| Preface | 7 |
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| Contents | 9 |
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| List of Contributors | 11 |
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| Trace theorem on the Heisenberg group on homogeneous hypersurfaces | 15 |
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| 1 Introduction | 15 |
| 2 A Hardy type inequality | 20 |
| 3 The proof of the trace and trace lifting theorem | 24 |
| 4 Concluding remarks | 28 |
| References | 28 |
| Strong unique continuation and finite jet determination for Cauchy Riemann mappings | 30 |
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| 1 Introduction | 30 |
| 2 Local coordinates | 31 |
| 3 Nondegeneracy conditions | 33 |
| 4 Necessary conditions and su.cient conditions for finite jet determination | 35 |
| 5 Lie group structures and jet parameterization | 37 |
| References | 40 |
| On the Cauchy problem for some hyperbolic operator with double characteristics | 42 |
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| 1 Introduction and statements | 42 |
| 2 The model operator | 44 |
| 3 Shibuya solutions | 45 |
| 4 Stokes multipliers | 47 |
| 5 Asymptotic analysis | 49 |
| 6 Final steps in the proof of the necessary condition | 55 |
| References | 57 |
| On the differentiability class of the admissible square roots of regular nonnegative functions | 58 |
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| 1 Introduction | 58 |
| 2 Regularity of well-chosen admissible roots | 59 |
| References | 66 |
| The Benjamin Ono equation in energy space | 67 |
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| 1 Introduction | 67 |
| 2 Bourgain spaces | 69 |
| 3 A priori estimate on weak solutions | 70 |
| 4 The gauge transformation | 71 |
| 5 The existence and uniqueness result | 73 |
| References | 74 |
| Instabilities in Zakharov equations for laser propagation in a plasma | 75 |
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| 1 Introduction | 75 |
| 2 The instability mechanism | 78 |
| 3 Scheme of the proof | 80 |
| 4 The linear instability | 84 |
| 5 The linear equation | 89 |
| 6 End of proofs | 92 |
| References | 93 |
| Symplectic strata and analytic hypoellipticity | 94 |
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| 1 Introduction | 94 |
| 2 The symplectic case | 95 |
| 3 The example of Baouendi Goulaouic | 95 |
| 4 Treves original conjecture | 96 |
| 5 The Poisson stratification of S | 97 |
| 6 Examples | 98 |
| 7 Treves conjecture | 98 |
| 8 Symplectic strata of codimension 2 | 99 |
| 9 Sketch of the proof | 100 |
| References | 103 |
| On the backward uniqueness property for a class of parabolic operators | 106 |
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| 1 Introduction, statements and remarks | 106 |
| 2 Proof of Theorem 1.1 | 110 |
| 3 Proof of Theorem 1.2 | 114 |
| References | 116 |
| Inverse problems for hyperbolic equations | 117 |
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| 1 Formulation of the problem and the main theorem | 117 |
| 2 Hyperbolic systems with Yang Mills potentials and domains with obstacles | 120 |
| 3 A geometric optics approach | 124 |
| References | 125 |
| On the optimality of some observability inequalities for plate systems with potentials | 127 |
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| 1 Introduction | 128 |
| 2 Preliminaries | 132 |
| 3 The sharp observability estimate | 135 |
| 4 Extension of Meshkov s construction to the bi-Laplacian equation | 138 |
| 5 Optimality of the observability constant for plate systems | 139 |
| 6 Further remarks and open problems | 141 |
| References | 141 |
| Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach | 143 |
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| Introduction | 143 |
| 1 A general setting and a motivating example | 144 |
| 2 Weak symplectic manifolds | 149 |
| 3 Right invariant weak Riemannian metrics on Lie groups | 157 |
| 4 The Hamiltonian approach | 165 |
| 5 Vanishing H0-geodesic distance on groups of diffeomorphisms | 171 |
| 6 The regular Lie group of rapidly decreasing diffeomorphisms | 179 |
| 7 The diffeomorphism group of S1 or R, and Burgers hierarchy | 188 |
| 8 The Virasoro Bott group and the Korteweg de Vries hierarchy | 195 |
| Appendix A Smooth calculus beyond Banach spaces | 212 |
| Appendix B Regular infinite-dimensional Lie groups | 216 |
| References | 223 |
| Non-effectively hyperbolic operators and bicharacteristics | 226 |
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| 1 Introduction | 226 |
| 2 Non-effectively hyperbolic symbols, elementary decomposition and a priori estimates | 227 |
| 3 Conditions for elementary decomposition | 231 |
| 4 Behavior of bicharacteristics and elementary decomposition | 240 |
| 5 Remarks | 254 |
| References | 254 |
| On the Fefferman Phong inequality for systems of PDEs | 256 |
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| 1 Introduction | 256 |
| 2 Background on the Weyl Hörmander calculus | 258 |
| 3 A proof by induction on the size of the system | 260 |
| References | 274 |
| Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations | 276 |
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| 1 Introduction | 276 |
| 2 Resonances for time-periodic potentials | 278 |
| 3 Strichartz estimates | 282 |
| 4 Non-trapping moving obstacles | 285 |
| 5 Trapping moving obstacles | 289 |
| References | 293 |
| An elementary proof of Fedili s theorem and extensions | 295 |
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| 1 Introduction | 295 |
| 2 Proof of the theorem | 296 |
| References | 298 |
| Outgoing parametrices and global Strichartz estimates for Schrödinger equations with variable coefficients | 299 |
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| 1 Introduction | 299 |
| 2 Outline of the proofs | 305 |
| References | 320 |
| On the analyticity of solutions of sums of squares of vector fields | 322 |
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| 1 Global Poisson stratification | 324 |
| 2 The analyticity conjectures | 328 |
| References | 335 |