NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 59
know that fτ0 L ∞ ( ˙S) ≤ T . We then obtain a complex structure jτ0 on ˙S by
jτ0(z) = πλTu0(z) −1 Tφfτ (z)(u0(z)) 0 −1 × J φfτ (z)(u0(z)) 0
Tφfτ (z) u0(z) 0 πλTu0(z).
By definition, the complex structure jτ0 is also of class L∞ and jτ (z) → j1(z)
pointwise. Our task is to improve the regularity of the limit fτ0 and the character of
the convergence fτ → fτ0 . We also have to establish convergence of the functions aτ
for τ ↗ τ0. The complex structures jτ are of course all smooth, but the limit jτ0 might
only be measurable.
3.1. The Beltrami equation
For the reader’s convenience, we briefly summarize a few classical facts from the
theory of quasiconformal mappings (see [8], [9]). The punctured surface ˙S carries
metrics gτ , also of class L ∞ for τ = τ0 and smooth otherwise, so that
In fact, gτ is given by
gτ (z) jτ (z)v, jτ (z)w = gτ (z)(v, w), for all v, w ∈ Tz ˙S.
gτ (z)(v, w) = dλ uτ (z) πλTuτ (z)v, J(uτ (z))πλTuτ (z)w .
In the case τ = τ0, we replace πλTuτ (z) by Tφfτ 0 (z)(u0(z))πλTu0(z). Wehave
sup τ gτ L ∞ ( ˙S) < ∞ and gτ → gτ0 pointwise as τ ↗ τ0. Our considerations about
the regularity of the limit are of local nature, so we may replace ˙S with a ball B ⊂ C
centered at the origin. Denoting the metric tensor of gτ by (g τ kl )1≤k,l≤2, wedefinethe
following complex-valued smooth functions:
and we note that
µτ (z) :=
1
2 (gτ 11 (z) − gτ 22 (z)) + igτ 12 (z)
1
2 (gτ 11 (z) + gτ 22 (z)) + gτ 11 (z)gτ 22 (z) − (gτ ,
12 (z))2
sup µτ L∞ ( ˙S) < 1
τ
and that µτ → µτ0 pointwise. We view the functions µτ as functions on the whole
complex plane by trivially extending them beyond B. Then they are also τ-uniformly
bounded in L p (C) for all 1 ≤ p ≤∞and µτ → µτ0 in Lp (C) for 1 ≤ p