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.

中文翻译：

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实球上的一系列尖锐不等式

我们在实球上证明了一系列尖锐的多线性积分不等式，这些函数涉及具有某些对称性的函数，这些对称性可以通过某些向量场集的湮灭来描述。所涉及的 Lebesgue 指数被认为与这些矢量场集的组合有关。此外，我们推导出一些欧几里得布拉斯坎普-李布不等式，这些不等式定位于半径为R的球，具有类型的爆炸因子$R^{\delta }$, 其中指数$\delta >0$与前面提到的 Lebesgue 指数有关，并证明在某些情况下δ是最优的。