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The proof of the $l^2$ Decoupling Conjecture

Jean Bourgain, Ciprian Demeter

Abstract

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

The proof of the $l^2$ Decoupling Conjecture

Abstract

We prove the Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to losses) implies the full range of expected Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

Paper Structure

This paper contains 14 sections, 27 theorems, 215 equations.

Table of Contents

  1. The $l^2$ Decoupling Theorem
  2. First applications
  3. The discrete restriction phenomenon
  4. Strichartz estimates for the classical and irrational tori
  5. The discrete restriction for lattice points on the sphere
  6. Additive energies and Incidence Geometry
  7. Additive energies of annular sets
  8. Counting solutions of Diophantine inequalities
  9. Norms and wave packet decompositions
  10. Parabolic rescaling
  11. Linear versus multilinear decoupling
  12. Proof of Theorem for the paraboloid
  13. Extension to other hypersurfaces
  14. Proof of Theorem

Key Result

Theorem 1.1

Let $S$ be a compact $C^2$ hypersurface in ${\mathbb R}^n$ with positive definite second fundamental form. If ${\operatorname{supp}}(\hat{f})\subset {\mathcal{N}}_\delta$ then for $p\ge\frac{2(n+1)}{n-1}$ and $\epsilon>0$

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: Discrete restriction: the lattice case (paraboloid)
  • Theorem 2.4: Strichartz estimates for irrational tori
  • Remark 2.5
  • Conjecture 2.6
  • Theorem 2.7
  • Theorem 2.8
  • ...and 25 more