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314 6. Introduction to Pseudodifferential Operators

then we may rewrite A v in the form

A v(x)=(2π) −n a(x, y, ξ)e i(x−y)·ξ v(y)dydξ . (6.1.25)

IR n Ω
Therefore, H¨ormander in [129] introduced the more general class of Fourier
integral operators of the form

Au(x)=(2π) −n e i(x−y)·ξ a(x, y, ξ)u(y)dydξ (6.1.26)
IR n Ω
n
m
with the amplitude function a ∈ S S S (Ω ×Ω ×IR ) and with the special phase
function ϕ(x, y, ξ)=(x − y) · ξ. The integral in (6.1.26) is understood in the
sense of oscillatory integrals by employing the same procedure as in (6.1.11),
(6.1.12). (See e.g. H¨ormander [129], Treves [306, Vol. II p. 315]). This class
m
of operators will be denoted by L (Ω).
Theorem 6.1.9. (H¨ormander [129, Theorem 2.1.1], Taira [301, Theorem
m
6.5.2]) Every operator A ∈L (Ω) can be written as
A = A 0 (x, −iD)+ R

m n −∞
where A 0 (x, −iD) ∈ OPS (Ω ×IR ) is properly supported and R ∈L (Ω)
where
(
−∞ m −∞ n
L (Ω):= L (Ω)= OPS (Ω × Ω × IR ) .
m∈IR
Proof: A simple proof by using a proper mapping is available in [301]
which is not constructive. Here we present a constructive proof based on the
presentation in Petersen [247] and H¨ormander [131, Prop. 18.1.22]. It is based
on a partition of the unity over Ω with corresponding functions {φ } ,φ ∈
C (Ω) , φ (x)=1. For every x 0 ∈ Ω and balls B ε (x 0 ) ⊂ Ω with fixed
∞
0
ε> 0, the number of φ with supp φ ∩ B ε (x 0 ) = ∅ is finite. Let us denote by
I 1 = {(j, ) | supp φ j ∩ supp φ = ∅} and by I 2 = {(j, ) | supp φ j ∩ supp φ =
∅} the corresponding index sets. Then

Au(x) = φ j (x)(Aφ u)(x)
j,

= φ j (x)Aφ u + φ j (x)Aφ u.
I 1 I 2
For every (j, ) ∈I 2 we see that the corresponding Schwartz kernel is given by

φ j (x)k(x, x − y)φ (y)
which is C (Ω × Ω). Moreover, for any pair (x 0 ,y 0 ) ∈ Ω × Ω, the sum
∞
0