Page 431 -

8.1 Pseudodifferential Operators on Boundary Manifolds 415

(−1)
χ r∗ v := v ◦ T r maps v given on O r to v ◦ χ r ,
afunctionon U r ;
(8.1.3)
χ u maps u given on U r to u ◦ χ r ,
∗
r := u ◦ χ r
afunctionon O r ⊂ Γ,
are defining the pushforward χ r∗ and pullback χ , respectively.
∗
r
k
Afunction v on Γ is said to be in the class C (Γ) if for every chart the
k
pushforward has the property v ◦ T r = χ r∗ v ∈ C (U r ).Hence,
: C (O r ) → C (U r ) ,
∞
∞
χ r∗
(8.1.4)
χ ∗ : C (U r ) → C (O r )
∞
∞
r
and
∞
∞
: C (O r ) → C (U r ) ,
χ r∗ 0 0
(8.1.5)
χ ∗ r : C (U r ) → C (O r ) .
∞
∞
0
0
In addition, we require the following smoothing property to be satisfied: If
ϕ, ψ ∈ C (Γ) with supp ϕ∩supp ψ = ∅ then the composition ϕAψ• extends
∞
0
∞
to a smoothing operator on Γ, i.e., for every u ∈ C (Γ) one has ϕAψu ∈
∞
C (Γ).
An immediate consequence of these definitions is the following theorem.
m
Theorem 8.1.1. Let A ∈L (Γ) and let A = {(O r ,U r ,χ r ) | r ∈ I} be an
atlas on Γ. Then for every pair of charts (O r ,U r ,χ r ) , (O t ,U t ,χ t ) and the
induced mapping Φ rt in (8.1.1), the induced local pseudodifferential operators
satisfy the compatibility relations
∗ ∗
r
t
χ A χ t χ t∗ = χ A χ r χ r∗ = A on D(O rt ) (8.1.6)
and

)
∗ ∗ ∗
r t rt
A χ t = χ t∗ χ A χ r χ r∗ χ = Φ rt∗ A χ r Φ =(A χ r Φ rt on D χ t (O rt ) ;
(8.1.7)
(see Remark 6.1.4).

Fig. 8.1.1. The local surface representation.