Proof of Theorem 2.6 getdoc1198. 429KB Jun 04 2011 12:04:06 AM

It is now seen that one should have lim T →∞ I 1 T = lim T →∞ I ′ 1 T = C 1+ β α,d Z R d Z R d p 1 x − y|x| −γ | y| −d−α1+β d x d y ‚Z R d ϕzdz Œ 1+ β χ 1+ β 3.97 Note that the integrals are finite by 3.21, 3.28 and 2.16. The justification of 3.97 requires some work, but we omit it for brevity. iii As d = γ, we must keep the measure µ γ in its original form 1.8. Arguing as in the proof of ii and taking into account 2.14, instead of 3.96 we obtain I ′ 1 T = 1 log T Z 1 Z R d Z R d p 1 x − y T sd α 1 + |x T s α | d ×   Z T 1−s −1 Z R d p u y − z T −sα ϕzχu + 1T s−1 dzdu   1+ β d y d x ds. Since lim T →∞ 1 log T Z R d p 1 x − y T sd α 1 + |x T s α | d d x = s 1 α σS d−1 p 1 y, it can be shown, with some effort, that lim T →∞ I 1 T = lim T →∞ I ′ 1 T = C 1+ β α,d 1 2 α σs d−1 Z R d p 1 y| y| −d−α1+β d y Z R d ϕzdz 1+ β χ 1+ β 0. Again, we omit the remaining parts of the proof.

3.7 Proof of Theorem 2.6

We only give an outline of the proof. The following lemma is constantly used. Lemma Let ϕ ∈ S R d , ϕ ≥ 0. a If d α2 + β1 + β, then the functions Gϕ, GGϕ 1+ β and GG ϕ 2 are bounded. b If d α1 + ββ, then additionally Gϕ 1+ β and GG ϕ 1+ β 1+ β are integrable and bounded. c If α γ ≤ d and d α2 + ββ − γβ, then additionally to the properties in a, Z R d GG ϕ 1+ β x 1 1 + |x| γ d x ∞. 1359 This Lemma follows easily from 1.12 and 3.25-3.28. Proof of part a of the theorem. As before we consider µ γ d x = |x| −γ d x. In 3.9 we substitute u ′ = s − u, then s ′ = T − sT and, finally, x ′ = x T 1 α s −1α , obtaining I 1 T = Z R d Z 1 Z R d p 1 x − ys −1α T −1α Z T 1−s T u ϕ yχ s + u T du 1+ β s −γα |x| −γ d y dsd x see 2.22. It is easily seen that by part b of the Lemma we have lim T →∞ I 1 T = Z R d p 1 x|x| −γ d x Z 1 s −γα χ 1+ β sds Z R d Gϕ y 1+ β d y. 3.98 For β 1 this is exactly log Eexp{−C〈 e X , ϕ ⊗ ψ〉}, where X is the limit process described in the theorem. Moreover, in this case 3.14 and boundedness of G ϕ easily imply I 2 T ≤ C T 1−γα1−21+β → 0. For β = 1 we use 3.10 and 3.4, obtaining I 2 T = I ′ 2 T − I ′′ 2 T − V 2 I ′′′ 2 T , where I ′ 2 T = H T F 2 T Z T Z R d Z R d p s x − y|x| −γ d x ϕ yχs Z T s T u−s ϕ yχ  u T ‹ dud y ds, I ′′ 2 T = H T F 2 T Z R d Z T Z R d p T −s x − yϕ yχ T T − s × Z s T s−u ϕχT − uv T ·, u y|x| −γ dud y dsd x I ′′′ 2 T = H T F 2 T Z R d Z T Z R d p T −s x − yϕ yχ T T − s Z s T s−u v 2 T ·, u y|x| −γ dud y dsd x. Substituting u ′ = u − s, s ′ = sT , and then x ′ = x T 1 α s 1 α and using part a of the Lemma and 2.22 we have lim T →∞ I ′ 2 T = Z R d p 1 x|x| −γ d x Z 1 s −γα χ 2 sds Z R d ϕ yGϕ yd y. 3.99 Applying 3.5, 2.22 and the Lemma above we get I ′′ 2 T ≤ C H T F 3 T Z R d Z T Z R d p s x − y|x| −γ ϕ yGϕGϕ yd y dsd x ≤ C 1 T −121−γα → 0, 1360 and, analogously, I ′′′ 2 T ≤ C H T F 3 T Z R d Z T Z R d p s x − y|x| −γ ϕ yGGϕ 2 yd y dsd x ≤ C 2 T −121−γα → 0. This and 3.98, 3.99 imply that for β = 1 the limit of V 2I 1 T + I 2 T is exactly log Eexp{−C〈 e X , ϕ ⊗ ψ}. Similar estimations, together with the Lemma, yield 3.16 and 3.17. This completes the proof of part a of the theorem. Proof of part b of the theorem. Following the general scheme one can show lim T →∞ I 1 T = Z R d p 1 x|x| −α d x χ 1+ β Z R d Gϕ 1+ β yd y, lim T →∞ I 2 T = c β Z R d p 1 x|x| −α d x χ 2 Z R d ϕ yGϕ yd y, and 3.16 and 3.17 recall that c β is defined by 2.24. This is accomplished by an argument similar to the one used in part a. Due to the criticality γ = α, the integrals R T . . . ds in 3.9- 3.11 require a different treatment. They are split into R 1 . . . ds + R T 1 . . . ds; the first summand converges to zero, and in the second one we use the substitution s ′ = log s log T . Here, again, we use repeatedly the Lemma above together with the easily checked fact that sup T 2 1 log T Z R d Z T T s hx|x| −α dsd x ∞ for any integrable and bounded function h recall that d α. We omit details. Proof of part c of the theorem. Recall that the case γ d has been proved in [BGT6]. For α γ ≤ d we use the Lemma part c is particularly important. We show lim T →∞ I 1 T = Z R d GG ϕ 1+ β xµ γ d xχ 1+ β 0, lim T →∞ I 2 T = c β Z R d G ϕGϕxµ γ d xχ 2 0, 3.16 and 3.17. Here µ γ is either given by 1.8 or, for γ d one can take µ γ d x = |x| −γ d x. Again, the details are omitted.

3.8 Proof of Theorem 2.8