NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 71
and we would still have the same inequality for a slightly smaller ε ′ k 0, and we define for z ∈ BR(0) the
functions
ξk(z) := (φτk ◦ ατk) xk + εk(z − xk) ,
which makes sense if k is sufficiently large. The transformation
: x ↦−→ xk + εk(x − xk)
satisfies (B1(xk)) = Bεk(xk) and (B1(y)) ⊂ Bεk(xk + εk(y − xk)) so that
|∇(φτk ◦ ατk)(x)|
Bε (xk) k p dx = ε 2
|∇(φτk k
◦ ατk)
B1(xk)
xk + εk(z − xk) | p dz
= ε 2
k ε −p
k |∇ξk(z)| p dz
and
B1(xk)
∇ξkL p (B1(xk)) = ε 1−(2/p)
k ∇(φτk ◦ ατk)L p (Bε k (xk)) (3.15)
= 1
by (3.14) and, for any y for which ξk|B1(y) is defined and for large enough k, wehave
∇ξkL p (B1(y)) ≤ ε 1−(2/p)
k ∇(φτk ◦ ατk)L p (Bε k (xk+εk(y−xk))) ≤ 1 (3.16)
by (3.13). The functions ξk satisfy the equation
¯∂ξk(z) = εk ˆFτk
xk + εk(z − xk) + εk ˆGτk
xk + εk(z − xk) =: Hτk(z) (3.17)