发表机构
The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究球面与格拉斯曼流形上各向同性测度的连续Brascamp--Lieb不等式，通过离散逼近与热半群保持两种方法给出证明，并概述混合秩情形。
AI 中文摘要
我们研究与球面$S^{n-1}$和格拉斯曼流形$G(n,k)$上的各向同性测度相关的连续Brascamp--Lieb不等式。我们给出了连续格拉斯曼不等式的两个证明。第一个证明通过离散各向同性测度逼近连续各向同性测度，然后应用等秩几何Brascamp--Lieb不等式。在秩一情形下，离散各向同性测度的构造遵循Barthe的工作，这是标准的；而在高秩情形下，有限近似各向同性测度的构造由Tchakaloff型定理给出。弱收敛、几何Brascamp--Lieb不等式和Fatou引理进而导出连续不等式。第二个证明将Barthe和Huet的热半群保持方法推广到$G(n,k)$上的各向同性测度。我们在格拉斯曼流形设定下证明了热半群保持定理，并表明该定理可导出格拉斯曼Brascamp--Lieb不等式。最后，我们还给出了连续Brascamp--Lieb不等式混合秩情形的概要。
英文摘要
We study continuous Brascamp--Lieb inequalities associated with isotropic measures on the sphere $S^{n-1}$ and on the Grassmannian $G(n,k)$. We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp--Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp--Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on $G(n,k)$. We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp--Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp--Lieb inequality.
Comments35 pages
