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A study guide for the $l^2$ Decoupling Theorem

Jean Bourgain, Ciprian Demeter

Abstract

This paper contains a detailed, self contained and more streamlined proof of our $l^2$ decoupling theorem for hypersurfaces.

A study guide for the $l^2$ Decoupling Theorem

Abstract

This paper contains a detailed, self contained and more streamlined proof of our decoupling theorem for hypersurfaces.

Paper Structure

This paper contains 11 sections, 16 theorems, 229 equations.

Table of Contents

  1. The theorem
  2. More notation
  3. A brief description of the argument
  4. A few useful inequalities
  5. An equivalent formulation
  6. $L^2$ decoupling
  7. Parabolic rescaling
  8. Linear versus multilinear decoupling
  9. From multilinear Kakeya to multilinear decouplings
  10. The iteration scheme
  11. The final argument

Key Result

Theorem 1.1

We have the following sharp (up to $\delta^{-\epsilon}$ losses) upper bound for ${\operatorname{Dec}}_n(\delta,p)$ if $2\le p\le \frac{2(n+1)}{n-1}.$ The implicit constant depends on $\epsilon,p,n$ but not on $\delta$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 4.1
  • Remark 4.2
  • Corollary 4.3
  • Remark 4.4
  • Theorem 5.1
  • Proposition 6.1: $L^2$ decoupling
  • Proposition 7.1
  • Definition 8.1
  • Theorem 8.2
  • ...and 11 more