**Contents**

1. Distributions and Sobolev spaces

- Distributions
- The Fourier transform
- Sobolev spaces on R
^{n} - The complex interpolation method
- Sobolev spaces on bounded domains and compact manifolds
- Sobolev spaces, L
^{p}style - Local solvability of constant coefficient PDE

2. Pseudodifferential Operators

- The Fourier integral representation and symbol classes
- The pseudolocal property
- Asymptotic expansions of a symbol
- Adjoints and products
- Coordinate changes: operators on a manifold
- L
^{2}and Sobolev space continuity - Families of pseudodifferential operators: Friedrichs’ mollifiers
- Garding’s inequality
- References to further work

3. Elliptic and hypoelliptic operators

- Elliptic operators
- Hypoelliptic opeators with constant strength
- Hypoelliptic operators with slowly varying strength

4. The initial value problem and hyperbolic operators

- Reduction to a first order system
- Symmetric hyperbolic systems
- Strictly hyperbolic equations
- Finite propagation speed: finite domain of dependence
- Quasilinear hyperbolic systems
- The vibrating membrane problem
- Parabolic evolution equations
- References to further work

5. Elliptic boundary value problems

- Reduction to first order systems and decoupling
- A priori estimates and regularity theorems
- Closed range and Fredholm properties
- Regular boundary value problems
- Reduction of a boundary value problem to a regular one

6. Wave front sets and propagation of singularities

- The wave front set of a distribution
- Propagation of singularities: the Hamilton flow
- Local solvability
- Systems: an exponential decay result

7. The sharp Garding inequality

- A multiple symbol
- Friedrichs’ symmetrization: proof of the sharp Garding inequality

8. Geometrical optics and Fourier integral operators

- Egorov’s theorem
- Propagation of singularities
- The geometrical optics construction
- Parametrix for elliptic evolution equations
- Fourier integral operators
- Operators with singular phase functions
- The fundamental asymptotic expansion lemma
- Egorov’s theorem for OPS
^{m}_{1/2,1/2}

9. Reflection of singularities

- Decoupling first order systems
- Elliptic evolution equations
- Reflection of singularities

10. Grazing rays and diffraction

- The ansatz
- Fourier-Airy integral operators
- The eikonal and transport equations
- Justification and analysis of the parametrix
- The Neumann operator
- The Kirchhoff approximation
- References to further work

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

- Fourier multipliers on L
^{p}and Holder spaces - L
^{p}and C^{a}behavior of operators in OPS^{m}_{1,0} - L
^{p}behavior of OPS^{0}_{1,d} - The algebras OPM
^{m}_{r}and OPN^{m}_{r}on L^{p} - Besov spaces and boundary regularity
- References to further work

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

- Functions of elliptic self-adjoint operators
- The asymptotic behavior of the spectrum
- Poisson-like kernels
- Convergence of eigenfunction expansions
- Eigenfunction expansions of measures
- Harmonic analysis on compact Lie groups
- Some Tauberian theorems

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

- L
^{2}continuity of OPS^{0}_{0,0}(R^{n}) - L
^{2}boundedness of OPS^{0}_{r,r}, 0 < r < 1 - L
^{2}continuity of other sets of operators - Hormander-Melin inequalities

14. Uniqueness in the Cauchy problem

- Carleman estimates
- Reduction to subelliptic estimates, and proof of UCP
- UCP, global solvability, and all that

15. Operators with double characteristics

- Hypoelliptic operators
- The subprincipal symbol and microlocal equivalence of operators
- Characteristics with involutive self-intersection
- Characteristics with noninvolutive self-intersection
- Characteristics with conical singularities and conical refraction