Index adjoint, 26 almost orthogonality, 78 asymptotic sum, 22 bicharacteristic (curve), 116 Cauchy problem, 93 change of variables, 37 characteristic (surface), 111 commutator, 40 continuity, L 2 , 30 convexity inequalities, 82 cotangent bundle, 29 density on a manifold, 74 Dirichlet problem, 124 elliptic (symbol, operator), 33 energy (inequality), 98 1-form, canonical, 39 2-form, symplectic, 39 factorization (of an operator), 103 Fourier distribution, 91 Fourier transform, 11 Garding inequality, 31 Gagliardo-Nirenberg inequality, 86 Hamiltonian (vector field), 40 Holder spaces, 80 Hormander's theorem, 102 hyperbolic (operator, system), 103 implicit function theorem, 121, 122 interpolation, 108 isometric embedding, 133 kernels, 9 Laplacian, 16 Littlewood-Paley decomposition, 77 local inversion theorem, 121 local symbol, 35 Meyer multiplier, 87 Morse's lemma, 69 Newton schenie, 146 non-stationary phase, 41 operator on a manifold, 37 oscillatory integral, 41 parametrix, 17 paraproduct, 86 partition of unity, 7, 36 Poisson bracket, 29 principal symbol, 27 propagation of singularities, 101 properly supported operator, 36 pseudo-differential operator, 15, 24 pseudo-local property, 19 quasilinear system, 128 Rauch's lemma, 113 regularization operator, 83 singular integral, 60 small divisors, 138 Sobolev space, 31

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