon the additional parameter γ. This can be done by the introduction of an appropriate pseudodifferential calculus. Assumption of the function ψ will be made in Section 6.1, namely, the strong pseudoconvexity conditions, to obtain a Carleman estimate. 5.1. Metric, symbols and Sobolev norms. Here, by ̺ and ̺ ′ we shall denote

̺ = (x, ξ, τ, γ) ∈ R n × R n × R + × R + and ̺ ′ = (x, ξ ′ , τ, γ) ∈ R n × R n-1 × R + × R + .
We set τ (x) = τ γϕ(x). Following [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] we consider the metrics on phase-space

g = γ 2 |dx| 2 + |dξ| 2 µ 2 , with µ 2 = µ 2 (̺) = |(τ (x), ξ)| 2 = τ (x) 2 + |ξ| 2 ,
and on tangent phase space

g T = γ 2 |dx| 2 + |dξ ′ | 2 µ 2 T , with µ 2 T = µ 2 T (̺ ′ ) = |(τ (x), ξ ′ )| 2 = τ (x) 2 + |ξ ′ | 2 ,
for τ ≥ 1 and γ ≥ 1. Below, the explicit dependencies of µ and µ T upon ̺ and ̺ ′ are dropped to ease notation. The metric g (resp. g T ) along with the order function µ (resp. µ T ) generates a (resp. tangential) Weyl-Hörmander pseudo-differential calculus as proven in [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF]Proposition 2.2]. Note that this uses the conditions 0 < C ≤ ψ and ψ ′ < ∞.

For a presentation of the Weyl-Hörmander calculus we refer to [START_REF] Lerner | Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators[END_REF], [START_REF]The analysis of linear partial differential operators[END_REF] and [START_REF]The Weyl calculus of pseudo-differential operators[END_REF]. Let a(x, ξ, τ, γ) ∈ C ∞ (R n × R n ), with τ, γ as parameter in [τ min , +∞) and [γ min , +∞), τ min > 0, γ min > 0, and m ∈ R, be such that for all multi-indices γ, β ∈ N n we have

(5.1) ∂ γ x ∂ β ξ a(̺) ≤ C γ,β γ γ µ m-|β| , ̺ ∈ R n × R n × [τ min , +∞) × [γ min , +∞).
With the notation of [START_REF]The analysis of linear partial differential operators[END_REF] we then have a(̺) ∈ S(µ m , g)(R n × R n ), which we write S m (g) for simplicity 6 . The associated class of pseudo-differential operators is denoted by

Ψ m (g) = Ψ(µ m , g)(R n × R n ).
Similarly we define tangential symbols and operators. Let

a(x, ξ ′ , τ, γ) ∈ C ∞ (R n + × R n-1
) and m ∈ R, be such that for all multi-indices γ, β ∈ N n we have

(5.2) ∂ γ x ∂ β ξ ′ a(̺ ′ ) ≤ C γ,β γ γ µ m-|β| T , ̺ ′ ∈ R n + × R n-1 × [τ min , +∞) × [γ min , +∞).
We then have a

(̺ ′ ) ∈ S m (g T ) = S(µ m T , g T )(R n + × R n-1 ). The associated class of tangential pseudo- differential operators is denoted by Ψ m (g T ) = Ψ(µ m , g T )(R n + × R n-1 ). Note that the condition ψ (k) < ∞, k ∈ N, is used to prove 7 that τ ∈ S(τ , g) ∩ S(τ , g T ). With ̺ = (x, ξ, τ, γ) ∈ R n × R n × R + × R + (resp. ̺ ′ = (x, ξ ′ , τ, γ) ∈ R n + × R n-1 × R + × R + ) we shall associate ̺ = (x, ξ, τ (x)) ∈ R n × R n × R + (resp. ̺′ = (x, ξ ′ , τ (x)) ∈ R n + × R n-1 × R + ). Note that if â(x, ξ, τ ) ∈ S m
τ , with the notation of Section 2.1, satisfying moreover, for all multi-indices

γ, β ′ , β ′′ ∈ N n , with β = β ′ + β ′′ , (5.3) ∂ γ x ∂ β ′ ξ ∂ β ′′ τ â(x, ξ, τ ) ≤ C γ,β ′ ,β ′′ |(ξ, τ )| m-|β| , x ∈ R n , ξ ∈ R n , τ ∈ [τ min , +∞), i.
e., differentiation w.r.t. τ yields the same additional decay as a differentiation w.r.t. ξ, then a(x, ξ, τ, γ) = â(x, ξ, τ (x)) ∈ S m (g), which we shall write a(̺) = â(̺). Similarly if â(x, ξ ′ , τ ) ∈ S m T,τ with the same additional property regarding differentiation w.r.t. τ we have a(

̺ ′ ) = â(̺ ′ ) ∈ S m (g T ).
In what follows we shall assume that symbols in S m τ and S m T,τ have this additional regularity property. We then say that a ∈ S m (g) (resp. S m (g T )) is homogeneous of degree m with respect to (ξ, τ ) (resp.

(ξ ′ , τ )) if we have a(̺) = â(̺)) (resp. a(̺ ′ ) = â(̺ ′ )) with â(x, ξ, τ ) ∈ S m τ (resp. â(x, ξ ′ , τ ) ∈ S m T,τ ) homogeneous of degree m in (ξ, τ ) (resp. (ξ ′ , τ )).
We shall also use the following classes of symbols S(τ

r µ m T , g T ) = τ r S m (g T ) on R n + × R n-1 , for r, m ∈ R. The associated class of tangential pseudo-differential operators is denoted by τ r Ψ m (g T ) = τ r Ψ(µ m T , g T )(R n + × R n-1
). We have the following lemma whose proof is similar to that of Lemma 2.7 in [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF].

Lemma 5.1. Let r, m ∈ R and a ∈ τ r S m (g T ). There exists C > 0 such that for τ sufficiently large

| (Op(a)u, v) ∂ ≤ C Op(τ r ′ µ m ′ T )u + Op(τ r ′′ µ m ′′ T )v + , u ∈ S (R n + ).
for r = r ′ + r ′′ , m = m ′ + m ′′ . 6 The dependence upon the metric g is kept explicit here as we shall actually have to face two calculi simultaneously, associated with the weight functions on both sides of the interface. Interactions between the two calculi will only occur at the interface where they coincide. See Section 5.2. 7 This condition was not written in [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] and [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF]. This is however made precise in [START_REF]A spectral inequality for the bi-laplace operator[END_REF], including the proof of τ ∈ S(τ , g) ∩

S(τ , g T ).

This contains the estimate

Op(τ s µ p T ) Op(a)u + ≤ C Op(τ s+r µ p+m T )u + , u ∈ S (R n + ), for s, p ∈ R.
Note also that we have

Op(τ r µ m T )u + ≍ Op(µ m T )τ r u + , (5.4)
for τ chosen sufficiently large.

Next we say that a(x, ξ ′ , τ, γ) ∈ τ r S m cl (g T ) on R n + ×R n-1 if there exists a sequence a (j) ∈ γ j τ r S m-j (g T ), with γ -j a (j) homogeneous of degree

m + r -j in (ξ ′ , τ ) for |(ξ ′ , τ )| ≥ r 0 , with r 0 ≥ 0, such that (5.5) a ∼ j≥0 a (j) , in the sense that a - N j=0 a (j) ∈ γ N +1 τ r S m-N -1 (g T ).
A representative of the principal part, denoted by σ(a), is then given by the first term in the expansion. Then we shall say that a(̺)

∈ τ r S m,σ cl (g) on R n + × R n-1 if a(̺) = m j=0 a j (̺ ′ )ξ j n , with a j ∈ τ r S m-j+σ cl (g T ).
The principal part is given by m j=0 σ(a j )(̺ ′ )ξ j n . With these symbol classes we associate classes of pseudodifferential operators,

τ r Ψ m cl (g T ) = τ r Ψ m cl (g T )(R n + × R n-1 ) and τ r Ψ m,σ cl (g) = τ r Ψ m,σ cl (g)(R n + × R n-1 ), as is done in Section 2.2.
We define the following semi-classical interior norm, for m ∈ N,

(5.6) u 2 m,τ = m j=0 Op(µ m-j T )D j n u 2 + , u ∈ S (R n + ).
We also set, for m ∈ N and σ ∈ R,

(5.7) u 2 m,σ,τ = Op(µ σ T )u 2 m,τ ∼ m j=0 Op(µ m-j+σ T )D j n u 2 + , u ∈ S (R n + ).
At the interface {x n = 0 + } we define the following norms, for m ∈ N and σ ∈ R,

(5.8) |tr(u)| 2 m,σ,τ = m j=0 Op(µ m-j+σ T ) tr j (u) 2 ∂ , u ∈ S (R n + ).

Transmission problem with two calculi.

In the present setting, using the system formulation of Section 1.6.2 we shall in fact work in {x n ≥ 0} with two weight functions, namely ϕr / ℓ = e γψr / ℓ . With each weight function we shall associate a pseudo-differential calculus, classes of symbols and pseudo-differential operators, and Sobolev norms, as introduced in the previous section.

We shall thus define τr / ℓ (x) = τ γϕr / ℓ (x),

µ 2 r / ℓ = τ 2 r / ℓ + |ξ| 2 , µ 2 T, r / ℓ = τ 2 r / ℓ + |ξ| 2 , the associated metrics gr / ℓ = γ 2 |dx| 2 + |dξ| 2 µ 2 r / ℓ , g T, r / ℓ = γ 2 |dx| 2 + |dξ ′ | 2 µ 2 T, r / ℓ ,
and the symbol classes S m (gr / ℓ ), S m (g T, r / ℓ ), S m cl (g T, r / ℓ ), S m,σ cl (g T, r / ℓ ) and the associated operator classes

Ψ m (gr / ℓ ), Ψ m (g T, r / ℓ ), Ψ m cl (g T, r / ℓ ), Ψ m,σ cl (g T, r / ℓ ).
Accordingly for a function defined in {x n ≥ 0} we denote by u m,τr / ℓ and u m,σ,τr / ℓ the associated norms as in (5.6)-(5.7).

Observe that the two calculi coincide at x n = 0, that is, on the interface, since ψ ℓ|x n=0 + = ψ r|x n=0 + , implying ϕ ℓ|x n=0 + = ϕ r|x n=0 + and µ ℓ|x n=0 + = µ r|x n=0 + . In particular we shall keep the notation 2 . We say that

G (w) = N s=1 A s ℓ w ℓ|x n=0 + + A s r w r|x n=0 + , B s ℓ w ℓ|x n=0 + + B s r w r|x n=0 + ∂ , with A s r / ℓ = a s r / ℓ (x, D, τ, γ) and B s r / ℓ = b s r / ℓ (x, D, τ, γ), is an interface quadratic form of type (m ℓ -1, m r -1, σ)
with C ∞ coefficients, if for each s = 1, . . . , N , we have

a s r / ℓ (̺), b s r / ℓ (̺) ∈ S mr / ℓ -1,σr / ℓ cl (g T, r / ℓ )(R n + × R n ), with σ ℓ + σ r = 2σ, ̺ = (̺ ′ , ξ n ) with ̺ ′ = (x, ξ ′ , τ, γ).
As in Section 3 we associate to G a bilinear symbol Σ G (̺ ′ , w, w).

We let W be an open conic set in R n-1 × R n-1 × R + . for any w = (z ℓ , z r ), z r / ℓ = (z r / ℓ 0 , . . . , z r / ℓ mr / ℓ -1 ) ∈ C mr / ℓ , and ̺′′ ∈ W , such that µ T,ℓ |xn=0 + = µ T,r |xn=0 + ≥ R, with ̺ ′′ = (x ′ , ξ ′ , τ, γ) and ̺′′ = (x ′ , ξ ′ , τ (x ′ , x n = 0 + )).

We have the following Gårding estimate.

Lemma 5.4. Let W be an open conic set in R n-1 × R n-1 × R + and let G be an interface quadratic form of type (0, m ℓ -1, m r -1, σ) that is positive definite in W . Let χ ∈ S 0 T,τ be homogeneous of degree 0, with supp( χ|xn=0 + ) ⊂ W and let N ∈ N. Then there exist

τ * ≥ 1, γ * ≥ 1, C > 0, C N > 0 such that Re G (Op(χ)u) ≥ C |tr(Op(χ ℓ )u ℓ )| 2 m ℓ -1,σ ℓ ,τ + |tr(Op(χ r )u r )| 2 mr-1,σr,τ -C N |tr(u ℓ )| 2 m ℓ -1,σ ℓ -N,τ + |tr(u r )| 2 mr-1,σr-N,τ for u = (u ℓ , u r ) ∈ S (R n + ) 2 , τ ≥ τ * , γ ≥ γ * , and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 T,τ , with ̺ ′ = (x, ξ ′ , τ, γ) and ̺′ r / ℓ = (x, ξ ′ , τr / ℓ (x)).
The proof is similar to that of Lemma 3.3 using that the two calculi, associated with ψ ℓ and ψ r respectively, coincide on the interface S = {x n = 0}. In particular note that χ ℓ|x n=0 + = χ r|x n=0 + as τℓ coincides with τr at the interface.

CARLEMAN ESTIMATE WITH TWO LARGE PARAMETERS

With a weight function of the form ϕ(x) = exp(γψ(x)), some condition on ψ can yield ϕ to fulfill the sub-ellipticity condition of Definition 1.1. Those are the strong pseudo-convexity conditions introduced by L. Hörmander (see [START_REF] Hörmander | On the uniqueness of the Cauchy problem[END_REF], [START_REF]Linear partial differential operators[END_REF]Section 8.6] and [START_REF]The analysis of linear partial differential operators[END_REF]Section 28.3]). We shall see that along with the transmission condition they are sufficient to derive Carleman estimates with an explicit dependency upon the additional parameter γ. In fact the strong pseudo-convexity condition is also necessary if one considers a weight function of this form; for such question we refer to [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF].

6.1. Strong pseudo-convexity. We recall the notion of strong pseudo-convexity and then adapt it to the geometry we consider.

As we restrict ourselves to elliptic operators in the present article, the classical notion of strong pseudoconvexity then reduces to the following one (the reader can compare with Section 28.3 in [START_REF]The analysis of linear partial differential operators[END_REF]). Definition 6.1 (strong pseudo-convexity up to a boundary). Let O be an open set. We say that a smooth function ψ is strongly pseudo-convex at x ∈ O w.r.t. p if ψ ′ (x) = 0 and if for all ξ ∈ R n and τ > 0, (s-Ψc) p(x, ξ + iτ ψ ′ (x)) = 0 and p, ψ (x, ξ + iτ ψ ′ (x)) = 0

⇒ 1 2i p(x, ξ -iτ ψ ′ (x)), p(x, ξ + iτ ψ ′ (x)) > 0.
Let

pr / ℓ ,ϕ (x, ξ, τ ) = pr / ℓ (x, ξ + iτ ϕ ′ r / ℓ (x)) = pr / ℓ (x, ξ + iτr / ℓ (x)ψ ′ r / ℓ (x)) = pr / ℓ ,ψ (x, ξ, τr / ℓ (x)) ∈ S m,0 cl (gr / ℓ ),
Similarly, the principal symbol of T j r / ℓ ,ϕ = e τ ϕr / ℓ T j r / ℓ e -τ ϕr / ℓ ∈ Ψ β k ,0 cl (gr / ℓ ), j = 1, . . . , m, is

t j r / ℓ ,ϕ (x, ξ, τ ) = t j r / ℓ (x, ξ + iτ ϕ ′ (x)) = t j r / ℓ (x, ξ + iτr / ℓ (x)ψ ′ (x)) = t j r / ℓ ,ψ (x, ξ, τr / ℓ (x)) ∈ S β k ,0 cl (gr / ℓ ).
The dependency upon γ is hidden either in ϕ or in τ .

Setting κr / ℓ ,ϕ = p + r / ℓ ,ϕ p 0 r / ℓ ,ϕ and κr / ℓ ,ψ = p + r / ℓ ,ψ p 0 r / ℓ ,ψ , we then find κr / ℓ ,ϕ (x, ξ, τ ) = κr / ℓ ,ψ (x, ξ, τr / ℓ (x)).

From these simple observations we thus conclude that {Pr / ℓ , T j r / ℓ , ϕ, j = 1, . . . , m} satisfies the transmission condition at (x 0 , ξ ′ 0 , τ 0 ), with x 0 ∈ S, if and only if {Pr / ℓ , T j r / ℓ , ψ, j = 1, . . . , m} satisfies the transmission condition at (x 0 , ξ ′ 0 , τ0 ) with τ0 = τℓ (x 0 ) = τr (x 0 ).

6.3.

Statement of the Carleman estimate with two large parameters. We shall prove the following theorem, counterpart of Theorem 1.6 in the case of a weight function of the form ϕ = exp(γψ), with an explicit dependency with respect to the second large parameter γ.

Theorem 6.3. Let x 0 ∈ S and let ψ ∈ C 0 (Ω) be such that

ψ k = ψ |Ω k ∈ C ∞ (Ω k ) for k = 1
, 2 and such that ψ has the strong pseudo-convexity property of Definition 6.2 with respect to P 1 and P 2 in a neighborhood of x 0 in Ω. Moreover, assume that P k , T j k , ψ, k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Then there exist a neighborhood W of x 0 in R n and three constants C, τ * > 0, and γ * > 0 such that for ϕ k = exp(γψ k ) and τk = τ γϕ k :

(6.1) k=1,2 τ -1/2 k e τ ϕ k u k 2 m k ,τ k + |e τ ϕ |S tr(u k )| 2 m k -1,1/2,τ ≤ C k=1,2 e τ ϕ k P k (x, D)u k 2 L 2 (Ω k ) + m j=1 |e τ ϕ |S (T j 1 (x, D)u 1 + T j 2 (x, D)u 2 ) |S | 2 m-1/2-β j ,τ ,
for all u k = w k|Ω k with w k ∈ C ∞ c (W ), τ ≥ τ * , and γ ≥ γ * . Here norms of defined on Ω k and S. They are locally equivalent to their counterpart defined on {x n > 0} and {x n = 0} above. 6.4. Preliminary estimates. The following lemma is the counterpart of Lemma 4.1, that is, an elliptic estimate. It will be applied on both the ℓ and r "sides". Hence, we formulate it for a weight funtion ϕ, τ and phase-space metric g in place of ϕr / ℓ , τr / ℓ , and gr / ℓ .

With

̺ ′ = (x, ξ ′ , τ, γ) ∈ R n + ×R n-1 ×R + ×R + we shall associate ̺′ = (x, ξ ′ , τ (x)) ∈ R n + ×R n-1 ×R + , with τ (x) = τ γϕ(x).
Lemma 6.4. Let h(̺) ∈ S k,0 cl (g), with ̺ = (x, ξ, τ, γ) and k ≥ 1, be polynomial in ξ n with homogeneous coefficients in (ξ ′ , τ ) and H = h(x, D, τ, γ). When viewed as a polynomial in ξ n the leading coefficient is 1. Let U be a conic open subset of V + × R n-1 × R + . We assume that all roots of h(̺ ′ , ξ n ) = 0 have negative imaginary part for ̺′ ∈ U . Letting χ(̺ ′ ) ∈ S 0 T,τ , ̺′ = (x, ξ ′ , τ ), be homogeneous of degree 0 and such that supp( χ) ⊂ U , and N ∈ N, there exist C > 0, C N > 0, τ * > 0 and γ * , such that

Op(χ)w 2 k,τ + |tr(Op(χ)w)| 2 k-1,1/2,τ ≤ C H Op(χ)w 2 + + C N w 2 k,-N,τ + |tr(w)| 2 k-1,-N,τ ,
for w ∈ S (R n + ) and τ ≥ τ * , γ ≥ γ * and χ(̺ ′ ) = χ(̺ ′ ) ∈ S 0 (g T ). We refer to [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] for a proof. The following proposition is the counterpart of Proposition 4.2, that is, an estimate exploiting the transmission condition, yielding an estimate of an interface norm. Proposition 6.5. Assume that the transmission condition for Pr / ℓ , T j r / ℓ , ψr / ℓ , j = 1, . . . , m is satisfied at (x 0 , ξ ′ 0 , τ0 ) ∈ S * T,τ (V ) with x 0 ∈ S ∩ V . Then there exists U , a conic open neighborhood of (x 0 , ξ ′ 0 , τ0 ) in V + × R n-1 × R + , such that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp( χ) ⊂ U , and there exist C > 0, τ * > 0, and γ * > 0 such that

C |tr(Op(χ ℓ )v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ r )v r )| 2 mr-1,1/2,τ ≤ m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ + P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + γ 2 v ℓ 2 m ℓ ,-1,τ ℓ + v r 2 mr ,-1,τr + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ , for τ ≥ τ * , γ ≥ γ * , v ℓ , v r ∈ S (R n + ) and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 (g T, r / ℓ ), with ̺ ′ = (x, ξ ′ , τ, γ) and ̺′ r / ℓ = (x, ξ ′ , τr / ℓ (x)).
Proof. The beginning of the proof is nearly identical to that of Proposition 4.2. In particular the neighborhood U is chosen similarly. Inequality (4.1) becomes

m ′ ℓ j=m+1 λ 2(m ℓ -1/2-β j ℓ ) T χ(̺ ′ )Σ e j ℓ,ψ (̺ ′ , z ℓ ) 2 + m ′ r j=m+1 λ 2(mr -1/2-β j r ) T χ(̺ ′ )Σ e j r,ψ (̺ ′ , z r ) 2 + m j=1 λ 2(m-1/2-β j ) T Σ t j ℓ,ψ (̺ ′ , z ℓ )+Σ t j r,ψ (̺ ′ , z r ) 2 ≥ C m ℓ -1 j=0 λ 2(m ℓ -1/2-j) T z ℓ j 2 + mr-1 j=0 λ 2(mr -1/2-j) T |z r j | 2 , for z r / ℓ ∈ C mr / ℓ and ̺′ = (x ′ , x n = 0 + , ξ ′ , τ ) ∈ U ∩ {x n = 0} with λ 2 T = |ξ| ′2 + τ 2 . Here, χ ∈ S 0 T,τ
is homogeneous of degree 0 and such that χ = 1 in a neighborhood of U . We set χr / ℓ (x, ξ ′ , τ, γ) = χ(x, ξ ′ , τr / ℓ ) ∈ S 0 (g T, r / ℓ ).

We then obtain, taking ̺′ = ̺′ = (x ′ , x n = 0 + , ξ ′ , τ (x)),

m ′ ℓ j=m+1 µ T 2(m ℓ -1/2-β j ℓ ) |xn=0 + χ(̺ ′′ )Σ e j ℓ,ϕ (̺ ′ , z ℓ ) 2 + m ′ r j=m+1 µ T 2(mr -1/2-β j r ) |xn=0 + χ(̺ ′′ )Σ e j r,ϕ (̺ ′ , z r ) 2 + m j=1 µ T 2(m-1/2-β j ) |xn=0 + Σ t j ℓ,ϕ (̺ ′ , z ℓ )+Σ t j r,ϕ (̺ ′ , z r ) 2 ≥ C m ℓ -1 j=0 µ T 2(m ℓ -1/2-j) |xn=0 + |z ℓ j | 2 + mr-1 j=0 µ T 2(mr -1/2-j) |xn=0 + |z r j | 2 ,
for all z r / ℓ ∈ C mr / ℓ and ̺ ′ = (x ′ , x n = 0+, ξ ′ , τ, γ) and ̺ ′′ = (x ′ , ξ ′ , τ, γ) such that ̺′ ∈ U ∩ {x n = 0}.

Here χ(̺ ′′ ) = χ(̺ ′′ , x n = 0 + ) with ̺′′ = (x ′ , ξ ′ , τ (x ′ , x n = 0 + )). We have χ(̺ ′′ ) = χℓ (̺ ′ ) |xn=0 + = χr (̺ ′ ) |xn=0 + . We set

G S (u) = m j=1 T j ℓ,ϕ u ℓ|x n=0 + + T j r,ϕ u r|x n=0 + 2 m-1/2-β j ,τ + m ′ ℓ j=m+1 E j ℓ,ϕ u ℓ|x n=0 + 2 m+m - ℓ +1/2-j,τ + m ′ r j=m+1 E j r,ϕ u ℓ|x n=0 + 2 m+m - r +1/2-j,τ .
with Er / ℓ ,ϕ = Op( χr / ℓ er / ℓ ,ϕ ). Then, according to the Gårding inequality of Lemma 5.4 for interface quadratic forms of type (m ℓ -1, m r -1, 1/2), there exists τ * > 0, γ * > 0, C > 0, and

C N > 0 such that G S (v) = Re G S (v) ≥ C |tr(v ℓ )| 2 m ℓ -1,1/2,τ + |tr(v r )| 2 mr-1,1/2,τ (6.2)
-C N |tr(v ℓ )| 2 m ℓ -1,1/2-N,τ + |tr(v r )| 2 mr -1,1/2-N,τ , with v = (v ℓ , v r ) and vr / ℓ = Op(χr / ℓ )vr / ℓ , for v = (v ℓ , v r ) ∈ S (R n + ) 2 , τ ≥ τ * ,
- ℓ -1 j=0 E j+m+1 ℓ,ϕ v ℓ |xn=0 + 2 m - ℓ -1/2-j,τ + m - r -1 j=0 E j+m+1 r,ϕ v r |xn=0 + 2 m - r -1/2-j,τ (6.3)
P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + γ 2 v ℓ 2 m ℓ ,-1,τ ℓ + v r 2 mr,-1,τr + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2
mr-1,-1/2,τ . Collecting (6.2) and (6.3) we obtain the result of Proposition 6.5, for τ and γ chosen sufficiently large, with an additional commutator argument as in the end of the proof of Proposition 4.2. 6.5. Proof of the Carleman estimate with two-large parameters. We prove a microlocal result, counterpart of that of Theorem 4.4. Patching microlocal estimates of this type, arguing as in Section 4.4 we can then obtain the local Carleman estimate of Theorem 6.3. The proof is left to the reader. Theorem 6.6. Let x 0 ∈ S ∩ V and let ψ ∈ C 0 (V ) be such that ψr / ℓ ∈ C ∞ (V + ) has the strong pseudoconvexity property of Definition 6.1 with respect to Pr / ℓ in a neighborhood of x 0 in V + . Moreover, assume that Pr / ℓ , ψr / ℓ , T j r / ℓ , j = 1, . . . , m satisfies the transmission condition at (x 0 , ξ ′ 0 , τ0 ) ∈ S * T,τ (V + ). Then there exists U a conic open neighborhood of (x 0 , ξ ′ 0 , τ0 ) in V + × R n-1 × R + such that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp( χ) ⊂ U , there exist C > 0, τ * > 0, and γ * > 0 such that, for ϕr / ℓ = exp(γψr / ℓ ) and τr / ℓ = τ γϕr / ℓ , (6.4)

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-β j -1/2,τ + γ 2 v ℓ 2 m ℓ ,-1,τ ℓ + v r 2 mr,-1,τr + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ ≥ C τ -1 2 ℓ Op(χ ℓ )v ℓ 2 m ℓ ,τ ℓ + τ -1 2 r Op(χ r )v r 2 mr ,τr + |tr(Op(χ ℓ )v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ r )v r )| 2 mr -1,1/2,τ , for all v ℓ , v r ∈ S (R n + ), τ ≥ τ * , γ ≥ γ * , and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 (g T, r / ℓ ), with ̺′ r / ℓ = (x, ξ ′ , τr / ℓ (x)) for ̺ ′ = (x, ξ ′ , τ, γ).
Proof. Applying Lemma 6.8 in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] we obtain that there exists U a conic open neighborhood of (x 0 , ξ ′ 0 , τ0 ) in V + × R n-1 × R + such that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp( χ) ⊂ U , there exist C > 0, τ 0 > 0, and γ 0 > 0 such that (6. With Proposition 6.5, making use of the transmission condition, we obtain for M chosen sufficiently large, in a possibly reduced neighborhood U ,

Re B a ℓ ,b ℓ (Op(χ ℓ )v ℓ ) + Re B ar,br (Op(χ r )v r ) + M m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-1/2-β j ,τ (6.6)
≥ C |tr(Op(χ ℓ )v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ r )v r )| 2 mr-1,1/2,τ -C ′ P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + γ 2 v ℓ 2 m ℓ ,-1,τ ℓ + v r 2 mr,-1,τr + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,
τ , Summing (6.5) ℓ , (6.5) r , and (6.6) we obtain the result of Theorem 6.6, by taking τ and γ sufficiently large. 6.6. Estimate with the simple characterisitic property. As in [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] and [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] a stronger estimate with two parameters can be achieved if one assumes that the operator P and the weight function ψ fulfills the so-called simple characterisitic property.

We introduce the map

ρ x,ξ : R + → C, τ → p(x, ξ + iτ ψ ′ (x)), (6.7)
where x ∈ Ω and ξ ∈ R n . Definition 6.7. Let U be an open subset of Ω. Given a weight function ψ and an operator P we say that the simple-characteristic property is satisfied in U if, for all x ∈ U , we have ξ = 0 and τ = 0 when the map ρ x,ξ has a double root.

Remark 6.8. In fact the simple-characteristic property implies the property of strong pseudo-convexity. We refer the reader to [START_REF] Rousseau | On Carleman estimates with two large parameters[END_REF] and [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF].

We have the following result.

Theorem 6.9. Let x 0 ∈ S and let ψ ∈ C 0 (Ω) be such that

ψ k = ψ |Ω k ∈ C ∞ (Ω k ) for k = 1
, 2 and such that ψ k and P k have the simple characteristic property of Definition 6.7 in a neighborhood of x 0 in Ω k . Moreover, assume that P k , T j k , ψ, k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 .

Then there exist a neighborhood W of x 0 in R n and three constants C, τ * > 0, and γ * > 0 such that, for ϕ k = exp(γψ k ) and τk = τ γϕ k , (6.8)

k=1,2 γ τ -1 2 k e τ ϕ k u k 2 m k ,τ k + |e τ ϕ |S tr(u k )| 2 m k -1,1/2,τ ≤ C k=1,2 e τ ϕ k P k (x, D)u k 2 L 2 (Ω k ) + m j=1 e τ ϕ |S (T j 1 (x, D)u 1 + T j 2 (x, D)u 2 ) |S 2 m-1/2-β j ,τ ,
for all u k = w k|Ω k with w k ∈ C ∞ c (W ), τ ≥ τ * and γ ≥ γ * . We prove the following microlocal result and the result of Theorem 6.9 can be deduced, arguing as in Section 4.4. Theorem 6.10. Let x 0 ∈ S ∩ V and let ψ ∈ C 0 (V ) be such that ψr / ℓ ∈ C ∞ (V + ) and such that ψr / ℓ and Pr / ℓ have the simple characteristic property of Definition 6.7 in a neighborhood of x 0 in V + . Moreover, assume that Pr / ℓ , T j r / ℓ , ψr / ℓ , j = 1, . . . , m satisfies the transmission condition at (x 0 , ξ ′ 0 , τ0 ) ∈ S * T,τ (V + ). Then there exists U a conic open neighborhood of (x 0 , ξ ′ 0 , τ0 ) in V + × R n-1 × R + such that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp( χ) ⊂ U , there exist C > 0, τ * > 0, and γ * > 0 such that, for ϕr / ℓ = exp(γψr / ℓ ) and τr / ℓ = τ γϕr / ℓ , (6.9)

P ℓ,ϕ v ℓ 2 + + P r,ϕ v r 2 + + m j=1 T j ℓ,ϕ v ℓ|x n=0 + + T j r,ϕ v r|x n=0 + 2 m-β j -1/2,τ + γ 2 v ℓ 2 m ℓ ,-1,τ ℓ + v r 2 mr,-1,τr + |tr(v ℓ )| 2 m ℓ -1,-1/2,τ + |tr(v r )| 2 mr-1,-1/2,τ ≥ C γ τ -1 2 ℓ Op(χ ℓ )v ℓ 2 m ℓ ,τ ℓ + γ τ -1 2 r Op(χ r )v r 2 mr,τr + |tr(Op(χ ℓ )v ℓ )| 2 m ℓ -1,1/2,τ + |tr(Op(χ r )v r )| 2 mr -1,1/2,τ , for all v ℓ , v r ∈ S (R n + ), τ ≥ τ * , γ ≥ γ * , and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 (g T, r / ℓ ), with ̺′ r / ℓ = (x, ξ ′ , τr / ℓ (x)) for ̺ ′ = (x, ξ ′ , τ, γ).
Proof. Applying Lemma 6.13 in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] we obtain that there exists U a conic open neighborhood of (x 0 , ξ ′ 0 , τ0 ) in V + × R n-1 × R + such that for χ ∈ S 0 T,τ , homogeneous of degree 0, with supp( χ) ⊂ U , there exist C > 0, τ 0 > 0, and γ 0 > 0 such that (6.10) Summing (6.10) ℓ , (6.10) r , and (6.6) we obtain the result of Theorem 6.10 by taking τ and γ sufficiently large.

C Pr / ℓ ,ϕ vr / ℓ 2 + -Re B ar / ℓ ,br / ℓ (Op(χr / ℓ )vr / ℓ ) ≥ C ′ γ τ -1 2 r / ℓ Op(χr / ℓ )vr / ℓ 2 mr / ℓ ,τr / ℓ -C ′′ γ 2 vr / ℓ 2 mr / ℓ ,-1,τr / ℓ + τ -1 2 r / ℓ tr(Op(χr / ℓ )vr / ℓ ) 2 mr / ℓ -1,1/2,τ + γ tr(Op(χr / ℓ )vr / ℓ ) 2 mr / ℓ -1,0,τ , for τ ≥ τ 0 , γ ≥ γ 0 , and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 (g T, r / ℓ ),
6.7. Shifted estimates. As in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] it may be interesting to consider shifted estimates in the Sobolev scales. Namely we may wish to have an estimate of the following form.

Corollary 6.11. Let x 0 ∈ S and let ψ ∈ C 0 (Ω) be such that

ψ k = ψ |Ω k ∈ C ∞ (Ω k ) for k = 1, 2
and such that ψ has the strong pseudo-convexity property of Definition 6.2 with respect to P 1 and P 2 in a neighborhood of x 0 in Ω. Moreover, assume that P k , T j k , ψ, k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Let ℓ ∈ N and s ∈ R. Then there exist a neighborhood W of x 0 in R n and three constants C, τ * > 0, and γ * > 0 such that for ϕ k = exp(γψ k ) and τk = τ γϕ k :

(6.11) k=1,2 τ s-1/2 k e τ ϕ k u k 2 ℓ+m k ,τ k + |τ s k e τ ϕ |S tr(u k )| 2 ℓ+m k -1,1/2,τ ≤ C k=1,2 τ s k e τ ϕ k P k (x, D)u k 2 ℓ,τ k + m j=1 |τ s k e τ ϕ |S (T j 1 (x, D)u 1 + T j 2 (x, D)u 2 ) |S | 2 ℓ,m-1/2-β j ,τ , for all u k = w k|Ω k with w k ∈ C ∞ c (W ), τ ≥ τ * , and γ ≥ γ * .
Corollary 6.12. Let x 0 ∈ S and let ψ ∈ C 0 (Ω) be such that

ψ k = ψ |Ω k ∈ C ∞ (Ω k ) for k = 1
, 2 and such that ψ k and P k have the simple characteristic property of Definition 6.7 in a neighborhood of x 0 in Ω k . Moreover, assume that P k , T j k , ψ, k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Let ℓ ∈ N and s ∈ R. Then there exist a neighborhood W of x 0 in R n and three constants C, τ * > 0, and γ * > 0 such that, for ϕ k = exp(γψ k ) and τk = τ γϕ k , (6.12)

k=1,2 γ τ s-1 2 k e τ ϕ k u k 2 ℓ+m k ,τ k + |τ s k e τ ϕ |S tr(u k )| 2 ℓ+m k -1,1/2,τ ≤ C k=1,2 τ s k e τ ϕ k P k (x, D)u k 2 ℓ,τ k + m j=1 τ s k e τ ϕ |S (T j 1 (x, D)u 1 + T j 2 (x, D)u 2 ) |S 2 ℓ,m-1/2-β j ,τ , for all u k = w k|Ω k with w k ∈ C ∞ c (W ), τ ≥ τ * and γ ≥ γ * .
The proofs of these two corollaries can be adapted from the proofs of their counterpart at a boundary, namely Corollaries 6.14 and 6.15 in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF].

APPLICATION TO UNIQUE CONTINUATION

With the Carleman estimates we have derived here we can obtain unique continuation results near an interface for high-order elliptic operators with the transmission condition, if we make a geometrical assumption, namely the strong pseudo-convexity condition. Result for a product of two operators can be obtained if additionaly the simple characteristic propery holds for one of them. 7.1. Uniqueness under strong pseudo-convexity and transmission condition. Theorem 7.1. Let P k and T j k , j = 1, . . . , m be given as in Section 1. Let x 0 ∈ S, f ∈ C 0 (Ω), and V be a neighborhood of x 0 , be such that f has the strong pseudo-convexity property of Definition 6.2 with respect to P 1 and P 2 in V . Moreover, assume that P k , f, T j k , k = 1, 2, j = 1, . . . , m satisfies the transmission condition at x 0 . Assume that u is such that u

k = u |Ω k ∈ H m k (Ω k ) and satisfies • |P k u k (x)| ≤ C |α|≤m-1 |D α u k (x)|, a.e. in V k = V ∩ Ω k , k = 1, 2; (7.13) • for j = 1, . . . , m and |α| ≤ m -β j , with α ∈ N n-1 , |D α T T j 1 u 1 (x) + T j 2 u 2 (x) | ≤ C k=1,2 |α ′ |≤|α| +β j k -1 |D α ′ u k (x)|, a.e. in V ∩ S; (7.14) • and u vanishes in {x ∈ V ; f (x) ≥ f (x 0 )}. V ′′ V ′ Σ {ϕ(x) = ϕ(x 0 ) -δ} {ϕ(x) = ϕ(x 0 ) -δ/2} {ϕ(x) = ϕ(x 0 )} {f (x) = f (x 0 )} B W u = 0 S x 0 FIGURE 1.
Local geometry for the unique continuation problem. The shaded region Σ contains the supports of [P, χ]u and [χ, D j

x k ]u. In the figure the funtions f and ϕ are continuous and yet only piecewise smooth. Here unique continuation is performed across a surface that is not related to S.

Then u vanishes in a neighborhood of x 0 .

Here D α

T denotes a familly of differential operators that act tangentially to the interface S and, in local coordinates near x 0 , where S = {x n = 0}, their principal symbol is ξ ′α .

Proof. Strong pseudo-convexity is a stable notion in C 2 (see Proposition 28.3.2 in [START_REF]The analysis of linear partial differential operators[END_REF]). Here the function f is continuous and piecewise smooth. The argument of [START_REF]The analysis of linear partial differential operators[END_REF] applies on both sides of the interface. For ε chosen sufficiently small, there exists a neighborhood V ′ of x 0 such that the function ψ(x) = f (x) -ε|xx 0 | 2 has the strong pseudo-convexity property of Definition 6.2 with respect to P 1 and P 2 in V ′ . Similary we saw in Section 1.6.4 for the proof of Proposition 1.8, that the transmission condition (or rather property (1.29) ) is robust upon perturbation of the weight function. Hence if ε is chosen sufficiently small P, ψ, B k , k = 1, . . . , µ will also satisfy this condition. We set ϕ = exp(γψ). As shown in Proposition 28.3.3 in [START_REF]The analysis of linear partial differential operators[END_REF] the strong pseudo-convexity of the function ψ with respect to P 1 and P 2 implies the sub-ellipticity condition for {P k , ϕ k } for γ chosen sufficiently large for both k = 1, 2 with ϕ k = ϕ |Ω k . Moreover, as seen in Section 6.2 P k , T j k , ϕ; k = 1, 2, j = 1, . . . , µ also satisfies the transmission condition at x 0 .

The geometrical situation we describre is illustrated in Figures 1 and2. The two figures show different unique continuation configuration: across an hypersuface that is not related to the interface S, or across the interface S. We call W the region {x ∈ V ; f (x) ≥ f (x 0 )} (region beneath {f (x) = f (x 0 )} in Figure 1) where u vanishes by assumption. We choose V ′′ a neighborhood of x 0 such that V ′′ ⋐ V ′ .

We pick a function χ ∈ C ∞ c (R n ) such that χ = 1 in V ′′ and supp(χ) ∩ V ⊂ V ′ . We observe that the Carleman estimate of Theorem 1.6 applies to χu by density (possibly by reducing the neighborhoods V and V ′ of x 0 ): We set Σ := V ′ \ (V ′′ ∪ W ) (see the shaded region in Figure 1). We have supp(M k,i u k ) ⊂ Σ, i ∈ I k and supp( Mk,j u k ) ⊂ Σ, i ∈ J k , as they are confined in the region where χ varies and u does not vanish.

We thus obtain For all δ > 0, we set V δ = {x ∈ V ; ϕ(x) ≤ ϕ(x 0 ) -δ}. There exists δ > 0 such that Σ ⋐ V δ . We then choose B a neighborhood of x 0 such that B ⊂ V ′′ \ V δ/2 and obtain, as χ ≡ 1 on B, for some τ 1 > 0. Since inf B ϕ > sup Σ ϕ + δ/2, letting τ go to ∞, we obtain u = 0 in B.

7.2. Uniqueness for products of operators. We now consider two sets of elliptic operators: P 1 and Q defined on Ω 1 and P 2 and Q 2 defined on Ω 2 . We denote their respective orders by m p 1 , m q 1 , m p 2 , and m q 2 . We assume that m p 1 = m p 2 = m p . We also consider interface operators T p,j k , k = 1, 2, j = 1, . . . , m p of order β p,j k and T q,j k , k = 1, 2, j = 1, . . . , m q of order β q,j k with m q = (m q 1 + m q 2 )/2. We assume that m pβ p,j 1 = m pβ p,j 2 = m pβ p,j and m q 1β q,j 1 = m q 2β q,j 2 = m qβ q,j . Then the operators P 1 , P 2 , T p,j k , k = 1, 2, j = 1, . . . , m p , allow one to define an elliptic transmission problem as presented in Section 1. The same is valid for Q 1 , Q 2 , T q,j k , k = 1, 2, j = 1, . . . , m q .

We observe that P 1 Q 1 , P 2 Q 2 , and the intreface operators T p,j k Q k , k = 1, 2, j = 1, . . . , m p , and T q,j k , k = 1, 2, j = 1, . . . , m q also allow to define an elliptic transmission problem. In fact, the operators P 1 Q and P 2 Q 2 are of respective order m 1 = m p +m q 1 and m 2 = m p +m q 2 . We set m = m p +m q = (m 1 +m 2 )/2. The interface operator T q,j k is of order β q,j k and we have m 1β q,j 1 = m 2β q,j 2 = mβ q,j . The interface operator T p,j k Q k is of order β p,j k + m q k and we have m 1 -(β p,j 1 + m q 1 ) = m 2 -(β p,j 2 + m q 2 ). One may possibly wonder about unique continuation for this product transmission problem, in particuler in the case no Carleman estimate of the type derived here can be achieved. Let us however assume that for a function ψ and the weigth function ϕ = exp(γψ) we can derive Carleman estimates for the transmission problems associated with P 1 , P 2 and Q 1 , Q 2 . More precisely we assume that the first problem satisfies the simple characteristic property while the second one only satisfies the strong pseudo-convexity condition. Theorem 7.2. Let P k , T p,j k , k = 1, 2, j = 1, . . . , m p , and Q k , T q,j k , k = 1, 2, j = 1, . . . , m q , be given as above. Let x 0 ∈ S, f ∈ C 0 (Ω), with f k = f |Ω k and V a neighborhood of x 0 , be such that (1) f k and P k have the simple characteristic property of Definition 6.7 in V ∩ Ω k ;

(2) f has the strong pseudo-convexity property of Definition 6.2 with respect to Q 1 and Q 2 in V ;

(3) P k , f, T p,j k , k = 1, 2, j = 1, . . . , m p satisfies the transmission condition at x 0 ; (4) Q k , f, T q,j k , k = 1, 2, j = 1, . . . , m q satisfies the transmission condition at x 0 . Assume that u is such that u k = u |Ω k ∈ H m p +m q k (Ω k ) and satisfies

• |P k Q k u k (x)| ≤ C |α|≤m p +m q k -1
|D α u k (x)|, a.e. in V k = V ∩ Ω k , k = 1, 2; (7.16)

• for j = 1, . . . , m p , |α| ≤ m pβ p,j , with α ∈ N n-1 ,

|D α T T p,j 1 Q 1 u 1 (x) + T p,j 2 Q 2 u 2 (x) | ≤ C k=1,2 |α ′ |≤|α|+m q k +β p,j k -1
|D α ′ u k (x)|, a.e. in V ∩ S; (7.17)

• for j = 1, . . . , m q , |α 1 | ≤ m p , |α 2 | ≤ m qβ q,j , with α 1 ∈ N n and α 2 ∈ N n-1 , |D α 1 D α 2 T T q,j 1 u 1 (x) + T q,j 2 u 2 (x) | ≤ C • and u vanishes in {x ∈ V ; f (x) ≥ f (x 0 )}.

Then u vanishes in a neighborhood of x 0 .

Here D α

T denotes a familly of differential operators that act tangentially to the interface S and, in local coordinates near x 0 , where S = {x n = 0}, their principal symbol is ξ ′α . Remark 7.3. Here we have assume that m p 1 = m p 2 . It would be interesting to know if such an assumption can be removed. This assumption is connected to the shifted estimate of Corollary 6.11, where the shift is the same on both sides of the interface. Having different Sobolev-scale shifts from one side to the other leads to technical difficulties with the transmission terms on the interface. It is not clear whether such estimates can be achieved without modifying the properties of the transmission operators.

Proof. The proof follows that of Theorem 7.1. We set ψ(x) = f (x) -ε|xx 0 | 2 and conditions (1)-( 4) in the statement of the theorem are also satisfied by ψ for ε chosen sufficiently small in a neighborhood V ′ ⊂ V of x 0 . We then set ϕ = exp(γψ).

We derive an estimate for P k Q k . We first write an estimate for P k . By Theorem 6.9, there exists V 1 neighborhood of x 0 in R n such that V 1 ⊂ V ′ and k=1,2 τ -1/2 e τ ϕ |S (T q,j 1 v 1 + T q,j 2 v 2 ) |S m p ,m q -1/2-β q,j ,τ , for all v k = w k|Ω k with w k ∈ C ∞ c (V 2 ), τ ≥ τ 2 and γ ≥ γ 2 , for τ 2 and γ 2 chosen sufficiently large. Letting V 3 = V 1 ∩ V 2 with the previous two estimates we obtain

(7.19) γ 1/2 k=1,2 τ -1 k e τ ϕ k v k m p +m q k ,τ k + τ -1/2 e τ ϕ |S tr(v k ) m p +m q k -1,1/2,τ k=1,2 e τ ϕ k P k Q k v k L 2 (Ω k ) + m p j=1 e τ ϕ |S (T p,j 1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S m p -1/2-β p,j ,τ + γ 1/2 m q j=1
τ -1/2 e τ ϕ |S (T q,j 1 v 1 + T q,j 2 v 2 ) |S m p ,m q -1/2-β q,j ,τ , for all v k = w k|Ω k with w k ∈ C ∞ c (V 3 ), τ ≥ τ 3 and γ ≥ γ 3 , for τ 3 and γ 3 chosen sufficiently large. We choose χ as in the proof of Theorem 7.1 and we apply estimate (7.19) to v k = χu k as can be done by a density argument. We now sketch how the remainder of the proof can be carried out.

We have

P k Q k (χu) = χP k Q k u + [P k Q k , χ]u. The term [P k Q k ,
χ]u is supported in the set Σ introduced in the proof of Theorem 7.1 and can be handle as it is done there. For the first term, with (7.16) we have

e τ ϕ χP k Q k u L 2 (Ω k ) |α|≤m p +m q k -1
e τ ϕ χD α u L 2 (Ω k )

|α|≤m p +m q k -1 e τ ϕ D α (χu) L 2 (Ω k ) + |α|≤m p +m q k -1

e τ ϕ [χ, D α ]u L 2 (Ω k ) .
The second term in the r.h.s. concerns functions with support located in Σ and their treatment is done as in the proof of Theorem 7.1. For the first term we have

|α|≤m p +m q k -1
e τ ϕ D α (χu) L 2 (Ω k ) e τ ϕ χu m p +m q k -1,τ k τ -1 k e τ ϕ χu m p +m q k ,τ k , which can be absorbed by the first term in (7.19) by chosing γ sufficiently large.

Next we have e τ ϕ |S (T p,j The terms [D α T T p,j k Q k , χ]u are supported in the set Σ and can be treated as in the proof of Theorem 7.1. For the first term, with (7.17) we have τ r e τ ϕ |S χD α T (T p,j The second term in the r.h.s. concerns functions with support located in Σ and their treatment is done as in the proof of Theorem 7.1. For the first term we have r + |α| ≤ m pβ p,j and |α ′ | ≤ |α| + m q k + β p,j k -1 and thus we write

1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S m p -1/2-β p,j ,τ ≤ e τ ϕ |S (T p,j 1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S m p -β p,j ,τ = r+|α| ≤m p -β p,j τ r D α T e τ ϕ |S (T p,j 1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S L 2 (S) r+|α| ≤m p -β p,j τ r e τ ϕ |S D α T (T p,j 1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S L 2 (S) . Writing D α T T p,j k Q k v k = χD α T T p,j k Q k u k + [D α T T p,j k Q k ,
1 Q 1 u 1 + T p,j 2 Q 2 u 2 ) |S L 2 (S)
τ r D α ′ (e τ ϕ k χu k ) |S L 2 (S)
tr(e τ ϕ k χu k ) m p +m q k -1,0,τ .

As γ 1/2 τ -1/2 e τ ϕ |S tr(χu k ) m p +m q k -1,1/2,τ ≥ γ 1/2 e τ ϕ |S tr(χu k ) m p +m q k -1,0,τ γ 1/2 tr(e τ ϕ k χu k ) m p +m q k -1,0,τ we see that the above terms can be absorbed by the second term in (7.19) by chosing γ sufficiently large. We finally write

CONTENTS 1 .

 1 Introduction and main result 1.1. Setting 1.2. Sub-ellipticity condition 1.3. Transmission conditions 1.4. Sobolev norms with a parameter 1.5. Statement of the main result 1.6. Local reduction of the problem near the interface 1.7. Examples 1.8. Notation 2. Pseudo-differential operators with a large parameter 2.1. Classes of symbols Date: May 7, 2016.

Definition 1 . 1 .

 11 Let k ∈ {1, 2} and let U be an open subset of Ω k and set ϕ k = ϕ |Ω k . The pair {P k , ϕ k } satisfies the sub-ellipticity condition on U if ϕ ′ k (x) := ∇ϕ(x) = 0 at every point in U and if

For

  m ∈ N and s ∈ R we introduce the following interface space H ,τ , u = (u 0 , . . . , u m ).

  Op(a) ∈ Ψ m τ (R n ) or simply Ψ m τ,cl (resp. Ψ m τ (R n ) or simply Ψ m τ,cl ). Here D denotes D x . The principal symbol of Op(a) is σ(Op(a)) = σ(a) ∈ S m τ /S m-1 τ (resp. S m τ,cl /S m-1 τ,cl ). Tangential operators are defined similarly. For a ∈ S m T,τ (R n

Proposition 2 . 3

 23 (composition). Let a ∈ S m T,τ (resp. S m T,τ,cl ) and b ∈ S m ′ T,τ (resp. S m ′ T,τ,cl ) be two tangential symbols. Then Op(a) Op(b) = Op(c) ∈ Ψ m+m ′ T,τ (resp. Ψ m+m ′ T,τ,cl ) with c ∈ S m+m ′ T,τ

Definition 2 . 6 .

 26 Let b ∈ S m,r τ (resp. S m,r τ,cl ), with b(x, D, τ ) = m j=0 b j (x, D ′ , τ )D j n , b j ∈ S m+r-j T,τ (resp. S m+r-j T,τ,cl ). We set b(x, D, τ ) * = m j=0 D j n b j (x, D ′ , τ ) * .

)

  We have a(x, D, τ ) * ∈ Ψ m,r τ (resp. Ψ m,r τ,cl ) and σ a(x, D, τ ) * ≡ m j=0 a j (̺ ′ )ξ j n ∈ S m,r τ /S m,r-1 τ (resp. S m,r τ,cl /S m,r-1 τ,cl

  Op(a) * -Op(a) ∈ Ψ m,r-1 τ (resp. Ψ m,r-1 τ,cl

Definition 3 . 1 (

 31 interface quadratic forms). Let w = (w ℓ , w r ) ∈ S (R n + )

  with Σ a s r / ℓ and Σ b s r / ℓ defined as in (3.1).

  ) for τ sufficiently large. Combining (3.4)-(3.5) with (3.3) yields the conclusion. Proposition 3.4. Assume that the transmission condition of Definition 1.4 holds at ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) ∈ S * T,τ (V ) with x 0 ∈ S (see also (1.25)-(1.26) and (1.29) for a formulation in the local setting). Then there exists U 1 a conic open neighborhood of ̺

4. 1 .

 1 Estimate with the transmission condition. Let us first consider a polynomial function with roots with negative imaginary parts in a microlocal region. Then, we have the following perfect microlocal elliptic estimate. We refer to Lemma 4.1 in Part I[START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] for a proof.

Lemma

4. 2 .

 2 Estimate with a positive Poisson bracket on the characteristic set. If we consider the case of two symbols a, b such that the Poisson bracket {a, b} is positive on the characterisitic set {a = b = 0}, an estimate with the control of a volume norm can be achieved. Lemma 4.3. Let U be an open set of V + . Let a ∈ S m,0 τ and b ∈ S m-1,1 τ be real symbols homogeneous of degree m in (τ, ξ), and set Q a,b (v) = 2 Re (Av, iBv) + , A = a(x, D, τ ), B = b(x, D, τ ).

.

  The sub-ellipticity condition of Definition 1.1 reads

5 . 3 .

 53 |tr(u)| m,σ,τ = |tr(u)| m,σ,τ ℓ = |tr(u)| m,σ,τr as in (5.8) for interface norms. Interface quadratic forms. Definition 5.2. Let w = (w ℓ , w r ) ∈ S (R n + )

Definition 5 . 3 .

 53 Let G be an interface quadratic form of type (m ℓ -1, m r -1, σ) associated with the bilinear symbol Σ G (̺ ′ , w, w). We say that G is positive definite in W if there exists C > 0 and R > 0 such that Re Σ G (̺ ′′ , x n = 0 + , w, w) ≥ C

τττ 1 2 1 2 2 τ

 112 FIGURE 2. Local geometry for the unique continuation problem. Here, unique continuation is performed across the interface S.

k=1, 2 τ

 2 -1/2 e τ ϕ k χu k m k ,τ + |e τ ϕ k tr(χu k )| m k -1,1/2,τ k=1,2 |α|≤m k -1 e τ ϕ k D α u k L 2 (Σ) + r+|α|≤m k -1 τ r (D α u k ) |S L 2 (Σ∩S) .

|α|≤m k - 1 D

 1 α u |S L 2 (Σ∩S , τ ≥ τ 1

k=1, 2

 2 |α ′ |≤|α 1 |+|α 2 | +β q,j k -1|D α ′ u k (x)|, a.e. in V ∩ S;(7.18)

γ 1 /2 τ - 1 2k 2 e 2 τ - 1 / 2 k

 112212 e τ ϕ k v k m p ,τ k + |e τ ϕ |S tr(v k )| m p -1,1/2,τ k=1,τ ϕ k P k v k L 2 (Ω k ) + m p j=1 e τ ϕ |S (T p,j 1 v 1 + T p,j 2 v 2 ) |S m p -1/2-β p,j ,τ , for all v k = w k|Ω k with w k ∈ C ∞ c (V 1), τ ≥ τ 1 and γ ≥ γ 1 , for τ 1 and γ 1 chosen sufficiently large. For Q k , by Corollary 6.11, there existsV 2 neighborhood of x 0 in R n such that V 2 ⊂ V ′ and k=1,2 τ -1 k e τ ϕ k v k m p +m q k ,τ k + τ -1/2 e τ ϕ |S tr(v k ) m p +m q k -1,1/2,τ k=1,e τ ϕ k Q k v k m p ,τ k + m q j=1

  χ]u we havee τ ϕ |S (T p,j 1 Q 1 v 1 + T p,j 2 Q 2 v 2 ) |S m p -1/2-β p,j ,τ r+|α| ≤m p -β p,j τ r e τ ϕ |S χD α T (T p,j 1 Q 1 u 1 + T p,j 2 Q 2 u 2 ) |S L 2 (S) + r+|α| ≤m p -β p,j τ r e τ ϕ |S ([D α T T p,j 1 Q 1 , χ]u k + [D α T T p,j 2 Q 2 , χ]v 2 ) |S L 2 (S) .

k=1, 2 |α ′ |≤|α|+m q k +β p,j k - 1 τττ

 21 r e τ ϕ |S (χD α ′ u k ) |S L 2 (S) r D α ′ (e τ ϕ k χu k ) |S L 2 (S) + r ([e τ ϕ |S χ, D α ′ ]u k ) |S L 2 (S)

  ∈ S we denote the roots of pr / ℓ ,ϕ (̺ ′ 0 , ξ n ), viewed as a polynomial in ξ n , by αr / ℓ ,1 , . . . , αr / ℓ ,nr / ℓ , with respective multiplicities µr / ℓ ,1 , . . . , µr / ℓ ,nr / ℓ satisfying µr / ℓ ,1 + • • • + µr / ℓ ,nr / ℓ = mr / ℓ , with m ℓ = m 1 and m r = m 2 . By [4, Lemma A.2], there exists a conic open neighborhood U of ̺ ′

	Naturally by (1.17)-(1.18) we have
	p2,ϕ (ω, ξ n ) = p r,ϕ (x, ξ, τ ),	tj 2,ϕ (ω, ξ n ) = t j r,ϕ (x, ξ, τ ).
	1.6.3. Symbol factorizations. For a fixed point ̺ ′ 0 = (x 0 , ξ ′ 0 , τ 0 ) ∈ S * T,τ (V ) (see the definition below in Section 1.8) with x 0 0 such that
	(1.22)
	(1.21)	).

  The operators Pr / ℓ ,2 and Pr / ℓ ,1 are thus formally self-adjoint. Their respective principal symbols ar / ℓ (x, ξ, τ ) ∈ are both real and homogeneous. We set pr / ℓ ,ϕ = ar / ℓ + ibr / ℓ . We then set Re(Ar / ℓ vr / ℓ , iBr / ℓ vr / ℓ ) + , Ar / ℓ = Op(ar / ℓ ), Br / ℓ = Op(br / ℓ ). Pr / ℓ ,ϕ = Ar / ℓ + iBr / ℓ + Rr / ℓ , Rr / ℓ ∈ Ψ

	S a,b (v) = 2 Note that we have mr / ℓ ,0 τ mr / ℓ τ and br / ℓ (x, ξ, τ ) ∈ S Q r / ℓ	-1,1
	(4.5)	mr / ℓ τ	,-1

1.6). We decompose the conjugated operator Pr / ℓ ,ϕ = e τ ϕr / ℓ Pr / ℓ e -τ ϕr / ℓ as Pr / ℓ ,ϕ = Pr / ℓ ,2 + iPr / ℓ ,1 , Pr / ℓ ,2 = 1 2 (Pr / ℓ ,ϕ + P * r / ℓ ,ϕ ), Pr / ℓ ,1 = 1 2i (Pr / ℓ ,ϕ -P * r / ℓ ,ϕ ).

1

  Ar / ℓ vr / With Proposition 4.2, making use of the transmission condition, we obtain for M chosen sufficiently large

ℓ 2 + + Br / ℓ vr / ℓ 2 + + tr(vr / ℓ ) 2 mr / ℓ -1,1/2,τ , with B ar / ℓ ,br / ℓ satisfying B ar / ℓ ,br / ℓ (vr / ℓ ) tr(vr / ℓ ) 2 mr / ℓ -1,1/2,τ .

  2 mr-1,1/2,τ , for τ chosen sufficiently large and for vr / ℓ = e τ ϕr / ℓ ur / ℓ with ur / ℓ = wr / ℓ |R n

+ with wr / ℓ ∈ C ∞ c (W ). (see the statement of Theorem 1.6). Observe then that, for any N ∈ N, vr / ℓ m,τ ≤ j∈J Op(χ j )vr / ℓ m,τ + Op(χ)vr / ℓ m,τ j∈J Op(χ j )vr / ℓ m,τ + vr / ℓ m,-N,τ , and tr(vr

  U be an open subset of O. The function ψ is said to be strongly pseudo-convex w.r.t. p in U up to the boundary if (s-Ψc) is valid for all x ∈ U . Definition 6.2 (strong pseudo-convexity at an interface). Let Ω, Ω 1 , Ω 2 , and S be as in Section 1. Let ψ be a continuous function such that ψ k = ψ |Ω k are smooth for k = 1, 2. Let U be an open subset of Ω that meets S. The function ψ is said to be strongly pseudo-convex w.r.t. P 1 and P 2 in U up to the interface if both ψ k , k = 1, 2, are strongly pseudo-convex w.r.t. P k in U k = U ∩ Ω k up to the boundary.Note in particular that for x ∈ S ∩ U (s-Ψc) is required to hold for both k = 1 and k = 2.6.2. Conjugated operators and transmission condition.Here we use directly the notation introduced in Section 1.6.2 with the weight functions of the form ϕr / ℓ = exp(γψr / ℓ ), which is sensible as the transmission is a coordinate invariant property. The principal symbol of Pr / ℓ ,ϕ = e τ ϕr / ℓ Pr / ℓ e -τ ϕr / ℓ ∈ Ψ m,0 cl (gr / ℓ ) in the present calculus is

  and γ ≥ γ * . Now, arguing as in the proof of Proposition 4.2 we write χr / ℓ pr / ℓ ,ϕ = χr / ℓ κr / ℓ ,ϕ p - r / ℓ ,ϕ = χr / ℓ χr / ℓ κr / ℓ ,ϕ pr / ℓ ,ϕ , where pr / ℓ ,ϕ denotes an extension of p - r / ℓ ,ϕ to the whole phase space. Then Op(χr / ℓ )Pr / ℓ ,ϕ = Op(p - r / ℓ ,ϕ ) Op(χr / ℓ ) Op( χr / ℓ κr / ℓ ,ϕ ) + Rr / ℓ , / ℓ ,ϕ = κr / ℓ ,ϕ ξ j n , j = 0, . . . , m - r / ℓ -1 in a neighborhood of U we have D j n Op(χr / ℓ ) Op( χr / ℓ κr / ℓ ,ϕ )vr / ℓ = E j+m+1 r / ℓ ,ϕ vr / ℓ + Rr / ℓ ,j vr / ℓ , with Rr / ℓ ,j ∈ γΨ

	+ |tr(v ℓ )| 2 m ℓ -1,-N,τ + |tr(v r )| 2 mr-1,-N,τ ,
	Recalling that e j+m+1
	mr / ℓ	-m -r / ℓ	+j,-1

r cl (gr / ℓ ). We thus obtain m

  5) C Pr / ℓ ,ϕ vr / ℓ 2 + -Re B ar / ℓ ,br / ℓ (Op(χr / ℓ )vr / ℓ ) ≥ C ′ τ -1 2 r / ℓ Op(χr / ℓ )vr / ℓ ≥ τ 0 , γ ≥ γ 0 , and χr / ℓ (̺ ′ ) = χ(̺ ′ r / ℓ ) ∈ S 0 (g T, r / ℓ ),where B ar / ℓ ,br / ℓ satisfies B ar / ℓ ,br / ℓ (Op(χr / ℓ )vr / ℓ ) tr(Op(χr / ℓ )vr / ℓ )

									2
									mr / ℓ	,τr / ℓ
	-C ′′ γ 2 vr / ℓ	2 mr / ℓ	,-1,τr / ℓ	+	τ -1 2 r / ℓ tr(Op(χr / ℓ )vr / ℓ )	2 mr / ℓ	-1,1/2,τr / ℓ	+ γ tr(Op(χr / ℓ )vr / ℓ )	2 mr / ℓ	-1,0,τr / ℓ	,
	for τ 2 mr / ℓ	-1,1/2,τ .

  where B ar / ℓ ,br / ℓ satisfies B ar / ℓ ,br / ℓ (Op(χr / ℓ )vr / ℓ ) tr(Op(χr / ℓ )vr / ℓ )

	2 mr / ℓ	-1,1/2,τ .

Here to simplify we consider the case S = {xn = 0}. Then ξn corresponds the (co)normal direction at the interface. In the main text we shall use change of variables to reach this configuration locally.

By abuse of notation, in the new local coordinates, we keep the notation P k and T j k , j = 1, . . . , m, k = 1, 2, for the operators introduced in the beginning of Section 1.

This interpretation makes sense in the case ϕ = ϕ(xn). Then γ k = ∂x n ϕ k . In Carleman estimates, "observation" region are associated with regions where the weight function is the largest.