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7.2 Coordinate Changes and Pseudohomogeneous Kernels 405

Theorem 7.2.6. (Kieser [156, Theorem 2.2.12])
m
For A ∈L (Ω) with m ∈ IN 0 let the Schwartz kernel of A be k(x, x−y) ∼
c

k κ+j (x, x − y) and let k κ+j satisfy the parity conditions (7.1.74) with
j≥0
σ j = j − m +1 for 0 ≤ j ≤ m. (7.2.31)
Then the finite part integral is invariant under change of coordinates; namely

p.f. k(x, x − y)u(y)dy =p.f. k(x ,x − y ) u(y )J(y )dy (7.2.32)

Ω Ω
where k is given by (7.2.21).

Remark 7.2.1: This class of operators includes all of the boundary inte-
gral operators generated by the reduction to the boundary of regular elliptic
boundary value problems based on Green’s formula. The proof of this the-
orem is delicate and will be presented after we establish some preliminary
results.
Let P denote the class of all polynomials in ε of the form
j j
℘(ω, ε)= a j (ω)ε with a j (−ω)=(−1) a j (ω)
j≥0
n
where ω ∈ IR with |ω| =1.
Lemma 7.2.7. If ℘ 1 ,℘ 2 ∈P then ℘ 1 + ℘ 2 ∈P and ℘ 1 ℘ 2 ∈P.

j k
Proof: Let ℘ 1 = a j (ω)ε and ℘ 2 = b k (ω)ε . Then
j≥0 k≥0

℘ 1 + ℘ 2 = c (ω)ε with c = a + b .
≥0

Clearly,

c (−ω)= a (−ω)+ b (−ω)=(−1) a (ω) − b (ω) =(−1) c (ω) .
Similarly, by the use of the Cauchy product,

℘ 1 ℘ 2 = c (ω)ε where c (ω)= a j (ω)b k (ω) .
≥0 j+k=
Hence,

c (−ω)= a j (−ω)b k (−ω)=(−1) a j (ω)b k (ω)=(−1) c (ω) .
j+k= j+k=