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322 6. Introduction to Pseudodifferential Operators

∞
with the Schwartz kernel k R ∈ C (Ω × Ω)of R. The transformed operator
∗
Φ ∗ RΦ has the Schwartz kernel
∂Φ
−1
(−1) (−1) ∞
k R Φ (x ) ,Φ (y ) det ∈ C (Ω × Ω ) .
∂y
∗
Hence, Φ ∗ RΦ is a smoothing operator.
ii) It remains to show that the properly supported part Φ ∗ A 0 Φ ∗ ∈
m
n

OPS (Ω × IR ). For any v ∈D(Ω ) we have the representation
(−1)

∗
(Φ ∗ A 0 Φ v)(x )=(A 0 Φ v) Φ (x )

∗

= (2π) −n e i(x−y)·ξ a(x, ξ)(v ◦ Φ)(y)dydξ
IR n Ω

= (2π) −n e i(x−y)·(Ξ (x ,y )ξ) a(x, ξ)v(y ) ×

n
IR Ω ∧|x −y |≤2ε
−1
x − y ∂φ

×ψ det dy dξ + Rv
ε ∂y
where

x − y

(Rv)(x)=(2π) −n e i(x−y)·ξ a(x, ξ)v(y ) 1 − ψ ×
ε
n
IR Ω
∂φ
−1

× det dy dξ
∂y
and y = Φ (−1) (y ) ,x = Φ (−1) (x ).

With the affine transformation of coordinates,

Ξ (x ,y )ξ = ξ ,
the first integral reduces to

(A Φ,0 v)(x )=(2π) −n e i(x −y )·ξ v(y )a Φ (x ,y ,ξ )dy dξ

n
IR Ω
n
m
with a Φ (x ,y ,ξ ) given by (6.1.46). Since a(x, ξ) belongs to S S S (Ω×, IR ),

with the chain rule one obtains the estimates
∂ ∂ ∂ α
γ
β

a Φ (x ,y ,ξ ) ≤ c(K, α, β, γ)(1 + |ξ |)
m−|α|

∂x ∂y ∂ξ
n

for all (x ,y ) ∈ K × K, ξ ∈ IR and any compact subset K Ω . Hence,
n
m

a Φ ∈ S S S (Ω × Ω × IR ).