NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 63
where the last estimate holds in view of
|ξ(ξ − z)| −p/(p−1) dξ d¯ξ ζ =z−1
ξ
=
C
If q solves q = Ɣ(µq + µ),then
C
=|z| 2−2p/(p−1)
¯∂(αn − α) = µn ∂(αn − α) − µ∂α+ µn ∂α
|z 2 ζ 2 − z 2 ζ | −p/(p−1) |z| 2 dζ d¯ζ
|ζ (ζ − 1)| −p/(p−1) dζ d¯ζ .
C
2πCp
= µn ∂(αn − α) + µn − µ + (µn − µ)Ɣ(µq + µ),
that is, the difference αn − α again satisfies an inhomogeneous Beltrami equation. By
Theorem 3.4, wehave
αn − α = A(µnqn + λn),
where λn = µn − µ + (µn − µ)Ɣ(µq + µ) and where qn ∈ L p (C) solves qn =
Ɣ(µnqn + λn). Combining this with (3.5)and(3.4), we obtain
|αn(z) − α(z)| ≤Cp µnqn + λnL p (C) |z| 1−2/p
≤ (Cp sup µnL
n
∞ (C) · c ′
pλnLp (C) + Cp λnLp (C)) |z| 1−2/p . (3.6)
Since µn − µLp (C) → 0 and (µn − µ)Ɣ(µq + µ)Lp (C) → 0 by Lebesgue’s
theorem we also have λnLp (C) → 0 and therefore αn → α uniformly on compact
sets. Since ¯∂(αn − α) = µn ∂(αn − α) + λn and αn − α = A(µnqn + λn), we verify
that
and
∂(αn − α) = Ɣ(µnqn + λn) = qn
¯∂(αn − α) = µnqn + λn.
Invoking (3.4) once again, we see that both ∂(αn − α)L p (C) and ¯∂(αn − α)L p (C)
can be bounded from above by a constant times λnL p (C) which converges to
zero.
We also need some facts concerning the classical case where µ ∈ C k,α (BR(0)),
BR(0) ={z ∈ C ||z|