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7.2 Coordinate Changes and Pseudohomogeneous Kernels 397

By substituting (7.2.11) into (7.2.10) we see that the first sum has the
form

log |x −y | p κ+ ,j (x ,x −y ) =: log |x −y |p +κ (x ,x −y ) .
≥1 −κ≤j≤ , ≥0
0≤j
(7.2.12)
The other two terms have an asymptotic expansion of the same form as
(7.2.4), namely

κ+p
|x − y | f κ+p (x ,Θ ) (7.2.13)
p≥0
which follows in the same manner as for the homogeneous expansion. Col-
lecting (7.2.11) and (7.2.13) yields

k(x, x − y) ∼ |x − y | f κ+p (x ,Θ )+ f κ+p (x ,Θ )
κ+p
p≥0

+ log |x − y |p +κ (x ,x − y ) .
≥0,
+κ≥0
as proposed.
7.2.1 The Transformation of General Hadamard Finite Part
Integral Operators under Change of Coordinates
Now we return to the transformation properties of the operator
m
A ∈L (Ω) in the form (7.1.8),
c

α
(Au)(x)= a α (x)D u(x)+ p.f. k(x, x − y)u(y)dy . (7.2.14)
|α|≤m
Ω
If m ∈ IN 0 then a α (x)=0. For m< 0, the integral operator is weakly singular
and its transformation was already discussed. For m ≥ 0, we regularize the
finite part integral in (7.2.14) and write

α
(Au)(x) = a α (x)+ d α (x) D u(x) (7.2.15)
|α|≤m

1 α α
+ k(x, x − y) u(y) − (y − x) D u(x) dy
α!
|α|≤m
Ω
where
1 α
d α (x)= p.f. k(x, x − y)(y − x) dy (7.2.16)
α!
Ω
(see also (7.1.5) and (7.1.6)). For the transformation of the derivatives due

to x = Ψ(x ), we need the identity (3.4.1), i.e.