Page 415 -

7.2 Coordinate Changes and Pseudohomogeneous Kernels 399

and
1 α
b α (x ) := p.f. k(x, x − y)(y − x) dy| x=Ψ(x )

α!
Ω

α

− p.f. k(x ,x − y )J(y ) Ψ(y ) − Ψ(x ) dy . (7.2.22)

Ω
For m< 0, the differential operator terms with the coefficients a α and b α do
not appear and

A u(x )= k(x ,y − x )J(y ) u(y )dy . (7.2.23)

Ω
In comparison with A in (7.2.14), we note that for m ≥ 0 the coefficients

of the differential operator part contain the extra terms b α (x ), which for
m ∈ IN 0 , in general, do not vanish. In the special case when m =0, see
Mikhlin and Pr¨ossdorf [215, Formulae (8) p.223 and (11) p.226]. This is a
very important difference from the coordinate transformation for operators
with weakly singular or regular kernel functions as in (7.2.23). Fore more
general m see also Sellier [281].
In the case 0 <m ∈ IN 0 we shall show that the b α (x ) in (7.2.22) will

vanish. For this purpose we first need the following lemma concerning the
coordinate transformation Ψ.

Lemma 7.2.3. Let ε (ω) denote the distance between x = Ψ(x ) and y =

Ψ(y ) where |y −x | = ε> 0 and y = x+ ε (ω)ω is the image of the sphere in

terms of polar coordinates ( ε ,ω) about x. Then, for sufficiently small ε> 0
and Ψ a C –diffeomorphism, one admits the asymptotic representation
∞
N
k N+1
ε (ω)= c k (ω)ε + O(ε ) (7.2.24)
k=1
for any N ∈ IN. The coefficients c k (ω) satisfy the parity conditions
c k (−ω)=(−1) k+1 c k (ω) . (7.2.25)

∞

Proof: With the C –diffeomorphism Φ(x)= x ,Φ(y)= y ,inverse to Ψ,

the Taylor expansion about x reads
1
α
α |α|

y − x = D Φ(x)ω .
α!
|α|≥1
Hence,
α α β β
2
2

ε = |y − x | = D Φ(x)ω · D Φ(x)ω |α|+|β| (7.2.26)
ε
|α|≥1,|β|≥1