xii Contents

48.2. Uniqueness and parabolic regularization 235

48.3. The Cauchy problem on a finite time interval 237

48.4. Strictly hyperbolic equations of second order 240

§49. Domain of dependence 243

§50. Propagation of singularities 247

50.1. The null-bicharacteristics 247

50.2. Operators of real principal type 247

50.3. Propagation of singularities for operators of real principal

type 249

50.4. Propagation of singularities in the case of a hyperbolic

Cauchy problem 255

§51. Problems 258

Chapter VII. Elliptic Boundary Value Problems and Parametrices 263

Introduction to Chapter VII 263

§52. Pseudodifferential operators on a manifold 264

52.1. Manifolds and vector bundles 264

52.2. Definition of a pseudodifferential operator on a manifold 265

52.3. Elliptic ψdo’s on a manifold 266

§53. Boundary value problems in the half-space 266

53.1. Factorization of an elliptic symbol 266

53.2. Explicit solution of the boundary value problem 268

§54. Elliptic boundary value problems in a bounded domain 270

54.1. The method of “freezing” coeﬃcients 270

54.2. The Fredholm property 273

54.3. Invariant form of the ellipticity of boundary conditions 276

54.4. Boundary value problems for elliptic systems of differential

equations 276

§55. Parametrices for elliptic boundary value problems 278

55.1. Plus-operators and minus-operators 278

55.2. Construction of the parametrix in the half-space 281

55.3. Parametrix in a bounded domain 284

§56. The heat trace asymptotics 285

56.1. The existence and the estimates of the resolvent 285

56.2. The parametrix construction 286

56.3. The heat trace for the Dirichlet Laplacian 288

56.4. The heat trace for the Neumann Laplacian 293

56.5. The heat trace for the elliptic operator of an arbitrary order 294

§57. Parametrix for the Dirichlet-to-Neumann operator 296