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418 8. Pseudodifferential and Boundary Integral Operators

∞
Then there exists a C –partition of unity subordinate to the finite covering
∞
{O r } r∈I . This means that there are nonnegative functions ϕ r ∈ C (O r ), i.e.
0
χ r∗ ϕ r ∈ C (U r ), such that
∞
0

ϕ r (x) = 1 for all x ∈ Γ. (8.1.13)
r∈I
For its construction we refer to Schechter [271, Section 9-4]. In addition, to the
partition of unity let {ψ r } r∈I be another system of functions ψ r ∈ C (O r ),
∞
0
i.e. χ r∗ ψ r ∈ C (U r ), with the property
∞
0
ψ r (x) = 1 for all x ∈ supp ϕ r .
Then A can be written as

A = A r1 + R 1 with A r1 := ϕ r Aψ r ,
r∈I
(8.1.14)

A = A r2 + R 2 with A r2 := ψ r Aϕ r ,
r∈I
and R 1 and R 2 both are smoothing operators mapping C (Γ)into C (Γ)
∞
∞
due to the smoothing property required in Definition 8.1.1.
defined in (8.1.2), A can be written as
In terms of the local operators A χ r

A = ∗ χ r∗ ψ r + R 1 = ∗ χ r∗ ϕ r + R 2 . (8.1.15)
r
r ψ r χ A χ r
ϕ r χ A χ r
r∈I r∈I
As a consequence of the mapping properties of pseudodifferential operators in
the parametric domains formulated in Theorem 6.1.12, we have the following
mapping properties for pseudodifferential operators on Γ.
m
Theorem 8.1.2. If A ∈L (Γ) then the following mappings are continuous:
∞
∞
A : C (Γ) → C (Γ) ,

A : E (Γ) → E (Γ) , (8.1.16)

s
A : H (Γ) → H s−m (Γ) .
s
Here H (Γ)and H s−m (Γ) are the standard trace spaces with norm and
topology defined in (4.2.27) for every s ∈ IR.
8.1.1 Ellipticity on Boundary Manifolds
Since the pseudodifferential operators on the manifold Γ are characterized
by their representations with respect to an atlas of Γ and its local charts
A = {(O r ,U r ,χ r ) | r ∈ I}, the concept of ellipticity in the domain given in
Definition 6.2.1 carries over to the pseudodifferential operators on Γ.