Hessian of Bellman functions and uniqueness of Brascamp--Lieb inequality

TL;DR

This paper investigates the conditions under which the Brascamp--Lieb inequality is unique by analyzing the Hessian of Bellman functions, providing sharp estimates for integrals involving these functions under specific assumptions.

Contribution

It establishes conditions on the Bellman function that lead to the uniqueness of the Brascamp--Lieb inequality, including sharp integral estimates.

Findings

Derived sharp integral estimates under certain assumptions.

Proved the uniqueness of the Brascamp--Lieb inequality in specific cases.

Identified conditions on the Bellman function's Hessian for inequality uniqueness.

Abstract

Under some assumptions on the vectors $a_{1},..,a_{n} \in\mathbb{R}^{k}$ and the function $B : \mathbb{R}^{n} \to \mathbb{R}$ we find the sharp estimate of the expression $\int_{\mathbb{R}^{k}} B(u_{1}(a_{1}\cdot x),..., u_{n}(a_{n}\cdot x))dx$ in terms of $\int_{\mathbb{R}}u_{j}(y)dy, j=1,...,n.$ In some particular case we will show that these assumptions on $B$ imply that there is only one Brascamp--Lieb inequality.