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A Universal Inequality for Stability of Coarse Lipschitz Embeddings

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Abstract

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* ∈ Lip0(X) there exist α, γ > 0 depending only on f and Qx* ∈ Lip0(Y) with \({\Vert{{Q_{{x^*}}}}\Vert_{{\rm{Lip}}}} \le \alpha {\Vert {{x^*}}\Vert_{{\rm{Lip}}}}\) such that

$$\Vert{{Q_{{x^*}}}f(x) - {x^*}(x)}\Vert\le \gamma {\left\| {{x^*}} \right\|_{{\rm{Lip}}}},\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;x \in X.$$

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.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Science, Jimei University, Xiamen, 361021, P. R. China

    Duan Xu Dai

  2. School of Science, Hubei University of Technology, Wuhan, 430068, P. R. China

    Ji Chao Zhang

  3. Key Laboratory of Applied Mathematics of Fujian Province University, Putian University, Putian, 351100, P. R. China

    Quan Qing Fang

  4. School of Mathematics and Physics, North China Electric Power University, Baoding, 071003, P. R. China

    Long Fa Sun

  5. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Ben Tuo Zheng

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  1. Duan Xu Dai
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  2. Ji Chao Zhang
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  3. Quan Qing Fang
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  4. Long Fa Sun
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Correspondence to Long Fa Sun.

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Additional information

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) and 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)

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Dai, D.X., Zhang, J.C., Fang, Q.Q. et al. A Universal Inequality for Stability of Coarse Lipschitz Embeddings. Acta. Math. Sin.-English Ser. 39, 1805–1816 (2023). https://doi.org/10.1007/s10114-023-2136-4

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  • DOI: https://doi.org/10.1007/s10114-023-2136-4

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