Cauchy Integral Decomposition of Multi-Vector Valued Functions on ...

120 R. Abreu Blaya, J. Bory Reyes, R. Delanghe and F. Sommen CMFT
(i) C Σ f extends to C ∞ (Ω), i.e.
C Σ f ∈ M(Ω) ∩ C ∞ (Ω).
(ii) If C + Σ f denotes the trace <strong>of</strong> C Σf to Σ, i.e. for x ∈ Σ, if
then
C + Σ f(x) =
lim C Σf(˜x),
Ω∋˜x→x
C + Σ f(x) = 1 2 (f(x) + H Σf(x)) .
(iii) H Σ f ∈ C ∞ (Σ) and C + Σ (C+ Σ f) = C+ Σ f.
Now let R > 0 be such that Ω ⊂ B(R) and put Ω ∗ = B(R) \ Ω. Then Ω ∗ is a
compact, connected manifold with C ∞ -boundary Σ ∗ = ∂B(R) ∪ Σ, whence for
F ∈ C ∞ (Σ ∗ ), Calderbank’s results [4, §§6–9] remain valid.
Putting for f ∈ C ∞ (Σ),
{
f(x), x ∈ Σ
F (x) =
0, x ∈ ∂B(R),
we thus have for the <strong>Cauchy</strong> and Hilbert transforms C Σ ∗F and H Σ ∗F that
C Σ ∗F ∈ M(Ω ∗ ) ∩ C ∞ (Ω ∗ ).
Moreover, straightforward arguments lead to the following results.
(iv) For x ∈ Ω ∗ ,
(v) For x ∈ Σ ∗
C Σ ∗F (x) = −C Σ f(x).
{
C + Σ ∗F (x) = 1 2 (F (x) + H Σ ∗F (x)) = −C Σ f(x), x ∈ ∂B(R)
1
(f(x) − H 2 Σf(x)), x ∈ Σ.
It thus follows from (iv) and (v) that for x ∈ Σ,
C − Σ f(x) =
lim C Σ f(˜x) = 1
R m \Ω∋˜x→x 2 (−f(x) + H Σf(x)) .
Putting Ω + = Ω and Ω − = R m \Ω, we obtain the following theorem by combining
the foregoing results.
Theorem 3.1. Let Ω ⊂ R m be an open bounded and connected domain with
C ∞ -boundary Σ such that R m \ Ω is connected. Furthermore, let for f ∈ C ∞ (Σ),
C Σ f and H Σ f be, respectively, the <strong>Cauchy</strong> and Hilbert transforms <strong>of</strong> f. Then
(i) C Σ f ∈ M(R m \ Σ) with C Σ f(∞) = 0;
(ii) H Σ f ∈ C ∞ (Σ);