LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 73we have C^2 = X z EB T(z,o)F and we see that the Banach space
V' := {! E W^1 ·P(D, C^2 ) I f(z) E X z for z E aD}
is a topological complement of Vin W^1 ·P(D, C^2 ).
We are left with the calculation of the tangent space of Bat some point u E B.
Elements in TuB are of the form!!:_,ti E w^1 ·P(D, c^2 )
dt t=Owhere ti-------. It is a smooth path in B c W^1 ·P(D, C^2 ) with lo = u. Pick z E aD.
Then ti-------. lt(z) is a path in F andd
dlt(z)I ETu(z)F
t t=O
which finishes the proof of the proposition. D
Proposition 3. 7. The Cauchy Riemann operator
[JJ: B ______, LP(D, C^2 )ui-------.us+J(u)ut
is a smooth map. The derivative at u 0
T := D[JJ(uo) : Tu 0 B ______, LP(D, C^2 )
can be written asT(h, k) = (hs + iht + ak, ks + ikt +bk)
where (h, k) E Tu 0 B and a, bare smooth maps from D into CR(C).
Proof. We consider the operator[JJ: {u E W^1 ·P(D,C^2 ) I u(D) c U}------; LP(D,C^2 )
u i-------. Us+ J(u)ut.It suffices to show that this is a smooth map. The derivatives
~ as' ~ at·. w^1 ·P(D , c^2 ) __, LP(D , c^2 )
are linear and continuous and therefore smooth. The map u i-------. J(u) has image
in W^1 ·P(D, C^2 ) and is smooth because J is smooth and its derivatives are bounded
on some closed ball containing U.
Moreover the product of the W^1 ·P-map z i-------. J(u(z)) with the LP-map z i-------. ~~(z)is again in LP. Indeed, if a. E W^1 ·P(D) , f3 E LP(D) and p > 2 then
fo 1 a./3 IP:::; lla.ll~o 11 /Jll~P :::; clla.llt'.in,p 11 /Jll~P ; c > 0
by the Sobolev embedding theorem. Now
W^1 ·P(D) x LP(D) ______, LP(D)(a., /3) i-------. a./3
is a bilinear continuous map and therefore smooth. This proves that [) J is smooth.