1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. FREDHOLM THEORY 161

The limit matrices

are symmetric and hence, modulo some compact perturbation, we may as well

assume that S is symmetric for all s and t. Associated to a symmetric matrix

valued function S : IR x IR/Z --) JR^2 nx^2 n is a symplectic matrix valued function

W : IR x IR--) Sp(2n) given by^1

(15) Jo8tllt+Sw=O, w(s,O)=Il.

Denote w±(t) = lims->±oo w(s, t).

Theorem 2.2. Suppose that det(Il - w±(l))-/= 0. Then the operator

D: W^1 ·P(JR x S^1 ; IR^2 n)--) LP(IR x S^1 ; IR^2 n)

given by (14) is Fredholm for 1 < p < oo. Its Fredholm index is given by the

difference of the Conley-Zehnder indices:

(16)

The index formula in terms of the Conley-Zehnder index is due to Salamon-
Zehnder [47], and an alternative proof was given by Robbin-Salamon [41]. The
proof of Theorem 2.2 and the relevant definitions occupy the next three sections.

2.3. .LP-estimates

For p = 2 the proof of the Fredholm property is fairly straight forward and details

have been carried out by several authors (cf [9, 40, 47, 49]). For p -/= 2 the
Fredholm property was proved in [29]. The case p = 2 is the Sobolev borderline

case, and the nonlinear Fredholm theory requires the case p > 2. Roughly speaking,

the reason is that, in dimension 2, the Sobolev space W^1 ·P embeds into the space of
continuous functions, while there are W^1 •^2 -functions on IR^2 which are discontinuous.

For p > 2 the proof of the Fredholm property relies on the following two lemmata.

We follow the line of argument in [3].

Lemma 2.3. There exists a constant c > 0 such that

(17)

Proof. This is essentially the Calderon-Zygmund inequality which asserts that

there exists a constant c 0 > 0 such that every compactly supported function u :

!Rm --) IR satisfies

where

m

L ll8i8jullLP(JRm) :::; C ll~ulb(JRm)

i ,j=l

m a2

~-'"'­- L.., 8x · 2

j=l J

(^1) Recall that the group of symplectic matrices is given by

Sp(2n) = {WE JR2nx2n : wT JoW =Jo}.

Here Jo E JR Zn x Zn denotes the standard complex structure given by ( x, y) ,..... ( -y, x) for x, y E IR n.