Conditions of Different Homothetic Sets and Laplace’s Method
Introduction to Homothetic Sets
We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤𝑐(𝑡)/𝑡𝛼}
This ensures that:
𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤𝑐(𝑡)} = 𝑡𝐴
1,𝑡
Verification of Conditions
Condition (5.3)
We verify this condition through rescaling:
1. Let 𝑡𝑞 ∈𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
Key result:
𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
Condition (5.5)
1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is uniformly bounded
below by a positive number for large 𝑡
Condition (5.6)
Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
Condition (5.7)
1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈ 𝐼(𝐴1)} 2. Show that 𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3.
Key result:
sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)=𝑡−𝛼/2√𝑐(𝑡)𝑂(1)=𝑜(1)
Condition (5.16)
Instead of checking (5.8), we verify (5.16):
𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
Conditions (5.9) - (5.13)
1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to homogeneity of 𝐼
3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12): Verified by
rescaling and using properties of 𝜋𝐴1
2. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
3. 𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
4. This ensures that:
5. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
6. Verification of Conditions
7. Condition (5.3)
8. We verify this condition through rescaling:
9. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 = 𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
10. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
11. Key result:
12. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
13. Condition (5.5)
14. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is uniformly
bounded below by a positive number for large 𝑡
15. Condition (5.6)
16. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
17. Condition (5.7)
18. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈ 𝐼(𝐴1)} 2. Show that 𝐾max(𝑡𝑝,𝑠)≤
𝑐/𝑡2 3. Key result:
19. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)=𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
20. Condition (5.16)
21. Instead of checking (5.8), we verify (5.16):
22. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
23. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
24. Conditions (5.9) - (5.13)
25. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to homogeneity
of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12): Verified by
rescaling and using properties of 𝜋𝐴1
26. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
27. 𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
28. This ensures that:
29. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
30. Verification of Conditions
31. Condition (5.3)
32. We verify this condition through rescaling:
33. 1. Let 𝑡𝑞 ∈𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
34. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
35. Key result:
36. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
37. Condition (5.5)
38. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is uniformly
bounded below by a positive number for large 𝑡
39. Condition (5.6)
40. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
41. Condition (5.7)
42. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈ 𝐼(𝐴1)} 2. Show that 𝐾max(𝑡𝑝,𝑠)≤
𝑐/𝑡2 3. Key result:
43. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)=𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
44. Condition (5.16)
45. Instead of checking (5.8), we verify (5.16):
46. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
47. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
48. Conditions (5.9) - (5.13)
49. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to homogeneity
of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12): Verified by
rescaling and using properties of 𝜋𝐴1
50. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
51. 𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
52. This ensures that:
53. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
54. Verification of Conditions
55. Condition (5.3)
56. We verify this condition through rescaling:
57. 1. Let 𝑡𝑞 ∈𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
58. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
59. Key result:
60. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
61. Condition (5.5)
62. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is uniformly
bounded below by a positive number for large 𝑡
63. Condition (5.6)
64. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
65. Condition (5.7)
66. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈ 𝐼(𝐴1)} 2. Show that 𝐾max(𝑡𝑝,𝑠)≤
𝑐/𝑡2 3. Key result:
67. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)=𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
68. Condition (5.16)
69. Instead of checking (5.8), we verify (5.16):
70. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
71. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
72. Conditions (5.9) - (5.13)
73. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to homogeneity
of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12): Verified by
rescaling and using properties of 𝜋𝐴1
74. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
75. 𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
76. This ensures that:
77. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
78. Verification of Conditions
79. Condition (5.3)
80. We verify this condition through rescaling:
81. 1. Let 𝑡𝑞 ∈𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
82. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
83. Key result:
84. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
85. Condition (5.5)
86. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is uniformly
bounded below by a positive number for large 𝑡
87. Condition (5.6)
88. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
89. Condition (5.7)
90. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈ 𝐼(𝐴1)} 2. Show that 𝐾max(𝑡𝑝,𝑠)≤
𝑐/𝑡2 3. Key result:
91. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)=𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
92. Condition (5.16)
93. Instead of checking (5.8), we verify (5.16):
94. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
95. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
96. Conditions (5.9) - (5.13)
97. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to homogeneity
of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12): Verified by
rescaling and using properties of 𝜋𝐴1
98. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
99. 𝐴
1,𝑡 = {𝑝 ∈𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
100. This ensures that:
101. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
102. Verification of Conditions
103. Condition (5.3)
104. We verify this condition through rescaling:
105. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
106. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
107. Key result:
108. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
109. Condition (5.5)
110. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
111. Condition (5.6)
112. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
113. Condition (5.7)
114. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
115. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
116. Condition (5.16)
117. Instead of checking (5.8), we verify (5.16):
118. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
119. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
120. Conditions (5.9) - (5.13)
121. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
122. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
123. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
124. This ensures that:
125. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
126. Verification of Conditions
127. Condition (5.3)
128. We verify this condition through rescaling:
129. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
130. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
131. Key result:
132. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
133. Condition (5.5)
134. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
135. Condition (5.6)
136. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
137. Condition (5.7)
138. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
139. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
140. Condition (5.16)
141. Instead of checking (5.8), we verify (5.16):
142. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
143. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
144. Conditions (5.9) - (5.13)
145. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
146. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
147. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
148. This ensures that:
149. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
150. Verification of Conditions
151. Condition (5.3)
152. We verify this condition through rescaling:
153. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
154. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
155. Key result:
156. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
157. Condition (5.5)
158. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
159. Condition (5.6)
160. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
161. Condition (5.7)
162. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
163. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
164. Condition (5.16)
165. Instead of checking (5.8), we verify (5.16):
166. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
167. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
168. Conditions (5.9) - (5.13)
169. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
170. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
171. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
172. This ensures that:
173. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
174. Verification of Conditions
175. Condition (5.3)
176. We verify this condition through rescaling:
177. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
178. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
179. Key result:
180. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
181. Condition (5.5)
182. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
183. Condition (5.6)
184. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
185. Condition (5.7)
186. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
187. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
188. Condition (5.16)
189. Instead of checking (5.8), we verify (5.16):
190. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
191. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
192. Conditions (5.9) - (5.13)
193. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
194. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
195. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
196. This ensures that:
197. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
198. Verification of Conditions
199. Condition (5.3)
200. We verify this condition through rescaling:
201. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
202. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
203. Key result:
204. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
205. Condition (5.5)
206. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
207. Condition (5.6)
208. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
209. Condition (5.7)
210. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
211. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
212. Condition (5.16)
213. Instead of checking (5.8), we verify (5.16):
214. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
215. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
216. Conditions (5.9) - (5.13)
217. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
218. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
219. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
220. This ensures that:
221. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
222. Verification of Conditions
223. Condition (5.3)
224. We verify this condition through rescaling:
225. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
226. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
227. Key result:
228. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
229. Condition (5.5)
230. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
231. Condition (5.6)
232. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
233. Condition (5.7)
234. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
235. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
236. Condition (5.16)
237. Instead of checking (5.8), we verify (5.16):
238. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
239. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
240. Conditions (5.9) - (5.13)
241. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
242. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
243. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
244. This ensures that:
245. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
246. Verification of Conditions
247. Condition (5.3)
248. We verify this condition through rescaling:
249. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
250. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
251. Key result:
252. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
253. Condition (5.5)
254. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
255. Condition (5.6)
256. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
257. Condition (5.7)
258. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
259. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
260. Condition (5.16)
261. Instead of checking (5.8), we verify (5.16):
262. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
263. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
264. Conditions (5.9) - (5.13)
265. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
266. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
267. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
268. This ensures that:
269. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
270. Verification of Conditions
271. Condition (5.3)
272. We verify this condition through rescaling:
273. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
274. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
275. Key result:
276. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
277. Condition (5.5)
278. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
279. Condition (5.6)
280. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
281. Condition (5.7)
282. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
283. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
284. Condition (5.16)
285. Instead of checking (5.8), we verify (5.16):
286. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
287. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
288. Conditions (5.9) - (5.13)
289. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
290. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
291. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
292. This ensures that:
293. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
294. Verification of Conditions
295. Condition (5.3)
296. We verify this condition through rescaling:
297. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
298. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
299. Key result:
300. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
301. Condition (5.5)
302. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
303. Condition (5.6)
304. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
305. Condition (5.7)
306. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
307. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
308. Condition (5.16)
309. Instead of checking (5.8), we verify (5.16):
310. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
311. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
312. Conditions (5.9) - (5.13)
313. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
314. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
315. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
316. This ensures that:
317. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
318. Verification of Conditions
319. Condition (5.3)
320. We verify this condition through rescaling:
321. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
322. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
323. Key result:
324. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
325. Condition (5.5)
326. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
327. Condition (5.6)
328. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
329. Condition (5.7)
330. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
331. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
332. Condition (5.16)
333. Instead of checking (5.8), we verify (5.16):
334. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
335. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
336. Conditions (5.9) - (5.13)
337. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
338. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
339. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
340. This ensures that:
341. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
342. Verification of Conditions
343. Condition (5.3)
344. We verify this condition through rescaling:
345. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
346. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
347. Key result:
348. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
349. Condition (5.5)
350. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
351. Condition (5.6)
352. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
353. Condition (5.7)
354. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
355. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
356. Condition (5.16)
357. Instead of checking (5.8), we verify (5.16):
358. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
359. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
360. Conditions (5.9) - (5.13)
361. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
362. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
363. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
364. This ensures that:
365. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
366. Verification of Conditions
367. Condition (5.3)
368. We verify this condition through rescaling:
369. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
370. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
371. Key result:
372. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
373. Condition (5.5)
374. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
375. Condition (5.6)
376. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
377. Condition (5.7)
378. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
379. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
380. Condition (5.16)
381. Instead of checking (5.8), we verify (5.16):
382. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
383. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
384. Conditions (5.9) - (5.13)
385. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
386. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
387. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
388. This ensures that:
389. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
390. Verification of Conditions
391. Condition (5.3)
392. We verify this condition through rescaling:
393. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
394. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
395. Key result:
396. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
397. Condition (5.5)
398. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
399. Condition (5.6)
400. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
401. Condition (5.7)
402. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
403. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
404. Condition (5.16)
405. Instead of checking (5.8), we verify (5.16):
406. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
407. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
408. Conditions (5.9) - (5.13)
409. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
410. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
411. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
412. This ensures that:
413. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
414. Verification of Conditions
415. Condition (5.3)
416. We verify this condition through rescaling:
417. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
418. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
419. Key result:
420. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
421. Condition (5.5)
422. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
423. Condition (5.6)
424. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
425. Condition (5.7)
426. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
427. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
428. Condition (5.16)
429. Instead of checking (5.8), we verify (5.16):
430. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
431. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
432. Conditions (5.9) - (5.13)
433. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
434. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
435. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
436. This ensures that:
437. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
438. Verification of Conditions
439. Condition (5.3)
440. We verify this condition through rescaling:
441. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
442. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
443. Key result:
444. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
445. Condition (5.5)
446. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
447. Condition (5.6)
448. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
449. Condition (5.7)
450. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
451. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
452. Condition (5.16)
453. Instead of checking (5.8), we verify (5.16):
454. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
455. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
456. Conditions (5.9) - (5.13)
457. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
458. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
459. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
460. This ensures that:
461. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
462. Verification of Conditions
463. Condition (5.3)
464. We verify this condition through rescaling:
465. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
466. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
467. Key result:
468. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
469. Condition (5.5)
470. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
471. Condition (5.6)
472. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
473. Condition (5.7)
474. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
475. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
476. Condition (5.16)
477. Instead of checking (5.8), we verify (5.16):
478. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
479. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
480. Conditions (5.9) - (5.13)
481. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
482. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
483. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
484. This ensures that:
485. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
486. Verification of Conditions
487. Condition (5.3)
488. We verify this condition through rescaling:
489. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
490. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
491. Key result:
492. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
493. Condition (5.5)
494. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
495. Condition (5.6)
496. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
497. Condition (5.7)
498. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
499. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
500. Condition (5.16)
501. Instead of checking (5.8), we verify (5.16):
502. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
503. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
504. Conditions (5.9) - (5.13)
505. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
506. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
507. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
508. This ensures that:
509. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
510. Verification of Conditions
511. Condition (5.3)
512. We verify this condition through rescaling:
513. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
514. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
515. Key result:
516. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
517. Condition (5.5)
518. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
519. Condition (5.6)
520. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
521. Condition (5.7)
522. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
523. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
524. Condition (5.16)
525. Instead of checking (5.8), we verify (5.16):
526. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
527. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
528. Conditions (5.9) - (5.13)
529. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
530. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
531. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
532. This ensures that:
533. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
534. Verification of Conditions
535. Condition (5.3)
536. We verify this condition through rescaling:
537. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
538. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
539. Key result:
540. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
541. Condition (5.5)
542. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
543. Condition (5.6)
544. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
545. Condition (5.7)
546. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
547. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
548. Condition (5.16)
549. Instead of checking (5.8), we verify (5.16):
550. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
551. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
552. Conditions (5.9) - (5.13)
553. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
554. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
555. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
556. This ensures that:
557. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
558. Verification of Conditions
559. Condition (5.3)
560. We verify this condition through rescaling:
561. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
562. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
563. Key result:
564. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
565. Condition (5.5)
566. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
567. Condition (5.6)
568. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
569. Condition (5.7)
570. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
571. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
572. Condition (5.16)
573. Instead of checking (5.8), we verify (5.16):
574. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
575. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
576. Conditions (5.9) - (5.13)
577. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
578. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
579. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
580. This ensures that:
581. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
582. Verification of Conditions
583. Condition (5.3)
584. We verify this condition through rescaling:
585. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
586. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
587. Key result:
588. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
589. Condition (5.5)
590. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
591. Condition (5.6)
592. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
593. Condition (5.7)
594. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
595. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
596. Condition (5.16)
597. Instead of checking (5.8), we verify (5.16):
598. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
599. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
600. Conditions (5.9) - (5.13)
601. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
602. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
603. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
604. This ensures that:
605. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
606. Verification of Conditions
607. Condition (5.3)
608. We verify this condition through rescaling:
609. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
610. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
611. Key result:
612. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
613. Condition (5.5)
614. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
615. Condition (5.6)
616. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
617. Condition (5.7)
618. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
619. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
620. Condition (5.16)
621. Instead of checking (5.8), we verify (5.16):
622. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
623. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
624. Conditions (5.9) - (5.13)
625. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
626. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
627. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
628. This ensures that:
629. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
630. Verification of Conditions
631. Condition (5.3)
632. We verify this condition through rescaling:
633. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
634. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
635. Key result:
636. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
637. Condition (5.5)
638. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
639. Condition (5.6)
640. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
641. Condition (5.7)
642. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
643. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
644. Condition (5.16)
645. Instead of checking (5.8), we verify (5.16):
646. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
647. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
648. Conditions (5.9) - (5.13)
649. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
650. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
651. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
652. This ensures that:
653. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
654. Verification of Conditions
655. Condition (5.3)
656. We verify this condition through rescaling:
657. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
658. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
659. Key result:
660. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
661. Condition (5.5)
662. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
663. Condition (5.6)
664. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
665. Condition (5.7)
666. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
667. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
668. Condition (5.16)
669. Instead of checking (5.8), we verify (5.16):
670. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
671. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
672. Conditions (5.9) - (5.13)
673. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
674. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
675. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
676. This ensures that:
677. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
678. Verification of Conditions
679. Condition (5.3)
680. We verify this condition through rescaling:
681. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
682. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
683. Key result:
684. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
685. Condition (5.5)
686. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
687. Condition (5.6)
688. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
689. Condition (5.7)
690. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
691. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
692. Condition (5.16)
693. Instead of checking (5.8), we verify (5.16):
694. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
695. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
696. Conditions (5.9) - (5.13)
697. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
698. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
699. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
700. This ensures that:
701. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
702. Verification of Conditions
703. Condition (5.3)
704. We verify this condition through rescaling:
705. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
706. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
707. Key result:
708. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
709. Condition (5.5)
710. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
711. Condition (5.6)
712. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
713. Condition (5.7)
714. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
715. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
716. Condition (5.16)
717. Instead of checking (5.8), we verify (5.16):
718. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
719. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
720. Conditions (5.9) - (5.13)
721. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
722. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
723. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
724. This ensures that:
725. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
726. Verification of Conditions
727. Condition (5.3)
728. We verify this condition through rescaling:
729. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
730. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
731. Key result:
732. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
733. Condition (5.5)
734. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
735. Condition (5.6)
736. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
737. Condition (5.7)
738. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
739. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
740. Condition (5.16)
741. Instead of checking (5.8), we verify (5.16):
742. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
743. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
744. Conditions (5.9) - (5.13)
745. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
746. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
747. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
748. This ensures that:
749. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
750. Verification of Conditions
751. Condition (5.3)
752. We verify this condition through rescaling:
753. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
754. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
755. Key result:
756. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
757. Condition (5.5)
758. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
759. Condition (5.6)
760. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
761. Condition (5.7)
762. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
763. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
764. Condition (5.16)
765. Instead of checking (5.8), we verify (5.16):
766. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
767. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
768. Conditions (5.9) - (5.13)
769. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
770. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
771. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
772. This ensures that:
773. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
774. Verification of Conditions
775. Condition (5.3)
776. We verify this condition through rescaling:
777. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
778. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
779. Key result:
780. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
781. Condition (5.5)
782. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
783. Condition (5.6)
784. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
785. Condition (5.7)
786. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
787. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
788. Condition (5.16)
789. Instead of checking (5.8), we verify (5.16):
790. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
791. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
792. Conditions (5.9) - (5.13)
793. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
794. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
795. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
796. This ensures that:
797. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
798. Verification of Conditions
799. Condition (5.3)
800. We verify this condition through rescaling:
801. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
802. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
803. Key result:
804. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
805. Condition (5.5)
806. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
807. Condition (5.6)
808. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
809. Condition (5.7)
810. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
811. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
812. Condition (5.16)
813. Instead of checking (5.8), we verify (5.16):
814. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
815. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
816. Conditions (5.9) - (5.13)
817. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
818. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
819. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
820. This ensures that:
821. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
822. Verification of Conditions
823. Condition (5.3)
824. We verify this condition through rescaling:
825. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
826. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
827. Key result:
828. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
829. Condition (5.5)
830. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
831. Condition (5.6)
832. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
833. Condition (5.7)
834. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
835. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
836. Condition (5.16)
837. Instead of checking (5.8), we verify (5.16):
838. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
839. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
840. Conditions (5.9) - (5.13)
841. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
842. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
843. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
844. This ensures that:
845. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
846. Verification of Conditions
847. Condition (5.3)
848. We verify this condition through rescaling:
849. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
850. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
851. Key result:
852. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
853. Condition (5.5)
854. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
855. Condition (5.6)
856. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
857. Condition (5.7)
858. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
859. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
860. Condition (5.16)
861. Instead of checking (5.8), we verify (5.16):
862. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
863. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
864. Conditions (5.9) - (5.13)
865. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
866. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
867. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
868. This ensures that:
869. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
870. Verification of Conditions
871. Condition (5.3)
872. We verify this condition through rescaling:
873. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
874. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
875. Key result:
876. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
877. Condition (5.5)
878. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
879. Condition (5.6)
880. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
881. Condition (5.7)
882. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
883. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
884. Condition (5.16)
885. Instead of checking (5.8), we verify (5.16):
886. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
887. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
888. Conditions (5.9) - (5.13)
889. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
890. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
891. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
892. This ensures that:
893. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
894. Verification of Conditions
895. Condition (5.3)
896. We verify this condition through rescaling:
897. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
898. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
899. Key result:
900. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
901. Condition (5.5)
902. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
903. Condition (5.6)
904. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
905. Condition (5.7)
906. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
907. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
908. Condition (5.16)
909. Instead of checking (5.8), we verify (5.16):
910. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
911. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
912. Conditions (5.9) - (5.13)
913. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
914. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
915. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
916. This ensures that:
917. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
918. Verification of Conditions
919. Condition (5.3)
920. We verify this condition through rescaling:
921. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
922. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
923. Key result:
924. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
925. Condition (5.5)
926. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
927. Condition (5.6)
928. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
929. Condition (5.7)
930. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
931. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
932. Condition (5.16)
933. Instead of checking (5.8), we verify (5.16):
934. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
935. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
936. Conditions (5.9) - (5.13)
937. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
938. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
939. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
940. This ensures that:
941. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
942. Verification of Conditions
943. Condition (5.3)
944. We verify this condition through rescaling:
945. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
946. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
947. Key result:
948. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
949. Condition (5.5)
950. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
951. Condition (5.6)
952. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
953. Condition (5.7)
954. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
955. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
956. Condition (5.16)
957. Instead of checking (5.8), we verify (5.16):
958. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
959. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
960. Conditions (5.9) - (5.13)
961. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
962. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
963. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
964. This ensures that:
965. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
966. Verification of Conditions
967. Condition (5.3)
968. We verify this condition through rescaling:
969. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
970. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
971. Key result:
972. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
973. Condition (5.5)
974. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
975. Condition (5.6)
976. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
977. Condition (5.7)
978. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
979. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
980. Condition (5.16)
981. Instead of checking (5.8), we verify (5.16):
982. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
983. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
984. Conditions (5.9) - (5.13)
985. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
986. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
987. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
988. This ensures that:
989. 𝐴𝑡,𝑀 ={𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
990. Verification of Conditions
991. Condition (5.3)
992. We verify this condition through rescaling:
993. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
994. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
995. Key result:
996. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
997. Condition (5.5)
998. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
999. Condition (5.6)
1000. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1001. Condition (5.7)
1002. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1003. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1004. Condition (5.16)
1005. Instead of checking (5.8), we verify (5.16):
1006. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1007. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1008. Conditions (5.9) - (5.13)
1009. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1010. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1011. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1012. This ensures that:
1013. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1014. Verification of Conditions
1015. Condition (5.3)
1016. We verify this condition through rescaling:
1017. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1018. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1019. Key result:
1020. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1021. Condition (5.5)
1022. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1023. Condition (5.6)
1024. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1025. Condition (5.7)
1026. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1027. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1028. Condition (5.16)
1029. Instead of checking (5.8), we verify (5.16):
1030. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1031. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1032. Conditions (5.9) - (5.13)
1033. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1034. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1035. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1036. This ensures that:
1037. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1038. Verification of Conditions
1039. Condition (5.3)
1040. We verify this condition through rescaling:
1041. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1042. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1043. Key result:
1044. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1045. Condition (5.5)
1046. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1047. Condition (5.6)
1048. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1049. Condition (5.7)
1050. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1051. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1052. Condition (5.16)
1053. Instead of checking (5.8), we verify (5.16):
1054. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1055. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1056. Conditions (5.9) - (5.13)
1057. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1058. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1059. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1060. This ensures that:
1061. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1062. Verification of Conditions
1063. Condition (5.3)
1064. We verify this condition through rescaling:
1065. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1066. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1067. Key result:
1068. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1069. Condition (5.5)
1070. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1071. Condition (5.6)
1072. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1073. Condition (5.7)
1074. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1075. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1076. Condition (5.16)
1077. Instead of checking (5.8), we verify (5.16):
1078. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1079. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1080. Conditions (5.9) - (5.13)
1081. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1082. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1083. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1084. This ensures that:
1085. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1086. Verification of Conditions
1087. Condition (5.3)
1088. We verify this condition through rescaling:
1089. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1090. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1091. Key result:
1092. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1093. Condition (5.5)
1094. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1095. Condition (5.6)
1096. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1097. Condition (5.7)
1098. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1099. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1100. Condition (5.16)
1101. Instead of checking (5.8), we verify (5.16):
1102. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1103. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1104. Conditions (5.9) - (5.13)
1105. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1106. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1107. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1108. This ensures that:
1109. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1110. Verification of Conditions
1111. Condition (5.3)
1112. We verify this condition through rescaling:
1113. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1114. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1115. Key result:
1116. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1117. Condition (5.5)
1118. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1119. Condition (5.6)
1120. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1121. Condition (5.7)
1122. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1123. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1124. Condition (5.16)
1125. Instead of checking (5.8), we verify (5.16):
1126. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1127. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1128. Conditions (5.9) - (5.13)
1129. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1130. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1131. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1132. This ensures that:
1133. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1134. Verification of Conditions
1135. Condition (5.3)
1136. We verify this condition through rescaling:
1137. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1138. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1139. Key result:
1140. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1141. Condition (5.5)
1142. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1143. Condition (5.6)
1144. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1145. Condition (5.7)
1146. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1147. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1148. Condition (5.16)
1149. Instead of checking (5.8), we verify (5.16):
1150. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1151. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1152. Conditions (5.9) - (5.13)
1153. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1154. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1155. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1156. This ensures that:
1157. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1158. Verification of Conditions
1159. Condition (5.3)
1160. We verify this condition through rescaling:
1161. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1162. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1163. Key result:
1164. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1165. Condition (5.5)
1166. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1167. Condition (5.6)
1168. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1169. Condition (5.7)
1170. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1171. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1172. Condition (5.16)
1173. Instead of checking (5.8), we verify (5.16):
1174. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1175. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1176. Conditions (5.9) - (5.13)
1177. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1178. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1179. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1180. This ensures that:
1181. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1182. Verification of Conditions
1183. Condition (5.3)
1184. We verify this condition through rescaling:
1185. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1186. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1187. Key result:
1188. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1189. Condition (5.5)
1190. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1191. Condition (5.6)
1192. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1193. Condition (5.7)
1194. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1195. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1196. Condition (5.16)
1197. Instead of checking (5.8), we verify (5.16):
1198. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1199. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1200. Conditions (5.9) - (5.13)
1201. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1202. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1203. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1204. This ensures that:
1205. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1206. Verification of Conditions
1207. Condition (5.3)
1208. We verify this condition through rescaling:
1209. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1210. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1211. Key result:
1212. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1213. Condition (5.5)
1214. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1215. Condition (5.6)
1216. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1217. Condition (5.7)
1218. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1219. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1220. Condition (5.16)
1221. Instead of checking (5.8), we verify (5.16):
1222. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1223. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1224. Conditions (5.9) - (5.13)
1225. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1226. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1227. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1228. This ensures that:
1229. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1230. Verification of Conditions
1231. Condition (5.3)
1232. We verify this condition through rescaling:
1233. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1234. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1235. Key result:
1236. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1237. Condition (5.5)
1238. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1239. Condition (5.6)
1240. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1241. Condition (5.7)
1242. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1243. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1244. Condition (5.16)
1245. Instead of checking (5.8), we verify (5.16):
1246. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1247. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1248. Conditions (5.9) - (5.13)
1249. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1250. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1251. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1252. This ensures that:
1253. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1254. Verification of Conditions
1255. Condition (5.3)
1256. We verify this condition through rescaling:
1257. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1258. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1259. Key result:
1260. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1261. Condition (5.5)
1262. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1263. Condition (5.6)
1264. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1265. Condition (5.7)
1266. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1267. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1268. Condition (5.16)
1269. Instead of checking (5.8), we verify (5.16):
1270. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1271. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1272. Conditions (5.9) - (5.13)
1273. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1274. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1275. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1276. This ensures that:
1277. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1278. Verification of Conditions
1279. Condition (5.3)
1280. We verify this condition through rescaling:
1281. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1282. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1283. Key result:
1284. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1285. Condition (5.5)
1286. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1287. Condition (5.6)
1288. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1289. Condition (5.7)
1290. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1291. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1292. Condition (5.16)
1293. Instead of checking (5.8), we verify (5.16):
1294. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1295. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1296. Conditions (5.9) - (5.13)
1297. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1298. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1299. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1300. This ensures that:
1301. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1302. Verification of Conditions
1303. Condition (5.3)
1304. We verify this condition through rescaling:
1305. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1306. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1307. Key result:
1308. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1309. Condition (5.5)
1310. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1311. Condition (5.6)
1312. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1313. Condition (5.7)
1314. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1315. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1316. Condition (5.16)
1317. Instead of checking (5.8), we verify (5.16):
1318. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1319. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1320. Conditions (5.9) - (5.13)
1321. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1322. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1323. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1324. This ensures that:
1325. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1326. Verification of Conditions
1327. Condition (5.3)
1328. We verify this condition through rescaling:
1329. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1330. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1331. Key result:
1332. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1333. Condition (5.5)
1334. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1335. Condition (5.6)
1336. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1337. Condition (5.7)
1338. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1339. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1340. Condition (5.16)
1341. Instead of checking (5.8), we verify (5.16):
1342. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1343. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1344. Conditions (5.9) - (5.13)
1345. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1346. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1347. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1348. This ensures that:
1349. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1350. Verification of Conditions
1351. Condition (5.3)
1352. We verify this condition through rescaling:
1353. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1354. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1355. Key result:
1356. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1357. Condition (5.5)
1358. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1359. Condition (5.6)
1360. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1361. Condition (5.7)
1362. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1363. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1364. Condition (5.16)
1365. Instead of checking (5.8), we verify (5.16):
1366. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1367. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1368. Conditions (5.9) - (5.13)
1369. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1370. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1371. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1372. This ensures that:
1373. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1374. Verification of Conditions
1375. Condition (5.3)
1376. We verify this condition through rescaling:
1377. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1378. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1379. Key result:
1380. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1381. Condition (5.5)
1382. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1383. Condition (5.6)
1384. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1385. Condition (5.7)
1386. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1387. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1388. Condition (5.16)
1389. Instead of checking (5.8), we verify (5.16):
1390. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1391. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1392. Conditions (5.9) - (5.13)
1393. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1394. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1395. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1396. This ensures that:
1397. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1398. Verification of Conditions
1399. Condition (5.3)
1400. We verify this condition through rescaling:
1401. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1402. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1403. Key result:
1404. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1405. Condition (5.5)
1406. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1407. Condition (5.6)
1408. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1409. Condition (5.7)
1410. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1411. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1412. Condition (5.16)
1413. Instead of checking (5.8), we verify (5.16):
1414. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1415. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1416. Conditions (5.9) - (5.13)
1417. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1418. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1419. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1420. This ensures that:
1421. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1422. Verification of Conditions
1423. Condition (5.3)
1424. We verify this condition through rescaling:
1425. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1426. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1427. Key result:
1428. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1429. Condition (5.5)
1430. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1431. Condition (5.6)
1432. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1433. Condition (5.7)
1434. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1435. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1436. Condition (5.16)
1437. Instead of checking (5.8), we verify (5.16):
1438. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1439. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1440. Conditions (5.9) - (5.13)
1441. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1442. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1443. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1444. This ensures that:
1445. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1446. Verification of Conditions
1447. Condition (5.3)
1448. We verify this condition through rescaling:
1449. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1450. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1451. Key result:
1452. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1453. Condition (5.5)
1454. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1455. Condition (5.6)
1456. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1457. Condition (5.7)
1458. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1459. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1460. Condition (5.16)
1461. Instead of checking (5.8), we verify (5.16):
1462. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1463. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1464. Conditions (5.9) - (5.13)
1465. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1466. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1467. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1468. This ensures that:
1469. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1470. Verification of Conditions
1471. Condition (5.3)
1472. We verify this condition through rescaling:
1473. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1474. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1475. Key result:
1476. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1477. Condition (5.5)
1478. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1479. Condition (5.6)
1480. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1481. Condition (5.7)
1482. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1483. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1484. Condition (5.16)
1485. Instead of checking (5.8), we verify (5.16):
1486. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1487. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1488. Conditions (5.9) - (5.13)
1489. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1490. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1491. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1492. This ensures that:
1493. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1494. Verification of Conditions
1495. Condition (5.3)
1496. We verify this condition through rescaling:
1497. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1498. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1499. Key result:
1500. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1501. Condition (5.5)
1502. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1503. Condition (5.6)
1504. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1505. Condition (5.7)
1506. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1507. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1508. Condition (5.16)
1509. Instead of checking (5.8), we verify (5.16):
1510. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1511. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1512. Conditions (5.9) - (5.13)
1513. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1514. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1515. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1516. This ensures that:
1517. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1518. Verification of Conditions
1519. Condition (5.3)
1520. We verify this condition through rescaling:
1521. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1522. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1523. Key result:
1524. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1525. Condition (5.5)
1526. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1527. Condition (5.6)
1528. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1529. Condition (5.7)
1530. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1531. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1532. Condition (5.16)
1533. Instead of checking (5.8), we verify (5.16):
1534. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1535. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1536. Conditions (5.9) - (5.13)
1537. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
1538. We begin by describing 𝐴𝑡,𝑀 through its rescaled version 𝐴
1,𝑡:
1539. 𝐴
1,𝑡 = {𝑝 ∈ 𝐼(𝐴1):𝜏𝐴1(𝑝)≤ 𝑐(𝑡)/𝑡𝛼}
1540. This ensures that:
1541. 𝐴𝑡,𝑀 = {𝑝 ∈ 𝐼(𝐴𝑡):𝜏𝐴𝑡(𝑝)≤ 𝑐(𝑡)} = 𝑡𝐴
1,𝑡
1542. Verification of Conditions
1543. Condition (5.3)
1544. We verify this condition through rescaling:
1545. 1. Let 𝑡𝑞 ∈ 𝐴𝑡,𝑀 and 𝑡𝑝 =𝜋𝐴𝑡(𝑡𝑞) 2. 𝑝 ∈ 𝐷𝐴1 3. Using Lemma 7.2:
1546. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)= 𝑡𝛼(𝜏𝐴1(𝑞)−𝜏𝐴1(𝑝))
1547. Key result:
1548. 𝜏𝐴𝑡(𝑡𝑞)−𝜏𝐴𝑡(𝑡𝑝)=𝑡𝛼
2|𝐷𝐼(𝑝)|⟨𝐺𝐴1(𝑝)exp𝑝
−1(𝑞),exp𝑝
−1(𝑞)⟩(1+𝑜(1))
1549. Condition (5.5)
1550. 1. Use Lemma 7.1.2: 𝜒𝐴𝑡
𝐹(𝑡𝑝)= 𝑡𝛼𝜒𝐴1
𝐹(𝑝) 2. Prove that inf𝑝∈𝐴
1,𝑡𝜒𝐴1
𝐹(𝑝) is
uniformly bounded below by a positive number for large 𝑡
1551. Condition (5.6)
1552. Our choice of 𝑐𝐴𝑡,𝑀 = 𝑐(𝑡) ensures this condition holds.
1553. Condition (5.7)
1554. 1. Define 𝑐 = sup{𝜆max(𝐷2𝐼(𝑞))/2|𝐷𝐼(𝑞)|2:𝑞 ∈𝐼(𝐴1)} 2. Show that
𝐾max(𝑡𝑝,𝑠)≤ 𝑐/𝑡2 3. Key result:
1555. sup
𝑡𝑝∈𝐴𝑡,𝑀√𝐾max (𝑡𝑝,𝑡0,𝑀(𝑡𝑝))𝑡0,𝑀(𝑡𝑝)= 𝑡−𝛼/2√𝑐(𝑡)𝑂(1)= 𝑜(1)
1556. Condition (5.16)
1557. Instead of checking (5.8), we verify (5.16):
1558. 𝐴𝜋𝐴𝑡
−1(𝑡𝑝)(𝑠,𝑣)=𝑠Id𝑇𝑡𝑝𝜋𝐴𝑡
−1(𝑡𝑝)+𝑜(1)
1559. as 𝑡 → ∞, uniformly for 𝑝 ∈ 𝐷𝐴1.
1560. Conditions (5.9) - (5.13)
1561. 1. (5.9): Easy to check using homogeneity of 𝐼 2. (5.10): Trivial due to
homogeneity of 𝐼 3. (5.11) and (5.13): Hold because 𝜏𝐴1(⋅) vanishes on 𝐷𝐴1 4. (5.12):
Verified by rescaling and using properties of 𝜋𝐴1
Conclusion
We have verified all necessary conditions for homothetic sets and the application of
Laplace’s method in this context.
