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

116 R. Abreu Blaya, J. Bory Reyes, R. Delanghe and F. Sommen CMFT
Notice also that if Y k is a k-vector, then e M Y k is an (m − k)-vector and that
T (k)
M
: R(k) 0,m → R (m−k)
0,m , defined by T (k)
M (Y k) = e M Y k , determines an isomorphism
between these spaces.
In a classical way, for any a, b ∈ R 0,m , let
[a, b] = ab − ba,
{a, b} = ab + ba
denote, respectively, the commutator and the anti-commutator between the elements
a and b.
Finally, let us recall the definition <strong>of</strong> the conjugation a ↦→ a on R 0,m . For each
i = 1, . . . , m, e i = −e i while for a, b ∈ R 0,m
ab = ba.
Notice that for any basic element e A with |A| = k,
e A = (−1) k(k+1)/2 e A .
2.2. The Dirac operator and its tangential part. Let Σ be a C ∞ -hypersurface
in R m and suppose that at x ⊥ ∈ Σ, a local coordinate system is given
by x ⊥ (ω) = (x 1 (ω), . . . , x m (ω)) where ω = (ω 1 , . . . , ω m−1 ) ∈ R m−1 and x i (ω),
i = 1, . . . , m, are smooth functions <strong>of</strong> ω.
For ɛ > 0 sufficiently small, the ɛ-normal open neighborhood Σ ɛ <strong>of</strong> Σ is determined
by
Σ ɛ = {x = n(x ⊥ )ν(x) + x ⊥ ∈ R m : x ⊥ ∈ Σ, |ν(x)| < ɛ}
where for x ⊥ ∈ Σ, the outward pointing unit normal at x ⊥ is denoted by n(x ⊥ )
and ν(x) = 〈n(x ⊥ ), x〉 = ±|x − x ⊥ |.
If (ε 1 , . . . , ε m−1 ) is an orthonormal frame for the tangent space at x ⊥ ∈ Σ and
n = n(x ⊥ ), then for x ∈ Σ ɛ
x = n〈n, x〉 +
m−1
∑
j=1
ε j 〈ε j , x〉.
Putting ν = 〈n, x〉, (ν, ω 1 , . . . , ω m−1 ) determines a coordinate system at x. As
( )
∂x ∂x
, . . . ,
∂ω 1 ∂ω m−1
is also a basis for the tangent space at x ⊥ , for each j = 1, . . . , m − 1, there ought
to exist C ∞ -functions Q ji (ν, ω 1 , . . . , ω m−1 ) such that
ε j =
m−1
∑
i=1
Q ji (ν, ω 1 , . . . , ω m−1 ) ∂x
∂ω i
.