非正算子的解耦、多重线性对偶性与分解
Disentanglement, Multilinear Duality and Factorisation for non-positive operators
AI总结：
本文首次将多重线性对偶与分解理论从正算子推广至赋范空间上的一般线性算子，通过向量值重述、拉德马赫型几何性质及p-稳定随机变量实现解耦。
摘要：
在先前的工作中，我们建立了定义在赋范格上的正线性算子的逐点加权几何均值的范数不等式的多重线性对偶与分解理论。本文首次将该理论的适用范围扩展至定义在赋范空间上的一般线性算子。该理论的适用范围包括多重线性傅里叶限制型不等式。我们还改进了先前关于正算子的理论。我们的结果都有一个共同的主题：线性算子的加权几何均值上的估计可以被解耦为对每个算子分别进行的定量估计。解耦的概念贯穿全文。先前工作中使用的方法——主要是凸优化——强烈依赖于正性。相比之下，本文使用了解耦的向量值重述、底层赋范空间的几何性质（拉德马赫型）以及与p-稳定随机变量相关的概率性考虑。
英文摘要：
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators. Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitative estimates on each operator separately. The concept of disentanglement recurs throughout the paper. The methods we used in the previous work - principally convex optimisation - relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.
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