Proof of Lemma 6.7 for N
1
≥ 1. First we recall that, by 6.3 and 5.40, B
N1
δ
b
N +1
, y
1
; ˜ C
N
≤ X
b= · ,y
1
P
N1−1
b
N +1
, b; ˜ C
N
p
b
, 6.87
where, by 5.39,
P
N1−1
b
N +1
, b; ˜ C
N
= P
b
N +1
, b; ˜ C
N
N
1
= 1, ≤
X
η
X
z
X
e
P b
N +1
, e; ˜ C
N
, ℓ
η
z p
e
P
N1−2
e, b; z N
1
≥ 2. 6.88
Then, by following the argument between 6.72 and 6.82 and using versions of 6.57 and 5.59, we obtain that, for N
1
≥ 2, X
b
N +1
p
b
N +1
M
N +1
b
N +1
1
V
t y1 −ǫ
b
N
,y
2
∩ {~x ∈˜C
N
}
P b
N +1
, e; ˜ C
N
, ℓ
η
z
≤ X
b
N +1
X
η
′
X
c
P
N
b
N +1
; ℓ
η
′
c, V
t
y 1
−ǫ
y
2
, ℓ~x
p
b
N +1
P b
N +1
, e; c,
ℓ
η
z
≤ P
N +1
e; V
t
y 1
−ǫ
y
2
,
ℓ~x , ℓ
η
z = R
N +1
e, y
2
;
ℓ~x , ℓ
η
z .
6.89 For N
1
= 1, we simply ignore ℓ
η
z and replace e by b, which immediately yields 6.45. For N
1
≥ 2, by a version of 5.59, we obtain
LHS of 6.45 ≤
X
b= · ,y
1
X
η
X
z
X
e
R
N +1
e, y
2
; ℓ~x , ℓ
η
z p
e
P
N1−2
e, b; z p
b
≤ X
b=
· ,y
1
R
N +N1
b, y
2
; ℓ~x
p
b
, 6.90
as required. The inequality 6.46 for N
1
≥ 1 can be proved similarly. This completes the proof of Lemma 6.7.
7 Bound on a
~x
J
From now on, we assume r ≡ |J| + 1 ≥ 3. The Fourier transform of the convolution equation 2.25
is ˆ
ζ
~t
J
~k
J
= ˆ A
~t
J
~k
J
+
t
X
• s=
ǫ
Û τ
s −ǫ
∗ p
ǫ
k ˆ a
~t
J
−s
~k
J
, 7.1
where t = t
J
≡ min
j ∈J
t
j
and k = P
j ∈J
k
j
. We have already shown in Proposition 5.1 and 5.79 that
ˆA
~t
J
~k
J
≤ kA
~t
J
k
1
≤ ǫO 1 + ¯t
r −3
, Û
τ
s −ǫ
∗ p
ǫ
k ≤ kτ
s −ǫ
k
1
kp
ǫ
k
1
≤ O1, 7.2
where ¯t ≡ ¯t
J
is the second-largest element of ~t
J
. To complete the proof of 2.37, we investigate the sum
P
• ǫ≤s≤t
|ˆa
~t
J
−s
~k
J
|. 871
First we recall 2.26 and 4.55 to see that a
N
~x
J
= a
N
~x
J
; 1 + X
∅6=IJ
a
N
~x
J \I
, ~x
I
; 2 +
X
y
1
a
N
y
1
, ~x
I
; 3 + a
N
y
1
, ~x
I
; 4 τ~x
J \I
− y
1
. 7.3
Let
∆
t
=
1 d
6, log1 + t
d = 6, 1 + t
1 ∧
6 −d
2
d 6.
7.4
The main estimates on the error terms are the following bounds:
Proposition 7.1 Bounds on the error terms. Let d 4 and λ = λ
ǫ
c
. For r ≡ |J| + 1 ≥ 3 and
N ≥ 0,
X
~x
J
a
N
~t
J
~x
J
; 1 ≤
δ
N ,0
X
j ∈J
δ
t
j
,0
+ ǫ
2
O β
1 ∨N
1 + t O 1 + ¯t
r −3
, 7.5
X
~x
J
a
N
~t
J \I
,~t
I
~x
J \I
, ~x
I
; 2 ≤ ǫ
O β
N
1 + β∆
¯t
1 + t O 1 + ¯t
r −3
, 7.6
X
~x
J
X
y
1
a
N
y
1
,
~x
I
; 3
τ~x
J \I
− y
1
≤ ǫ O
β
N +1
1 + t ∆
¯t
1 + ¯t
r −3
, 7.7
X
~x
J
X
y
1
a
N
y
1
, ~x
I
; 4 τ~x
J \I
− y
1
≤ ǫ O
β
1 ∨N
1 + t ∆
¯t
1 + ¯t
r −3
. 7.8
For d ≤ 4 and λ = λ
T
, the same bounds with β replaced by ˆ
β
T
≡ β
1
T
−α
hold. The bounds 7.5–7.8 are proved in Sections 7.1–7.4, respectively.
Proof of 2.37 and 2.39 assuming Proposition 7.1. By 7.3 and 7.5–7.8, we have that, for d
4, |ˆa
~t
J
~k
J
| ≤ X
N ≥0
X
~x
J
a
N
~t
J
~x
J
≤ O 1 + ¯t
r −3
X
j ∈J
δ
t
j
,0
+ ǫ
1 + β∆
¯t
1 + t ,
7.9 hence, for any
κ 1 ∧
d −4
2
,
t
X
• s=
ǫ
|ˆa
~t
J
−s
~k
J
| ≤ O 1 + ¯t
r −3
1 + ǫ
t
X
• s=
ǫ
1 + β∆
¯t
1 + t − s
≤ O 1 + ¯t
r −3
log1 + ¯t 1 +
β∆
¯t
≤ O 1 + ¯t
r −2−κ
, 7.10
872
which implies 2.37, due to 7.1–7.2. For d
≤ 4, β in 7.10 is replaced by ˆ β
T
, and ∆
¯t
= 1 + ¯t. Therefore, for any κ α,
t
X
• s=
ǫ
|ˆa
~t
J
−s
~k
J
| ≤ O 1 + ¯t
r −2
log1 + ¯t ˆ
β
T
≤ OT
r −2−κ
, as T
↑ ∞. 7.11
This completes the proof of 2.39.
7.1 Proof of 7.5
