度量空间与图上的偏心 p-求和 Lipschitz 算子及积分不等式
Eccentric $p-$summing Lipschitz operators and integral inequalities on metric spaces and graphs
AI总结：
本文针对度量空间上的 Lipschitz 算子，利用度量对偶单位球中的特殊函数提出偏心 p-可和性概念及积分控制，并将其用于分析满足可和性与对称性条件的度量图数值指标。
摘要：
近年来，将线性算子的 p-可和性概念推广到度量空间上的 Lipschitz 算子情境，已得到广泛研究。这类研究主要利用由相关 Arens-Eells 空间所提供的度量空间 M 的线性化，以及 M 与由 M 上实值 Lipschitz 函数定义的度量对偶空间 M^# 之间的对偶性。然而，对于度量度量空间元素序列之间距离的测度（本质上涉及 p-可和性的定义），还存在其他方法。其中一种方法是考虑使用 M^# 单位球的特定子集来计算序列之间的距离，例如通过度量从两个点到一个固定点的取值之差所得到的实 Lipschitz 函数。我们引入了涉及此类函数的 Lipschitz 算子可和性的新概念，其特征在于这些算子的积分控制。为展示所得结果的适用性，我们在论文最后一部分将第一部分获得的理论工具用于度量图分析。特别地，我们展示了在这些图上定义的数值指标在满足某些可和性与对称性条件时其行为的新结果。
英文摘要：
The extension of the concept of $p-$summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space $M$ afforded by the associated Arens-Eells space, along with the duality between $M$ and the metric dual space $M^\#$ defined by the real-valued Lipschitz functions on $M.$ However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of $p-$summability) exist. One approach involves considering specific subsets of the unit ball of $M^\#$ for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference of the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, we use in the last part of the paper the theoretical tools obtained in the first part to the analysis of metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.
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