164 D. SALAMON, FLOER HOMOLOGYfor every ( E C 0 (JR x S^1 , JR^2 n). Since C 0 is dense in W^1 ,P, this estimate continues
to hold for all ( E W^1 ,P. It follows that D: wl,p---) LP is injective and has a closed
range. Thus it suffices to prove that the range is dense. Let'T/ E LP(JR x S^1 , JR^2 n) n L^2 (JR x S1, JR^2 n)
be given. Then, by Step 1, there exists a ( E W^1 ,^2 such that D( = 'T/· By Step 3,
( E W^1 ,P and this proves that 'T/ is in the range of D : W^1 ,P---> LP. Hence D has a
dense range, and hence it is onto. This proves the lemma in the case p :2:: 2. DExercise 2.5. Prove that Lemma 2.4 continues to hold with p replaced by q S 2.Hint: Choose p :2:: 2 such that l/p + l/q = 1. Define w-^1 ,q(JR x S1,JR^2 n) as the
dual space of W^1 ,P(JR x S1,JR^2 n). Prove that the operator D satisfies an estimate
of the formll(llLQ Sc llD(llw-1,q ·
To see this interprete D : Lq ---> w-l,q as the functional analytic adjoint of the
operatorD* =-Os+ JoOt + s: wl,p---) LP.
Now prove that ll8s( llw-1,q S ll(llLQ and use this to establish an estimate of the
formll(llw1,q Sc llD(llLQ ·
Finally, prove that D : W^1 ,q ---> Lq has a dense range. DProof of Theorem 2.2 (the Fredholm property) If det(ll - w±(l)) =f. 0, then
it follows from Lemma 2.4 that there exist constants T > 0 and c > 0 such that,
for every ( E W^1 ,P(JR x S^1 ,JRn),(19) ( ( s, t) = 0 for Is I S T - 1
Now choose a smooth cutoff function (3 : JR---> [O, 1] such that (3(t) = 0 for ltl :2:: T
and (3(t) = 1 for ltl S T - 1. Using the estimate (17) for (3( and (19) for (1 - (3)(
we obtain
ll(llw^1 .p < llf3(llw^1 .p + 11(1 - (3)(llw1,p
< C1 (llf3(11LP + llD(f30lb + llD((l - (3)()11LP)
< C2 (ll(llLP[-T,T] + llD(llLP) ·
Thus we have established an estimate of the form
(20) ll(llwi,p SC (ll(llLP[-T,T] + llD(llLP)
for ( E W^1 ,P(JR x S1,JR^2 n). Since the restriction operator
W^1 'P(JR x S^1 ,JRn)---> LP([-T,T] x 81 ,JRn)is compact it follows from Exercise 2.1 that D has a finite dimensional kernel
and a closed range. That D also has a finite dimensional cokernel, follows from
elliptic regularity. Namely, suppose that 'T/ E Lq(JR x S^1 , JR^2 n) (with l/p + l/q = 1)
annihilates the image of D. Then it follows from local elliptic regularity that
'T/ E w 1 ;'~ and there is a constant c > 0 such that
ll'rfllw1,q([k,k+I]xS') < c (11D*ryllLq([k-l,k+2JxS') + ll'rfllLq([k-l,k+2]xS'))
C llTJllLq([k-l,k+2JxS') ·