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where A is a smooth map taking value into L(Ê2n ,Ê2n ). Since u(s, t − 1) converges to x1(t) for
s → −∞, u(s, t) converges to x2(t) for s → −∞, and u(s, 2t − 1) converges to y(t) for s → +∞,
for any t ∈ [0, 1], the L(Ê2n ,Ê2n )-valued function A has the following asymptotics:
A(s + (t − 1)i) → A − 1 (t), A(s + ti) → A− 2 (t) for s → −∞, A(s + (2t − 1)i) → A+ (t), for s → +∞,
for any t ∈ [0, 1], where A − 1 (t), A− 2 (t), and A+ (t) are symmetric matrices such that the solutions
of the linear Hamiltonian systems
d
dt Ψ− 1 (t) = iA− 1 (t)Ψ− 1 (t),
d
dt Ψ− 2 (t) = iA− 2 (t)Ψ− 2 (t),
d
dt Ψ+ (t) = 2iA + (t)Ψ + (t), Ψ − 1 (0) = Ψ− 2 (0) = Ψ+ (0) = I,
are conjugated to the differential of the Hamiltonian flows along x1, x2, and y:
Ψ − 1 (t) ∼ Dxφ H1 (1, x1(0)), Ψ − 2 (t) ∼ Dxφ H2 (1, x2(0)), Ψ + (t) ∼ Dxφ H1#H2 (1, y(0)).
Then, by the definition of the Maslov index µ Ω in terms of the relative Maslov index µ, we have
µ Ω (x1) = µ(Ψ − 1 iÊn , iÊn ) − n
2 , µΩ (x2) = µ(Ψ − 2 iÊn , iÊn ) − n
2 , µΩ (y) = µ(Ψ + iÊn , iÊn ) − n
2 .
We claim that the linear operator
is Fredholm of index
D + G : W 1,p
iÊn(Σ Ω Υ,�n ) → Ω 0,1
L p(ΣΩ Υ,�n ).
ind (D + G) = µ Ω (x1) + µ Ω (x2) − µ Ω (y).
(128)
In order to deduce this claim from Theorem 5.9, we show that the operator D + G is conjugated
to a linear perturbed Cauchy-Riemann operator on a strip with jumping Lagrangian boundary
conditions, in the sense of section 5.3.
Indeed, given v : Σ Ω Υ →�n let us consider the�2n -valued map ˜v on Σ = {0 ≤ Im z ≤ 1}
defined as
The map v ↦→ ˜v gives us an isomorphism
where
˜v(z) := (v(z), v(z)).
W 1,p
iÊn(Σ Ω Υ,�n ) ∼ = X 1,p
S ,V ,V ′(Σ,�2n ),
S = {0}, V = ((0), ∆Ên), V ′ = (0),
∆Ên being the diagonal subspace ofÊn ×Ên , and (0) being the zero subspace ofÊ2n . This follows
from comparing the norm (126) to the X 1,p
S
the other hand, by comparing the norm (127) to the X p
S
w ↦→ 2 � w[∂s] gives us an isomorphism
norm by means of (87), (88), (90) and (91). On
norm by (89), we see that the map
Ω 0,1
L p(ΣΩ Υ ,�n ) ∼ = X p
S (Σ,�2n ).
It is easily seen that composing the operator D + G by these two isomorphisms produces the
operator
∂ Ã
: X1,p
S ,V ,V ′(Σ,�2n ) → X p
S (Σ,�2n ), u ↦→ ∂u + Ãu,
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