Fourier Restriction, Decoupling, and Applications: (Series Number 184 Cambridge Studies in Advanced Mathematics)

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.

Table of Contents:
Background and notation; 1. Linear restriction theory; 2. Wave packets; 3. Bilinear restriction theory; 4. Parabolic rescaling and a bilinear-to-linear reduction; 5. Kakeya and square function estimates; 6. Multilinear Kakeya and restriction inequalities; 7. The Bourgain–Guth method; 8. The polynomial method; 9. An introduction to decoupling; 10. Decoupling for the elliptic paraboloid; 11. Decoupling for the moment curve; 12. Decouplings for other manifolds; 13. Applications of decoupling; References; Index.

About the Author :
Ciprian Demeter is Professor of Mathematics at Indiana University, Bloomington. He is one of the world's leading experts in Fourier restriction theory and its applications to number theory, which he teaches regularly at the graduate level. He received the Sloan fellowship in 2009 and was an invited speaker at the 2018 International Congress of Mathematicians in Rio de Janeiro.

Review :
'The topic of decoupling is now a major active area of research in both harmonic analysis and analytic number theory. There are a number of survey articles and lecture notes already on these topics, but this book - by one of the leading contributors to the field - is more comprehensive than any of these, being almost completely self-contained and detailing a number of older results as well as the more recent ones. It also has a number of exercises and insightful commentary. This text is an excellent resource, both for students and for existing researchers in the field.' Terence Tao, University of California, Los Angeles 'This book gives a self-contained introduction to some major recent developments in Fourier analysis. The restriction conjecture, raised by Stein in the 1960s, is still open and looks very difficult, but there has been fundamental progress in the area over the last decade, leading to striking applications in PDE and analytic number theory. This book is written by one of the main players in those developments. Demeter has worked hard to present the key points clearly and to minimize the technical issues involved by starting with simple cases of each new idea and pausing to give heuristics and examples.' Larry Guth, Massachuetts Institute of Technology 'Restriction theory, which has long occupied a central place in Euclidean harmonic analysis, has gained new urgency and impetus with the development of decoupling. Ciprian Demeter is one of the main contributors to this area as well as an excellent expositor of it. New researchers have already been clamouring for instructional materials on the subject. Experienced harmonic analysts will still want to read it for new insights from one of the main players in the field. Starting from classical results re-interpreted in light of the current understanding of the subject, the book then proceeds to the more recent topics culminating in the Bourgain-Demeter-Guth proof of Vinogradov's conjecture. I have been awaiting this book eagerly, and it did not disappoint. I expect it to be a constant presence on my desk, from graduate teaching to my own research, for many years to come.' Izabella Laba, University of British Columbia 'This book deals with the spectacular recent developments in modern Fourier analysis, with an emphasis on restriction theory and decoupling. Some of the results are new and many are just a few years old, notably the breakthrough theorems of Bourgain and Demeter on decoupling and their many applications. It is wonderful that this material is available in book form so soon, especially as the author has succeeded admirably in his goal of bringing forth the central ideas without obscuring them with too many technical details. Thus the presentation is accessible to non-experts and the book will be valuable for a wide readership, including graduate students.' Pertti Mattila, University of Helsinki 'This book … is well explained with insightful comments and many examples and exercises to further explain and extend the material.' Andreas Nilsson, Mathematical Reviews Clippings 'Undoubtedly, this book will become an indispensable reference in the subject.' Pablo Alejandro Rocha, European Mathematical Society