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408 7. Pseudodifferential Operators as Integral Operators

α
I = c χ (0) k q (x, −ω)ω dω = I r ,
|ω|=1
which implies the proposed result (7.2.34) in this case.
Next, we consider the case −p := q + n + |α| < 0. Then the right–hand
side in (7.2.34) takes the form
⎧ ⎫
2 0
0 −p−1 ε α
⎨ −p −p ⎬
= p.f. + r χ(r)dr + k q (x, −ω)ω dω
I r −
p
ε→0 ⎩ p ⎭
0 |ω|=1

α
= c χ (p) k q (x, −ω)ω dω
|ω|=1
where
2 0
−p
c χ (p)= r −p−1 χ(r)dr − 0 .
p
0
Similarly,

1 α
α
I = c χ (p) k q (x, −ω)ω dω + p.f. P(ε, ω)k q (x, −ω)ω dω
p ε→0
|ω|=1 |ω|=1
with
−p

P(ε, ω)= ε (ω) .
By inserting the expansion (7.2.24) of ε (ω), we obtain
N −p
k N+1
P(ε, ω) = c k (ω)ε + O(ε )
k=1
, −p

N
N
ε
= c 1 (ω) −p −p 1+ c k (ω) ε k−1 + O(ε )
c 1 (ω)
k=2
, p

∞ N c k (ω)
N

ε
= c 1 (ω) −p −p (−1) ε k−1 + O(ε ) .
c 1 (ω)
=0 k=2
We note that by setting ℘ 1 := N c k (ω) k−1 that ℘ 1 ∈P since (7.2.25) holds.
ε
c 1 (ω)
k=2
Now, apply · p times Lemma 7.2.7; then it follows that
N
ε
P(ε, ω)= c 1 (ω) −p −p ℘ + O(ε )
with ℘ ∈P.If N = p + 1, then