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Small cap decoupling for the paraboloid in $\mathbb{R}^n$

Larry Guth, Dominique Maldague, Changkeun Oh

Abstract

We extend the small cap decoupling program established by Demeter, Guth, and Want to paraboloids in $\mathbb{R}^n$ for some range of $p$.

Small cap decoupling for the paraboloid in $\mathbb{R}^n$

Abstract

We extend the small cap decoupling program established by Demeter, Guth, and Want to paraboloids in for some range of .
Paper Structure (6 sections, 14 theorems, 114 equations)

This paper contains 6 sections, 14 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Reduction to a multilinear small cap decoupling
  3. Proof of the multilinear decoupling
  4. Reduction to incidence estimate
  5. Proof of the incidence estimate
  6. Proof of Proposition \ref{['0526.prop14']}

Key Result

Theorem 1.1

Let $n \geq 2$ and $\frac{n-1}{2} \leq |\vec{\alpha}| \leq \frac{n}{2}$. Then for $2 \leq p \leq 2+\frac{2}{|\vec{\alpha}| }$ and $\epsilon>0$, for all functions $F: \mathbb{R}^n \rightarrow \mathbb{C}$ whose Fourier transform is supported on $\mathcal{N}_{\mathbb{P}^{n-1}}(R^{-1})$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Conjecture 1.4: Restriction conjecture, MR545235
  • Proposition 1.5
  • Theorem 2.1: Multilinear small cap decoupling
  • Proposition 2.2
  • Lemma 3.1: Refined flat decoupling, Corollary 4.2 of MR4153908
  • Lemma 3.2: Refined decoupling
  • Corollary 3.3: cf. Corollary 5.7 of MR4153908
  • ...and 8 more