Fourier Decoupling Theory

A Concise Course
Erscheinungsjahr 2026
ISBN: 978-3-032-38326-6
Verlag: Springer

Buch, Englisch, 141 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: Graduate Texts in Mathematics

Originating in the study of wave equations, Fourier decoupling theory is connected to partial differential equations, geometric measure theory, analytic number theory, combinatorics, and dynamical systems. It is a rapidly growing area of harmonic analysis, and its major applications include Bourgain and Demeter’s resolution of the Discrete Restriction Conjecture and subsequent breakthroughs on Vinogradov’s mean value theorem and the Lindelöf Hypothesis.

This book offers a concise introduction to decoupling theory. To keep the presentation accessible, it focuses on decoupling inequalities for two-dimensional parabolas and three-dimensional cones, whose proofs illustrate the central ideas behind modern decoupling methods. It explores applications of decoupling theory to analytic number theory and geometric measure theory, as well as its connections to the Kakeya and Nikodym problems, Fourier restriction theory, the Bochner–Riesz problem, local smoothing estimates, and square function estimates, emphasizing the simplest and most representative settings. The only prerequisite is an undergraduate course in Fourier analysis, making the book suitable for graduate students and advanced undergraduates.

Zielgruppe

Graduate

Autoren/Hrsg.

Guo, Shaoming

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

Chapter 1. Preliminaries in Fourier analysis.- Chapter 2. Decouplings for the parabola.- Chapter 3. Connections to problems in analytic number theory.- Chapter 4. Besicovitch–Kakeya sets and Nikodym sets.- Chapter 5. Decouplings for cones in dimension three.- Chapter 6. Bourgain's circular maximal function.- Chapter 7. Wolff's circular maximal function.- Chapter 8. The Kakeya problem in dimension two.- Chapter 9. The Nikodym problem in dimension two.- Chapter 10. The Bochner-Riesz problem in dimension two.- Chapter 11. Local smoothing estimates in dimension three.- Chapter 12. Square function estimates in dimension three.

Über Autor(innen)

Shaoming Guo is currently a professor at Chern Institute of Mathematics, Nankai University, working in harmonic analysis, in particular, in Fourier decoupling theory, Fourier restriction theory, and related areas, including analytic number theory and geometric measure theory.