Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Algebraic Brascamp-Lieb Inequality

Jennifer Duncan

TL;DR

This work proves a global nonlinear Brascamp–Lieb inequality for a broad class of quasialgebraic maps, incorporating an affine-invariant weight that neutralizes local degeneracies and yields constants depending only on the degrees and exponents. The authors extend local nonlinear BL bounds to a global setting by leveraging endpoint multilinear Kakeya-type inequalities for algebraic varieties (via Zhang, Zorin-Kranich) and a discrete-to-continuum limiting procedure, including a careful discretization of fibres and a Fremlin tensor-product framework. The main result covers polynomial, rational, and algebraic maps, yielding a polynomial BL inequality as a corollary and connecting to Young-type inequalities on algebraic groups through a controlled degree analysis. The approach highlights a robust degree-based, affine-invariant structure for nonlinear Brascamp–Lieb inequalities and broadens the toolkit for nonlinear harmonic analysis on quasialgebraic data.

Abstract

We prove a global nonlinear Brascamp-Lieb inequality for a general class of maps, encompassing polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that compensates for local degeneracies, and yields a uniform constant that depends only on the 'degree' of the maps involved.

An Algebraic Brascamp-Lieb Inequality

TL;DR

This work proves a global nonlinear Brascamp–Lieb inequality for a broad class of quasialgebraic maps, incorporating an affine-invariant weight that neutralizes local degeneracies and yields constants depending only on the degrees and exponents. The authors extend local nonlinear BL bounds to a global setting by leveraging endpoint multilinear Kakeya-type inequalities for algebraic varieties (via Zhang, Zorin-Kranich) and a discrete-to-continuum limiting procedure, including a careful discretization of fibres and a Fremlin tensor-product framework. The main result covers polynomial, rational, and algebraic maps, yielding a polynomial BL inequality as a corollary and connecting to Young-type inequalities on algebraic groups through a controlled degree analysis. The approach highlights a robust degree-based, affine-invariant structure for nonlinear Brascamp–Lieb inequalities and broadens the toolkit for nonlinear harmonic analysis on quasialgebraic data.

Abstract

We prove a global nonlinear Brascamp-Lieb inequality for a general class of maps, encompassing polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that compensates for local degeneracies, and yields a uniform constant that depends only on the 'degree' of the maps involved.

Paper Structure

This paper contains 12 sections, 20 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Brascamp-Lieb Inequalities: Linear and Nonlinear
  3. Preliminary Definitions and Notation
  4. Main Results
  5. Endpoint Multilinear Kakeya-type Inequalities
  6. Setup for the proof of Theorem \ref{['genalgbras']}
  7. Reductions
  8. Central Constructions
  9. Heuristic Explanation of Proof Strategy
  10. Lemmas
  11. Proof of Proposition \ref{['discretealgbras']}
  12. Appendix

Key Result

Theorem 1.2

For each $j\in\{1,...,m\}$, let $B_j:U\rightarrow M_j$ be a $C^2$ submersion defined on an open neighbourhood of a point $x_0\in\mathbb{R}^n$. For each $\varepsilon>0$, there exists a $\delta>0$ such that

Figures (3)

  • Figure 1: The specific case when $\mathcal{V}_j=\{V_j^{(1)},V_j^{(2)},V_j^{(3)}\}$
  • Figure 2: Picture of $H_j$
  • Figure 3: overlapping $\delta^{\beta}$-tubes

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2: Bennett, Bez, Buschenhenke, Cowling, Flock 2018
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Quasialgebraic Brascamp--Lieb Inequality
  • Corollary 1.6: Polynomial Brascamp-Lieb Inequality
  • Corollary 1.7
  • Proposition 1.8
  • proof
  • Theorem 1.9: Endpoint Multilinear Kakeya Inequality, Guth (2009)
  • ...and 26 more