Contents
1. Distributions and Sobolev spaces
- Distributions
- The Fourier transform
- Sobolev spaces on Rn
- The complex interpolation method
- Sobolev spaces on bounded domains and compact manifolds
- Sobolev spaces, Lp 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
- L2 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 OPSm1/2,1/2
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. Lp and Holder space theory of pseudodifferential operators
- Fourier multipliers on Lp and Holder spaces
- Lp and Ca behavior of operators in OPSm1,0
- Lp behavior of OPS01,d
- The algebras OPMmr and OPNmr on Lp
- 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
- L2 continuity of OPS00,0 (Rn)
- L2 boundedness of OPS0r,r, 0 < r < 1
- L2 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