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7.2 Coordinate Changes and Pseudohomogeneous Kernels 401

since the integrand is continuous. To show that b α (x ) = 0 it therefore suffices

to prove that

α
p.f. k q (x, x − y)(y − x) χ(|y − x|)J(y )dy

Ω

α
= p.f. k q (x, x − y)(y − x) χ(|y − x|)dy (7.2.28)
Ω
for q = κ + j ∈ IN 0 and j =0,...,L, in view of the asymptotic expansion

of k. In the integral on the left–hand side set x = Ψ(x )and y = Ψ(y ). The

C ∞ cut–off function χ( ) has the properties χ( )=1 for 0 ≤ ≤ 0 and
χ( )=0 for 2 0 ≤ with some fixed 0 > 0. For ε> 0, the integral on the
right–hand side is given by

α
p.f. k q (x, x − y)(y − x) χ(|y − x|)dy
Ω
2 0

α
= p.f. r q+|α|+n−1 χ(r)drk q (x, −ω)ω dω
ε→0
|ω|=1 r=ε
q+|α|+n 2 0
0 q+|α|+n−1 ε
q+|α|+n
= p.f. + r χ(r)dr − ×
ε→0 q + |α| + n q + |α| + n
0

α
× k q (x, −ω)ω dω
|ω|=1

α
= c χ k q (x, −ω)ω dω
|ω|=1
where
2 0
q+|α|+n q+|α|+n−1
c χ = + r χ(r)dr
q + |α| + n
0
is a constant.
In the same manner, for the integral on the left–hand side in (7.2.28), and
by employing the transformation, we obtain