NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 89
(3) If µ(E,F) < 0, thenDF is injective, while if µ(E,F) + 2χ(M0) > 0, the
operator DF is surjective.
As an application, we obtain the following (here and in what follows, H m (M0) :=
W m,2 (M0)).
COROLLARY 2.2.3
(i) For q > 1 and k ∈ N0, letω∈ W k,q (M0,T∗ 0,1M0). Then there exists u ∈
W k+1,q (M0) real-valued on Ɣ0 such that ¯∂u = ω.
(ii) For m>1/2, letf∈ H m (∂M0) be a real-valued function. Then there exists
a holomorphic function v ∈ H m+(1/2) (M0) such that Re(v)|Ɣ0 = f .
(iii) For k ∈ N and q>1, the space of W k,q (M0) holomorphic functions on M0
which are real-valued on Ɣ0 is infinite-dimensional.
Proof
(i) Let L ∈ N be arbitrarily large, and let Ɣ be a connected nonempty open segment
of one connected component ∂1M0 = S 1 of ∂M0. ThenƔ can be defined in a
coordinate θ (respecting the orientation of the boundary) by Ɣ ={θ ∈ S 1 |
0 0.SinceL can be taken as
large as we want, this achieves the proof of (i).
(ii) Let w ∈ H m+(1/2) (M0) be a real function with boundary value f on ∂M0.Then
by (i), there exists R ∈ H m+(1/2) (M0) such that i ¯∂R =−¯∂w and R purely real
on Ɣ0; thus v := iR + w is holomorphic such that Re(v) = f on Ɣ0.
(iii) Taking the subbundle F as in the proof of (i), we have that dim ker DF =
χ(M0) + 2L if L satisfies 2χ(M0) + 2L >0, and since L can be taken as large
as we like, this concludes the proof.