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7.2 Coordinate Changes and Pseudohomogeneous Kernels 409

ε
p.f. P(ε, ω)= p.f. c 1 (ω) −p −p ℘(ε, ω) .
ε→0 ε→0
p+1
j
j
Therefore, employing ℘(ε, ω)= a j (ω)ε with a j (−ω)=(−1) a j (ω), we
j=0
find
1 α
p.f. P(ε, ω)k q (x, −ω)ω dα
p ε→0
|ω|=1
p+1
1 j−p α

= p.f. a j (ω)ε k q (x, −ω)ω dω
p ε→0
j=0
|ω|=1
1 α
= a p (ω)k q (x, −ω)ω dω .
p
|ω|=1
The latter integral as an integral over the unit sphere satisfies
1 α
a p (ω)k q (x, −ω)ω dω
p
|ω|=1
1 α
= a p (−ω)k q (x, ω)(−ω) d(−ω)
p
|ω|=1
1 α
= (−1) p−q−n−1+|α| a p (ω)k q (x, −ω)ω dω
p
|ω|=1
1 α
= − a p (ω)k q (x, −ω)ω dω
p
|ω|=1
and, hence, vanishes. This implies again

α
I = c χ (p) a p (ω)k q (x, −ω)ω dω = I r .
|ω|=1
This completes the proof of Lemma 7.2.8.

We are now able to prove Theorem 7.2.6.
Proof of Theorem 7.2.6: With the asymptotic pseudohomogeneous ex-
pansion of the Schwartz kernel of A,
L

k(x, x − y)= k κ+j (x, x − y)+ k R (x, x − y) ,
j=0