Contents

1. Distributions and Sobolev spaces

1. Distributions
2. The Fourier transform
3. Sobolev spaces on Rⁿ
4. The complex interpolation method
5. Sobolev spaces on bounded domains and compact manifolds
6. Sobolev spaces, L^p style
7. Local solvability of constant coefficient PDE

2. Pseudodifferential Operators

1. The Fourier integral representation and symbol classes
2. The pseudolocal property
3. Asymptotic expansions of a symbol
4. Adjoints and products
5. Coordinate changes: operators on a manifold
6. L² and Sobolev space continuity
7. Families of pseudodifferential operators: Friedrichs’ mollifiers
8. Garding’s inequality
9. References to further work

3. Elliptic and hypoelliptic operators

1. Elliptic operators
2. Hypoelliptic opeators with constant strength
3. Hypoelliptic operators with slowly varying strength

4. The initial value problem and hyperbolic operators

1. Reduction to a first order system
2. Symmetric hyperbolic systems
3. Strictly hyperbolic equations
4. Finite propagation speed: finite domain of dependence
5. Quasilinear hyperbolic systems
6. The vibrating membrane problem
7. Parabolic evolution equations
8. References to further work

5. Elliptic boundary value problems

1. Reduction to first order systems and decoupling
2. A priori estimates and regularity theorems
3. Closed range and Fredholm properties
4. Regular boundary value problems
5. Reduction of a boundary value problem to a regular one

6. Wave front sets and propagation of singularities

1. The wave front set of a distribution
2. Propagation of singularities: the Hamilton flow
3. Local solvability
4. Systems: an exponential decay result

7. The sharp Garding inequality

1. A multiple symbol
2. Friedrichs’ symmetrization: proof of the sharp Garding inequality

8. Geometrical optics and Fourier integral operators

1. Egorov’s theorem
2. Propagation of singularities
3. The geometrical optics construction
4. Parametrix for elliptic evolution equations
5. Fourier integral operators
6. Operators with singular phase functions
7. The fundamental asymptotic expansion lemma
8. Egorov’s theorem for OPS^m_1/2,1/2

9. Reflection of singularities

1. Decoupling first order systems
2. Elliptic evolution equations
3. Reflection of singularities

10. Grazing rays and diffraction

1. The ansatz
2. Fourier-Airy integral operators
3. The eikonal and transport equations
4. Justification and analysis of the parametrix
5. The Neumann operator
6. The Kirchhoff approximation
7. References to further work

11. L^p and Holder space theory of pseudodifferential operators

1. Fourier multipliers on L^p and Holder spaces
2. L^p and C^a behavior of operators in OPS^m_1,0
3. L^p behavior of OPS⁰_1,d
4. The algebras OPM^m_r and OPN^m_r on L^p
5. Besov spaces and boundary regularity
6. References to further work

12. Spectral theory and harmonic analysis of elliptic self-adjoint operators

1. Functions of elliptic self-adjoint operators
2. The asymptotic behavior of the spectrum
3. Poisson-like kernels
4. Convergence of eigenfunction expansions
5. Eigenfunction expansions of measures
6. Harmonic analysis on compact Lie groups
7. Some Tauberian theorems

13. The Calderon-Vaillancourt theorem and Hormander-Melin inequalities

1. L² continuity of OPS⁰_0,0 (Rⁿ)
2. L² boundedness of OPS⁰_r,r, 0 < r < 1
3. L² continuity of other sets of operators
4. Hormander-Melin inequalities

14. Uniqueness in the Cauchy problem

1. Carleman estimates
2. Reduction to subelliptic estimates, and proof of UCP
3. UCP, global solvability, and all that

15. Operators with double characteristics

1. Hypoelliptic operators
2. The subprincipal symbol and microlocal equivalence of operators
3. Characteristics with involutive self-intersection
4. Characteristics with noninvolutive self-intersection
5. Characteristics with conical singularities and conical refraction