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A study guide for the $\ell^2$ decoupling theorem for the paraboloid

Ataleshvara Bhargava, Tiklung Chan, Zi Li Lim, Yixuan Pang

Abstract

This article serves as a study guide for the $\ell^2$ decoupling theorem for the paraboloid originally proved by Bourgain and Demeter. Given its popularity and importance, many expositions about the $\ell^2$ decoupling theorem already exist. Our study guide is intended to complement and combine these existing resources in order to provide a more gentle introduction to the subject.

A study guide for the $\ell^2$ decoupling theorem for the paraboloid

Abstract

This article serves as a study guide for the decoupling theorem for the paraboloid originally proved by Bourgain and Demeter. Given its popularity and importance, many expositions about the decoupling theorem already exist. Our study guide is intended to complement and combine these existing resources in order to provide a more gentle introduction to the subject.
Paper Structure (35 sections, 44 theorems, 250 equations, 3 figures)

This paper contains 35 sections, 44 theorems, 250 equations, 3 figures.

Table of Contents

  1. Introduction
  2. What is decoupling?
  3. Main theorem and examples
  4. Notation
  5. Decoupling Properties
  6. Inductive structure
  7. General decoupling results
  8. Local and weighted versions of decoupling
  9. Interpolation
  10. Parabolic rescaling
  11. Multilinear decoupling
  12. The Two-dimensional Proof
  13. Overview
  14. Bilinear-to-linear reduction
  15. Bilinear decoupling estimates
  16. ...and 20 more sections

Key Result

Theorem 1.2.1

For all $\varepsilon>0$ the following results hold:

Figures (3)

  • Figure 1: Subcritical case
  • Figure 2: Lower dimensional decoupling
  • Figure 3: Transverse wave packets at scales $R$ and $R^{1/2}$

Theorems & Definitions (114)

  • Definition 1.1.1
  • Remark 1.1.1
  • Remark 1.1.2
  • Example 1.1.1: Necessity of $p\geq 2$; Exercise 9.8 in demeter_fourier_2020
  • Example 1.1.2: Littlewood-Paley decomposition
  • Definition 1.1.2: Decoupling constant
  • Remark 1.1.3
  • Theorem 1.2.1: bourgain_proof_2015
  • Remark 1.2.1
  • Example 1.2.1
  • ...and 104 more