viii Contents §2.5. Precise speed estimate 79 §2.6. Local Cauchy problems 83 Appendix 2.I. Constant coefficient hyperbolic systems 84 Appendix 2.II. Functional analytic proof of existence 89 Chapter 3. Dispersive Behavior 91 §3.1. Orientation 91 §3.2. Spectral decomposition of solutions 93 §3.3. Large time asymptotics 96 §3.4. Maximally dispersive systems 104 3.4.1. The L1 L∞ decay estimate 104 3.4.2. Fixed time dispersive Sobolev estimates 107 3.4.3. Strichartz estimates 111 Appendix 3.I. Perturbation theory for semisimple eigenvalues 117 Appendix 3.II. The stationary phase inequality 120 Chapter 4. Linear Elliptic Geometric Optics 123 §4.1. Euler’s method and elliptic geometric optics with constant coefficients 123 §4.2. Iterative improvement for variable coefficients and nonlinear phases 125 §4.3. Formal asymptotics approach 127 §4.4. Perturbation approach 131 §4.5. Elliptic regularity 132 §4.6. The Microlocal Elliptic Regularity Theorem 136 Chapter 5. Linear Hyperbolic Geometric Optics 141 §5.1. Introduction 141 §5.2. Second order scalar constant coefficient principal part 143 5.2.1. Hyperbolic problems 143 5.2.2. The quasiclassical limit of quantum mechanics 149 §5.3. Symmetric hyperbolic systems 151 §5.4. Rays and transport 161 5.4.1. The smooth variety hypothesis 161 5.4.2. Transport for L = L1(∂) 166 5.4.3. Energy transport with variable coefficients 173 §5.5. The Lax parametrix and propagation of singularities 177 5.5.1. The Lax parametrix 177 5.5.2. Oscillatory integrals and Fourier integral operators 180
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