ℓu ℓ ℓ~x getdoc6cef. 1092KB Jun 04 2011 12:04:29 AM

First we investigate the contribution to 7.7 from the sum over u in 7.60, which is, by 7.43, 7.46 and Lemma 5.6, X ~x J X u ,v ,y 1 t u ≤t v X η X c X b N +1 P N b N +1 ; ℓ η

c, ℓu

p b N +1 P N ′ b N +1 , v ; c | {z } ≤ P N +N ′+1 v ; ℓu p v ,y 1 τ~x J \I − y 1 × P {u −→ ~x I ′ } ◦ {u −→ ~x I \I ′ } . 7.61 Note that |I| ≥ 2. By 5.79 and 5.89 and using P y 1 p v ,y 1 = O1 and t v t J \I , we can perform the sums over ~x J and y 1 to obtain 7.61 ≤ ǫ O 1 + ¯t J \I |J\I|−1 1 + ¯t I |I|−2 | {z } ≤ 1+¯t |J|−3 X u ,v t u ≤t v t J \I , t u ≤t I P N +N ′+1 v ; ℓu . 7.62 Then, by 1 + ¯t −1 ≤ 1 + t −1 for |J| ≥ 2 and using 5.18, we obtain 7.62 ≤ ǫ O ˆ β T N +N ′ 1 + t 1 + ¯t r −3 X • s ′ t J \I ǫ O β T 1 + s ′ d 2 X • s ≤s ′ ∧t I ǫ 1 + s ≤ ǫ O ˆ β T N +N ′ 1 + t 1 + ¯t r −3 X • s ′ t ǫ O β T 1 + s ′ d−42 + X • t I s ′ t J \I ǫ O β T 1 + t I 2 1 + s ′ d 2 , 7.63 where the first sum is readily bounded by O ˆ β T ∆ ¯t . The second sum is bounded as X • t I s ′ t J \I ǫ O β T 1 + t I 2 1 + s ′ d 2 ≤ Oβ T 1 + t I 2 ×    1 + t I −d−22 d 2, log1 + t J \I d = 2, 1 + t J \I 2−d2 d 2, 7.64 which is further bounded by O ˆ β T ∆ ¯t , using |I| ≥ 2 and t I ≤ ¯t. Therefore, 7.63 ≤ ǫ O ˆ β T N +N ′ +1 ∆ t J \I 1 + t 1 + ¯t r −3 . 7.65 Next we investigate the contribution to 7.7 from the sum over z in 7.60, which is, by 7.43, a version of 7.48 and 6.46, X ~x J X v ,z,y 1 t z t v X η X c X b N +1 P N b N +1 ; E t v

z, ℓ

η

c, ℓ~x

I ′ p b N +1 P N ′ b N +1 , v ; c | {z } ≤ Q N +N ′+1 v ,z; ℓ~x I ′ × p v ,y 1 τ~x J \I − y 1 τ~x I \I ′ − z. 7.66 881 By 5.79 and P y 1 p v ,y 1 = O1 and using the fact that t v t z ≤ t I \I ′ and t v t J \I , we can perform the sums over ~x J \I ′ and y 1 to obtain 7.66 ≤ O 1 + ¯t J \I |J\I|−1 1 + ¯t I \I ′ |I\I ′ |−1 | {z } ≤ 1+¯t |J\I′|−2 X ~x I ′ X v ,z t v t z ≤t I \I′ , t v t J \I Q N +N ′+1 v , z; ℓ~x I ′ . 7.67 By repeatedly applying 5.18 to 6.20, we have X v,z Q N +N ′+1 s,s ′ v, z; ℓ~t I ′ ≤ Oβ T 2 O ˆ β T N +N ′ ˜b 2 s,s ′ 1 + s ′ ∧ max i ∈I ′ t i 1 + ¯ s ′ ~t I ′ |I ′ |−1 . 7.68 Since s ′ ≤ t I \I ′ , we have ¯ s ′ ~t I ′ ≤ ¯t. Therefore, by 7.56, 7.67 ≤ O ˆ β T N +N ′ 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 ≤ ǫ O ˆ β T N +N ′ +2 ∆ t I \I′ 1 + t 1 + ¯t r −3 . 7.69 When d 4, the above ˆ β T is replaced by β. Summarising 7.60, 7.65 and 7.69, we now conclude that 7.7 for |I| ≥ 2 also holds. This together with 7.57 completes the proof of 7.7. Proof of Lemma 7.3. As we have done so far, β T and ˆ β T below are both replaced by β when d 4. First we prove 7.54. By 1 + t ∨2−d2 ≤ ∆ t and t ≤ ¯t for |J| ≥ 2 and using 7.27, we obtain X • s ≤t ˜b 2 s,t j δ s,t j β T ≤ ǫ 1 + t j ∨2−d2 log1 + t j δ d,2 1 + t j d−22 δ t ,t j β T = ǫ 1 + t 4 −d 2 ∨3−d log1 + t δ d,2 1 + t β T ≤ ǫ 1 + t ∨2−d2 1 + t O ˆ β T ≤ ǫ ∆ ¯t 1 + t O ˆ β T . 7.70 For d 2, we use 1 + t −d−22 ≤ 1 + t −1 1 + t j ∨4−d2 and 7.27 if d ∈ 2, 4], so that X • s ≤t ˜b 2 s,t j β 2 T ≤ X • s ≤t ǫ 2 −δ s,t j 1 + s d−22 1 + t j − s d−22 β 2 T ≤ ǫ 1 + t j ∨4−d2 log1 + t j δ d,4 1 + t d−22 O β 2 T ≤ ǫ 1 + t j ∨4−d log1 + t j δ d,4 1 + t O β 2 T ≤ ǫ O ˆ β T 2 1 + t . 7.71 882 For d ≤ 2, on the other hand, we use 7.27 and 1 + t j −1 ≤ 1 + t −1 to obtain X • s ≤t ˜b 2 s,t j β 2 T ≤ 1 + t j 2−d2 log1 + t j δ d,2 X • s ≤t ǫ 2 −δ s,t j 1 + s d−22 β 2 T ≤ ǫ O ˆ β T 1 + t j 2−d2 β T ≤ ǫ O ˆ β T 1 + t 1 + t j 4−d2 β T ≤ ǫ O ˆ β T 2 1 + t . 7.72 Since ∆ ¯t ≥ 1, this completes the proof of 7.54. To prove 7.55, we simply use 7.27 and t ≤ ¯t to obtain X • s ≤t ˜b 2 s,s β T ≤ X • s ≤t ǫ 2 −δ s,2 ǫ 1 + s ∨2−d2 log1 + s δ d,2 1 + s d−22 β T ≤ ǫ O ˆ β T 1 + t ∨2−d2 ≤ ǫ O ˆ β T ∆ ¯t , 7.73 and use t ≤ t I ≤ ¯t for |I| ≥ 2 and use 7.27 twice to obtain X • s ≤t s ≤s ′ ≤t I ˜b 2 s,s ′ β 2 T ≤ ǫ O ˆ β T X • s ≤t ǫ 1 −δ s,2 ǫ 1 + s d−22 β T ≤ ǫ O ˆ β T 2 . 7.74 This completes the proof of 7.55. Finally we prove 7.56, for d 2 and d ≤ 2 separately the latter is easier. For brevity, we introduce the notation T I ′ = max i ∈I ′ t i . 7.75 Note that t I \I ′ ∧ T I ′ ≤ ¯t since I ′ and I \ I ′ are both nonempty. Then, for d 2, X • s ≤t J \I s ≤s ′ ≤t I \I′ 1 + s ′ ∧ T I ′ ˜b 2 s,s ′ β 2 T = X • s ′ ≤t I \I′ 1 + s ′ ∧ T I ′ X • s ≤s ′ ∧t J \I ǫ 3 −δ s,s′ −δ s,2 ǫ δ s′,2 ǫ 1 + s d−22 1 + s ′ − s d−22 β 2 T ≤ ǫ O ˆ β T X • s ′ ≤t I \I′ ǫ 1 −δ s′,2 ǫ 1 + s ′ ∧ T I ′ 1 + s ′ d−22 β T ≤ ǫ O ˆ β T X • s ′ ≤¯t ǫ 1 −δ s′,2 ǫ β T 1 + s ′ d−42 + X • T I ′ ≤s ′ ≤t I \I′ ǫ 1 −δ s′,2 ǫ 1 + T I ′ β T 1 + s ′ d−22 , 7.76 where the second sum in the last line is interpreted as zero if T I ′ t I \I ′ . The first sum is readily bounded by O ˆ β T ∆ ¯t , whereas the second sum, if it is nonzero so that, in particular, T I ′ ≤ ¯t, is bounded by X • T I ′ ≤s ′ ≤t I \I′ ǫ 1 −δ s′,2 ǫ 1 + T I ′ β T 1 + s ′ d−22 ≤ Oβ T 1 + T I ′ ×    1 + T I ′ −d−42 d 4 log1 + t I \I ′ d = 4 1 + t I \I ′ 4−d2 d 4 ≤ O ˆ β T ∆ ¯t . 7.77 883 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