金字塔与扩展度量测度空间
Pyramids and Extended Metric Measure Spaces
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中文总结 AI 辅助
本文证明金字塔可通过1-利普希茨序由emm-空间表示，首次实现所有金字塔的具体几何空间构造，还引入集中emm-空间并建立相关刻画，为研究emm-空间收敛性与测度集中现象提供新几何方法。
中文摘要 AI 辅助
金字塔是M.格罗莫夫（Birkhäuser 1999）引入的度量测度空间（mm-空间）的推广，用于建立测度集中问题的几何框架；扩展度量测度空间（emm-空间）是安布罗西奥-吉利-萨瓦雷（Invent. Math. 2014）引入的mm-空间的另一推广，旨在将索伯列夫微积分与最优传输理论扩展至更广泛范围。本文证明，每个金字塔都可通过1-利普希茨序由emm-空间表示；若金字塔是集中的，则该对应在同构意义下唯一，这首次表明所有金字塔都可由具体几何空间实现。基于此表示，本文引入集中emm-空间的新概念，定义集中emm-空间的可观测距离，并建立浓度的三个等价刻画，涉及金字塔、利普希茨可观测量与可观测距离。此外，本文通过发展纤维化方法，证明与emm-空间相关的金字塔弱收敛下，切格能量的Γ-上极限不等式，以及对数索伯列夫不等式、庞加莱不等式的稳定性。本文结果为研究emm-空间的收敛性，以及广泛类无穷维模型（目前大多超出现有几何方法的处理范围）的测度集中现象提供了新的几何方法，这类模型的重要例子出现在概率论中，包括维纳空间、构型空间、高斯场（如质量高斯自由场）、空间白噪声与质量双拉普拉斯场。
英文摘要
A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkhäuser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio--Gigli--Savaré (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent. We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This shows, for the first time, that all pyramids can be realised by concrete geometric spaces. Based on this representation, we introduce a new notion, a concentrated emm-space. We then define the observable distance for concentrated emm-spaces and establish three equivalent characterisations of concentration in terms of pyramids, Lipschitz observables and the observable distance. Furthermore, by developing an fibration approach, we show the $Γ$-$\limsup$ inequality of Cheeger energies as well as the stability of the log-Sobolev inequality and the Poincaré inequality under the weak convergence of pyramids associated with emm-spaces. Our results provide a new geometric approach for studying both the convergence of emm-spaces and concentration-of-measure phenomena across a broad class of infinite-dimensional models that have so far remained largely beyond the reach of existing geometric methods. Many of the significant examples arise in probability theory, including the Wiener space, the configuration space, Gaussian fields such as massive Gaussian free fields, spatial white noise and massive bi-Laplacian fields.
