Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Remarks on Gaussian-stability for Brascamp--Lieb Inequalities

Jonathan Bennett, Michael Christ

TL;DR

This work establishes a quantitative Gaussian-stability result for the general Euclidean Brascamp–Lieb inequality in the regime $1<p_j<2$, showing that near-extremizers must be close to (complex) Gaussian inputs. The authors combine a Fourier-invariance property of BL constants with a stable version of Beckner’s sharp Hausdorff–Young inequality to derive a deficit bound that scales quadratically with the distance to Gaussians. They further extend the stability to tuples of factors, proving that, for simple data, near-extremizers are close to a Gaussian tuple that realizes the optimal constant, including extensions to nonnegative Gaussian inputs. These results provide a robust framework for identifying near-extremizers as Gaussians in multilinear Euclidean inequalities, with implications for harmonic analysis and related stability questions.

Abstract

We establish a stable form of the general Euclidean Brascamp-Lieb inequality in all cases in which the Lebesgue exponents are strictly between 1 and 2, asserting that all near-extremizers are nearly Gaussian.

Remarks on Gaussian-stability for Brascamp--Lieb Inequalities

TL;DR

This work establishes a quantitative Gaussian-stability result for the general Euclidean Brascamp–Lieb inequality in the regime , showing that near-extremizers must be close to (complex) Gaussian inputs. The authors combine a Fourier-invariance property of BL constants with a stable version of Beckner’s sharp Hausdorff–Young inequality to derive a deficit bound that scales quadratically with the distance to Gaussians. They further extend the stability to tuples of factors, proving that, for simple data, near-extremizers are close to a Gaussian tuple that realizes the optimal constant, including extensions to nonnegative Gaussian inputs. These results provide a robust framework for identifying near-extremizers as Gaussians in multilinear Euclidean inequalities, with implications for harmonic analysis and related stability questions.

Abstract

We establish a stable form of the general Euclidean Brascamp-Lieb inequality in all cases in which the Lebesgue exponents are strictly between 1 and 2, asserting that all near-extremizers are nearly Gaussian.
Paper Structure (4 sections, 8 theorems, 57 equations)

This paper contains 4 sections, 8 theorems, 57 equations.

Table of Contents

  1. introduction
  2. Proof of Theorem \ref{['thm:sharpenedBL']}
  3. Remarks and examples
  4. Stability for tuples; proof of Theorem \ref{['thm:main_variant']}

Key Result

Theorem 1.2

If $\operatorname{BL}({\mathbf{B},\mathbf{p}})$ is finite and $1<p_j<2$ for all $j$ then there are constants $c_j=c(p_j,d_j)>0$ such that for all tuples ${\mathbf{f}} = (f_j: 1\le j\le m)$ of nonzero complex-valued $f_j\in L^{p_j}(\mathbb{R}^{d_j})$.

Theorems & Definitions (25)

  • Definition 1.1: Gaussian-stability
  • Theorem 1.2: A sharpened Brascamp--Lieb inequality below $L^2$
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 2.1: Fourier invariance of Brascamp--Lieb constants BBBCF
  • Theorem 2.2: Stable Hausdorff--Young Inequality C:stable HY
  • ...and 15 more