New sharp Gagliardo-Nirenberg-Sobolev inequalities and an improved Borell-Brascamp-Lieb inequality

TL;DR

This paper introduces a new Borell-Brascamp-Lieb inequality that yields sharp Gagliardo-Nirenberg-Sobolev inequalities, bridging geometric and functional perspectives and unifying previous results in the field.

Contribution

It presents a novel Borell-Brascamp-Lieb inequality that leads to sharp Euclidean inequalities, unifying and simplifying existing results in geometric and functional analysis.

Findings

New Borell-Brascamp-Lieb inequality established

Sharp Gagliardo-Nirenberg-Sobolev inequalities derived

Unified framework connecting geometry and analysis

Abstract

We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n and in the half-space R^n\_+. This gives a new bridge between the geometric pont of view of the Brunn-Minkowski inequality and the functional point of view of the Sobolev type inequalities. In this way we unify, simplify and results by S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret.