CHAPTER 1
Tempered distributions and the Fourier transform
Microlocal analysis is a geometric theory of distributions, or a theory of geometric distributions. Rather than study general distributions which are like general
continuous functions but worse we
CHAPTER 11
Hochschild homology
11.1. Formal Hochschild homology
The Hochschild homology is dened, formally, for any associative algebra. Thus
if A is the algebra then the space of formal k -chains, for k N0 is the (k + 1)-fold
tensor product
A(k+1) = A A
CHAPTER 10
K-theory
This is a brief treatment of K-theory, enough to discuss, and maybe even prove,
the Atiyah-Singer index theorem. I am starting from the smoothing algebra discussed earlier in Chapter 4 in order to give a smooth treatment of K-theory (t
CHAPTER 9
The wave kernel
Let us return to the subject of good distributions as exemplied by Dirac
delta functions and the Schwartz kernels of pseudodierential operators. In fact
we shall associate a space of conormal distributions with any submanifold of
CHAPTER 8
Elliptic boundary problems
Summary
Elliptic boundary problems are discussed, especially for operators of Dirac type.
We start with a discussion of manifolds with boundary, including functions spaces
and distributions. This leads to the jumps for
CHAPTER 7
Scattering calculus
7.1. Scattering pseudodierential operators
There is another calculus of pseudodierential operators which is smaller than
the traditional calculus. It arises by taking amplitudes in (2.2) which treat the
base and bre variables
CHAPTER 6
Pseudodierential operators on manifolds
In this chapter the notion of a pseudodierential on a manifold is discussed.
Some preliminary material on manifolds is therefore necessary. However the discussion of the basic properties of dierentiable ma
CHAPTER 5
Microlocalization
5.1. Calculus of supports
Recall that we have already dened the support of a tempered distribution in
the slightly round-about way:
(5.1)
if u S (Rn ), supp(u) = cfw_x Rn ; S (Rn ), (x) = 0, u = 0 .
Now if A : S (Rn ) S (Rn ) i
CHAPTER 4
Isotropic calculus
The algebra of isotropic pseudodierential operators on Rn has global properties very similar to the algebra of pseudodierential operators on a compact manifold discussed below. There are several reasons for the extensive discu
CHAPTER 3
Residual, or Schwartz, algebra
The standard algebra of operators discussed in the previous chapter is not really
representative, in its global behaviour, of the algebra of pseudodierential operators
on a compact manifold. Of course this can be a
CHAPTER 2
Pseudodierential operators on Euclidean space
Formula (1.92) for the action of a dierential operator (with coecients in
C (Rn ) on S (Rn ) can be written
P (x, D)u = (2 )n
ei(xy) P (x, )u(y )dyd
(2.1)
= (2 )n
eix P (x, )( )d
u
where u( ) = F u(
CHAPTER 12
The index theorem and formula
Using the earlier results on K-theory and cohomology the families index theorem of Atiyah and Singer is proved using a variant of their embedding proof. The
index formula in cohomology (including of course the form