Effective Brascamp-Lieb inequalities
Timothée Bénard, Weikun He
Abstract
We establish an effective upper bound for the Brascamp-Lieb constant associated to a weighted family of linear maps.
Table of Contents
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Effective Brascamp-Lieb inequalities
Authors
Abstract
We establish an effective upper bound for the Brascamp-Lieb constant associated to a weighted family of linear maps.
Paper Structure
This paper contains 16 sections, 15 theorems, 81 equations.
Table of Contents
- Introduction
- Bounds for Brascamp-Lieb constants
- Bounds for localised regularised Brascamp-Lieb constants.
- Applications
- Essential rank and perceptivity
- Essential rank
- Perceptivity
- Upper bounds for localised regularised data
- Lieb's theorem
- Proof of \ref{['pr:rlub']}
- Lower bound for localised regularised data
- Bounds on Brascamp-Lieb constants
- Brascamp-Lieb constants as limits
- Proof of Theorems \ref{['thm:ub']}, \ref{['thm:lb']}
- Visual inequality
- ...and 1 more sections
Key Result
Theorem 1.1
Let ${\mathscr D} = \bigl((\ell_j)_{j \in J}, (q_j)_{j \in J}\bigr)$ be a Brascamp-Lieb datum. Assume ${\mathscr D}$ globally critical (i.e. eq:gc holds) and $\boldsymbol{\alpha}$-perceptive for some $\boldsymbol{\alpha}\in \mathbb{R}_{> 0}^J$. Then writing $d=\dim H$, we have
Theorems & Definitions (33)
- Definition : Essential rank
- Definition : Essential acuity
- Definition : Metric perceptivity
- Theorem 1.1: Upper bound
- Theorem 1.2: Lower bound
- Example
- Corollary 1.3: Joint local boundedness in $(\ell_j)_j$ and $(q_j)_j$ near simple data
- Definition : Metric perceptivity 2
- Theorem 1.4: Upper bound
- Theorem 1.5: Lower bound
- ...and 23 more