Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions
Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions
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DOI：
10.1016/j.jfa.2021.108968
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发表时间：
2018-12
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影响因子：
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通讯作者：
L. Ambrosio;Shouhei Honda;J. Portegies;David Tewodrose
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文献类型：
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作者：
L. Ambrosio;Shouhei Honda;J. Portegies;David Tewodrose
本文利用特征映射研究了紧RCDΦ(K，N)空间(X，d，m)到L 2(X，m)的嵌入⁎族t。推广了已知的闭黎曼流形的部分经典结果[10]、[11]，证明了由↓t诱导的重定标拉回度量Φt⁎g L 2在L 2(X，m)中的收敛为tΦ0.此外，我们讨论了Φt⁎g L 2关于测量的Φ收敛和t的行为.应用包括对所有p<；Alexandrov空间在非折叠环境下的定量L p收敛，这一结果甚至对闭黎曼流形和亚历山大空间也是新的.
In this paper we study the family of embeddings Φ t of a compact RCD⁎(K, N) space (X, d, m) into L 2 (X, m) via eigenmaps. Extending part of the classical results [10],[11] known for closed Riemannian manifolds, we prove convergence as t↓ 0 of the rescaled pull-back metrics Φ t⁎ g L 2 in L 2 (X, m) induced by Φ t. Moreover we discuss the behavior of Φ t⁎ g L 2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative L p-convergence in the noncollapsed setting for all p<∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.