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5.14. Proposition. Let 1 < p < ∞. Assume that the paths of symmetric matrices A(±∞, ·)
satisfy the assumptions of Theorem 5.9. Then
∂A : X 1,p
S ,V ,V ′(Σ,�n ) → X p
S (Σ,�n )
is semi-Fredholm with ind∂A := dimker∂A − dimcoker∂A < +∞.
Proof. We claim that there exist c ≥ 0 and s ≥ 0 such that, for any u ∈ X 1,p
S ,V ,V ′(Σ,�n ), there
holds
�u�X1,p (Σ) ≤ c � �
�(∂ + A)u�Xp (Σ) + �χsu�X p (Σ) , (97)
where χs is the characteristic function of the set {z ∈ Σ | |Re z| ≤ s}.
By Theorem 5.11 (ii), the asymptotic operators
∂ + A(−∞, ·) : W 1,p
(Σ,�n ) → L p (Σ,�n ),
N ∗V0,N ∗V ′
0
∂ + A(+∞, ·) : W 1,p
N ∗Vk,N ∗V ′
k ′
p
(Σ,�n
) → L (Σ,�n
),
are invertible. Since invertibility is an open condition in the operator norm, there exist s >
max |Re S | + 2 and c1 > 0 such that for any u ∈ X 1,p
S ,V ,V ′(Σ,�n ) with support disjoint from
{|Rez| ≤ s − 1} there holds
�u� X 1,p (Σ) = �u� W 1,p (Σ) ≤ c1�(∂ + A)u� L p (Σ) = c1�(∂ + A)u� X p (Σ). (98)
By Proposition 5.10, there exists c2 > 0 such that for every u ∈ X 1,p
S ,V ,V ′(Σ,�n ) with support in
{|Rez| ≤ s} there holds
�u� X 1,p (Σ) ≤ c2(�u� X p (Σ) + �∂u� X p (Σ))
≤ (c2 + �A�∞)�u� X p (Σ) + c2�(∂ + A)u� X p (Σ).
The inequality (97) easily follows from (98) and (99) by writing any u ∈ X 1,p
S ,V ,V ′(Σ,�n ) as
u = (1 − ϕ)u + ϕu, for ϕ a smooth real function on Σ having support in {|Re z| < s} and such
that ϕ = 1 on {|Re z| ≤ s − 1}.
Finally, by Proposition 5.13 the linear operator
X 1,p
S ,V ,V ′(Σ,�n ) → X p
S (Σ,�n ), u ↦→ χsu,
is compact. Therefore the estimate (97) implies that ∂A has finite dimensional kernel and closed
range, that is it is semi-Fredholm with index less than +∞.
It would not be difficult to use the regularity of weak solutions of the Cauchy-Riemann operator
to prove that the cokernel of ∂A is finite-dimensional, so that ∂A is Fredholm. However, this will
follow directly from the index computation presented in the next section.
5.5 A Liouville type result
Let us consider the following particular case in dimension n = 1:
k = 1, k ′ = 0, S = {0}, V0 = (0), V1 =Ê, V ′
0 =Ê, A(z) = α,
with α a real number. In other words, we are looking at the operator ∂ +α on a space of�-valued
maps u on Σ such that u(s) is purely imaginary for s ≤ 0, u(s) is real for s ≥ 0, and u(s + i) is
real for every s ∈Ê. Notice that Φ − (t) = Φ + (t) = e iαt , so
e iα iÊ∩Ê=(0) ∀α ∈Ê\(π/2 + π�), e iαÊ∩Ê=(0) ∀α ∈Ê\π�,
so the assumptions of Theorem 5.9 are satisfied whenever α is not an integer multiple of π/2. In
order to simplify the notation, we set
X p (Σ) := X p
{0} (Σ,�), X 1,p (Σ) := X 1,p
{0},((0),Ê),(Ê) (Σ,�).
We start by studying the regularity of the elements of the kernel of ∂α:
63
(99)