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320 6. Introduction to Pseudodifferential Operators

Coordinate Changes

Let Φ : Ω → Ω be a C –diffeomorphism between the two open subsets
∞
n

Ω and Ω in IR , i.e. x = Φ(x). The diffeomorphism is bijective and x =
Φ −1 (x ).

If v ∈D(Ω ) then

∗
u(x):=(v ◦ Φ)(x)=:(Φ v)(x) (6.1.43)
∗

defines u ∈D(Ω) and the linear mapping Φ : D(Ω ) →D(Ω) is called the
pullback of v by Φ.
Correspondingly, the pushforward of u by Φ is defined by the linear map-
ping Φ ∗ : D(Ω) →D(Ω ) given by

v(x ):= u ◦ Φ −1 (x )=:(Φ ∗ u)(x ) . (6.1.44)

m
Now we consider the behaviour of a pseudodifferential operator A ∈L (Ω)
under changes of coordinates given by a C –diffeomorphism Φ.
∞
m
Theorem 6.1.16. (H¨ormander [131, Th. 18.1.17]) Let A ∈L (Ω) then
m
A Φ := Φ ∗ AΦ ∈L (Ω ). Moreover, if A = A 0 +R with A 0 properly supported

∗
and R a smoothing operator then in the decomposition
∗
∗
A Φ = Φ ∗ A 0 Φ + Φ ∗ RΦ ,

∗
∗
Φ ∗ A 0 Φ is properly supported and Φ ∗ RΦ is a smoothing operator on Ω .The
complete symbol class of A Φ has the following asymptotic expansion
1 ∂ ∂ α
α

a Φ (x ,ξ ) ∼ − i a Φ (x ,y ,ξ ) (6.1.45)
α! ∂ξ ∂y | y =x
0≤|α|
where
∂Φ x − y
−1 −1

a Φ (x ,y ,ξ )= det (y) det Ξ (x ,y ) ψ

∂y ε
−1

a x, Ξ (x ,y ) ξ (6.1.46)
with x = Φ −1 (x ) ,y = Φ −1 (y ) and where the smooth matrix–valued function

Ξ(x ,y ) is defined by

Ξ(x ,y )(x − y )= Φ −1 (x ) − Φ −1 (y ) (6.1.47)

which implies
−1
∂Φ

Ξ(x ,x )= . (6.1.48)

∂x
−1

The constant ε> 0 is chosen such that Ξ(x ,y ) exists for |x −y |≤ 2ε.

(6.1.45) yields for the principal symbols the relation