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The symbol C∞ S ,c indicates bounded support.
Given p ∈ [1, +∞[, we define the Xp norm of a map u ∈ L1 loc (Σ,�n ) by
�
|u(z)| p |z − w| p/2−1 ds dt,
�u� p
Xp (Σ) := �u�p Lp �
(Σ\Br(S)) +
w∈S
Σ∩Br(w)
where r < 1 is less than half of the minimal distance between pairs of distinct points in S . This is
just a weighted L p norm, where the weight |z − w| p/2−1 comes from the identities (88), (89), and
(91) of the last section. Note that when p > 2 the X p norm is weaker than the L p norm, when
p < 2 the X p norm is stronger than the L p norm, and when p = 2 the two norms are equivalent.
The space X p
S (Σ,�n ) is the space of locally integrable�n -valued maps on Σ whose X p norm
is finite. The X p norm makes it a Banach space. We view it as a real Banach space.
The space X 1,p
S (Σ,�n ) is defined as the completion of the space C ∞ S ,c (Σ,�n ) with respect to
the norm
�u� p
X1,p (Σ) := �u�p Xp (Σ) + �Du�p Xp (Σ) .
It is a Banach space with the above norm. Equivalently, it is the space of maps in X p (Σ,�n )
whose distributional derivative is also in X p . The space X 1,p
S ,V ,V ′(Σ,�n ) is defined as the closure
in X 1,p
S (Σ,�n ) of the space of all u ∈ C ∞ S ,c (Σ,�n ) such that
u(s) ∈ N ∗Vj ∀s ∈ [sj, sj+1], j = 0, . . .,k,
u(s + i) ∈ N ∗V ′
j ∀s ∈ [s ′ j , s′ j+1 ], j = 0, . . .,k′ .
Equivalently, it can be defined in terms of the trace of u on the boundary of Σ.
Let A :Ê×[0, 1] → L(Ê2n ,Ê2n ) be continuous and bounded. For every ∈ [1, +∞[, the linear
operator
∂A : X 1,p
S (Σ,�n ) → X p
S (Σ,�n ), ∂Au := ∂u + Au,
is bounded. Indeed, ∂ is a bounded operator because of the inequality |∂u| ≤ |Du|, while the
multiplication operator by A is bounded because
�Au� X p (Σ) ≤ �A�∞�u� X p (Σ).
We wish to prove that if p > 1 and A(z) satisfies suitable asymptotics for Re z → ±∞ the
operator ∂A restricted to the space X 1,p
S ,V ,V ′(Σ,�n ) of maps satisfying the boundary conditions
(92) is Fredholm.
Assume that A ∈ C 0 (Ê×[0, 1], L(Ê2n ,Ê2n )) is such that A(±∞, t) ∈ Sym(2n,Ê) for every
t ∈ [0, 1]. Define Φ + , Φ − : [0, 1] → Sp(2n) to be the solutions of the linear Hamiltonian systems
Then we have the following:
(92)
d
dt Φ± (t) = iA(±∞, t)Φ ± (t), Φ ± (0) = I. (93)
5.9. Theorem. Assume that Φ− (1)N ∗V0 ∩ N ∗V ′
0 = (0) and Φ+ (1)N ∗Vk ∩ N ∗V ′
k ′ = (0). Then
the boundedÊ-linear operator
is Fredholm of index
− 1
2
∂A : X 1,p
S ,V ,V ′(Σ,�n ) → X p
S (Σ,�n ), ∂Au = ∂u + Au,
ind∂A = µ(Φ − N ∗ V0, N ∗ V ′
0) − µ(Φ + N ∗ Vk, N ∗ V ′
k ′)
k�
(dim Vj−1 + dimVj − 2 dimVj−1 ∩ Vj) − 1
k
2
′
�
(dim V ′
j−1 + dimV ′
j − 2 dimV ′
j−1 ∩ V ′
j ).
j=1
60
j=1
(94)