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Regularized Brascamp-Lieb inequalities via Optimal Transport and Study of Equality Cases

Bader Ammari

Abstract

We consider regularized Brascamp-Lieb inequalities using the theory of optimal transportation, more precisely an anisotropic version of Caffarelli's contraction theorem. Furthermore, we provide a full picture concerning the issues of finiteness of the Brascamp-Lieb constant and of the existence of Gaussian extremizers. Finally, we find all optimizers for these regularized Brascamp-Lieb inequalities by employing heat flow methods that were already used to settle this question for the non-regularized Brascamp-Lieb inequality, and give some interesting applications.

Regularized Brascamp-Lieb inequalities via Optimal Transport and Study of Equality Cases

Abstract

We consider regularized Brascamp-Lieb inequalities using the theory of optimal transportation, more precisely an anisotropic version of Caffarelli's contraction theorem. Furthermore, we provide a full picture concerning the issues of finiteness of the Brascamp-Lieb constant and of the existence of Gaussian extremizers. Finally, we find all optimizers for these regularized Brascamp-Lieb inequalities by employing heat flow methods that were already used to settle this question for the non-regularized Brascamp-Lieb inequality, and give some interesting applications.
Paper Structure (6 sections, 28 theorems, 357 equations)

This paper contains 6 sections, 28 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. Proof of Regularized Brascamp-Lieb Inequalities
  3. Finiteness and Structure of Regularized Brascamp-Lieb Inequalities
  4. Optimizers of Regularized Brascamp-Lieb Inequalities
  5. Some Applications
  6. A general Caffarelli's Contraction Theorem

Key Result

Theorem 1.1

Let $\mathbb{R}^{n \times n} \ni \mathcal{Q} \geq 0$, $C_i: \mathbb{R}^n \rightarrow \mathbb{R}^{n_i}$ surjective linear transformations for $i=1,\cdots,m$ and $p_1,\cdots,p_m>0.$ Then, the largest constant $C$ such that holds for all non-negative functions $f_1,\cdots,f_m$ is also the largest constant such that the inequality holds for centered Gaussian functions and is usually called the Brasca

Theorems & Definitions (51)

  • Theorem 1.1: Lieb Lieb
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2
  • ...and 41 more