Page 423 -

7.2 Coordinate Changes and Pseudohomogeneous Kernels 407

2 0

α
I =p.f. log 0 + r −1 χ(r)dr − log ε (ω) k q (x, −ω)ω dω .
ε→0
0 |ω|=1
In terms of the expansion (7.2.24) of ε (ω) in Lemma 7.2.3, we have
N
,
k N+1
− log ε (ω)= − log c k (ω)ε + O(ε )
k=1
N

c k (ω) k−1 N
= − log ε − log c 1 (ω) − log 1+ ε + O(ε )
c 1 (ω)
k=2
from which we obtain

p.f. − log ε (ω)
ε→0
N
,
∞
1

N
= p.f. − log c 1 (ω) − log ε + c k k−1 + O(ε )
ε
ε→0 c 1
=1 k=2
= − log c 1 (ω) .
Hence,

α
α

I = c χ (0) k q (x, −ω)ω dω − log c 1 (ω) k q (x, ω)ω dω .
|ω|=1 |ω|=1
Since the integration is taken over the unit sphere, we have

α
− log c 1 (ω) k q (x, −ω)ω dω
|ω|=1

α
= − log c 1 (−ω)k q (x, +ω)(−ω) d(−ω)
|ω|=1

α
= − log c 1 (ω)(−1) −q−n−1+|α| k q (x, −ω)ω dω
|ω|=1

α
= log c 1 (ω)k q (x, −ω)ω dω
|ω|=1
which implies

α
log c 1 (ω)k q (x, −ω)ω dω =0 .
|ω|=1
Consequently,