NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

70 CASIM ABBAS
so that, for z ∈ B, wehave
∂(φτ ◦ ατ )(z) = ∂jτ φτ ατ (z) ◦ ∂sατ (z)
= u ∗
0λ ◦ jτ − i(u ∗
0λ)ατ (z) ◦ ∂sατ (z)
− (α ∗
τ γτ )(z) · ∂
+ i(α∗ τ
∂s γτ )(z) · ∂
∂t
and
=: ˆFτ (z) + ˆGτ (z)
=: ˆHτ (z),
(3.8)
sup ˆFτ L
τ
p (B) < ∞ for some p>2 (3.9)
since ατ → ατ0 in W 1,p (B) and supτ jτ L∞ < ∞. Wealsohave
sup ˆGτ C
τ
k (B ′ ) < ∞ (3.10)
for any ball B ′ ⊂ B ′ ⊂ B and any integer k ≥ 0 in view of Proposition 3.9 (the
proposition asserts uniform convergence after passing to a suitable subsequence, but
uniform bounds on all derivatives are established in the proof). We claim now that,
for every ball B ′ ⊂ B ′ ⊂ B, there is a constant CB ′ > 0 such that
∇(φτ ◦ ατ )L p (B ′ ) ≤ CB ′, ∀ τ ∈ [0,τ0). (3.11)
Arguing indirectly, we may assume that there is a sequence τk ↗ τ0 such that
Now define
∇(φτk ◦ ατk)L p (B ′ ) →∞ for some ball B ′ ⊂ B ′ ⊂ B. (3.12)
εk := inf ε>0 ∃ x ∈ B ′ : ∇(φτk ◦ ατk)L p (Bε(x)) ≥ ε 2/p−1 ,
which are positive numbers since ε (2/p)−1 →+∞. Because we assumed (3.12) we
must have infk εk = 0, hence we will assume that εk → 0. Otherwise, if ε0 =
(1/2) infk εk > 0, then we cover B ′ with finitely many balls of radius ε0, andwe
would get a k-uniform L p -bound on each of them, contradicting (3.12). We claim that
∇(φτk ◦ ατk)L p (Bε k (x)) ≤ ε (2/p)−1
k , ∀ x ∈ B ′ . (3.13)
Otherwise, we could find y ∈ B ′ so that
∇(φτk ◦ ατk)Lp (Bε (y)) >ε k (2/p)−1
k ,