The H\"older-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized H\"older-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.

中文翻译：

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Hölder-Brascamp-Lieb 不等式的变体

H\"older-Brascamp-Lieb 不等式是一组多线性不等式，概括了 Young 和 Loomis-Whitney 不等式的卷积不等式。Bennett 等人 (2008) 对指数的全部范围进行了分类。在类似于在 Ivanisvili 和 Volberg (2015) 的研究中，我们为这些不等式引入了一个大小概念，它概括了 $L^p$ 范数。在这个新设置下，我们确定了广义 H\"older-Brascamp-Lieb 的充分必要条件输入不等式以保持并建立当底层线性映射与 Young 的卷积不等式匹配时极端化器存在的充分条件。