A family of sharp inequalities on real spheres

TL;DR

This paper establishes a family of sharp multilinear inequalities on real spheres involving symmetric functions and extends these results to localized Euclidean Brascamp--Lieb inequalities with optimal blow-up factors.

Contribution

It introduces new sharp multilinear inequalities on spheres linked to symmetry properties and derives localized Euclidean inequalities with optimal blow-up behavior.

Findings

Proved sharp multilinear inequalities on real spheres.

Derived localized Euclidean Brascamp--Lieb inequalities with optimal blow-up factors.

Connected Lebesgue exponents to combinatorics of symmetry sets.

Abstract

We prove a family of sharp multilinear integral inequalities on real spheres involving functions that possess some symmetries that can be described by annihilation by certain sets of vector fields. The Lebesgue exponents involved are seen to be related to the combinatorics of such sets of vector fields. Moreover we derive some Euclidean Brascamp--Lieb inequalities localized to a ball of radius $R$, with a blow-up factor of type $R^\delta$, where the exponent $\delta>0$ is related to the aforementioned Lebesgue exponents, and prove that in some cases $\delta$ is optimal.