Fourier Restriction, Decoupling, and Applications

Description

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. Cover......Page 1 Front Matter ......Page 2 CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS......Page 3 Fourier Restriction, Decoupling, and Applications......Page 5 Copyright ......Page 6 Contents ......Page 7 Preface......Page 11 Acknowledgments......Page 15 Background and Notation......Page 17 1 Linear Restriction Theory......Page 19 2 Wave Packets......Page 44 3 Bilinear Restriction Theory......Page 53 4 Parabolic Rescaling and a Bilinear-to-Linear Reduction......Page 97 5 Kakeya and Square Function Estimates......Page 104 6 Multilinear Kakeya and Restriction Inequalities......Page 123 7 The Bourgain–Guth Method......Page 148 8 The Polynomial Method......Page 171 9 An Introduction to Decoupling......Page 201 10 Decoupling for the Elliptic Paraboloid......Page 215 11 Decoupling for the Moment Curve......Page 253 12 Decouplings for Other Manifolds......Page 282 13 Applications of Decoupling......Page 302 References ......Page 340 Index......Page 347