A family of sharp inequalities on real spheres

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 δ is optimal.

Keywords:

• Multilinear inequalities
• homogeneous spaces
• heat flow

AMS Subject ClassificationS:

• 26D15
• 43A15
• 43A85
• 52A40

Acknowledgements

The author would like to thank the referees for recommending various improvements in the exposition of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).