A Universal Inequality for Stability of Coarse Lipschitz Embeddings  

作　　者：Duan Xu Dai Ji Chao Zhang Quan Qing Fang Long Fa Sun Ben Tuo Zheng 

机构地区：[1]School of Science,Jimei University,Xiamen,361021,P.R.China [2]School of Science,Hubei University of Technology,Wuhan,430068,P.R.China [3]Key Laboratory of Applied Mathematics of Fujian Province University,Putian University,Putian,351100,P.R.China [4]School of Mathematics and Physics,North China Electric Power University,Baoding,071003,P.R.China [5]Department of Mathematical Sciences,University of Memphis,Memphis,TN,38152,USA

出　　处：《Acta Mathematica Sinica,English Series》2023年第9期1805-1816,共12页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant Nos.12126329,12171266,12126346,12101234);Simons Foundation(Grant No.585081);Educational Commission of Fujian Province(Grant No.JAT190589);Natural Science Foundation of Fujian Province(Grant No.2021J05237);the research start-up fund of Jimei University(Grant No.ZQ2021017);the research start-up fund of Putian University(Grant No.2020002);the Natural Science Foundation of Hebei Province(Grant No.A2022502010);the Fundamental Research Funds for the Central Universities(Grant No.2023MS164);the Natural Science Foundation of Fujian Province(Grant No.2023J01805)。

摘　　要：Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for each x^(*)∈Lip_(0)(X)there existα,γ>0 depending only on f and Q_(x)*∈Lip_(0)(Y)with‖Q_(x)*‖_(Lip)≤α‖x^(*)‖_(Lip)such that|Q_(x)*f(x)-x^(*)(x)|≤γ‖x^(*)‖_(Lip),for all x∈X.Coarse stability for a pair of metric spaces is studied.This can be considered as a coarse version of Qian Problem.As an application,we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual.Indeed,we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract.If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space,then the problem also has a negative answer for a separable space.

关 键 词：Lindenstrauss Problem coarse Lipschitz embedding coarse stability Banach space 

分 类 号：O178[理学—数学]