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410 7. Pseudodifferential Operators as Integral Operators

to prove (7.2.32), it suffices to consider the coordinate transformation for a
typical term in the expansion since k R is not singular any more. To this end
consider k q = k κ+j in Lemma 7.2.8 and we have the parity condition

σ j
k q (x, x − y)= k κ+j (x, x − y)=(−1) k κ+j (x, y − x)
with σ j = j − m + 1. Hence,

k q (x, x − y)=(−1) j−m+1 k q (x, y − x)
= (−1) q+n+1 k q (x, y − x)
= (−1) −q−n−1 k q (x, y − x)
as assumed in (7.2.33) and k q ∈ Ψhf q . Therefore, with L sufficiently large,

L

p.f. k(x, x − y)u(y)dy = p.f. k κ+j (x, x − y)u(y)dy
j=0
Ω Ω

+ k R (x, x − y)u(y)dy
Ω
L

= p.f. k κ+j (x ,x − y ) u(y )J(y )dy

j=0
Ω

+ k R (x ,x − y ) u(y )J(y )dy

Ω

= p.f. k(x ,x − y ) u(y )J(y )dy ,

Ω
which completes the proof of Theorem 7.2.6.
In fact, the parity conditions required in Theorem 7.2.6 are in-
variant under the change of coordinates since that is a special case of
the following theorem.
m
Theorem 7.2.9. Let A ∈L (Ω) with m ∈ Z having the pseudohomogeneous
c
kernel expansion of the Schwartz kernel

k A (x, x − y) ∼ k κ+j (x, x − y) .
j≥0
Suppose that the parity conditions
k κ+j (x, x − y)=(−1) m−j+σ 0 k κ+j (x, x − y) for 0 ≤ j ≤ L (7.2.35)
with a fixed σ 0 ∈ IN 0 are satisfied. Let Φ be a diffeomorphism defining by

x = Φ(x), a change of coordinates, and let