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7.1 Pseudohomogeneous Kernels 385

1
α
α
k κ+j (x, tz)=(2π) −n a 0 (x, ξ){e iξ·tz − ξ (itz) }dξ
m−j
α!
|α|<κ+j
|ξ|≤R
1 1
α 0 α
− R −κ−j+|α| Θ a m−j (x, Θ)dω(Θ)(itz)
κ + j −|α| α!
|α|<κ+j
|Θ|=1

a
+ e iξ·tz 0 m−j (x, ξ)dξ
|ξ|≥R
Now set t =1 and R = 1, which gives the form of k κ+j (x, z). Next, set

R = t −1 and change the coordinates ξ = tξ which yields the identity
k κ+j (x, tz)= t κ+j k κ+j (x, z) .
Again, k κ+j ∈ ψhf κ+j .
For m ∈ IN 0 , let us consider first the case κ + j = 0. Then (7.1.63) reads
α
k 0 (x, z)=(2π) −n/2 F −1 a 0 (x, ξ) − a α,0 (x)(−D) δ(z)
ξ →z −n
|α|≤2k
where a 0 (x, ξ)= |ξ| −n 0 (x, Θ). In this case Lemma 7.1.2 implies
a
)
−n −n
k 0 (x, z)= c(x){a 0 + a 1 log |z|} (7.1.66)
,
1 iπ Θ · z 0

−n
0
)

+(2π) a (x, Θ) − c(x) log + dω(Θ)
)
)
−n
|Θ · z 0 | 2 |Θ · z 0 |
)
)
| ) Θ|=1
where z 0 = z/|z| and c(x)= 1 a 0 −n (x, Θ)dω(Θ). The constants a 0
)
)
ω n
| ) Θ|=1
and a 1 are explicitly given by Lemma 7.1.2. This shows that k 0 (x, z) ∈
n
ψhf 0 (Ω × IR ) and for x = y, the function k 0 (x, x − y) is the Schwartz
kernel of a 0 (x, −iD).
−n
For the remaining case κ + j ∈ IN, i.e. j> m + n, we have again the
representation (7.1.65) for the Schwartz kernel of a m−j (x, −iD). For the ho-
mogeneity of k κ+j , we see that

1
α
k κ+j (x, tz)=(2π) −n a 0 (x, ξ) e iξ·tz − ξ (itz) α dξ
m−j
α!
|α|≤κ+j
|ξ|≤R
1 1
−κ−j+|α| ) α 0 α
− R Θ a m−j (x, Θ)dω(Θ)(itz)
)
)
κ + j −|α| α!
|α|<κ+j
|Θ|=1
1
α iξ·tz 0
) α 0
+ (log R)Θ a m−j (x, Θ)dω(Θ)(itz) + e a m−j (x, ξ)dξ
)
)
α!
|α|=κ+j
|Θ|=1 R≤|ξ|
(7.1.67)