Page 420 -

404 7. Pseudodifferential Operators as Integral Operators

Hence,

1 α
b α (x )= p.f. k κ+j (x, x − y)(y − x) dy

α!
Ω\{|x−y|<ε}

α
− k κ+j (x, x − y)(y − x) dy .
Ω\{|x−y|<εR(ω)}
Since κ + j = −m − n + j ∈ IN 0 , the kernel function k κ+j is positively
homogeneous without log–terms. Then, in terms of polar coordinates y −x =
ω,wehave

εR(ω)
1 α κ−m+|α|+j−1

b α (x )= p.f. k κ+j (x, −ω)ω r drdω .
α!
|ω|=1 r=ε
Since κ = −m − n, we have to distinguish two cases:
|α| = m − j:

1 |α|−m+j 1
b α (x )= p.f. ε ×

α! ε→0 m −|α| + j

|α|−m+j α
× R(ω) − 1 k κ+j (x, −ω)ω dω =0
|ω|=1
since |α|− m + j =0.

|α| = m − j:

1 α

b α (x )= k κ+j (x, −ω)ω log R(ω)dω .
α!
|ω|=1
This gives with j = m −|α| and κ = −n − m the desired formula (7.2.30).

7.2.2 The Class of Invariant Hadamard Finite Part Integral
Operators under Change of Coordinates
For m ∈ IN 0 , we shall see that the extra terms b α in Theorem 7.2.2 also
vanish for a large class of operators whose kernel functions satisfy the parity
conditions.
We now state the following crucial result concerning the transformation
of finite part integral operators.