180 D. SALAMON, FLOER HOMOLOGYProposition 3.9. Suppose that HE Hreg, x,y,z E P(H), u E M(y,x;H,J), and
v E M(z, y; H, J) .-Then there exist constants c > 0 and Ro > 0 such that, for
every R > Ro and every rJ E W^2 ·P (JR x S^1 , ( v# RU) *TM),
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Proof. Let us denoteVR ( S, t) _ { - v#Ru(s, y(t), t) ,s ::::; 0,
s::::: 0,UR(s, t) = { y(t), S s:; 0,
v#Ru(s, t), s :'.'.'. 0.Note that VR(s, t) = v(s + R, t) for s ::::; -R/2 - 1 and uR(s, t) = u(s - R, t) for
s :'.'.'. R/2+ 1. By Proposition 1.21, the difference between VR(s, t) and v(s+R, t) (in
the Ce-norm for any .e) is exponentially small as R ---+ oo, and so is the difference
between uR(s, t) and u(s - R, t). Hence there exist constants Ro > 0, co > 0, andc 1 > 0 such that, for every R :'.'.'.Ro and every T/u E W^1 ·P(JR x S^1 ,uR*TM),
Similar inequalities hold with Dun replaced by Dvn.Now, for R > 2, we have
# ( t)
= { VR(S, t),
V RU S, UR ( s, t) ,if s ::::; 0,
ifs::::: 0.Note that v#Ru(s, t) = y(t) for -R/2::::; s::::; R/2. In order to establish the required
estimate for DR= Dv#nu we fix a vector field rJ E W^1 ·P(JR x S^1 , (v#Ru)*TM) and
definerJu(s, t) = fJR(s)rJ(s, t) E Tun(s,t)M,
rJv(s, t) = (1 - fJR(s))rJ(s, t) E Tvn(s,t)M,where fJR(s) = (J(s/ R + 1/2) is a smooth cutoff function such that
fJR(s) = { ~:
if s s:; -R/2,
ifs;:::: R/2,-R-^1 < - (J R "(s) < - R-^1 '
for R :'.'.'.Ro. Note that DRT/u =Dun T/u and DRT/v = Dvn T/v· Hence we obtain
the following inequality:llrJllwi.P < llrJullw1,p + llrJvllw1,p
< Co (llDun T/ullLP + llDvn rJvllLP)
Co (IJDR(fJRrJ)llLP + llDR((l - fJR)rJ)lb)
< 2co llDR*rJllLP +
4
~0
llrJllLP.The last inequality follows from the fact that DR(fJRTJ) = fJRDRT/ - fJR^1 rJ and
lfJR'(s)I s:; 2/R for alls. With 4co/R::::; 1/2 we obtain an inequality
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for all rJ E W^1 ·P(JR x 8^1 , (v#Ru)*TM). Now observe that