We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of constant curvature. In the latter settings, our intrinsic dimensional functional inequalities improve on a series of known results and lead to new Hamilton-type matrix inequalities. Our proofs rely on scaling, tensorization, and stochastic methods.

中文翻译：

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模型空间上的内在维函数不等式

我们启动了对经典函数不等式的内在维度版本的系统研究，它捕获了底层对象的精细属性。我们关注模型空间：欧几里得空间、汉明立方和常曲率流形。在后一种情况下，我们的内在维函数不等式改进了一系列已知结果，并导致新的哈密顿型矩阵不等式。我们的证明依赖于缩放、张量化和随机方法。