Page 402 -

386 7. Pseudodifferential Operators as Integral Operators

By choosing first R = 1 one finds representations of k κ+j (x, tz)andof

k κ+j (x, z) where t = 1. Transforming ξ = tξ and afterwards choosing
R = t −1 , we finally obtain
1 α

) α 0

κ+j
k(x, tz)= t k κ+j (x, z)−log t Θ a m−j (x, Θ)dω(Θ)(iz) .
)
)
α!
|α|=κ+j
|Θ|=1
n
Hence, k κ+j ∈ Ψhf κ+j (Ω × IR ) can be written explicitly as

a
k κ+j (x, z) = |z| κ+j (2π) −n p.f. e iξ·z 0 0 (x, ξ)dξ − (log |z|)p κ+j (x, z)
m−j
IR n
with
1 α
) α 0
p κ+j (x, z) = Θ a m−j (x, Θ)dω(Θ)(iz) .
)
)
α!
|α|=κ+j
|Θ|=1
It remains to establish that
n
k(x, z)= k κ+j is in C (Ω × IR )for 0 ≤ <κ + J.
0≤j<J
From the definitions of k and k κ+j we obtain
α

∂ ∂ β
−i k(x, z) − k κ+j (x, z)
∂z ∂x
0≤j<J
∂ β

iξ·z α
−n
= (2π) e ξ a(x, ξ) − a m−j (x, ξ) dξ
∂x
IR n j<J

∂ β

= (2π) −n 1 − χ(ξ) e iξ·z α a − a m−j dξ
ξ
∂x
j<J
|ξ|≤1

∂ β
+(2π) −n χ(ξ)e iξ·z α a − a m−j dξ
ξ
∂x
1 j<J
|ξ|≥ 2
where |α|≤ <κ + J. The first of the integrals on the right–hand side is
n
in C (Ω × IR ) due to the Paley–Wiener–Schwartz theorem 3.1.3. The inte-
∞
|α|+m−J
grand of the second integral can be estimated by c(α, β , m, J)|ξ|
where |α|+m−J< κ+J +m−J = −n. Therefore, this integral is continuous
for all these α and any β.
(ii) Now let k ∈ ψhk κ with κ = −m − n for m ≥ 0 be given. Then k admits

the pseudohomogeneous expansion k κ+j with
j≥0