Fourier Restriction, Decoupling, and Applications

Comprehensive coverage of recent, exciting developments in Fourier restriction theory, including applications to number theory and PDEs.

Ciprian Demeter (Author)

9781108499705, Cambridge University Press

Hardback, published 2 January 2020

345 pages, 22 b/w illus. 115 exercises
23.4 x 15.7 x 2.2 cm, 0.6 kg

'Undoubtedly, this book will become an indispensable reference in the subject.' Pablo Alejandro Rocha, European Mathematical Society

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.

Subject Areas: Combinatorics & graph theory [PBV], Numerical analysis [PBKS], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Number theory [PBH]