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Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

Jean Bourgain, Ciprian Demeter

Abstract

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to $ε$ losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus. Our second main result is an $l^2$ decoupling for non degenerate curves which has implications for Vinogradov's mean value theorem.

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

Abstract

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from \cite{BD3}. As a consequence of this we obtain sharp (up to losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus. Our second main result is an decoupling for non degenerate curves which has implications for Vinogradov's mean value theorem.

Paper Structure

This paper contains 8 sections, 22 theorems, 202 equations.

Table of Contents

  1. Statements of results
  2. $l^p$ decouplings for hypersurfaces
  3. Linear and multilinear decoupling
  4. The bootstrapping argument
  5. The proof of Theorem
  6. Multilinear versus linear decoupling for curves
  7. The $l^2$ decoupling for curves
  8. A more general perspective on decouplings for curves

Key Result

Theorem 1.1

Let $n\ge 2$. If ${\operatorname{supp}}(\hat{f})\subset {\mathcal{N}}_\delta$ then for $p\ge\frac{2(n+1)}{n-1}$ and $\epsilon>0$

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Strichartz estimates for irrational tori: the hyperbolic case
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 15 more