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398 7. Pseudodifferential Operators as Integral Operators

n
∂ ∂Ψ j m ∂

= (x )g (x ) =: D j , (7.2.17)

∂x j ∂x m ∂x
m, =1
where the Riemanian metric is given by the tensor (3.4.2), i.e.
∂Ψ ∂Ψ

g ik = ·
∂x ∂x
j k

and the g m (x ) are given by its inverse, see (3.4.4).
Both, the differential operators and the regularized integral in (7.2.15)

can now be transformed in the usual way. With u(x ):= u Ψ(x ) , we obtain

α

(Au)(x)=(A u)(x )= a α Ψ(x ) + d α Ψ(x ) D u(x )

|α|≤m

α
α
1

+ k(x ,x − y )J(y ) u(y ) − Ψ(y ) − Ψ(x ) D u(x ) dy .

α!
|α|≤m
Ω
(7.2.18)
From Lemma 7.2.1 we know that the transformed kernel k ∈ Ψhk κ with

κ = −n − m. Therefore, we may write (7.2.18) in terms of a finite part
integral and obtain

(A u)(x ) = a α Ψ(x ) + d α Ψ(x )

|α|≤m
1 α α

− p.f. k(x ,x − y )J(y ) Ψ(y ) − Ψ(x ) dy D u(x )

α!
Ω

+p.f. k(x ,x − y )J(y ) u(y )dy

Ω

= (Au) Ψ(x ) . (7.2.19)

We summarize these results in the following theorem.
m
Theorem 7.2.2. Let m ∈ IR. The operator A ∈L (Ω) in the form (7.2.14)
c
m

under change of coordinates x = Ψ(x ) becomes A ∈L (Ω ) and has the
c

form

α
α

(Au) Ψ(x ) =(A u)(x )= a α Ψ(x ) D u(x )+ b α (x )D u(x )

|α|≤m |α|≤m

+p.f. k(x ,x − y )J(y ) u(y )dy (7.2.20)

Ω
where

k(x ,x − y ):= k Ψ(x ),Ψ(x ) − Ψ(y ) (7.2.21)