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390 7. Pseudodifferential Operators as Integral Operators

Hence, by taking the limit 0 <t →∞,wehave

a 0 (x, −ξ) = lim t n+q a ψ (x, −tξ)
−n−q
t→∞
σ
= (−1) lim t n−q a ψ (x, tξ)
t→∞
σ 0
= (−1) a (x, ξ) .
−n−q
ii) Conversely, the kernel k q (x, x − y) can be expressed in the form (6.1.12),
i.e.

−2N −n N 0 i(x−y)·ξ
k q (x, x − y)= |y − x| (2π) (−∆ ξ ) a (x, ξ)e dξ .
−n−q
IR n
Then
σ
k q (x, x − y)=(−1) k q (x, x − y)
σ 0
follows immediately from a 0 −n−q (x, −ξ)=(−1) a −n−q (x, ξ).
As a consequence of Lemma 7.1.9, the parity conditions will provide us
with a criterion when the local differential operator in the representation
(7.1.60) will vanish.
m
Theorem 7.1.10. Let A ∈L (Ω) ,m ∈ IN 0 and suppose the parity condi-
c
tions
a 0 m−j (x, −ξ)=(−1) m−j+1 0 m−j (x, ξ) for 0 ≤ j ≤ m (7.1.76)
a
for the homogeneous terms in the symbol expansion of A.Then

(Au)(x)= p.f. k(x, x − y)u(y)dy . (7.1.77)

Ω
Proof: From the representation (7.1.60) we observe that

β β
A (•− x) ψ(|•−x|) | x =0 + (k(x, x − y)(y − x) ψ(|y − x|)dy
Ω
m
for any |β| >m and A ∈L (Ω). This implies
c
m

β β
a m−j (x, D) (•− x) ψ(|•−x|) + A R (•− x) ψ(|•−x|) (x)
j=0
m
α β
= a α (x)D y (y − x) ψ(|y − x|) | y=x
j=0 |α|=m−j

β
+p.f. k κ+j (x, x − y)(y − x) ψ(|y − x|)dy
Ω

β
+ k R (x, x − y)(y − x) ψ(|y − x|)dy (7.1.78)