56 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
Proof. Using a finite covering argument it suffices to show that for f.^1 > 0 small
enough and c' > 0 large enough
llwllwk+1,p(D,,,1C2) ::; c'.Arguing indirectly we find a sequence (we) such that
(we)s + J(we)(we)t = 0
lwe(O)I ::; 1
llwe II wk.P(D.,C2) ::; c
llwe II Wk+'-P(D,e Pl ---> +oo
Ee '\. 0
we(O) ---> a.Denoting by 8 the Cauchy-Riemann operator associated to J(a) we have
0 = fJwe + (J(we) - J(a))(we)t.
Take a smooth map (3: D, ---> [O, 1] with (3 = 1 near 0 and supp((3) C B,. Define
(3 0 (x) = (3(~) for 8 > 0. Now
fJ(f3owe) = (fJf3o)we + f3o8we
= (fJf3o)we - (J(we) - J(a))f3o(we)t
= (fJf3o)we - (J(we) - J(a))(f3owe)t + (J(we) - J(a))(f3o)twe.
With a constant O"( 8), which has the property that O"( 8) ---> 0 for 8 ---> 0 we find for
8 « 1 (some thinking and the product rule when taking the derivative is required!!)
ck,p llf3owe llk+1,p< collwellk,p,D, + a(8)llf3owellk~l,p
+c'(8)llf3owellk,p + c"(8)llwellk,p
< a(8)llf3owellk+1,p + d(8)llwellk,p,D,·
For 8 « 1 we have a(8) ::; ~ck,p· Hence
1
2ck,p llf3owe II k+I,p ::; d( 8) llwe II k,p,D,.Therefore, for f.^1 > 0 close to 0
oo +--llwellk+1,p,D,, ::; llf3owellk+1,p
This contradiction proves the result.
The following is a good exercise.::; d'(8)llwellk,p,D,::; d'(8). c =: c < 00.
DExercise 2.2. l. Find the Holder estimates for the linear Cauchy-Riemann
problem in the literature.- Prove a version of the above theorem for the Holder norms.
- Either construct yourself or find in the literature examples, which show that
there are no apriori estimates for the linear Cauchy Riemann operator of
the form