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1 | (28) |
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1.1 Newton Polyhedra Associated with φ, Adapted Coordinates, and Uniform Estimates for Oscillatory Integrals with Phase φ |
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5 | (5) |
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1.2 Fourier Restriction in the Presence of a Linear Coordinate System That Is Adapted to φ |
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10 | (1) |
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1.3 Fourier Restriction When No Linear Coordinate System Is Adapted to φ---the Analytic Case |
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11 | (6) |
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1.4 Smooth Hypersurfaces of Finite Type, Condition (R), and the General Restriction Theorem |
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17 | (6) |
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1.5 An Invariant Description of the Notion of r-Height |
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23 | (1) |
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1.6 Organization of the Monograph and Strategy of Proof |
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24 | (5) |
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Chapter 2 Auxiliary Results |
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29 | (21) |
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2.1 Van der Corput---Type Estimates |
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30 | (1) |
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31 | (3) |
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2.3 Integral Estimates of van der Corput Type |
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34 | (4) |
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2.4 Fourier Restriction via Real Interpolation |
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38 | (2) |
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2.5 Uniform Estimates for Families of Oscillatory Sums |
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40 | (6) |
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2.6 Normal Forms of φ under Linear Coordinate Changes When hlin (φ) < 2 |
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46 | (4) |
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Chapter 3 Reduction to Restriction Estimates near the Principal Root Jet |
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50 | (7) |
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Chapter 4 Restriction for Surfaces with Linear Height below 2 |
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57 | (18) |
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4.1 Preliminary Reductions by Means of Littlewood-Paley Decompositions |
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58 | (6) |
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4.2 Restriction Estimates for Normalized Rescaled Measures When 22j δ3 < 1 |
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64 | (11) |
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Chapter 5 Improved Estimates by Means of Airy-Type Analysis |
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75 | (30) |
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5.1 Airy-Type Decompositions Required for Proposition 4.2(c) |
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76 | (9) |
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5.2 The Endpoint in Proposition 4.2(c): Complex Interpolation |
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85 | (11) |
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5.3 Proof of Proposition 4.2(a), (b): Complex Interpolation |
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96 | (9) |
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Chapter 6 The Case When hlin (φ) ≥ 2: Preparatory Results |
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105 | (26) |
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6.1 The First Domain Decomposition |
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107 | (2) |
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6.2 Restriction Estimates in the Transition Domains El When hlin (φ) ≥ 2 |
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109 | (6) |
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6.3 Restriction Estimates in the Domains Dl, l < lpr, When hlin (φ) ≥ 2 |
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115 | (8) |
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6.4 Restriction Estimates in the Domain Dpr When hlin (φ) ≥ 5 |
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123 | (2) |
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6.5 Refined Domain Decomposition of Dpr: The Stopping-Time Algorithm |
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125 | (6) |
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Chapter 7 How to Go beyond the Case hlin (φ) ≥ 5 |
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131 | (50) |
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7.1 The Case When hlin (φ) ≥ 2: Reminder of the Open Cases |
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132 | (3) |
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7.2 Restriction Estimates for the Domains D'y(1) Reduction to Normalized Measures vδ |
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135 | (3) |
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7.3 Removal of the Term y2 B--1 bB--1 (y1) in (7.7) |
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138 | (3) |
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7.4 Lower Bounds for hr (φ) |
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141 | (2) |
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7.5 Spectral Localization to Frequency Boxes Where |ξ| ~λi The Case Where Not All λi s Are Comparable |
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143 | (8) |
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7.6 Interpolation Arguments for the Open Cases Where m = 2 and B = 2 or B = 3 |
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151 | (18) |
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7.7 The case where λ1 ~λ2 ~λ3 |
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169 | (9) |
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178 | (1) |
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7.9 Collecting the Remaining Cases |
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178 | (1) |
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7.10 Restriction Estimates for the Domains D'(l), 1 ≥2 |
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179 | (2) |
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Chapter 8 The Remaining Cases Where m = 2 and B = 3 or B = 4 |
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181 | (63) |
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183 | (2) |
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8.2 Refined Airy-Type Analysis |
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185 | (4) |
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8.3 The Case Where λρ(δ) < 1 |
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189 | (8) |
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8.4 The Case Where m = 2, B = 4, and A = 1 |
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197 | (7) |
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8.5 The Case Where m = 2, B = 4, and A = 0 |
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204 | (8) |
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8.6 The Case Where m= 2, B= 3, and A= 0: What Still Needs to Be Done |
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212 | (5) |
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8.7 Proof of Proposition 8.12(a): Complex Interpolation |
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217 | (16) |
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8.8 Proof of Proposition 8.12(b): Complex Interpolation |
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233 | (11) |
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Chapter 9 Proofs of Propositions 1.7 and 1.17 |
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244 | (7) |
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9.1 Appendix A: Proof of Proposition 1.7 on the Characterization of Linearly Adapted Coordinates |
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244 | (1) |
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9.2 Appendix B: A Direct Proof of Proposition 1.17 on an Invariant Description of the Notion of r-Height |
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245 | (6) |
| Bibliography |
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251 | (6) |
| Index |
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257 | |
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