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400 7. Pseudodifferential Operators as Integral Operators

which implies, if we define

<

α α β β |α|−|β|−2
F(ε, , ω):= −ε + D Φ(x)ω • D Φ(x)ω
|α|≥1,|β|≥1
for | |≤ 0 and |ε|≤ ε 0 , |ω| = 1, that the equation
F(ε, , ω)=0

admits a unique C –solution
∞

= f(ε, ω), i.e. F ε, f(ε, ω),ω =0
since =
∂F >
β
D Φ(x)ω α · D Φ(x)ω β > 0
α
| =0 = >
∂ ?
|α|=1
|β|=1
for all ω on the unit sphere, because Φ is diffeomorphic. Moreover, f(ε, ω)is
also defined for ε< 0 where f(ε, ω) < 0. In particular, we see that
−f(−ε, −ω)= f(ε, ω) (7.2.27)
since
F(−ε, − , ω)= −F(ε, , ω)=0 .
Since F is C ,sois f the asymptotic expansion (7.2.24) for any N ∈ IN.
∞
Hence, the parity condition (7.2.25) then is an immediate consequence of
(7.2.27).

Lemma 7.2.4. For 0 <m ∈ IN 0 , the coefficients b α (x )=0 in (7.2.20).

m
Proof: Since A ∈L (Ω), it follows from Theorem 7.1.8 that the kernel k
c
belongs to Ψhk κ (Ω) with κ = −n − m. Hence,
L

k(x, x − y)= k κ+j (x, x − y)+ k R (x, x − y)
j=0
where k R ∈ C κ+L−δ (Ω × Ω) with some δ ∈ (0, 1) and k κ+j ∈ Ψhf κ+j .For L
sufficiently large,

α
p.f. k R (x, x − y)(y − x) dy| x=Ψ(x )
Ω

α
= k R (x, x − y)(y − x) dy| x=Ψ(x )
Ω

α

= k R (x ,x − y )J(y ) Ψ(y ) − Ψ(x ) dy

Ω