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416 8. Pseudodifferential and Boundary Integral Operators

one has
For the principal symbols of the local operators A χ r

,m χ t (x),ξ (8.1.8)
σ A χ r ,m χ r (x),ξ = σ A χ t

where x = T r ( )= T t (τ ) ∈ O rt ⊂ Γ and the variables ξ and ξ in IR n−1 \{0}

are related by the equations
n−1 n−1
∂T t ∂T r

(τ )ξ ι = ( )ξ . (8.1.9)
ι
∂τ ι ∂ ι
ι=1 ι=1
m
∈L (U r )
Conversely, if a family of local pseudodifferential operators A χ r
is given satisfying (8.1.7) for the whole atlas A and satisfying the smooth-
ing property in Definition 8.1.1, then (8.1.6) defines a pseudodifferential
m
operatorA ∈L (Γ).
Proof: The equations (8.1.6) and (8.1.7) are immediate consequences of the
previous definitions.
For the transformation proposed in (8.1.8) we employ the coordinate
(−1)
transformation Φ rt = χ t ◦ χ r given in (8.1.1), apply (6.1.49) to (8.1.7)
and obtain

(−1) ∂Φ rt
,m (τ ,ξ )= σ A χ r ,m Φ rt (τ ) , ξ .

∂
σ A χ t
The argument in the right–hand side can be expressed component–wise as
n−1 n−1 n

∂Φ rt ∂τ ∂τ ∂x
λ
ξ =
ξ λ := ξ = λ ι ξ ,λ =1,...,n − 1 .
ι
∂ λ ∂ ∂x ∂
ι=1 ι ι=1 =1 ι
Inserting (3.4.27) with g nλ = 0 and the inverse γ µλ to γ tµλ = n ∂x ∂x on
t
t
∂τ ∂τ
=1 µ λ
Γ (c.f. (3.4.2), (3.4.4)) we find with (3.4.24), (3.4.28):
n−1 n−1 n
∂x ∂x µλ

ξ λ = γ t ξ ι for λ =1,...,n − 1 . (8.1.10)
∂τ ∂ ι

µ
ι=1 µ=1 =1
On the other hand, if (8.1.9) is satisfied, the scalar multiplication of both
sides of (8.1.9) by ∂τ gives
∂T t
µ
n−1 n−1 n
∂x ∂x
γ tµι ξ ι = ξ .
ι
∂ ∂τ

ι=1 ι=1 =1 ι µ
µλ
Multiplication by γ yields
t
n−1 n−1 n
∂x ∂x

ξ λ = γ t µλ ι
ξ ,
∂ ∂τ

µ=1 ι=1 =1 ι µ