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A new proof of an inequality of Bourgain

Polona Durcik, Joris Roos

TL;DR

The paper provides a new proof of Bourgain's trilinear smoothing inequality by importing recent additive combinatorics techniques. Central to the approach are a degree-lowering scheme via PET-induction and a dual difference interchange that converts a Gowers-type uniformity control into a single oscillatory integral bound, all while exploiting curvature in the phase $t^2$. The authors establish key lemmas—degree lowering from $u^3$ to $u^2$, a duality-based reduction, a bilinear bound with curvature, and a pivotal smoothing bound (the 'miracle' lemma)—to assemble the final smoothing inequality with explicit dependence on Sobolev norms. This synthesis bridges real harmonic analysis with combinatorial multilinear-methods, offering a pathway to generalizations and connections to related curvature-induced operators. The work highlights the potential of global, macro-scale Cauchy–Schwarz and Gowers-norm techniques in proving smoothing inequalities and their quantitative Roth-type consequences. $\mathcal{I}(f_0,f_1,f_2)$ serves as the focal object, encapsulating how curvature and multilinearity interact under smoothing averages.

Abstract

The purpose of this short note is to demonstrate how some techniques from additive combinatorics recently developed by Peluse and Peluse-Prendiville can be applied to give an alternative proof for a trilinear smoothing inequality originally due to Bourgain.

A new proof of an inequality of Bourgain

TL;DR

The paper provides a new proof of Bourgain's trilinear smoothing inequality by importing recent additive combinatorics techniques. Central to the approach are a degree-lowering scheme via PET-induction and a dual difference interchange that converts a Gowers-type uniformity control into a single oscillatory integral bound, all while exploiting curvature in the phase . The authors establish key lemmas—degree lowering from to , a duality-based reduction, a bilinear bound with curvature, and a pivotal smoothing bound (the 'miracle' lemma)—to assemble the final smoothing inequality with explicit dependence on Sobolev norms. This synthesis bridges real harmonic analysis with combinatorial multilinear-methods, offering a pathway to generalizations and connections to related curvature-induced operators. The work highlights the potential of global, macro-scale Cauchy–Schwarz and Gowers-norm techniques in proving smoothing inequalities and their quantitative Roth-type consequences. serves as the focal object, encapsulating how curvature and multilinearity interact under smoothing averages.

Abstract

The purpose of this short note is to demonstrate how some techniques from additive combinatorics recently developed by Peluse and Peluse-Prendiville can be applied to give an alternative proof for a trilinear smoothing inequality originally due to Bourgain.
Paper Structure (7 sections, 6 theorems, 63 equations)

This paper contains 7 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Degree lowering
  4. Proof of Lemma \ref{['lem:pet']}
  5. Dual difference interchange: Proof of Lemma \ref{['lem:ddi1']}
  6. Bilinear case: Proof of Lemma \ref{['lem:bilin']}
  7. Proof of Lemma \ref{['lem:miracle']}

Key Result

Theorem 1

Let $K$ be a compact interval and suppose that $f_0\in L^\infty$ is supported in $K$ and $f_1, f_2\in L^2$. There exists an absolute constant $\sigma > 0$ such that where $C$ depends only on $K$ and $\chi$.

Theorems & Definitions (13)

  • Theorem 1: Bourgain
  • proof : Proof of Theorem \ref{['thm:main']} using \ref{['eqn:penult']}
  • Remark
  • Lemma 1
  • Remark
  • Lemma 2: Dual difference interchange
  • Remark
  • Lemma 3: Bilinear case
  • Lemma 4
  • Remark
  • ...and 3 more