Fourier Decoupling Theory: A Concise Course - Shaoming Guo

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.