On decoupling and restriction estimates

TL;DR

The paper shows that the restriction conjecture for the hyperbolic paraboloid in $\mathbb{R}^d$ implies an $l^p$-decoupling theorem for the hyperbolic paraboloid in $\mathbb{R}^{2d-1}$, thereby linking restriction estimates to decoupling through a multi-scale, frequency-decomposition framework. It begins from Guth's reformulation of the restriction problem and develops an iterative decoupling scheme across coordinate blocks, using anisotropic scales and a pivotal lemma to trade between them. The result yields a concrete decoupling bound for $2 \le p \le 2+\frac{4}{2d-2}$ with $\vec{\alpha}=(\tfrac{1}{2},\dots,\tfrac{1}{2})$, culminating in a direct proof of the $l^p$-decoupling theorem for the hyperbolic paraboloid in $\mathbb{R}^3$. This work forges a conceptual link between restriction conjectures and decoupling, providing a practical route to higher-dimensional decoupling results from restriction hypotheses.

Abstract

In this short note, we prove that the restriction conjecture for the (hyperbolic) paraboloid in $\mathbb{R}^d$ implies the $l^p$-decoupling theorem for the (hyperbolic) paraboloid in $\mathbb{R}^{2d-1}$. In particular, this gives a simple proof of the $l^p$ decoupling theorem for the (hyperbolic) paraboloid in $\mathbb{R}^3$.

Theorems & Definitions (7)

• Conjecture 1.1: Restriction conjecture, MR545235
• Theorem 1.2: MR3736493
• Theorem 1.2: MR3736493
• Theorem 1.3
• Proposition 2.1: Proposition 1.5 of guth2024smallcapdecouplingparaboloid
• Lemma 2.2
• proof : Proof of Lemma \ref{['0224.lem22']}