  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.
