Fourier Restriction, Decoupling, and Applications Cambridge Studies in Advanced Mathematics Series

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.

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.

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.