NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

88 GUILLARMOU and TZOU
where F ⊂ E |∂M0 is a totally real subbundle (i.e., a subbundle such that JF ∩ F is
the zero section) and where DF is the restriction of D to the Lq-based Sobolev space
with ℓ derivatives and boundary condition F
W ℓ,q
F (M0,E):= ξ ∈ W ℓ,q (M0,E) ξ(∂M0) ⊂ F .
When q = 2, we will use the standard notation H ℓ (M0) := W ℓ,2 (M0) for L 2 -based
Sobolev spaces. The boundary Maslov index for a totally real subbundle F ⊂ E∂M0
of a complex vector bundle is defined in generality in [26, Appendix C.3]; we only
recall the definition for our setting.
Definition 2.2.1
Let E = M0 × C and ∂M0 = m j=1 ∂iM0 be a disjoint union of m circles. The
boundary Maslov index µ(E,F) is the degree of the map ρ ◦ : ∂M0 → ∂M0,
where
|∂iM0 : S 1 ∂iM0 → GL(1, C)/GL(1, R)
is the natural map assigning to z ∈ S 1 the totally real subspace Fz ⊂ C, where
GL(1, C)/GL(1, R) is the space of totally real subbundles of C, and where ρ :
GL(1, C)/GL(1, R) → S 1 is defined by ρ(A.GL(1, R)) := A 2 /|A| 2 .
In this setting, we have the following boundary-value Riemann-Roch theorem stated
in [26, Theorem C.1.10].
THEOREM 2.2.2
Let E → M0 be a complex line bundle over an oriented compact Riemann surface
with boundary, and let F ⊂ E |∂M0 be a totally real subbundle. Let D be a smooth
Cauchy-Riemann operator on E acting on W ℓ,q (M0,E) for some q ∈ (1, ∞) and
ℓ ∈ N. Then we have the following.
(1) The following operators are Fredholm:
DF : W ℓ,q
F (M0,E) → W ℓ−1,q (M0,T ∗
0,1 M0 ⊗ E),
D ∗
F
ℓ,q ∗
: WF (M0,T0,1M0 ⊗ E) → W ℓ−1,q (M0,E).
(2) The real Fredholm index of DF is given by
Ind(DF ) = χ(M0) + µ(E,F),
where χ(M0) is the Euler characteristic of M0 and where µ(E,F) is the
boundary Maslov index of the subbundle F .