The last fifteen years have seen a flurry of exciting developments in Fourier restriction theory, leading to significant new applications in diverse fields. This timely text brings the reader from the classical results to state-of-the-art advances in multilinear restriction theory, the Bourgain–Guth induction on scales and the polynomial method. Also discussed in the second part are decoupling for curved manifolds and a wide variety of applications in geometric analysis, PDEs (Strichartz estimates on tori, local smoothing for the wave equation) and number theory (exponential sum estimates and the proof of the Main Conjecture for Vinogradov's Mean Value Theorem). More than 100 exercises in the text help reinforce these important but often difficult ideas, making it suitable for graduate students as well as specialists. Written by an author at the forefront of the modern theory, this book will be of interest to everybody working in harmonic analysis.
Author(s): Ciprian Demeter
Series: Cambridge Studies in Advanced Mathematics (184)
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 349
Contents......Page 6
1 Linear Restriction Theory......Page 18
1.1 The Restriction Problem for Manifolds......Page 20
1.2 The Restriction Conjecture for the Sphere and the Paraboloid......Page 23
1.3 Proof of the Restriction Conjecture for Curves in R2......Page 27
1.4 The Stein–Tomas Argument......Page 29
1.5 Constructive Interference......Page 35
1.6 Local and Discrete Restriction Estimates......Page 36
1.7 Square Root Cancellation and the Role of Curvature......Page 40
2 Wave Packets......Page 43
3 Bilinear Restriction Theory......Page 52
3.1 A Case Study......Page 53
3.2 Biorthogonality: The C´ ordoba–Fefferman Argument......Page 54
3.3 Bilinear Interaction of Transverse Wave Packets......Page 58
3.4 Proof of Theorem 3.1 When n = 2......Page 60
3.5 A General Bilinear Restriction Estimate in L4(R3)......Page 61
3.6 From Point-Line to Cube-Tube Incidences......Page 68
3.7 A Bilinear Incidence Result for Lines......Page 72
3.8 Achieving Diagonal Behavior for Tubes......Page 76
3.9 Induction on Scales and the Proof of Theorem 3.1......Page 80
3.10 Weighted Wave Packets: Another Proof of Theorem 3.1......Page 85
4 Parabolic Rescaling and a Bilinear-to-Linear Reduction......Page 96
5.1 A Few Kakeya-Type Conjectures......Page 103
5.2 Square Function Estimates......Page 112
5.3 Square Functions and the Restriction Conjecture......Page 118
6 Multilinear Kakeya and Restriction Inequalities......Page 122
6.1 The Brascamp–Lieb Inequality......Page 124
6.2 Plates and Joints......Page 129
6.3 The Multilinear Kakeya Inequality......Page 131
6.4 The Multilinear Restriction Theorem......Page 136
7 The Bourgain–Guth Method......Page 147
7.1 From Bilinear to Linear for Hypersurfaces......Page 148
7.2 From Multilinear to Linear for the Moment Curve......Page 151
7.3 From Multilinear to Linear for Hypersurfaces......Page 156
8 The Polynomial Method......Page 170
8.1 Polynomial Partitioning......Page 171
8.2 A Discrete Application of Polynomial Partitioning......Page 174
8.3 A New Linear Restriction Estimate......Page 177
8.4 The Induction Hypothesis......Page 180
8.5 Cellular Contribution......Page 184
8.6 Two Types of Wall Contribution......Page 187
8.7 The Transverse Contribution......Page 190
8.8 Tubes Tangent to a Variety......Page 192
8.9 Controlling the Tangent Contribution......Page 195
8.10 Putting Things Together......Page 197
9.1 The General Framework......Page 200
9.2 Local and Global Decoupling......Page 206
9.3 A Few Basic Tools......Page 211
10 Decoupling for the Elliptic Paraboloid......Page 214
10.1 Almost Extremizers......Page 216
10.2 A Detailed Proof of the Decoupling for the Parabola......Page 220
10.3 Decoupling for Pn−1......Page 233
10.4 Another Look at the Decoupling for the Parabola......Page 237
11 Decoupling for the Moment Curve......Page 252
11.1 Decoupling for the Twisted Cubic......Page 253
11.2 Rescaling the Neighborhoods Ŵ(δ)......Page 255
Contents vii 11.3 The Trilinear-to-Linear Reduction......Page 256
11.4 A Multiscale Inequality......Page 258
11.5 Iteration and the Proof of Theorem 11.1......Page 260
11.6 Two Trilinear Kakeya Inequalities......Page 263
11.7 Ball Inflations......Page 264
11.9 Proof of the Multiscale Inequality......Page 271
11.10 Decoupling for the Higher-Dimensional Moment Curve......Page 278
12.1 Hypersurfaces with Positive Principal Curvatures......Page 281
12.2 The Cone......Page 283
12.3 lp Decouplings: The Case of Nonzero Gaussian Curvature......Page 287
12.4 A Refined lp Decoupling for the Parabola......Page 291
12.5 Arbitrary Curves with Nonzero Torsion......Page 295
12.6 Real Analytic Curves......Page 297
13.1 Decoupling and Exponential Sums......Page 301
13.2 A Number Theoretic Approach to Lower Bounds......Page 303
13.3 Strichartz Estimates for the Schr¨ odinger Equation on Tori......Page 308
13.4 Eigenfunction Estimates for the Laplacian on Tori......Page 311
13.5 Vinogradov’s Mean Value Theorem for Curves with Torsion......Page 313
13.6 Exponential Sums for Curves with Torsion......Page 317
13.7 Additive Energies......Page 320
13.8 lth Powers in Arithmetic Progressions......Page 324
13.9 Decoupling and the Restriction Conjecture......Page 327
13.10 Local Smoothing for the Wave Equation......Page 329
13.11 Problems for This Chapter......Page 334
References......Page 339
Index......Page 346
