A family of sharp inequalities on real spheres

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.