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402 7. Pseudodifferential Operators as Integral Operators

α

I (ε) := k q (Ψ(x ),Ψ(x ) − Ψ(y ))(Ψ(y ) − Ψ(x )) ×

Ω \{|y −x |<ε}

×χ(|Ψ(y ) − Ψ(x )|)J(y )dy

α
= k q (x, x − y)(y − x) χ(|y − x|)dy
Ω\{|y−x|< ε (ω)}
2 0

α
= q+n−1+|α| χ( )d k q (x, −ω)ω dω
|ω|=1 = ε (ω)
⎧ ⎫
q+|α|+n 2 0
0 q+|α|+n−1 ε (ω)

⎨ q+|α|+n ⎬
= + r χ(r)dr − ×
⎩q + |α| + n q + |α| + n ⎭
|ω|=1 0
α
× k q (x, −ω)ω dω .
Hence, with the expansion of ε (ω) in Lemma 7.2.3,

p.f. I (ε)
ε→0
1
α
α
k q (x, −ω)ω dα − p.f. P(ε, ω)k q (x, −ω)ω dω
= c χ
ε→0 q + |α| + n
|ω|=1 |ω|=1
with
N q+|α|+n
k N+1
P(ε, ω):= c k (ω)ε + O(ε ) .
k=1
By using the geometric series, P(ε, ω) can be written in the form
N
q+n+|α|
q+n+|α| c k k−1 N
P(ε, ω)= −ε q+n+|α| c 1 (ω) 1+ (ω)ε + O(ε )
c 1
k=1
= −ε q+n+|α| c 1 (ω) q+n+|α| ×
∞ N
q + n + |α| c k k−1 N
× (ω)ε + O(ε )
c 1
=0 k=2
q+n+|α|+ q+n+|α|+N
= − ε c (ω)+ O(ε )
≥0
where q + n + |α| + = 0 for all ∈ IN 0 . Hence,

q+n+|α|+ α

I (ε) = − ε c (ω)k q (x, −ω)ω dω
≥0
|ω|=1

α
+ c χ (ω)k q (x, −ω)ω dω + O(ε q+n+|α|+N ) .
|ω|=1