The nonlinear Brascamp-Lieb inequality for simple data

TL;DR

This paper extends the classical Brascamp-Lieb inequality to a nonlinear setting for simple data, and shows the Euclidean constant approximation in Young's convolution inequality near the identity in Lie groups.

Contribution

It introduces a nonlinear generalisation of the Brascamp-Lieb inequality and applies it to analyze the behavior of Young's convolution inequality on Lie groups.

Findings

Nonlinear Brascamp-Lieb inequality established for interior exponents.

Best constant in Young's inequality approaches Euclidean constant near identity.

Proof uses induction on scales with Gaussian extremisers.

Abstract

We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant as the size of the neighbourhood approaches zero, answering a question of Cowling, Martini, M\"uller and Parcet. Our proof consists of running an efficient, or "tight", induction on scales argument which uses the existence of gaussian extremisers to the underlying linear Brascamp-Lieb inequality in a fundamental way.