NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

68 CASIM ABBAS
be the conformal transformations as in Section 2, that is,
Tατ (z) ◦ i = jτ (ατ (z)) ◦ Tατ (z), z ∈ B.
The L ∞ -bound (3.1) on the family of functions (fτ ) and the above L 2 -bound imply
convergence of the harmonic forms α ∗ τ γτ after maybe passing to a subsequence.
PROPOSITION 3.9
Let τ ′ k be a sequence converging to τ0, and let B ′ = Bε ′(0) with B′ ⊂ B. Then there is
a subsequence (τk) ⊂ (τ ′ k ) such that the harmonic 1-forms α∗ τk γτk converge in C∞ (B ′ ).
Proof
First, the harmonic 1-forms α ∗ τ γτ satisfy the same L 2 -bound as in Proposition 3.8:
α ∗
τ γτ 2
L2 (B) =
=
=
B
B
Uτ
α ∗
τ γτ ◦ i ∧ α ∗
τ γτ
α ∗
τ (γτ ◦ jτ ) ∧ α ∗
τ γτ
γτ ◦ jτ ∧ γτ
≤u ∗
0 λL 2 ,jτ
≤ C,
where C is a constant depending only on λ and u0 since
We write
sup jτ L∞ ( ˙S) < ∞.
τ
α ∗
τ γτ = h 1
τ ds + h2
τ dt,
where hk τ , k = 1, 2 are harmonic and bounded in L2 (B) independent of τ. Ify ∈ B
and BR(y) ⊂ BR(y) ⊂ B, then the classical mean-value theorem
h k 1
τ (y) =
πR2
h k
τ (x)dx
implies that, for any ball Bδ = Bδ(y) with Bδ ⊂ Bδ ⊂ B, we have the rather generous
estimate
h k
τ C0 (Bδ(y)) ≤ 1
√ h
πδ k
τ L2 √
C
(B) ≤ √ .
πδ
BR(y)