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394 7. Pseudodifferential Operators as Integral Operators

α
form c α (x)D . Then the coefficients c α (x) can be recovered by for-
|α|=m−j
mula (7.1.9), i.e.
1 α i(x−y)·ξ 0
−n
c α (x)=(2π) (y − x) ψ(y)e dya m−j (x, ξ)dξ
α!
IR n Ω
(7.1.86)

α
− p.f. k κ+j (x, x − y)(y − x) ψ(y)dy for |α| = m − j.
Ω
If the decomposition
α
a 0 (x, ξ)= c α (x)(iξ) + a 00 (x, ξ)
m−j m−j
|α|=m−j
is known then in formula (7.1.86) one can simply replace a 0 m−j by a 00 .
m−j
7.2 Coordinate Changes and Pseudohomogeneous
Kernels

As we have seen in Theorems 7.1.1 and 7.1.8, all the pseudodifferential op-
m
erators A ∈L (Ω) have the general form (7.1.8) with a kernel function
c
k ∈ Ψhk κ (Ω). In particular, for m< 0 the integral operator is weakly sin-

gular and the change of coordinates x = Φ(x) with Φ a diffeomorphism and

x = Φ (−1) (x )= Ψ(x ) results in the traditional transformation formula

k(x, x − y)u(y)dy = k Ψ(x ) ,Ψ(x ) − Ψ(y ) u Ψ(y ) J(y )dy (7.2.1)
Ω Ω
∂Ψ

with the Jacobian J(y )= det . The new kernel k(x ,x − y ) is still

∂y

weakly singular in Ω .
In the case m ≥ 0, however, k is strongly singular and the traditional
transformation rule (7.2.1) needs to be modified.
In what follows, we first examine the properties of the pseudohomogeneous
kernel under the change of coordinates.
Lemma 7.2.1. Let

k(x, x − y) ∼ k κ+j (x, x − y) (7.2.2)
j≥0
(see (7.1.2)) be the pseudohomogeneous asymptotic expansion of k ∈ Ψhk κ .
Then the transformed kernel

k(x ,x − y ):= k Ψ(x ) ,Ψ(x ) − Ψ(y ) ∼ k κ+p (x ,x − y ) (7.2.3)

p≥0
is in Ψhk κ ,too.