NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

62 CASIM ABBAS
Recalling our original situation, we have the following result which shows that
there is some sort of conformal mapping for j1 on the ball B.
THEOREM 3.5 ([9, Theorem 4])
Let µ : C → C be an essentially bounded measurable function with µ|C\B ≡ 0 and
p>2 such that cpµL ∞ (C) < 1. Then there is a unique map α : C → C with
α(0) = 0 such that
∂α = µ∂α
in the sense of distributions with ∂α − 1 ∈ L p (C).
The desired map α is given by
α(z) = z + u(z),
where u ∈ Bp solves the equation ∂u = µ∂u + µ. In particular, α ∈ W 1,p (B).
Lemma 8 in [9] states that α : C → C is a homeomorphism. We can apply the
theorem to all the µτ , 0 2 such that cp sup n µnL ∞ (C) < 1 and any compact set
B ⊂ C.
Proof
We first estimate with g ∈ Lp (C) and z = 0
|Ag(z)| = 1
z
g(ξ) dξ d¯ξ
2π C ξ(ξ − z)
≤ |z|
2π gLp
1
(C)
ξ(ξ − z)
≤ CpgLp 2
1−
(C) |z| p ,
L p
p−1 (C)
(3.5)