1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. FREDHOLM THEORY 171

e^68 f(s) is nonincreasing and this implies f(s) :::; e-o(s-sa) f(s 0 ) for s ;::: s 0. The

argument for s ---+ -oo is similar, and this proves an estimate of the form

11 l~(s, t)12 dt:::; ce-61sl

for a ll s. To get the pointwise inequality, apply the operator 88 - J 0 8t to the

equation D~ = 0 to obtain

~~ = Jo8t(S0 - 8s(S0,

where ~ = 82 / 8s^2 + 82 / 8t^2. This implies that there is a constant c > 0 such that

~1~1

2

2: -cl~l^2

for all ~ E ker D. This inequality in turn can be used to derive a mean value
inequality of the form

l~(s, t)l^2 :::; c2 r 1~1^2

r j Br(s,t)

for r > 0 ands, t E JR. With r = 1, say, we obtain the required pointwise exponential

~~ 0

Proof of Proposition 1.21: (ii)===? (iii): First it follows from standard elliptic
estimates that every finite energy solution u : IR x IR/Z ---+ M of (7) satisfies

lim sup (1'Vs8su(s, t)I + l'Vt8su(s, t)I) = 0, sup l'Vt8tu(s, t)I < oo.

s_,±oo O:S:t'.'01 s, t

It then follows by inspection that the matrix function S(s, t) in (14) satisfies the

requirements of Lemma 2.11. Since Du8 8 u = 0, it follows from Lemma 2.11 that
88 u converges to zero exponentially as s ---+ ±oo. This proves the proposition. D