?. Next, We have Op w (a)u ? S (R n ) and

H. We and . Op-w, ?(1, g) For any ?, r ? R, with Proposition 2.5, the operator H maps H ?,r (R n ) into itself continuously and is invertible, for ? sufficiently large, arguing as in the proof of Lemma 2.3. We thus set?uset? set?u = H ?1 u ? H ?,r (R n )

A. Op-w, R n ) the injectivity of Op w (? ? ?s µ ?k ) from L 2 into H k,s (R n ) (see Lemma 2.3) we obtain Op w (? ? s µ k )? u = v. In particular, as ?, r ? R are chosen arbitrarily we find v ? H k ? ,s ? (R n ) for any k ? , s ? ? R, if ? is chosen sufficiently large and we have by (B.17) v k ? ,s ? = Op w (? ? s ? µ k ? )

?. As-v, we also have u k+k ? ,s+s ? = Op w (? ? s+s ? µ k+k ? )u L 2 = Op w (? ? s+s ? µ k+k ? ) Op w (? ? ?s µ ?k )v L 2 v k ? ,s ? . References [Ali83] S. Alinhac, Non-unicité duprobì eme de Cauchy, Ann. of Math, issue.2 1, pp.117-77, 1983.

]. V. Bar00 and . Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim, vol.42, pp.73-89, 2000.

[. Bony and J. Chemin, Espaces fonctionnels associ??s au calcul de Weyl-H??rmander, Espaces fonctionnels associés au calcul de Weyl-Hörmander, pp.77-118, 1994.
DOI : 10.24033/bsmf.2223

M. [. Bukhgeim and . Klibanov, Global uniqueness of class of multidimensional inverse problems, Soviet Math. Dokl, vol.24, pp.244-247, 1981.

J. [. Bellassoued, C. Le-rousseau, G. Bardos, J. Lebeau, and . Rauch, Carleman estimates for elliptic boundary value problems, Preprint (2013), 50 pages Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim, vol.30, pp.1024-1065, 1992.

]. R. Bru90 and . Brummelhuis, A counterexample to the Fefferman-Phong inequality for systems, C. R. Acad. Sci. Paris Sér. I Math, issue.3, pp.310-95, 1990.

M. [. Bellassoued and . Yamamoto, Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases Uniqueness in the Cauchy problem for partial differential equations., Amer, Appl. Anal. J. Math, vol.91, issue.1, pp.35-67, 1958.

]. T. Car39 and . Carleman, Sur uneprobì eme d'unicité pour les systèmes d'´ equations aux dérivées partiellesàpartiellesà deux variables indépendantes, Ark. Mat. Astr. Fys, vol.26, issue.17, pp.1-9, 1939.

D. [. Colombini and . Santo, Condition Is Not sufficient for uniqueness in the cauchy problem, Communications in Partial Differential Equations, vol.33, issue.11-12, pp.2113-2128, 1995.
DOI : 10.1016/0022-0396(74)90094-1

F. Colombini, D. Santo, and C. Zuily, The Fefferman-Phong inequality in the locally temperate Weyl calculus, Osaka J. Math, vol.33, pp.847-861, 1996.

]. B. Deh84 and . Dehman, Unicité duprobì eme de cauchy pour une classe d'opérateurs quasi-homogènes, J. Math. Kyoto Univ, vol.24, pp.453-471, 1984.

D. Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Inventiones mathematicae, vol.41, issue.1, pp.119-171, 2009.
DOI : 10.1007/s00222-009-0196-4

V. [. Eller and . Isakov, Carleman estimates with two large parameters and applications, Contemporary Math, pp.117-136, 2000.

]. M. Ell00 and . Eller, Carleman estimates with a second large parameter, Journal of Mathematical Analysis and Applications, vol.249, pp.491-514, 2000.

E. [. Fernández-cara and . Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.17, issue.5, pp.583-616, 2000.
DOI : 10.1016/S0294-1449(00)00117-7

O. [. Fursikov and . Yu, Imanuvilov, Controllability of evolution equations Lecture notes, Proc. Nat. Acad. Sci, pp.75-4673, 1978.

]. L. Hör58 and . Hörmander, On the uniqueness of the Cauchy problem, Math. Scand, vol.6, pp.213-225, 1958.

[. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, vol.8, pp.21-64, 1983.

]. L. Hör85a and . Hörmander, The Analysis of Linear Partial Differential Operators The Analysis of Linear Partial Differential Operators The Analysis of Linear Partial Differential Operators, 1985.

O. Yu, V. Imanuvilov, M. Isakov, and . Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Comm. Pure Appl. Math, vol.56, pp.1366-1382, 2003.

N. [. Isakov and . Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Applicationes Mathematicae, vol.35, issue.4, pp.447-465, 2008.
DOI : 10.4064/am35-4-4

]. V. Isa93 and . Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, vol.105, pp.217-238, 1993.

D. [. Koch and . Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Communications on Pure and Applied Mathematics, vol.2, issue.3, pp.339-360, 2001.
DOI : 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D

[. Rousseau and N. Lerner, Carleman estimates for anisotropic elliptic operators with jumps at an interface, Anal. PDE, to appear On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, pp.712-747, 2010.

J. Lebeau and L. Robbiano, Contr??le Exact De L??quation De La Chaleur, Communications in Partial Differential Equations, vol.52, issue.1-2, pp.335-356, 1995.
DOI : 10.1016/0022-0396(87)90043-X

[. Rousseau and L. Robbiano, 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, Arch. Rational Mech. Anal, vol.105, pp.953-990, 2010.

]. C. Sog89 and . Sogge, Oscillatory integrals and unique continuation for second order elliptic differential equations, J. Amer. Math. Soc, vol.2, issue.3, pp.491-515, 1989.

]. C. Zui83 and . Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem, Birkhauser, 1983.

J. Le and R. , Laboratoire de Mathématiques -Analyse, Probabilités, Modélisation -Orléans, CNRS UMR 7349