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6.1 Basic Theory of Pseudodifferential Operators 321

A
D(Ω) E(Ω)

Φ ∗ Φ ∗

AΦ ∗
Φ ∗
D(Ω ) E(Ω )

A Φ
Figure 6.1.1: The commutative diagram of a pseudodifferential operator under
change of coordinates
∂Φ

(−1)
σ A Φ ,m (x ,ξ )= σ A,m Φ (x ) , ξ . (6.1.49)

∂x
Here, ψ(z) is a C 0 ∞ cut–off function with ψ(z)=1 for all |z|≤ 1 2 and
ψ(z)=0 for all |z|≥ 1.
Remark 6.1.4: The relation between A Φ and A can be seen in Figure 6.1.1
Proof: i) If A 0 is properly supported, so is Φ ∗ A 0 Φ due to the following.
∗
With Lemma 6.1.4, we have that to any compact subset K y Ω the subset

Φ (−1) (K y )= K y ⊂ Ω is compact. Then to K y Ω there exists the compact
subset K x Ω such that for every u ∈D(Ω) with supp u ⊂ K y there holds

supp A 0 u ⊂ K x . Consequently, for any v ∈D(Ω ) with supp v ⊂ K y we have
∗
supp Φ v ⊂ Φ (−1) (K y )= K y and supp A 0 Φ v ⊂ K x . This implies
∗
∗
supp Φ ∗ A 0 Φ v = supp A 0Φ v = Φ(supp A 0 Φ v) ⊂ Φ(K x )=: K x .
∗
On the other hand, if K x Ω is a given compact subset then K x :=

Φ (−1) (K x ) is a compact subset in Ω to which there exists a compact subset
K y Ω with the property that for every u ∈D(Ω) with supp u ∩ K y = ∅
there holds supp A 0 u ∩ K x = ∅.Now let K y := Φ(K y ) which is a compact

subset of Ω .Thus,if v ∈D(Ω ) with supp v ∩ K y = ∅ then
supp Φ v ∩ K y =(Φ (−1) supp v) ∩ K y = Φ (−1) (supp v ∩ K y )= ∅
∗
and, since A 0 is properly supported, supp A 0 Φ v ∩ K x = ∅. Hence,
∗

∗
Φ (−1) Φ(supp A 0 Φ v) ∩ Φ(K x ) = ∅ which yields
∗
supp Φ ∗ A 0 Φ v ∩ K x = ∅ .
∗
According to Lemma 6.1.4, this implies that Φ ∗ A 0 Φ is properly supported.

Clearly, for v ∈D(Ω )wehave

∂Φ −1

∗

(Φ ∗ RΦ v)(x )= k R Φ (−1) (x ) ,Φ (−1) (y ) v(y ) det dy
∂y
Ω