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7.2 Coordinate Changes and Pseudohomogeneous Kernels 395

Proof: Since with Φ, also Ψ(y ) is a smooth diffeomorphism, the Taylor

expansion of z = x − y about x can be written as the asymptotic expansion

x − y = Ψ(x ) − Ψ(y ) ∼ π α (x ,x − y ) . (7.2.4)
|α|≥1
The homogeneous polynomials π α are given explicitly as
(α) α
(−1) ∂ Ψ α
π α (x ,x − y )= (x )(x − y ) . (7.2.5)

α! ∂x α
y − x

By using the homogeneity of π α and setting Θ = , the relation

|y − x |
(7.2.4) yields

|α|−1
|x − y | π α (x , −Θ ) . (7.2.6)
|x − y| = |x − y |
|α|≥1
Hence,
|α|−1

y − x |α|=1 π α (x , −Θ )+ |α|≥2 |x − y | π α (x , −Θ )
Θ = = −
|y − x| π α (x , −Θ )+ |α|−1 π α (x , −Θ )

|α|=1 |α|≥2 |x − y |

holds for every |x − y | > 0. This implies with |x − y |→ 0,

π α (x , −Θ )
|α|=1
.
Θ = Θ(Θ )= − (7.2.7)

|α|=1
π α (x , −Θ )
∂Ψ
The denominator does not vanish since ∂x is invertible. In fact, Θ(Θ ) defines

a diffeomorphic mapping of the unit sphere onto itself.
Now we consider first the positively homogeneous terms of the asymptotic
expansion of k(x, x − y), namely

f(x, x − y) ∼ f κ+j (x, x − y) . (7.2.8)
j≥0
In terms of the transformation and homogeneity of f κ+j , this reads
κ+j

f κ+j (x, x − y)= |x − y| f κ+j Ψ(x ),Θ(Θ )
j≥0 j≥0
κ+j
|α|−1

= |x − y | |x − y | f κ+j Ψ(x ),Θ(Θ ) .
κ+j
π α (x , −Θ )
j≥0 |α|≥1
The second factor can be rewritten in the form