NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

50 CASIM ABBAS
defined as
H 1
j (S) := γ ∈ E 1 (S) dγ = 0, d(γ ◦ j) = 0
(2.12)
and where E 1 (S) denotes the space of all (smooth) real-valued 1-forms on S, and
we write E 0,1 (S) = E 0,1
j (S) for the space of complex antilinear 1-forms on S with
respect to j, that is, complex-valued 1-forms σ such that iσ + σj = 0. Note that
our definition coincides with the set of closed and co-closed 1-forms on S. Moreover,
by elliptic regularity, we may also consider Sobolev forms. We identify H 1 (S,R)
with R 2g , and we consider the following parameter-dependent section in the bundle
L p (T ∗ S ⊗ C) 0,1 → W 1,p (S,C)
F : W 1,p (S,C) × R 2g −→ L p (T ∗ S ⊗ C) 0,1
F (b + if, σ ):= ¯∂jf (b + if ) − u ∗
0λ ◦ jf + i(u ∗
0λ) + ¯∂jf a0 + ψjf (σ ) + i
ψjf (σ ) ◦ jf
(2.13)
with jf as in (2.8). Recalling that z ↦→ jf (z) may not be differentiable, we interpret
the equation d(γ ◦ jf ) = 0 in the sense of weak derivatives. The solution set of (2.10)
is then the zero set of F . We consider the real parameter δ which we dropped from
the notation, fixed at the moment. For g ≡ 0 and b + if small in the W 1,p -norm,
we consider the composition ˆF (b + if, σ ) = fg(F (b + if, σ )). Its linearization
in the point (b + if, σ ) = (0,σ0), where σ0 is defined by ψj0(σ0) = γ0 and where
F (0,σ0) = 0,is
where
D ˆF (0,σ0) :W 1,p (S,C) × R 2g −→ L p (T ∗ S ⊗ C) 0,1
j0
D ˆF (0,σ0)(ζ,σ) = ¯∂j0ζ + ψj0(σ ) + i(ψj0(σ ) ◦ j0) + Lζ,
L : W 1,p (S,C) → W 1,p (T ∗ S ⊗ C) 0,1
j0 ↩→ L p (T ∗ S ⊗ C) 0,1
j0
is the compact linear map
where
Lζ =− 1
2 u∗
0 λ ◦ (Aζ + j0Aζj0) + i
2 u∗
0 λ ◦ (j0Aζ − Aζj0) + Bζ + iBζj0,
Bζ = d
dτ
τ=0
ψjτ k(σ0), ζ = h + ik,