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7.2 Coordinate Changes and Pseudohomogeneous Kernels 403

Consequently, for N sufficiently large, we have

α
p.f. I (ε)= c χ (ω)k q (x, −ω)ω dω . (7.2.29)

ε→0
|ω|=1
Hence,
1 α

p.f. P(ε, ω) k q (x, −ω)ω =0
ε→0 q + |α| + n
|ω|=1
and, consequently,

α
p.f. I (ε)= c χ k q (x, −ω)ω dω
ε→0
|ω|=1
which proves (7.2.28).
Note that c χ (ω) is independent of the coordinate transformation, therefore
this integral is independent of the special choice of Ψ, and, hence, is invariant
with respect to the change of coordinates.

α
In general, for m ∈ IN 0 , the extra differential operator b α (x )D u(x )

|α|≤m
is unpleasant for m> 2 in view of (7.2.17). However, if the transformation
is a linear diffeomorphism Ψ L (x ) then this extra differential operator can be

represented in a more simplified manner (see Kieser [156, Theorem 2.2.9]).
Theorem 7.2.5. If k ∈ Ψhk κ with κ = −n − m, m ∈ IN 0 ,and Ψ L (x ) is

a bijective linear transformation then the coefficients b α (x ) can be written

explictly in the form
1 α

b α (x )= k −n−|α| (x, −ω)ω log R(ω)dω for |α|≤ m, x = Ψ L (x ) .
α!
|ω|=1
(7.2.30)
J κ+J−δ
Here, k = k κ+j + k R where k R ∈ C (Ω × Ω) with some δ ∈ (0, 1)
j=0
and k κ+j ∈ Ψhf κ+j (Ω). Moreover, R(ω):= |L −1 ω| −1 where L is the matrix

representation for the linear transformation Ψ L : y − x = L(y − x ).

Proof: From (7.2.22) it is clear, that the difference defining b α (x ) is deter-
mined by the singular behavior of the integrals near to y = x only. In view
of (3.2.18) for the finite part integrals, we now consider

α

k κ+j (x ,x − y )J(y ) Ψ L (y ) − Ψ L (x ) dy

Ω \{|x −y |<ε}

α
= k κ+j (x, x − y)(y − x) dy .
Ω\{|x−y|<εR(ω)}