NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

74 CASIM ABBAS
W 1,p
loc (C) since the sequence (ξk − ξk) depends on the choice of the ball BR(0). Our
sequence ξk − ξk(0) has a convergent subsequence on any ball.
3.4. Convergence in W 1,p (B ′ )
Pick a sequence τk ↗ τ0. We claim that the sequence ( ˆFτk) converges in L p (B) maybe
after passing to a suitable subsequence (recall that, so far, we only have the uniform
bound (3.9)). The functions ˆFτk converge pointwise almost everywhere after passing
to some subsequence. Indeed, the sequence {(u∗ 0λ ◦ jτk − i(u∗ 0λ))ατ (z)} converges
k
already pointwise since jτk and ατk do (recall that the sequence (ατk) converges in
W 1,p (B) and therefore uniformly). The sequence (∂sατk) converges in Lp (B) and
therefore pointwise almost everywhere after passing to a suitable subsequence. Then
by Egorov’s theorem, for any δ>0, there is a subset Eδ ⊂ B with |B\Eδ| ≤δ so that
the sequence ˆFτk converges uniformly on Eδ. Letαbe the Lp-limit of the sequence
(∂sατk), andletε>0. We introduce
C := 2 sup (u
0≤τ0 sufficiently small such that
αL p (B\Eδ) ≤ ε
3 C .
Now choose k0 ≥ 0 so large that, for all k ≥ k0, wehave
∂sατk − αL p (B) ≤ ε
3 C
Then, if k, l ≥ k0, wehave
and ˆFτk − ˆFτlL ∞ (Eδ) ≤ ε
3 |B| .
ˆFτk − ˆFτlL p (B) ≤ˆFτk − ˆFτlL p (Eδ) +ˆFτk − ˆFτlL p (B\Eδ)
proving the claim.
≤|Eδ|ˆFτk − ˆFτlL ∞ (Eδ) + 2 sup ˆFτkL
k≥k0
p (B\Eδ)
≤|B|ˆFτk − ˆFτl p
L ∞ (Eδ)
≤ ε
+ C · sup ∂sατkL
k≥k0
p (B\Eδ)
Recalling that φτ = aτ + ifτ and that the family fτ satisfies a uniform L ∞ -bound, we
have
sup Im(φτ ◦ ατ )L
τ
∞ (B) < ∞.