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406 7. Pseudodifferential Operators as Integral Operators

Lemma 7.2.8. Let k q ∈ Ψhf q (Ω). In the case when q ≤−n is an integer
then we require in addition that k q satisfies the parity condition
k q (x, y − x)=(−1) −q−n+1 k q (x, x − y) for y = x. (7.2.33)

Then the following invariance properties hold:

α

p.f. k q (x, x − y)(y − x) χ(|y − x|)J(y )dy
Ω

α
= p.f. k q (x, x − y)(y − x) χ(|y − x|)dy . (7.2.34)
Ω

In the integral on the left–hand side it is understood that x = Ψ(x ) and
y = Ψ(y ).The C ∞ cut–off function χ( ) has the properties χ( )=1 for

0 ≤ ≤ 0 ,χ( )=0 for 2 0 ≤ for some fixed 0 > 0. In formula (7.2.34)

(x) for all sufficiently small
we assume that B 2 0 ⊂ Ω and Ψ B ε (x ) ⊂ B 0
ε> 0.
Proof: With the polar coordinates y−x = rω, the right–hand side of (7.2.34)
is given by
2 0

α
I r =p.f. k q (x, −ω)ω χ(r)r q+|α|+n−1 drdω .
ε→0
|ω|=1 r=ε
For the left–hand side of (7.2.34), since for ε> 0 the integrand is regular, we
have similarly

2 0

α
I =p.f. k q (x, −ω)ω χ(r)r q+|α|+n−1 drdω .
ε→0
|ω|=1 r=ε(ω)
Now we first consider the case q + n + |α| = 0. Then,
2 0

−1 α
I r = p.f. log 0 + r χ(r)dr − log ε k q (x, −ω)ω dω
ε→0
|ω|=1
0

α
= c χ (0) k q (x, −ω)ω dω
|ω|=1
with the constant c χ (0) := log 0 + 2 0 r −1 χ(r)dr.
0
For the left–hand side of (7.2.34) we have