Therefore, the right-hand side of 7.76 is bounded by ǫO ˆ
β
T
2
∆
¯t
, as required. For d
≤ 2, we use 7.27 twice and 1 + t
I \I
′
∧ T
I
′
≤ 1 + ¯t = ∆
¯t
to obtain X
• s
≤t
J \I
s ≤s
′
≤t
I \I′
1 + s
′
∧ T
I
′
˜b
2
s,s
′
β
2
T
≤ ǫ O ˆ β
T
∆
¯t
X
• s
′
≤t
I \I′
ǫ
1 −δ
s′,2 ǫ
β
T
1 + s
′ d−22
≤ ǫ O ˆ β
T
2
∆
¯t
. 7.78
This completes the proof of 7.56 and hence of Lemma 7.3.
7.4 Proof of 7.8
Recall the definition 4.53 of a
N
y
1
, ~x
I
; 4
±
and denote by a
N ,N1,N2
y
1
, ~x
I
; 4
±
the contribution from B
N1
δ
b
N +1
, y
1
; C
N
and A
N2
e, ~x
I
; ˜ C
e
N
, i.e., − a
N ,N1,N2
y
1
, ~x
I
; 4
±
= X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
in ˜ C
e N
}
B
N1
δ
b
N +1
, y
1
; C
N
A
N2
e, ~x
I
; ˜ C
e
N
. 7.79
Compare 7.79 with φ
N ,N1,N2
y
1
, y
2
±
in 6.39 and note that the only difference is that A
N2
e, ~x
I
; ˜ C
e
N
in 7.79 is replaced by B
N2
δ
e, y
2
; ˜ C
e
N
in 6.39 cf., Figure 8. Similarly to the proof of 7.6 in Section 7.2, we discuss the following three cases separately: i
|I| = 1, ii
|I| ≥ 2 and N
2
= 0, and iii |I| ≥ 2 and N
2
≥ 1. i Let I =
{ j} for some j ∈ J. Then, by the similarity of 7.79 and 6.39, we can follow the same proof of Lemma 6.3 and obtain
a
N ,N1,N2
y
1
, x
j
; 4
±
≤ X
u
1
R
N +N1,N2
u
1
, x
j
+
1
{N≥1}
Q
N +N1,N2
u
1
, x
j
p
u
1
,y
1
. 7.80
By 5.79 and 6.19–6.20, we obtain X
N
1
,N
2
≥0
X
~x
J
X
y
1
a
N ,N1,N2
y
1
, x
j
; 4
±
τ~x
J
j
− y
1
≤ O 1 + ¯t
J
j
|J
j
|−1
| {z
}
≤ 1+¯t
|J j |−1
X
N
1
,N
2
≥0
X
• s
≤t
X
u
1
,x
j
R
N +N1,N2
s,t
j
u
1
, x
j
+
1
{N≥1}
Q
N +N1,N2
s,t
j
u
1
, x
j
≤ O ˆ β
T
∨N−1
O 1 + ¯t
r −3
X
• s
≤t
˜b
2
s,t
j
δ
s,t
j
+ β
T
β
T
. 7.81
By 7.54, we conclude that, for I = { j},
X
~x
J
X
y
1
a
N
y
1
, x
j
; 4
±
τ~x
J
j
− y
1
≤ ǫ O ˆ
β
T
1 ∨N
∆
¯t
1 + t 1 + ¯t
r −3
. 7.82
884
ii Let |I| ≥ 2 and N
2
= 0. Then, by 5.68 and following the argument around 6.89, we have a
N ,N1,0
y
1
,
~x
I
; 4
±
≤ X
v
X
e
p
e
X
∅6=I
′
I
δ
v
,e
P {e −→ ~x
I
′
} ◦ {e −→ ~x
I \I
′
} +
X
z
6=e
P e, z; v ,
ℓ~x
I
′
τ~x
I \I
′
− z
× X
b
N +1
6=e
p
b
N +1
Bound on ˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
∩ {v∈˜C
N
} in ˜C
e N
}
B
N1
δ
b
N +1
, y
1
; C
N
. 7.83
By Lemmas 6.5–6.7 and following the proof of Lemma 6.3 for N
2
= 0 in Section 6.3.1, we obtain X
b
N +1
p
b
N +1
˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
∩ {v∈˜C
N
} in ˜C
e N
}
B
N1
δ
b
N +1
, y
1
; C
N
≤ X
η
X
u
R
N +N1
u
, e; ℓ
η
v +
1
{N≥1}
Q
N +N1
u
, e; ℓ
η
v
p
u ,y
1
. 7.84
Therefore, similarly to 7.23, we have a
N ,N1,0
y
1
,
~x
I
; 4
±
≤ X
∅6=I
′
I
˜ a
N ,N1,0
y
1
,
~x
I
′
,
~x
I \I
′
; 4
1
+ ˜ a
N ,N1,0
y
1
,
~x
I
′
,
~x
I \I
′
; 4
2
, 7.85
where ˜
a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
1
= X
u
,e
R
N +N1
u
, e; ℓe
+
1
{N≥1}
Q
N +N1
u
, e; ℓe
× p
u ,y
1
p
e
P {e −→ ~x
I
′
} ◦ {e −→ ~x
I \I
′
} ,
7.86 and cf., 6.55–6.56
˜ a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
2
= X
u ,v
R
N +N1,1
u , v ;
ℓ~x
I
′
+
1
{N≥1}
Q
N +N1,1
u , v ;
ℓ~x
I
′
× p
u ,y
1
τ~x
I \I
′
− v,
7.87 First, we estimate the contribution to 7.8 from ˜
a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
1
. By 5.79 and 5.89 and following the argument around 7.39, we obtain
X
~x
J
X
y
1
˜ a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
1
τ~x
J \I
− y
1
≤ X
• s
t s
≤s
′
t
I
sup
w
X
u,v
R
N +N1
s,s
′
u, v; ℓw, s
′
+ ǫ
+
1
{N≥1}
Q
N +N1
s,s
′
u, v; ℓw, s
′
+ ǫ
× ǫ O 1 + ¯t
I |I|−2
1 + ¯t
J \I
|J\I|−1
≤ ǫO 1 + ¯t
|J|−3
X
• s
t s
≤s
′
t
I
Bound on X
u,v
R
N +N1
s,s
′
u, v +
1
{N≥1}
Q
N +N1
s,s
′
u, v ,
7.88
885
where we have used ¯t
I
≤ ¯t for |I| ≥ 2 and 1 + ¯t
J \I
|J\I|−1
= 1 if J \ I = { j} and t
j
= max
i ∈J
t
i
otherwise we use ¯t
J \I
≤ ¯t. By 7.55, we obtain 7.88
≤ ǫ
2
O ˆ β
T
1 ∨N+N
1
∆
¯t
1 + t 1 + ¯t
r −3
. 7.89
Next, we estimate the contribution to 7.8 from ˜ a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
2
. By 5.79 and repeatedly applying 5.18 to 6.19–6.20, we obtain
X
~x
J
X
y
1
˜ a
N ,N1,0
y
1
, ~x
I
′
, ~x
I \I
′
; 4
2
τ~x
J \I
− y
1
≤ X
• s
t
J \I
s ≤s
′
t
I \I′
X
u,v
R
N +N1,1
s,s
′
u, v; ℓ~t
I
′
+
1
{N≥1}
Q
N +N1,1
s,s
′
u, v; ℓ~t
I
′
× O 1 + ¯t
I \I
′
|I\I
′
|−1
1 + ¯t
J \I
|J\I|−1
| {z
}
≤ 1+¯t
|J\I′|−2
≤ O ˆ β
T
∨N+N
1
−1
O 1 + ¯t
|J|−3
X
• s
t
J \I
s ≤s
′
t
I \I′
1 + s
′
∧ max
i ∈I
′
t
i
˜b
2
s,s
′
β
2
T
. 7.90
By 7.56, we arrive at 7.90
≤ ǫ O ˆ
β
T
1 ∨N+N
1
+1
∆
¯t
1 + t 1 + ¯t
r −3
. 7.91
Summarizing 7.89 and 7.91 yields that, for |I| ≥ 2 and N
2
= 0, X
~x
J
X
y
1
a
N ,N1,0
y
1
, ~x
I
; 4
±
τ~x
J \I
− y
1
≤ ǫ O ˆ
β
T
1 ∨N+N
1
∆
¯t
1 + t 1 + ¯t
r −3
. 7.92
iii Let |I| ≥ 2 and N
2
≥ 1. By 5.68 and 7.84, we have a
N ,N1,N2
y
1
,
~x
I
; 4
±
≤ X
v ,z
X
e
p
e
X
∅6=I
′
I
P
N2
e, z; v τ~x
