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 ...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.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.12126329,12171266,12126346,12101234)Simons Foundation(Grant No.585081)+6 种基金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)。
文摘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.
